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by: Destany Daniel


Marketplace > Rice University > Economcs > ECON 502 > MACROECONOMICMONETARY THEORY I
Destany Daniel
Rice University
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This 266 page Class Notes was uploaded by Destany Daniel on Monday October 19, 2015. The Class Notes belongs to ECON 502 at Rice University taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/225023/econ-502-rice-university in Economcs at Rice University.

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Date Created: 10/19/15
REAL BUSINESS CYCLE MODELS In this chapter we look at two modifications of stochastic growth models to account for two fea tures of business cycles The first is a complication to investment to introduce more short run dynamics into the model The second involves changing household preferences to allow for vari able labor supply 1 Time to build investment Kydland and Prescott 1982 introduced the ideathat it takes time to build new productive capital structures They assumed that household utility takes a modi ed CobbDouglas form This can be motivated by appealing to the long run evidence that labor supply is relatively inelastic Spe cifically in the CobbDouglas form with share parameter 9 the fraction of hours supplied to the market tends to be close to 9 and thus relatively constant over time Kydland and Prescott generalize the CobbDouglas function however by allowing the curva ture of the overall function to change as e 17 e 1 7 tct Zt Y 2 B 1 4 1 t 0 In 1 the parameter 7 can be interpreted as the intertemporal elasticity of substitution for both consumption c and leisure or nonmarket uses of time 2 Kydland and Prescott 1982 showed how by allowing for time to build capital one can pro duce a simple model that has investment more volatile than consumption has most aggregates moving procyclically and allows productivity shocks to account for about 50 of post1950 business cycle variability Specifically the productive capital stock kt is assumed to evolve according to 134 kt11 5ktslt 2 where sJt j 1 J is new capital that isj periods from completion Thus sJt evolves according to sJt sJHt Each of the sJt become additional state variables for the dynamic programming problem Let the investment required at stage j of the process be pj per unit of capital Then the total fixed investment at time t will be J 2 ltpJsJt 3 1 Kydland and Prescott also allow for inventory investment If yt is the inventory stock at the be ginning of period t then inventory investment during period t will be yt1 7 yt Total investment in period t will thus be J It Z PjsjtYt1 Yt j1 4 Output is taken to be constant elasticity of substitution CES between current production and the beginning of period inventory 1 1 7 7v v 1 7w8tkf 1 72 0 WM 5 where l1V is the elasticity of substitution Using 4 and 5 the aggregate resource con straint now becomes J 1 7V 7 ct 2 ltpjsjtyt1 7y 1 iwetkf a 72 1 WWW j 1 6 135 Kydland claims that by adding the time to build technology and inventory using reasonable parameter values from the microeconomic evidence and identifying productivity shocks with the Solow residual the model can account for key properties ofthe US business cycle It can match the relative volatility of investment and consumption and the procyclicality of most ag gregates The model can also account for about half the volatility in US output The major prob lem with the model however is that it cannot explain employment variation Kydland therefore turned to intertemporal substitution as a solution 2 Intertemporal Substitution in Labor Supply To account for the fact that output variation over the business cycle is associated with a variation in the level of employment we need to drop the growth model assumption of an inelastic labor supply Consider again the static problem ma iUc h subject to pc wh 7 c The consumer chooses to allocate time to market or nonmarket activity based on the real wage As we noted when we examined the simple model of labor supply to be consistent with the long run evidence on changes in average hours worked or lifetime labor supply as real wages change the utility function needs to produce a labor supply curve that is relatively inelastic But this then makes it difficult to use the same model to account for significant employment variation over business cycles We look to intertemporal substitution to provide a source of large short run uctuations Casual observation such as that holidays or work hours over the week can be shifted with little or no compensation required suggests that the elasticity of the distribution of work supplied over time is high Life cycle participation decisions the timing of education and training or investment in a new job via search can also be in uenced by intertemporal wage movements 136 To illustrate further consider again the two period model we examined at the beginning of the course max Uc0 ho c1 hl subject to p0c0 p101 woho w1h1 8 c0h0c1h1 gtllt W gtllt wherep1 p1 andw1 1 lr lr Using this model we want to nd a U consistent with a the evidence that labor supply is inelastic in the long run b the evidence that employment variation can be considerable in the short run Solely to facilitate a graphical analysis we make three simplifying assumptions i Impose the assumption of a long run inelastic labor supply by assuming total labor supply over the two periods is fixed at h ho h This is equivalent to implicitly imposing restrictions on the utility function U ii Assume utility is separable between consumption and labor supply That is suppose the util ity obtained from c1 versus c2 is independent of the amount of work done in each period iii Define y as real income in present value terms and assume all income is consumed in either of the two periods Let Go cOy and 91 c1y be the shares of consumption in the two periods Define a price index P by P 90130 e1P1 9 The intertemporal budget constraint can then be written in terms of P as copo c1P1 90130 91131 y Py Woho W1h1 10 Now split the original maximization problem into two subproblems We can first consider the problem of maximizing utility by choosing consumption taking labor supplies in the periods as 137 given We can then maximize the overall level of utility by choosing the two levels of labor sup ply We introduce y as a new choice variable along with a second budget constraint Thus consider the problem 11 max Uc0 ho c1 hl subject to p0c0 p101 Py cOc1 With utility separable between consumption and labor supply we can solve this problem for con sumption as a function of real prices in the two periods and real income y 12 7 P0 P1 7 P0 P1 c0 7 c0 FFy and c1 7 c1 FFy De ne the indirect utility function that incorporates the maximizing consumption decisions by P0 P1 P0 P1 p0 p1 Vh0 hp Y39aF UCOF F y ho 01F F y 111 13 The labor supply decision can then be represented by the maximization problem max Vh h 39E sub39ect to the constraint Vih V2h l4 ho hl y 0y pngyP J P 0 P 1 y l and the constraint that follows from the assumption that long run labor supply elasticity is zero h0h1h 15 Definehh0 andput h hh1 139The implicit restrictions on U involved in this assumption can be imposed more form ally by setting the prob lem up as a ee stage maximization At the first stage choose h Then choose h and y as outlined above and finally choose the consumption levels c0 and cl We then want U to be such that h is very inelastic with respect to a change in prices over the two periods that keeps real wages constant 138 To focus on intertemporal allocation assume that real wages are constant over the two periods by letting wi Dpi As an aside we should note that although real wage variation might be im portant in some business cycle episodes the empirical evidence does not show a consistent re lationship between real wages and cyclical variations in employment The labor supply maximization can then be written as maxVhhihyI subjectto hhih 16 P 5 Assuming individuals are fairly indifferent to the allocation of time across the two periods in difference curves for the indirect utility function V in hy space will be as illustrated in the fol lowing diagram It indicates that individuals object to supplying almost all the time in one period yin x r i i i i i i i i i i h h or the next but there is a wide range of distribution of work time across the two periods that leaves them more or less indifferent Higher real income y of course makes individuals better off Note that even though V is a function of the allocation of consumption across the two periods and so responds to changes in poP and plP as long as individuals have access to capital markets and can borrow and lend at the same interest rate r in the first period as we are assuming we would expect this general pattern for the indifference curves to remain valid 139 The position of the budget line depends on pOP and p1P and thus future relative to current pric es In particular a decrease in real interest rates r will tend to raise p1 relative to p0 We can represent the maximization as below As the diagram shows if we raise p1 relative to p0 current labor supply h decreases Of course in general the same indifference curve need not be tangent to the two budget lines that is the change in p0 relative to p1 is likely in general to change maximized utility The size of the labor supply response depends however on the at ness of the indifference curves If they are very at over a broad range of values of h then a small change in p0 relative to p1 can produce a large change in labor supply in the current period h As a result of intertemporal substitution the shortrun labor supply curve might be much more elastic that is the opportunity cost of not working is much less than analyses based on long run labor supply would suggest Further fixed costs of employment might also imply a more elastic labor supply schedule for a representative individual than for any one individual In particular we conclude that as a result of intertemporal substitution current labor supply re sponds positively to the real interest rate If substitution effects dominate wealth effects in the laborleisure choice we would also find that current labor supply responds positively to an in crease in the real wage 140 In the two period model we have taken utility of labor supply to be nonseparable over time The utility cost of working today depends on the amount of labor supplied in other periods Such a formulation would appear to be difficult to incorporate into a dynamic programming framework 3 Intertemporal substitution in a dynamic programming model Kydland and Prescott 1982 also showed how intertemporal substitution can be incorporated into a stochastic growth model Specifically they modified household preferences by introducing another stock variable to the model We can let current leisure contribute to a stock of recent leisure just like current investment contributes to the stock of physical capital Furthermore there can be a depreciation rate on this stock so that as periods of nonmarket time fade into the past they have less of an effect on current decisions The desire to take leisure in any given period is then lessened if the current stock of past leisure recently enjoyed is higher Specifically the utility function is assumed to take the form x e 1 7 e 1 77 tct Zt 2 B 1 Y 17 t 0 but where now zt is not current leisure or nonmarket work but rather a stock variable zt kzt71u t 18 and current nonmarket time is denoted it With this model the dynamic programming problem has zt as an additional state variable When combined with the time to build and inventory features discussed above the model accounted for additional features of the business cycle including procyclical movements in employment Benhabib Rogerson and Wright 1991 further modified the utility function 17 to allow ct to be a CBS function between homeproduced and marketed goods and services Home production l4l then is allowed to have its own productivity shock Home production allows hours in the consumption sector to respond positively to productivity shocks In addition to increasing hours to accumulate more capital while productivity is high there is an incentive to substitute market for home production of consumption goods when mar ket productivity is high In fact the addition of a household production sector could result in la bor owing from the household sector to all market sectors during upswings rather than a transfer of labor from the consumption goods to the investment goods sector Another problem with the standard model that home production can alleviate is that it introduces another source of shocks to the system The standard model yields a very high positive correla tion between productivity and output or productivity and hours This is inconsistent with the data and can be alleviated by introducing productivity shocks in the household sector that are less than perfectly correlated with market productivity shocks However the model does not include the time to build or inventory accumulation features of the earlier Kydland and Prescott model While the Benhabib Rogerson and Wright model can match some business cycle characteristics better than the Kydland and Prescott model it is less success ful at matching others Another problem with the household production model is that it is very difficult to get independent confirmation of reasonable values for the parameters governing home production This makes it difficult to verify how well the model truly can account for busi ness cycle facts 4 Fixed Costs of Employment The intertemporal substitution model might seem to be more suited to explaining variations in hours worked over the business cycle than variations in numbers of employees To explain the latter we need to introduce fixed costs of employment An interesting attempt to account for employment uctuations over the business cycle is the in 142 divisible labor model of Gary Hansen JME 1985 Hansen starts with household preferences of the form BtUct l iht 19 t 0 where ct is consumption in period t and ht is hours worked in period t However he assumes that labor is supplied in indivisible units either households work an exogenously xed number of J IIJ hours in period t or they do not work at all This 39 a non v lity into the 39 maximization problem More generally we would like to look at a model where individuals can substitute leisure across time to account for variations in overtime and average weekly hours worked over the cycle but where there are also xed costs of employment As a result of the latter individuals have limi tations on their ability to vary hours and to some extent face employment lotteries with relatively xed hours of work but a variable probability of being employed A completely general model is dif cult to analyze because of nonconvexities Hansen and Rogerson make simplifying as sumptions in order to make the objective function for the household concave More recent papers have followed up this work by using more exible but also more complicated speci cations that again cannot be solved analytically but instead can only be simulated using reasonable pa rameter values Hansen and Rogerson assume that households choose lotteries rather than hours worked Spe ci cally each period instead of choosing hours households choose a probability It of working a xed number ho of hours and a lottery then determines whether or not any particular household works Since the contract is being traded the household is paid whether or not it works In effect the rm is providing complete unemployment insurance to the workers Since all households are identical they all choose the same contract ithat is the same 0 How 143 ever ex post a fraction 0 of the households will work and l0 will be ex post involuntarily un employed Per capita hours worked in period t will be given by xtho Each household however faces an uncertain prospect of supplying hours to market activity and so chooses cc to maximize expected utility Hansen assumes that utility in each period is separable in consumption and lei sure Uct lht uct vlht 20 Then consumption in each period will be independent of whether or not the household works Expected utility will be given by EUct lht It uctvlh0 lOLt uctvl A uct Boat 21 for constants A and B In particular we can treat the representative household as if it had pref erences which were linear in the probability of working 0 For maximizing utility we can ig nore the constant A in the perperiod utility function Since households are paid for the expected amount of time they spend working the household budget constraint becomes ct it wtoqho rtkt 22 instead of the straightforward generalization of the CassKoopmans model we might otherwise expect Thus the problem solved by the typical household is 0 max Btuct Boat subject to ct it wtoqho rtkt and km 15kt it 23 0 t with kg and ho given The representative firm will choose labor input ht by varying the number 144 of workers actually employed rather the hours each employee works to equate the marginal product of labor to the real wage as in the competitive version of the simple CassKoopmans model The rm also demands capital services from households to equate the marginal product of capital to the real rental rate rt which the rm takes as given Note that since 0 htho we can equivalently model the representative household as choosing ht to maximize an intertemporal utility function which is linear in ht The elasticity of substitution between leisure in different periods for the aggregate economy is in nite and independent of the willingness of individuals to substitute leisure across time Hansen shows that this model is able to more closely mimic features of aggregate time series data in particular the response of hours worked to a change in productivity than a similar model without indivisible labor 145 OVERLAPPTNG GENERATIONS BUSNESS CYCLE MODEL To discuss equilibrium business cycle models we need to introduce a labor supply decision by households and we will compensate by simplifying the consumption side of the model Let pref erences be Uc0n Vc1 Throughout this discussion we shall consider only stationary solu tions to the relevant first order conditions although the above discussion of nonstationary solutions can be readily extended to the models discussed in this section of the notes With labor supply in the first period the demand for cash at the end of the first period is m p0 nco and this is spent on goods next period In p1c1 1 Example 1 Constant Money Supply As above assume money balances are constant per capita at m The stationary solution will be the constant price one We get equilibrium c0 c1 n as solutions to max Uc0nVc1 1 c0 c1 n subject to c1 n c0 2 Again p will be given by m p 61 2 Example 2 Transfers to the Old Generation Now assume the quantity of m is changed via transfers to the old generation Then balances next period are given by m x Let the transfer x be selected randomly from some known distribution Each period looks like any other except the quantity of money has changed The state of the sys tem is described completely by the quantity of money The strategy is to seek solutions pm 00011 c1111 nIII 213 The current price is assumed to be known by traders when they trade Use p for tomorrow s price Traders do not know x and p but are assumed to know their probability distributions As sume traders maximize expected utility maxNEXY pUc0 n 3 n 00 m subject to m pn c0 4 Since each person is getting a transfer proportional to his holdings there is no in ation tax the extra cash received by each indiVidual matches the decrease in the value of money so intuition suggests c1 will be unchanged and p will uctuate with m Thus we conjecture that and p n 5 where 61 is the solution in example 1 Assume rational expectations so that if these are the solutions for p and p then we can also use them to model consumer expectations about the determinants of p and p39 In particular the dis tribution of p can be obtained from Prltp39 13 Pr X 13 6 1 Then the problem becomes max E Uc0 n VCIiicl 7 n co m mX 214 and there are no random variables in the objective function We get the same solution for c0 mm and c1 as in example 1 and the conjectured solution for prices is indeed a solution Note The consumer does not choose 31 in 7 to maximize utility It is what he knows the equilib rium consumption will be but the consumer is not responsible for maintaining equilibrium in markets by himself Refer back to the competitive equilibrium version of the CassKoopmans model for further discussion of a related point 3 Example 3 Transfers to the Young Generation Now suppose there is a lump sum transfer to the young in quantity mz where 2 has a known dis tribution The young tomorrow will also get a gift which is unknown today When the current young get mz the old have m so the money supply is mlz The young person s budget con straint when young becomes m pncmz 8 Next period when the current young will be old consumption will be m p39c1 The current young generation39s maximum problem is max Uc n 9 n c m P subject to m pncmz 10 Here p is unknown by the current young generation since it will depend on next periods as yet unknown distribution to the young generation 215 Strategy for solving this problem Our aim is to eliminate as many variables as possible to simplify the problem We attack the problem in 2 steps 1 Given m find the maximizing cn 2 Find the equilibrium behavior of prices To do the latter we guess a solution for p as a function of the state of the economy P 1301152 11 Then if our guess is correct and expectations are rational a good model for the expected price next period would be P39 1301131 Pm1ZZ39 12 with the distribution of 239 known and 2 known to the young Cagying out this strategy For given In the problem is1 max Ucn 13 subject to cnmzmp 14 Consumption minus real labor income c n can be either positive or negative depending on m mz 7 m p will be the intercept of the budget constraint with the c axis in the diagram on the next page An upward shift in the constraint follows from an increase in mz 7 m p 1Note that EVm p39 is constant for given In 216 III quot Assume neither leisure nor consumptlon are 1nferlor goods 5 Solve the conditional maximization problem for c 4 with c gt 0 n 116 with n lt 0 Now eliminate c and n from the problem to get a function of m De ne mzim P szp l a Uc f1 n 1 The young individual thus chooses m to Differentiate 18 to get FONC for a maX 217 15 16 17 18 0 f lm Z mllil lv lglil lt19 P P P P Now impose the equilibrium conditions In equilibrium money demand money supply so mlz m 20 Substitute the market clearing condition 20 into 19 to get f f i l lv39 LVUQ Equation 21 is then a market equilibrium condition giving current price as a function of expect ed future price The state variables are m and 2 so we guess a solution p pmz We further know that m is the known historical quantity of money If this is doubled we d expect all future prices to double Hence we guess pmz m z Substitute our guess into the equilibrium condition E ZFG ZJEhlmdfgg dmaimwd 0 where we have assumed expectations are rational so that if pmz m z is the equilibrium pricing function then p m39 z39 m1z z39 will model indiVidual expectations Since m is known to the young m can be taken through the expectations operator and we can rewrite 22 l l l l f Elv v vl 23 agt a agtlt1m agt Also 12 is known to the young this period and therefore also is a constant with respect to the 218 distribution of next period prices Hence it too can be taken through the expectations operator in 23 to give iTizif l q E A 24 for some constant A Solve the equation consisting of the leftmost and rightmost parts of 24 for I lZA2 Then insert 1 zA into the expression for A on the right hand side of 24 to get a nonlinear equation to be solved for A using the functions I and V and the distribution for z39 A Emu um 25gt Then substitute A into 1 zA to get 1 So our guess was correct and we indeed have a solution ofthe form p m z Now look at the behaVior of the solution with respect to shocks 2 Differentiate 24 with respect to z if l lj 1Z f q39lgt l 0 26 Simplify 26 to get from which we can conclude 2Note that this solution will depend on the known functional form for f which in turn depends on the orig inal utility function 219 gt 0 since fquot lt 0 fquot gt 0 28 z f f d From 28 the elasticity of I with respect to 12 is 1 f z f f e 0 l 29 7 3 From this analysis we conclude that p does not move proportionately to 12 The young hold over some cash for consumption tomorrow The motion of economy through time is given by mt1 mt1Zt 30 Pt mt Zt 31 nt n 7 32 M20 7 1 t i 0 qTZt 33 with zt an ii d shock We have a shortrun nonneutrality of money here in that p does not move proportionately to 12 However we also have a quantity theory result in as far as is constant Pt Zt Also if m0 Bmo then mt Bmt and pt Bpt Note that the effects on employment and consumption are income effects When the young get extra balances they increase their consumption of goods and leisure We introduce substitution effects by postulating agents have to solve an information processing problem 220 4 Example 4 Transfers to Old Generation with Unobserved Real Shocks This example is a simplified version of the paper Expectations and the Neutrality of Money Lucas JET 3972 Lucas takes the idea from Phelps of introducing islands all affected by monetary policy in the same way but with different real effects in each Let 9t number of traders in my market relative to the average and assume the distribution of this real shock is In at N0 12 34 In each market the old demand goods and the young choose cn and money to carry over to next period To eliminate distributional effects assume the money supply in each market is the same mm mtxt with In xt N N00392 35 As in example 3 solve for cn in terms of m The maximization problem is maxUc n 36 n c m p subject to m pn c 37 Given m the problem is rrcraerc n 38 subject to n 7 c I 39 Solve the constrained maximization 3 8 and 39 for c cm p and n nm p and de ne fmp E Ucm p nmp Individuals then choose m to maX f 40 IE P P The FONC for a maX is 0 f 31 Ev mf 41 P P In equilibrium money demand money supply so am mX orm 1 42 De ne FX p39lmp as the conditional distribution of X39p given m and p Also de ne the func tion h E f39 Then from 41 and 42 we get l mXX39 X 1113 v dFx39 39 m 43 GP p I 9p p p l p The state variables here are m X and 9 Individuals know m but don t know X or 9 Current price movements will give some noisy information on X and 9 Solution strategy Guess a solution p pmX9 and solve for it Lucas shows that any solution to 43 is in fact monotonic in XG Then as in eXample 3 he guesses pmX9 m X9 He proves there is a unique continuous solution z to43 with z z bounded I strictly positive and continuously differentiable Lucas then eXamines two polar cases i 9 l with probability one so trading takes place in a single market and no nonmonetary dis 222 turbances are present Lucas has two markets with fraction 92 9 6 02 of the young allocated to one market and 192 of the young allocated to the other market Then p mxy where y is the unique solution to hy V y This is the classical neutrality of money result ii x l with probability one so the only disturbance is exclusively real The money supply is fixed The disturbances have real consequences Those of the young who nd themselves in a market with few of their cohorts a low 9 obtain in effect a lower price of future consumption They attempt to distribute this gain to the future by holding higher cash balances This attempt is partially frustrated by a rise in the current price level raising current expenditure The solu tion here is p m l9 where I is continuously differentiable with an elasticity between zero and one Returning to the general case Lucas notes that the current price informs agents only of xG Lu cas did not require lnGt N N0 l72 and lnxt N N00392 as I have written above and will use short ly He did however impose restrictions on the distributions of x and 9 In particular he required3 Pr 9 S Q I fx9 z is an increasing function ofz Pr x S x I fx9 z is an decreasing function ofz With these restrictions on the distributions of x and 9 Lucas shows that the equilibrium price is 13011929 WINK9 where I is continuously differentiable with an elasticity between zero and one The monetary changes x have real consequences because agents cannot discriminate perfectly between real and monetary demand shifts 339Also he assumed 9 has a continuous symmetric density on 02 and as we noted above in Lucas paper there are two markets with 92 of young allocated to one market and 1 92 allocated to the other 223 5 A parametric example Take lnet N N0 l72 1n Xt N N00392 and assume Uc n c1B anB 44 with 1 lt B lt 0 and kl1Bgt cn over the relevant range ofc and n so that U is concave and 17a Vltygt La 45 with B lt 0L lt 1 For the U function 44 the solution to rrcrariiUc n 46 subjectto nc1ngtocgto 47 is c and 48 so that we may write with H k1B1 k1B B hy HyB 49 for a constant H and 1ltBlt 0 Guess a solution pmX9 mAX9B for constants A and B If this guess is correct the equi librium condition 43 becomes 224 B v 7a He 1 m E 9p p 9p p39 Also if the guess for the solution price is correct and agents have rational expectations then III P 50 their expectation of next period s price will be given by y B p39 m A 51 In addition their expectation of the relationship between this period s price and x and 9 will be given by B p mAC g 52 and hence knowing p reveals xG but not the individual x and 9 Let E value of x assumed by the individual and V value of 9 assumed by the individual Then 1 IniAjlB 53 with In E N00392 and In w N0172 54 From 50 and using the assumption that mxG is known we get HB El my pl 1 lt5 Then if we de ne 225 1 K1 Elx39lialglwmll HBHW 56 and use the facts i m and p give no information about next period s shocks X and 939 and ii the conditional distribution of E given m and p is independent of the distributions of X and 939 we can rewrite 55 as w 1 B 0171 3 15 11701 P a Elx 1115491 0 X1 130171 mX l3 B 15 1171 9 m H 9 1 lt 1 1 17 i 7 L 7 mX 15ltmA1BEX1a3B 11 BE ai1 myp 1BH1B 0L 3 7 L 1 mp HBHW 57 1 1 mp IBEaa11mp11BH1B 1 1 1T K1 BltmAgt BE 1lmyp B Now we use signal processing to derive the conditional distribution of Under the rational eX pectations assumption knowing m and p is equivalent to knowing EW xG Hence ElnE0quot1 lmp 011 E1nE1nE 111111 58 Now use the projection theorem to get lnE alnE lnw 8 59 where E8lnE lnV 0 Then ElnE anln E a ElnE lnw2 0 so 0392 a392 172 that is 02 60 GZMZ Substitute 59 and 60 into 58 to get 226 2 ElnE quot1lmp 0L71026 121n 61 l 2 Now use the fact4 that if ln y N Nu0392 then E y yelL where y e26 Then a71a EE 1mp1 193 62 where recall the parameter a is given by the variance ratio 60 Substitute 62 into 57 to con clude that for the constant B given by a 63 p K1nrmA K3 g7 15 64 B B 1 Now consider some special cases i Suppose 172 0 and price movements result only from monetary disturbances Then from 60 and 63 a l and B 1 Prices increase in proportion to the monetary shock X ii Suppose 0392 0 so there is a smooth monetary policy but real shocks Then a 0 and B OLB1B 6 01 Price responds to the real shock but less than proportionately because of substitution effects A low 9 is a good event and that causes you to work harder today and consume more tomorrow iii If neither 0392 0 nor 172 0 individuals respond in part as if there is a real shock and in part 4Check a probability or statistics text 227 6 as if there is a monetary shock A Phillips Curve For the special utility function considered above we have mtXt lnnt9 X 70 ln 65 etpt where no is given by lnl klB Substitute the solution 64 for pt Xt B pt K4mt 66 t into 65 to nd Xt lnnt9X K517Bln 67 t Also from 66 Kt lnpt9 X K6 lnmt Bln 68 t Now average 67 and 68 over markets and use E In 9 0 to obtain expressions for aggregate employment and average nominal prices given by In Nt K5 lB 1n Xt 69 lnPtK6lnmtBlnXt 70 The aggregate price level follows a random walk as does mt Aggregate employment uctuates around a constant level and prices and employment are perfectly correlated 228 If the distribution of X is changed to N060392 without people knowing that a big X will be mis taken for a low 9 and employment will be increased However people will catch on They know mt with a oneperiod lag and hence have a history on Xt There is no longrun tradeoff between employment and in ation even though 69 would indicate that such a tradeoff eXists It has been observed that in the model Lucas discussed in Expectations and the Neutrality of Money consumption moves countercyclically When a large positive money shock is mistaken for a low 9 current output by the young is higher but current prices are also higher and current real consumption lower This resulted from the special structure of preferences Lucas assumed to make the model easier to solve Ifwe use Uc0 c1 n0 n1 for preferences we can obtain a pos itive correlation between c0 no and the rst period price p0 To do so we want rst period con sumption to be more responsive to expected discounted permanent income than to anticipated real interest rate movements 229 MONEY AND GROWTH 1 Why introduce money into our models There are several issues we are interested in that can only be discussed in a monetary economy i We want to discuss in ation as a macroeconomic phenomenon in its own right and separately from either growth or business cycles Since the nominal price level is nothing but the rate of exchange between money and goods and services it is clear we need to introduce money into our model in order to study in ation ii We are also interested in the interactions between money and other variables It is a centuries old ideathat in ation stimulates economic activity This can be seen in the writings of Hume and the quantity theorists it is worthwhile looking at Hume s essay Also note that the tendency of nominal prices to move procyclically in preWWH data and the tendency of the rate of in ation to vary procyclically in at least some cycles in post WWII data are consistent with the ideathat money shocks are a causal factor in some business cycle uctuations 2 The relationship between money and other variables The graphs we previously examined showed that there appeared to be some relationship in the postwar US data between changes in money growth and subsequent movements in real GDP growth However the relationship also appears to have weakened in recent decades as money growth rates have become more variable while GDP has become less so Over longer periods of time money growth rates are more closely related to in ation rates than to real GDP growth rates The graphs are repeated below for convenience Apart from the recent US data international evidence also suggests that severe monetary shocks particularly currency crises and severe contractions in money growth rates can have substan tial real effects There is perhaps a hint in the evidence both from the current developed econo mies in earlier eras as well as currently underdeveloped economies that monetary shocks have larger real effects when financial markets are less developed 168 Chart 1 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 dev GDPgr 7 PCE inf 7 M2gr Chart 2 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 dev GDPgr 7 Mlgr 7 Mbasegr 3 Money in the Cass K 00pmans Model A monetary analogue to the CassKoopmans model is the model considered by Sidrauski A rep resentative consumer has preferences 2 B Um mt 1 169 where mt represents real money balances It is assumed that the transaction services provided by money depend on the rate of exchange between money and goods If the quantity of nominal bal ances doubles but the nominal price of goods also doubles then the transaction services supplied by the money are unchanged Thus real balances m appear in the utility function Technology is as in the CassKoopmans model ct km fkt t 15kt 2 with new money received as cash transfers of real value Vt w 3 where Pt denotes the price of the output in terms of money The individual budget constraint now becomes Ptct km Mt1 Pt fkt 1 5ktl Mt PtVt 4 with the values of M chosen by the individual to be distinguished from the values of ll the per capita money supplied by the government In equilibrium money demand will equal money sup ply so an equilibrium condition will be Mt M 5 but we cannot use this condition to cancel out M and v from the individual budget constraint We assume the individual maximizes utility by choosing money balances and consumption and we want equilibrium prices to be such that he chooses to hold the quantity of money supplied Now if we define mt MtPt and Tct PM PtPt we get 170 Mt1Pt Mt1Pt1Pt1Pt mt11 771 6 and the budget constraint can be written in real terms as ctkt1 mt1l ntfkt 1 5ktmtvt 7 Assume the representative individual has perfect foresight so that all future TET and vT are known at time t lt 17 As in the Cass Koopmans model we introduce Lagrange multipliers tht for the budget constraint at time t Speci cally we define the Lagrangian L 2 BtUctmt Qtfkt 1 51ltt111tVt Ct kt1 mt1175tl 8 and set equal to zero the derivatives of L with respect to ct km and1 mm to get for t 2 0 0 Ucctmt qt 9 0 Bt1Umctlamt1 qut tht1 m 10 0 BtIQt1fykt11 5l Bt lt 11 ctkt1 mt1l rctfkt1 5ktmtvt 12 with initial conditions k0 M0 and a TVC for q From these first order conditions we get difference equations in q k and m which are solved us ing the two initial conditions and the transversality condition The solution will depend on the path assumed for the exogenous money supply process vt 4 Stationary States We can study the stationary point of the set of difference equations by looking for a set of values 139Since the individual takes the price level as given choosing real balances is equivalent to choosing nominal balances l7l of the capital stock real money balances and shadow price of capital q which remain constant over time It is clear that there is no hope of getting a stationary solution for an arbitrary money supply process vt However suppose we have vt V is constant that is the real change in the money supply is constant Then we must have VtMt1PtMtPtmtl17T tmtV 13 and at a stationary point mm mt m so that V mTE Then the rst order conditions 9 10 11 and 12 become Ucc 111 q 14 BUmCmBqQ1TE 15 Bf39k15l 1 16 c fk 5k using v mm 17 Hence f39k 1 5 lB l p for p the rate oftime discounting or f39k p 5 18 so that stationary k depends only on technology and preferences as in the CassKoopmans mod el Given the stationary value of k we obtain the stationary value of c from c fk 5k To nd the stationary value for real balances we use the 14 and 15 to get Ucp 7 p711 Um 19 which with v mTE is solved for the stationary values of real money balances and in ation given v and c Speci cally if we substitute 7 vm into 19 we get Um Ucp lpvm 20 172 which can be solved for m mv Note that we can interpret 19 as saying that the ratio of the marginal utility of real money balances to the marginal utility of consumption in a stationary state must equal a nominal rate of interest the rate of time discount p plus the rate of in ation 7 plus an interaction term pic In stationary equilibrium there is no interaction between the mon etary and real sectors A change in the real rate of money supply increase V will affect the sta tionary level of real money balances and the stationary in ation rate but will not affect the stationary values of c or k Also observe that the demand for money is a demand for real money balances a onceover change in the quantity of money supplied will merely produce a once over increase in the nominal price level to restore equilibrium between the demand and supply of money Now observe that comparing stationary states with different rates of real money growth v we have dTE i ivm v dV m m2 21 We can then use 20 to determine m v l p Uc U l m mV lt1 gt m lt1 gt J p 22 p v p v U U Pf m lUchr m2 U0 Umminm1pVUc C The term in braces is negative if we assume neither c nor m is an inferior good2 Thus starting from v near 0 m v lt 0 and increases in v will decrease stationary money demand and increase the stationary rate of in ation As v is increased mv decreases Suppose we as 239 That is if we consider the imaginary problem of choosing the two goods 0 and m to maximize Uc m subject to pee pmm y and impose the requirement that dcdy gt 0 and dmdy gt 0 then we must have Umm UmUcmUC lt 0 173 sume that at some level of v say v the denominator in 22 becomes zero while it remains neg ative for all V lt v Then we would nd m v gt 700 as V gt v However if m0 gt 0 there must then be some level of v lt V such that m 0 and hence from 22 m v 0 which contradicts the original assumption Thus we conclude that we must have m gt 0 and m v gt 0 as v in creases Furthermore as m gt 0 TE vm increases without bound Stationary equilibrium in a tion rates enter into the hyperin ationary range and people do what they can to avoid using money to nance their transactions 5 Behavior out 0fstati0nary states While the stationary values of c and k are independent of v there will in general be an interaction between the real and monetary sectors of the economy out of the stationary state This results from changes in real money balances altering the marginal utility of consumption as can be seen if we consider the special additivelyseparable utility function Uc m Wc Zm 23 In this case the marginal utility of consumption is independent of m and under the policy vt v constant over time the rst order conditions 9 10 l l and 12 become W Ct qt 24 Z mt qt 175t71qt71B 25 qt f kt 05 qHB 26 ct kH1 fkt 1 51ltt 27 Equations 24 26 and 27 are just the CassKoopmans equations which are solved for ct and kt independently of monetary policy For the additively separable utility case we can obtain the demand for money for periods prior 174 to long run stationary equilibrium from 25 mt Z39 1 ltTEHNHB qt 28 Then Z concave implies Z 1 E G has G lt 0 Now rewrite the argument of G as EMMAB lt 1H 1P75t71PTEti1 QtQt71l 1H P1P75t71 Qt Qt71Qt71l 29 The demand for money depends negatively on a nominal interest rate and the shadow value of capital qH The term in square brackets is a nominal interest rate since qt qHqt1 is the cap ital gain on real capital k and p is a rate of time preference so p qt qHqH is a real interest rate and lpTEt1 is an in ation premium Since G lt 0 the demand for money falls as the nom inal interest rate rises The term qH falls as wealth k increases so the demand for money rises as real wealth increases 6 Transactions Model of Money Demand Barro discusses a simple model of the transactions demand for money balances that I believe you have already studied in your previous macroeconomics courses However it may be worth brief ly reviewing the model here Those students unfamiliar with the model should carefully read the relevant chapters in Barro Consider an individual who wishes to use money to finance a ow of transactions uniformly distributed over time Instead of holding noninterest bearing money this individual could hold bonds yielding a nominal interest rate of Rt which are however unacceptable as a medium of exchange The bonds can be converted to money in order to finance transactions but it costs the individual a real amount 7 for each conversion3 Following a conversion the money balances will be exhausted gradually over time Let T be the time interval fraction of the unit time inter 339Barro defines transactions costs as a nominal amount 7 However since these costs are likely to involve the expenditure of labor time more than a nominal withdrawal fee it makes more sense to define them in real terms 175 val between conversions of bonds to money Assume the desired expenditure is an instanta neous ow at rate Pc The total money balances required to fund expenditure for an interval of length T will then be PcT The holdings of money balances will then follow a sawtoothed path as in the following diagram The number of conversions per period will be UT and this will impose a real transactions cost of y T 30 The average money balances under this scenario will be m 31 R3 32 The total cost 17 of nancing transactions using money is therefore 176 17 will be minimized by choosing T to solve the first order condition 71R9 0 T2 2 thatis 2Y T Rc From 31 and 35 average real money balances will then be gcv 2 2R wlBl 33 34 35 36 Barro observes that while this equation indicates economies of scale in holding real money bal ances since real balances rise less than proportionately with real transactions c the cost of time and thus the real costs of transactions 7 will be positively correlated with c This will tend to make average real money balances move more closely to proportional to c More generally we can write the demand for average real transactions balances as Mqu 37 with R affecting Q negatively c affecting Q positively and real transactions costs 7 affecting Q positively If the conversion times are then distributed randomly across individuals the economywide per capita real money balances will equal the average money balances held by each individual as 177 given in 37 Equivalently we can write aggregate real money demand as 1L ltIgtR c y 38 7 Money Neutrality Suppose there is a once and for all change in the level of the money stock M Since the remainder of the model the goods and labor market clearing conditions and the constituent demand and supply functions do not involve M or P none of the real variables in 38 will be affected by a change in M or P Hence equilibrium will require a proportionate movement in P equal to the original movement in M As with stationary states in the Sidrauski model changes in the money supply will be neutral with a change in M merely resulting in an equiproportionate change in nominal prices P An increase in M holding P fixed will produce an incipient increase in household wealth The positive wealth effect would tend to raise consumption demand and reduce labor supply The in cipient excess demand for goods would bid prices up until real money balances decline to their original level and the initiating wealth effect disappears While increases in money supply will raise nominal prices they are not the only source of price changes in the model In particular any other shocks that effect the equilibrium real interest rate the level of real transactions or real transactions costs 7 will affect the demand for money and thus average nominal prices 7 even if the nominal quantity of money is unchanged 8 In ation In ation is a continuing increase in average nominal prices While real shocks can affect average nominal prices by altering the demand for money these shocks are unlikely to produce a persis tent rise in prices 7 particularly of any more than a negligible magnitude For example we could imagine continuing improvements in transactions technology producing modest declines in 178 money demand of one or two percent a year with a corresponding low in ation rate The only feasible source of substantial persistent in ation however is substantial persistent growth of money supply This observation is borne out by both time series and crosssectional data A nominal interest rate R re ects a promise to pay a lender R currency units in the future for each unit of currency lent today When there is persistent in ation however lenders will antic ipate that tomorrow s currency will be worth less in real terms than today s currency They will demand compensation not only for foregoing consumption for a period of time this is the real interest rate but also for the expected loss in the purchasing power of money The real return on a bond with a nominal interest rate of R when the expected in ation rate is Tie will be r 7l 39 Alternatively for a real interest rate of r and an expected in ation rate of rte the nominal interest rate will be Rr7cerTEe 40 Note that for both r and Tie small the product rTEe will be very small and we can write the expected real interest rate which is the variable that affects c i nS and so on as approximately Ri e A change in the rate of growth of the money supply will alter the in ation rate in long run equi librium Since this in turn will alter the nominal interest rate R the return on holding money and the demand for real money balances will be affected Again however if R fully adjusts to the expected in ation rate no real variables will be affected The change in the demand for real mon ey balances associated with a change in the in ation rate can be accommodated by an additional change in the price level For example suppose the money supply is growing constantly at the rate u and then the rate of 179 growth is increased to u and maintained at that higher level Assume individuals are fully aware of the increased money growth and immediately adjust their forecast of in ation accordingly With an unchanged real interest rate r the nominal interest rate will immediately rise to re ect the difference between u and u The demand for real money balances will decline and the aver age level of nominal prices will jump immediately the new higher money growth rate is instigat ed The situation can be represented in the following diagram prices 7 Time 9 Expectations and Hyperin ation Cagan argued that in a hyperin ation which he defined as an in ation with prices rising more than 50 per month by far the most dominant in uence on the demand for money will be changes in the expected rate of in ation While hyperin ationary episodes will often be associ ated with changes in real income wealth and real rates of return on bonds and other assets the effects of such changes on the demand for money will be small in comparison with changes in Tie Cagan used the above idea to explain the drop in real money balances characteristic of hyper in ations It is typically observed in hyperin ations that prices actually rise faster than the rate of money growth so that real balances fall Cagan argued that this fall in real balances can be explained by the rise in the cost of holding money balances caused by higher expected rates of 180 in ation Cagan hypothesized that the expected rate of in ation could be modeled as an exponentially weighted distributed lag on past actual rates of in ation Equivalently he assumed expectations were adapted or updated to account for the error made in the preVious period Expressing his model in discrete time we get 51 TEEBTErTEEBgt0 41 Note that the equation 41 for expected in ation can be written NEH 17B7rf357ct 42 and using the lag operator we can rewrite 42 as TEE1 L7 43 1 7 1 7 BL We then conclude that if prices were constant at some period t 0 in the past and expectations for in ation in period t 1 were also for zero in ation then the first order difference equation 41 has the solution til 7 6217W HNT quot51 44 Thus the current expected in ation rate is an exponentially weighted average of past actual in ation rates To focus on the effect of expected in ation on money demand Cagan postulated a demand for real money balances of the form 1nMtPt ouste y 45 181 We can also write the money demand function 45 as In Mt ln Pt mte y 46 Now take the quasidifference of 46 and use 42 to eliminate the terms in TEE3 to get In Mt1 1Bln Mt1n Pt1 1Bln Pt lt1qu BY 47 or In Mt11Bln Mt1n PM 1B 1n Pt lt1qu BY 48 which is an equation solely in terms of the observable variables M P and TE Hence estimates of 5 0L and y can be obtained4 Note that we in fact get two estimates of The parameter 5 is said to be over identi ed Cagan found that with the above method of relating expectations to observed data on in ation rates this equation performs very well in explaining the movement in real money balances He notes that in several cases however that observations in months toward the end of the hyper in ationary period do not fit the equation very well and he ascribes this to his expectational equa tion If individuals come to believe that the hyperin ation may be coming to an end they may no longer form their expectations about future rates of in ation solely on the basis of past actual rates He also notes that the different values of 5 he finds for different countries suggests that B is not constant but depends on the recent history of in ation experienced by individuals in the economy In economies where rapid in ation has more recently been experienced in ationary expectations appear to respond more rapidly to innovations in the in ation rate in the sense that B is closer to 1 439This is not the way Cagan originally obtained the estimates instead he formed 15 for different values of B and performed a grid search to determine which values of B worked best 182 Sargent The Ends of Four Big In ations focused attention on the closing months of hyper in ationary episodes Sargent makes the point that as the in ation rate subsides at the end of the episode the demand for real money balances will rise As a result large increases in the supply of money can be accommodated by increasing money demand without in ation having to rise The ends of hyperin ations are often characterized by falling in ation rates despite the fact that the supply of money continues to grow at a rapid pace Sargent notes that in the episodes he stud ies the expansion of money at the end of the hyperin ation only occurs as individuals exchange gold foreign exchange or other assets for domestic money In this way the central bank can be sure that the increase in the domestic money supply is all demanded by the public and so will be nonin ationary We are lead to ask why the hyperin ation comes to an end if the money supply continues to grow at a rapid rate Equivalently why does money demand start to rise again Why should in dividuals believe the hyperin ation is over Sargent argues along with Cagan that the money expansion is used primarily as atax to finance government expenditure in excess of explicit tax receipts In ation is atax on the holding of real money balances The base of the tax is the level of real balances the tax rate is the rate of depreciation in the real value of money or the rate of rise in prices Sargent and Cagan both note that if a government begins to finance its expenditure by simply printing more money and using the new issue to purchase goods and services the sub sequent rise in in ation will further worsen the government budget deficit Explicit taxes are lev ied on income or consumption from previous years With an increase in the in ation rate the real value of tax collections falls and the government turns to everhigher rates of in ation to finance its expenditure Sargent argues that believable scal reform and a credible commitment to only expand the money supply in line with money demand are the necessary prerequisites to ending a hyperin ation Once individuals come to believe the government is no longer going to finance its expenditure with the in ation tax the money supply can continue to expand for a time without in ation being reignited because of increases in money demand 183 RATTONAL EXPECTATIONS The idea behind rational expectations is to propose that in forming their expectations of the fu ture behavior of a relevant economic variable individuals use information about the actual path of that variable they experience More particularly we propose that individuals are not system atically biased in their estimates of expected values Any mistakes individuals make in their fore casts will be immediately corrected the moment they are perceived The situation is analogous to the microeconomic model of a consumer maximizing utility As microeconomists we do not propose that every time a consumer enters a store he looks at the relative price of A to B then asks himself what his utility function is forms the relevant Lagrangian and does a few quick calculations before putting the preferred item in the shopping basket Rather the idea is that utility maximization is a way of modeling the consumer as having a welldefined set of goals and reacting intelligently to his environment He can experiment by varying his consumption away from its current levels and if the change makes him worse off he can go back to what he was consuming before The system has a feedback mechanism and by reacting to the feedback the consumer can alter his choices until they maximize his utility Sim ilarly we postulate that individuals will avoid having biased expectations if they can They react to the observed outcomes of the variables they are interested in If they find their expectations are consistently in error they will change the way they form expectations until no systematic bias remains From this it follows that if we as economists get an accurate model of the actual statistical pro cess followed by economic variables a fortiorz39 we shall also have an accurate model of individ ual expectations Note that this does not assume that individuals know our model of the economy We as economists are trying to build a model of their choice behavior and the way they form expectations If we think we have a good model of the behavior of economic variables relevant to an individual s choices and we think the individual has unbiased expectations then our model ought also to provide a model of the individual s expectations This does not mean we 190 are assuming the individual is using our model to form his expectations 1 Example 1 Independently Identically Distributed Variables Suppose the variables of interest is iid yt N61 039 with mean and variance assumed known Also assume yt is not autocorrelated that is covyt ys 0 for s i t If this is a good model of the process the individual faces then we would expect a good model of his expectations to be the mean of the random variable In that case we would have 7 37 for i21 E 7 l tyt yt for i 0 2 Example 2 Antocorrelated Variables Now suppose yt is autocorrelated Yt1P PYt13t0ltPlt 1 8tN6 2 The yr PYt1 8t yti 7 piyt 37 linear combination of 8 s and taking expectations we will get ypiy 7y for i21 Etym t 3 yt for 1 0 191 3 Example 3 Signal Processing Consider a set ofrandom variables y X1 X2 Xn with population means uy m m un all lt co and second moments about the origin Ey2 6EX 03912EX 5 03913 alllteltgt 4 Then it can be shown for those who know some analysis use the CauchySchwartz inequality l X r l S that all the cross second moments EXiX39 E Xi exist and are nite y y J y Now consider estimating y on the basis of knowing values only for the random variables Xi and the first and second moments of all the random variables in the set More speci cally we shall look for the best linear estimate of y ya0a1X1a2X2aan 5 where by best we mean Ey y 2 is minimized A necessary and sufficient condition for a0 a1 a2 an to minimize Ey a0 a1X1 aan2 6 is Ey a0 aiXi aanXi 0 i 0 n 7 or in other words y y is orthogonal to each Xi i 0 1 n M 2 Proof Put J Ey 7 aixi and differentiate J with respect to each a1 1 0 192 H gia ZEHy 7 Z aixijxk 0 for aminimum each k 0 l n 8 k i 0 from which we can conclude the orthogonality principle H E y Zaixijxk 0 k01n 9 i 0 To check the second order condition for a minimum use vector notation Put X X0 X1 X2 X11 and a a0 a1 a2 an39 and we can write 9 as EX39y EX Xa 10 But the variancecovariance matrix EX39X is nonnegative de nite so J is minimized by choosing a to satisfy the first order condition 9 2 aiXi is called the projection of y on 1 X1 X2 Xn and denoted PM 1 X1 X2 Km 11 If y and each of the X1 are normal random variables then the conditional distribution of y given the X1 is normal with mean given by Pyll X1 X2 Xn That is for normal random vari ables the projection is also the conditional expectation of y given the variables Xi see for eX ample Rao Linear Statistical Inference and its Applications pages 441442 Put 8 y 2 ain forecast error Then we can write y2aixi8 12 and from 9 E8Xi Ofori 0 l n 13 193 so that EeEaiXi 0 14 Also Ey2 EXaiXi2 E82 15 Suppose we want to estimate a signal s but see s along with noise n That is X s n 16 is observed with Esn 0 Es2 En2 lt co Es En 0 Then the best linear forecast ofthe signal given the observation of X is EsllXa0a1X l7 and 9 in this case becomes writing s a0 a1X 8 multiplying through by l and X and taking eXpectations Es l EX 30 18 ESX EX EX2 a1 that is 0 1 0 a0 19 ESX 0EX2 al which can be solved for a0 0 and a E E Ws l E52 20 1 EX2 Es n2 Es2 En2 194 Since the means of s and n are zero there is no constant term in the forecast of s If the variance of the noise goes to zero movements in X completely reveal s so the coef cient of X is 1 If the variance of s goes to zero movements in X tell us nothing about s and the coef cient of X is zero 4 Example 4 Combined autocorrelated and White Noise Now suppose yt follows the process Yt vtut 21 where u is iid N05u2 independent of v at all leads and lags and v is autocorrelated vt va wt 22 with w iid N0039W2 Assume individuals know current and all past values of y but do not know the decomposition of y into u or v in any period We need to determine Et ym when yt is given by the process described above First we will have Etyt yt For i gt 0 Ethi EtVti 23 EtVti piEtVt 24 Now observe that vt vawt so Etvt pEthJrEtwt However consumers will lea1n nothing new at t about vH since it depends on shocks from til and earlier Hence Eth EHVH and we conclude that I PLEtVt Etwt 25 195 Now use signal processing Project wt onto Ytii thiiil a P Wt7iut7i7putiiila i 2 0 26 to obtain Wt Zuiwtiiutiiiputiiil8t 27 i 0 where 8 is the forecast error orthogonal to all terms in the information set Now multiply through by each term on the right side of 27 and take expectations to obtain the normal equations 6 OLOO39V2V1pz03937061po3931 28 0 euiilpo w oev lpzgto leoci1po 29 Equations 29 are a set of difference equations in the projection coefficients Xi The comple mentary equation corresponding to 29 is kzpo iMGVZ r p2o pog 0 30 DiVide equation 30 through by 0393 and use R for the variance ratio Gav0393 pkziMRlpp0 an The roots of 31 are x R1p2iAR1p2274p2 2P R1p2iAR217p222R1p2 2P 32 Thus both roots will be real We can write the quadratic equation 31 as 196 127Kk10forKgt0 33 Hence if lt 0 W 7 K l gt 0 so both roots must be positive Finally since the coefficient of W is l and the constant term is l the product of the roots is 1 Therefore we may write the two roots as 7 and UK where 7 lt 1 We conclude that Xi All Bl7ti for constants A and B But then we must have B 0 if we are to have minimum forecast error variance essentially we have a transversality condition for the variance minimization problem The constant A will be deter mined from the first equation 28 above Using the variance ratio R 28 can be written R a0R1p27oc1p ARlp27Ap7 34 Hence using 12 lkl first term in 32 plus the square root of the discriminant R A Rk 35 Rlp27pk and we conclude that 7 11 7 R7 EtwtiRzk wutrput1immipmyraipm 36 i0 5 Example 5 Combined Random Walk and White Noise The key result from 36 is that adaptive expectations can be equivalent to rational expectations We obtain a similar result when vt follows a random walk yt0Lvtut 37 with ut and wt vt vt1 independent random variables both serially uncorrelated utN0039 and wtN0039 V 38 197 The shock vt is a random walk The key feature of a random walk is that there is no tendency for the variable to return to a particular level We shall assume that 0L and the variances of u and w are known to the consumer Assuming that 0c is known is equivalent to assuming forecasts are not biased Assuming the variances are known is a way of modeling the assumptions i The agent believes y will vary from period to period and ii The agent has some idea that shocks that have implications for longterm movements in y tend to have one magnitude given by the standard deviation of w while those that are the result of temporary shocks that last just one period tend to have a different magnitude given by the standard deviation of u This split into two shocks is undoubtedly a simplification but may be close enough to yield a satisfactory explanation of observed behavior We need to determine Etyti when yt is given by the process described above We have EtYti 0c EtVti um 06 EtVti 39 since ut is serially uncorrelated Also we can write Vti VtiVti1 Vti1Vti2 Vt139Vt Vt 40 and EtWtjj 0 0 S j lt i since wt is serially uncorrelated Therefore Etvti Etvt 4l Etyti Etvt 0c for i Z l 42 The random walk v is like the permanent part of shocks to y so to forecast future y the consumer needs to estimate v Of course current y is observed at time t so if i 0 Etyti yt 198 We again assume that the agent does not observe the random walk V and noise part u of sep arately Rather he had to form an expectation for the current value of V given his past observa tions of y Actual past y values are noisy signals on the shocks w of interest to the consumer The shocks u have no intrinsic value for forecasting future y They affect the consumer only be cause they can be confused with the shocks w to v 1 Thus we wish to calculate EthYt yt1 Yt2 43 Following our discussion of signal processing we might think of using the orthogonality princi ple This will enable us to write vt as a sum of a the best linear forecast of vt using past incomes and b the forecast error vt a0 ytoc a1yt10L 8t 44 with E yti0L8t 0 for i 0 1 2 45 We would then multiply through by each of the right hand side variables take expectations and solve the resulting equations for the coefficients a1 However we cannot use that method directly here since the variance of v and hence also of y is in nite Instead we need to alter the problem so that we are projecting a stationary variable that is a variable with finite variance onto a set of stationary variables Now observe that 139 Some students might have noticed that the proof of the orthogonality principle used a finite set of variables as the information set whereas here we are using an infinite number of variables Technically the theorem above can be proved for an infinite set of variables in the information set by using Banach Space theory 199 EtVt EtVt1 EtWt and no new information is obtained at t that will enable the individual to improve on the forecast of vt1 made at t l The information about vt1 contained in yt was already available from yt1 and previous values of y Therefore EtVtl Etthl Etth2 EtIWtl 47 and continuing in this fashion we can conclude that EtVt EtWt EHWH Et2Wt2 48 Information about wti in period ti can be obtained from the change in y from period til to pe riod ti However this change in y also contains all previous w shocks and we need to purge yti yti1 of the effects of these previous w shocks Formally we project wt against the current and previous changes in y to obtain Wt a0ytyt1 a1yt1yt2 8t 49 with the forecast error 8t orthogonal to the right hand side variables E8tyti yti1 0 for all i 2 0 50 Then note that from 48 and 49 EtVt 2 aiytii ytiiil 2 31Yt7171 Yt7172 2 31Yt7172 Yt7173 51 10 10 10 2 aimii 0 2 31Yt7171 0 2 31Yt7171 0 2 31Yt7172 0 i0 i0 i0 i0 and if we telescope the intermediate terms we find 200 EtVt Z an00 52 i 0 Now multiply 49 through by each of the right hand side variables and take expectations to get the equations called the normal equations 0 Eytryti1wtl Z aJEytjyt7j71ytryti1li0 1 25 53 j0 To evaluate the expectations in 53 use Yti Ytil Wti uti util 54 so that 2 39f 39 0 EWtwtiiutii7utiiil 6W 1 1 55 0 if i 2 1 while 6 203 if i j EYtjYtj1Yti39Yti1EWtjutjutj1Wtiutiuti1 GLzl if i 7 j 1 56 0 otherwise Substitute these expressions into the normal equations 53 to nd For i 0 63V a06 V 263 7 alo 57 For i 2 l 201 0 io aiil0393V2039 aiio39 ail 58 Equation 58 is a second order difference equation for the coefficients a1 Rewrite the second order difference equation as 52 262 0 ail W 62 Jamal 59 L1 with the corresponding complementary equation 52 20392 0 1 x M2 60 L1 The quadratic 60 has roots O2 262 2 W ZL 4 61 on Note that the discriminant is positive so both roots are real Also we may write the quadratic equation 60 as 01K7t73forKgt0 62 Hence if lt 0 l K W gt 0 so both roots must be positive Finally since the coef cient of W is l and the constant term is l the product of the roots is 1 Therefore we may write the two roots as 7 and 1 where 7 lt 1 so 62 x137 RIBwhereR W 63 2 4 53 202 The graph of 7 as a function of the ratio of the variance of w to the variance of u is as follows 7 08 07 06 05 04 03 02 01 The solution to the second order difference equation is a1 AN B1Li for constants A and B 64 Now if B i 0 y values from further into the past will have a bigger in uence in determining the current expected value of the permanent shock vt In effect we again have to appeal to a trans versality condition the relevant minimization problem here is the minimization of the squared forecast error to give us a second boundary condition forcing B 0 Thus we conclude that a1 AN 65 with A chosen to satisfy the boundary condition at i 0 2 0w Am 203 7 Mag 66 or dividing through by 5 203 RAR27AXA1I R1 67 Therefore A7 R and is graphed as a function of the variance of w relative to the variance of u as follows Increases in the relative variance of permanent shocks decrease the effect of lagged y on the cur rent estimate of v Current y shocks will reveal the current value of w more accurately making it less important to use lagged y to filter out the effect of noise ut Substituting our solution 65 for ai back into 52 we nd EtVt AYt00 MYtrOL MGR23900 68 or using lag operator notation Em m 06 69 Returning to our expression 42 for Etym i Z l we find 204 A l 7 7M 7 A EtyHi myt 70 6 Rational Expectations with Several Variables Thus far we have used the idea of rational expectations in a somewhat limited sense We have argued that in forming expectations of a variable of interest maximizing agents will use avail able information about the evolution of that variable over time In this section we extend the idea to observable relationships between alz39jfkrent variables To model expectations we need to model how agents view their own situation We cannot read their minds but we can hope that so far as their situation can be ascertained on the basis of ob jective evidence we a1rive at the same analysis of their situation as they do In short if agents exploit readily observed stable relationships between variables in forming their expectations then any model of such relationships will automatically also be a model of expectations This position is most reasonable for repeated events It may also be worth noting that in many situations we may not have to postulate that all agents form expectations rationally For equilibrium prices to behave as if all traders used all publicly available information there need only be a su iciently elastic supply of speculators who form expectations rationally For example nancial markets are often said to be informationally ef ficient in the sense that stock and bond prices incorporate all relevant publicly available infor mation In a competitive economic system with free entry into markets by speculators there should be no pro ts available for trading on the basis of publicly available information So long as there are enough speculators forming expectations rationally this will be true even if some traders ignore publicly available information Muth s Model of an Agricultural Market As an example of a rational expectations equilibrium we shall look at a variant of Muth39s model of an agricultural market in Rational Expectations and the Theory of Price Movements re 205 printed in the Lucas amp Sargent readings Muth postulated that demand depends negatively on the current price of the product x9 5p 71 while supply is given by X YEt1Pt ut 72 Falmers must decide at time tl on their planting of the crop or their decision to retain animals for future sale They make their decision on the basis of their forecast of the likely future price for time t the time of sale using information available at time t l The random error term ut rep resents a shock to supply such as uctuations in the weather Suppose that ut evolves according to ut put1 8t 0 lt p lt l and that farmers know p and the distribution of 8 at N0 03982 and Eetes 0 ifs t 73 As special cases if p 0 the ut shocks are independent and the farmers have no advance infor mation about future weather on the basis of past weather and as p gt 1 ut becomes a random walk and there is no tendency for the weather shocks to return to some normal level Guessing the Equilibrium Pricing Function We postulate that the equilibrium price is determined to equate demand to the available supply The equilibrium price will therefore be a random variable since supply is random We want to solve for the equilibrium price as a function of the shocks Since both the supply and demand curves are linear and ut is first order so only ut1 observed at time tl will help forecast ut we guess the equilibrium price will be 206 pt out Thu 74 If our guess is correct and expectations are rational this will also provide a model of expecta tions Etlpt TEoEt1Ut T iEt1Ut1 75 Since we have assumed ut1 is observed at time t we have Etlut PEtlutl Et18t Put1 76 so that Et1pt7c0p 71ut1 Verifying the Guess is Correct Now equate supply to demand and substitute our guess for pt and therefore Et1pt into the result ing equation to get B out BMUtl YTEoPUt1 With1 ut 77 Hence our guess for the equilibrium price function will be correct if we choose 570 1 78 5751 YITOP t W1 79 The equations 78 and 79 can be solved for 1 no 7 7 and 717 B 80 pi BltY B Observe that as p gt 0 so that farmers have no advance information about the weather 7E1 gt 0 so that supply and therefore equilibrium price is unaffected by weather at tl 207 7 Identification in Rational Expectations Models Modify the supply curve above so that while planting depends on Empt harvests depend on pt Also consider the model from the point of view of an econometrician and add error terms to rep resent measurement error in the variables Xtd 39 Bpttgn 81 X Olpt t YEt1Pt t ut t 82t 82 where 8 and 82t are white noise Now we have a pair of simultaneous equations determining the two endogenous variables pt and Xt as a function of the exogenous variable weather Since the exogenous variable ut appears in the supply curve but not the demand curve it would appear that only the demand slope coefficient 5 is identi ed The idea of identi cation is illus trated in the following diagram pA Identification Movements in the S curve will lead to equilibrium p Q combinations on the D curve Movements in the D curve lead to equilibrium p Q combinations on the S curve When we collect data from this market all we see is a set of p Q points To estimate the simul taneous equation model we need shift variables so we can distinguish the S and D curves Shift variables for the D but not the S curve enable us to identify the slope of the S curve while shift 208 variables for the S curve but not the D curve enable us to identify the slope of the D curve Now suppose there is a demand for an inventory of the product in addition to current consump tion Modify the demand curve in addition to the supply curve considered above Xtd 51 VEtpt1 Pt t 8n 83 X 113W YEt1Pt t ut t 82t 84 Now the current weather will in uence Et pm and hence the demand for the product We can no longer use an exclusion restriction to identify the demand curve This applies more generally Rational expectations implies that all exogenous variables will be relevant to the formation of expectations anal therefore all exogenous variables will appear in all equations where expec tational variables appear Exclusion restrictions cannot be used to identify parameters This is a strong criticism of the econometric methods which have been used to estimate ISLM type models Such models are a set of simultaneous equations with expectational variables in virtually every behavioral equation Typically exclusion restrictions have been used to identify and esti mate the parameters of those models While the assumption that expectations are formed rationally vitiates the use of exclusion restric tions as ameans of identifying parameters rational expectations provides crossequation restric tions on parameter values which can be used for identification instead For example in the case of the permanent income hypothesis consumption function the assumption of rational expecta tions gave us restrictions relating the parameters of the consumption function to those of the in come process In the present model we can also use crossequation restrictions to identify parameters For example returning to the equations Xtd Bpt 8n 85 209 th 113W YEt1Pt ut 82t 86 and assuming now that the 8 shocks are in uences on supply and demand which can be observed by market participants but are not observable to the econometrician we can guess an equilibrium price function Pt 7E011t int1 75281t 75382t 87 yielding the equilibrium relationship BTE0ut TElut1 7128 E3820 8n 06739E0ut T lutq 7128 E3820 YTE0p 7E1 ut 8288 from which we can conclude that our guess is correct if we choose no 751W752 5753i 5 89 Thus the econometrician could estimate the reduced form equations ut put1 8t 90 Pt 7E011t TE1Ut 1 75281t 75382t 91 Kt B out BMW1 t 1575281t 575382t 92 From90 we can get an estimate of p and from 91 estimates of no and E1 The estimates of no and m can then be used with the estimates of 570 and 571 obtained from 92 to yield an estimate of B in fact 5 is overidentifzed the model implies a testable parameter restriction Given esti mates of B p no and E1 estimates ofOL and Ycan be recovered Contrary to our impression based on looking at exclusion restrictions alone all parameters are identi ed in this model Rational expectations implies that the exogenous variables should enter the equation in a particular way and this leads to nonlinear restrictions on the parameters which can be used to test the validity 210 of the model 8 Using Error Variance and Covariancesfor Identi cation Now suppose the coef cient of ut in the supply equation is an unknown parameter 5 which we also need to estimate Then we nd 7E0 7 5 instead of 7 1 93 on 5 0c and we cannot identify 5 0L and y from the estimated 7 coef cients but we can still identify There is no simple rule such as counting the number of included and excluded exogenous vari ables that can be used to determine which parameters if any are identi ed in a rational expec tations model However restrictions on the error terms can often be used to identify parameters when coef cients alone are insuf cient In the above example suppose we impose E81t82t 0 ie the two measurement errors are uncorrelated Then the theoretical model will imply restrictions on the variances and covariances of the error terms in 91 and 92 l Var 291t 382t ioi 53 XB2Gi6 94 l var17BTE281tBTE382t szoiwzozi 95 l COVTE281tTE382t17Bn281t7BTE382t XB2x0391275039 96 Thus estimates of the variances and covariance of the error terms in 91 and 92 can be used along with our estimate of B to recover estimates of 0c and the variances of the structural distur bances 81 and 82 Then the estimates of no and 7E1 can be used to recover estimates ofy and 5 211 The main point here is that we have to use new methods to identify and estimate economic mod els when expectations are formed rationally If we attempt to use exclusion restrictions in the presence of rational expectations we might obtain incorrect parameter estimates and make in valid inferences It is important that we specify models where the structural parameters can be identi ed for es sentially the same reason we gave for having a correctly speci ed model for econometric policy evaluation When policy is changed we might have some hope that structural parameters will remain fixed or change in predictable ways However reduced form parameters are almost guar anteed to change following a change in the policy regime 212 A SIMPLE REPRESENTATIVE AGENT ECONOMY Throughout this discussion we ignore the question of exactly when it makes sense to use a rep resentative agent The first semester microeconomics course will discuss some conditions that suffice to allow one to aggregate preferences As you shall see these are rather restrictive One area of focus for the second semester macroeconomics course is models of heterogeneous agents where aggregation is not possible This is currently an active area of research in macro economics Later in this course we shall also discuss a couple of simple models where heterogeneity results in aggregate outcomes that cannot be explained in a representative agent model It is easier how ever to discuss many of the basic ideas of macroeconomics in representative agent models Production Technology The model has one economic unit which can be thought of as a combined household and firm We assume there is just one factor of production the labor services of the representative house hold For the time being we take the amount of capital as fixed Using the per capita labor input of n units per period the firm produces c units of consumer goods per capita We assume the production function relating output to labor input is given by 0 Km 1 is continuous increasing and concave Since f is increasing or f gt 0 the marginal product of labor is always positive Since f is concave so that f lt 0 the marginal product of labor decreas es as n increases We shall also assume that f0 0 so that if there is no labor input there is also no output The production function is graphed in Figure 1 cfn V I1 Figure 1 Production function in a simple representative agent economy 2 Household Preferences 0 We assume the representative household places a positive value on additional units of consump tion 0 but dislikes supplying additional labor services We use the utility function Ucn to rep resent the utility of 0 units of consumption together with n units of labor supply We assume that U is continuous in both arguments Since more c is preferred to less U1cn gt 0 Since less work is preferred to more U2cn lt 0 We shall also assume there is decreasing marginal utility of consumption U11 lt 0 and increas ing marginal disutility of work U22 lt 0 More precisely we shall also assume that U is strictly concave Indi erence curves for the representative household are graphed in Figure 2 cll UcnU2 UC nU1 Uc nU v 0 V I1 Figure 2 Indifference curves for market versus nonmarket activity 0 Observe that the slope of an indifference curve can be obtained by differentiating Uc n 2 U0 2 with respect to n BUdc BU a a a 0 3 Thus the slope is given by dc 3U BU E a ail 4 0 Suppose the representative household receives profits of the representative firm together with la bor income It will therefore choose 0 and n to maXUc 11 subject to c 1 v Vn 5 C H p p Note that this budget constraint is a straight line with intercept rtp and slope w p in cn space as illustrated in Figure 3 ell slope y P 1139 Figure 3 Choosing labor time and consumption to maximize utility Use the budget constraint to eliminate c The household will then choose n to maxUC E v Vn n 6 H P P This maximization problem leads to a first order necessary condition FONC for a maximum U1 U2 0 7 That is the slope of an indifference curve is set equal to the slope of the budget constraint wp or the marginal rate of substitution between consumption and non market use of time is set equal to the real wage The second order sufficient condition SOSC for a maximum is given by W W W Dz U U U U lt0 8 p llp 2l 12p 22 0 An increase in per capita real firm profits holding the real wage rate fixed will result in a pure income effect 5ll Figure 4 The income effect on labor supply and consumption 0 In algebraic terms we have that the partial derivatives U1 and U2 are functions of c and n with c an implicit function of n given by the budget constraint in 5 Totally differentiating the first order condition 7 with respect to rtp we get dn w D7 U U 0 9 dltnpgt 13 12 O so that even if the SOSC 8 holds the effect on n is ambiguous If both consumption and leisure are normal goods the income effect will increase c and reduce n as illustrated in Figure 4 A substitution effect is measured by the change in the maximizing c and n holding the level of utility xed 0 An increase in the real wage holding per capita real profits fixed will rotate the budget line anti clockwise around its intercept on the c axis As shown in Figure 5 this will lead to both a sub stitution effect and an income effect income effect 1139 Figure 5 Income and substitution effects from a change in the real wage Algebraically we again differentiate the FONC but now with respect to wp noting that c is also a function of wp via the budget constraint Thus we obtain dn w D7U nU U 0 10 dWp 1 11 p 12 If leisure is a normal good the term in square brackets will be negative But U1 gt 0 so the effect on n is still ambiguous If both consumption and leisure are normal goods the positive income effect will involve the household doing less work taking more leisure and consuming more goods The substitution effect will always involve a move around the indifference curve to the right to where the slope of the indifference curve matches the new higher slope of the budget line the higher real wage Thus the substitution effect will involve the household being encouraged by the higher real wage to supply more effort 11 increases and consume more output in exchange Basically the higher real wage makes satisfying one s desires through market activity as opposed to non mar ket activity leisure more attractive so the household substitutes the market for non market 14 activity Thus the substitution effect of a higher real wage encourages more work and more con sumption Both the income and the substitution effects of a higher real wage favor increased consumption However while the income effect encourages less work the substitution effect encourages more so the net effect on work effort of an increase in the real wage is ambiguous 3 Equilibrium and optimal allocations We are interested in two ways of allocating resources in this economy i competitive equilibrium with p and w given to the consumer and the firm where p is the price of the output c and w the price of the input n ii choose c and n to maxUc 11 subject to c S fn 11 c n 0 The fundamental welfare theorems applied to this example state that if we have a competitive equilibrium pwcn then cn will be a Pareto optimum and conversely if we have a Pareto optimum then we can find a price vector pw so that pwcn is a competitive equilibrium 0 Suppose we have a competitive equilibrium as illustrated in Figure 6 The representative firm will choose output c and labor input to maximize profits TEpC wnpfn wn 12 taking prices as given Note that real profits rtp will be given by rtp c wpn 13 which is the vertical distance between the production function and the straight line in Figure 6 The firm will thus choose n to maximize 12 which leads to the first order condition f n wp 14 or the marginal product of labor is set equal to the real wage slope wp wpn TLp TLp gt Household problem n Firm problem n Figure 6 A competitive equilibrium 0 The lines in these two diagrams in Figure 6 both have slope wp At a level of w p which rep resents equilibrium in the labor and also the product market the points of tangency in the two diagrams will be identical households and firms will choose the same levels of n in an equilib rium We can therefore combine the two maximization problems on a single diagram as in Figure 7 slope wp TLp Figure 7 The competitive equilibrium on one diagram 16 0 Now we can erase the common price line from Figure 7 and get the solution of maxUc 11 subject to c S fn 15 c n that is the solution to the Pareto Optimal problem graphed in Figure 8 C Figure 8 The Pareto Optimal problem 0 To go in the other direction suppose we have a Pareto optimum as illustrated in Figure 8 Then we can put a straight line through this equilibrium point and have it neither inside the preferred consumption set nor the production possibility set the price line separates these two convex sets 0 Note that corners on the production or indifference curves do not affect these results so long as the sets remain convex The purpose of these results from our point of view is that it is simpler to maximize U subject to the technological constraint than to solve for the competitive equilibrium particularly in the dy namic models we will be discussing 4 A more general algebraic formulation of the welfare theorems We can demonstrate the results algebraically as follows 0 Definition A competitive equilibrium inthis economy is a set of four numbers c0 no p w such that c0 S Km and i Ucn gt Uc0 n0 gt pc wn gt pco wno ii pc wn gt pco wno gt c gt fn Question What happened to firm profits in specification i of the consumer39s maximization problem 0 Definition c0 no is an optimum if Ucn gt Uc0 no gt c gt fn Let c0 n0 p w be a competitive equilibrium Suppose Ucn gt Uc0 no Then from i pc wn gt pco wno and from ii c gt fn Note that the steps do not depend on dimensionality or assumptions about U and f The converse requires more mathematical machinery In particular we need the separation the orem in n dimensions This theorem says one can always separate convex sets by a hyperplane More specifically Theorem Let A and B be two closed convex sets in n dimensional space such that A has at least two points and no interior point of A is in B Then there is p1 p2 pH not all 0 and a number c such that x E A gt21pixi 2 c i xe Bgtzpixi Sc 16 i 0 In our example we take A cn Ucn 2 Uc0 no 18 B cn c S fn 17 Since we also have n 2 n 2 0 and c 2 0 under our assumptions on U and f both A and B are closed convex sets Question What happens if A and B are not convex sets By our choice of co no as a point of tangency between an indifference curve and c fn there are no interior points ofA in B Applying the separation theorem El p1 p2 a such that if cn E A then p1c p2n 2 a and if cn E B then p1c p2n S a That is if Ucn 2 Uc0 no then p1c p2n 2 a and if c S fn then p1c p2n S a Now c0 no E A so that p1c0 p2n0 2 a and c0 n0 E B so that plco p2n0 S a Hence a plco p2n0 and we get Ucn 2 Uc0 n0 gt p1c p2n 2 plco p2n0 and C S fI1 gt P1c P211 S P100 P2I10 From the last statement if p1c p2n gt plco p2n0 then c gt fn But to show we have a compet itive equilibrium we want the inequalities in the first statement to be strict This is the difficult part of the theorem and I will leave further discussion to your micro courses One use of this theorem is to demonstrate the existence and uniqueness of the competitive equi librium We know the optimum exists because it results from the maximization of a continuous function over a closed and bounded set this is a result from analysis and I again leave further discussion to your micro courses The optimum is also easily shown to be unique under our as sumptions on U and f If it were not unique there would be two equally preferred optimal points 19 Any point on a line joining these will be interior to the production set because it is a convex set These points will be preferred to the two original points by the strict concavity of U contradict ing the assumed optimality of the original points It then follows from the theorem that the equi librium also exists and is unique The equilibrium prices aren t unique because if c n p w is a competitive equilibrium then so also will be c n 5p 5w for any 5 gt 0 5 Application Keynes criticism of classical economics 0 0 Prior to The General Theory of Employment Interest and Money economists had focused on two major explanations for business cycles i Many classical economists such as Hume had noted that business cycle fluctuations ap peared to be associated with monetary disturbances However their economic theories generally implied that money supply changes should be neutral Microeconomic theory implied that sup pliers and demanders were concerned with relative prices so that the nominal price level should be irrelevant for determining the allocation of resources Hume argued that the empirical evi dence on the long run effect of money supply changes was consistent with the implication of the theory that once over money supply changes will change the level of all nominal prices while leaving resource allocation unaffected While Hume and other quantity theorists conceded on the basis of empirical evidence that short run money supply changes did not appear to be neutral with respect to real variables Keynes and other economists argued that such short run non neu tralities were not consistent with the formal models of classical economics ii The Austrian school of economists in particular but also Wicksell in Sweden argued that disturbances to the aggregate level of investment played an important part in business cycle fluc tuations As we shall see later in the course when we examine the evidence this seems to be a reasonable hypothesis Durable goods output moves more than the output of non durable goods and services over the business cycle Furthermore there is no doubt that technological change can produce high variance stochastic movements in the level of output corresponding to a given level of inputs and hence the marginal products of physical and human capital Classical micro 20 economic theory would then imply that technological change could produce random variation in the desired level of investment The Austrian economists Keynes in his earlier writings Wicksell Fisher and others had in the early part of this century attempted to link the two ideas of the source of business cycle fluctu ations by analyzing the way money supply changes might in uence the financing of investment The Austrian economists in particular argued against The General Theory ofEmployment Inter est and Money on the grounds that postulating autonomous fluctuations in investment avoided the central issue of macroeconomics of explaining the connection between monetary disturbanc es and investment and other real variables The main innovations in The General Theory were i It explicitly attempted to explain the levels of output and employment at each point of time rather than uctuations in variables Keynes did not even discuss the trade cycle until chapter 22 of his book and then did so only to point out that his theory of the determination of the levels of employment and output automatically accounted for the business cycle ii Keynes focused attention on the level of employment of labor in addition to including a role for investment and monetary phenomena in his discussion It was a major innovation to lead off discussion of macroeconomic phenomena by concentrating on the labor market As we shall see employment does not appear to be the most cyclical of variables iii Keynes denied that macroeconomic phenomena were inherently short run in nature and suggested instead that prolonged periods of unemployment were a likely outcome in a market economy he introduced the notion of an unemployment equilibrium Keynes positioned himself vis a vis a classical theory of employment We can caricature his argument to some extent by adding a quantity theory money demand equation to the simple representative agent general equilibrium model model we have discussed above 21 c y Given real output determined from the real economy slope wP will determine nominal p and w but cannot affect y wp or n TLp In this model we have pf n w which is Keynes postulate I Also the marginal utility of the wage U1wp and the marginal disutility of employment 2 U2 These are equal from the household maximum problem and this yields Keynes39 postulate II 0 The variables one could observe here are y n w and p Keynes focuses on changes in employ ment n He lists the forces which can affect n given the theory p7 of the General Theory Apart from sectoral features the forces are shifts in U and f Are shifts in these sufficient to explain large scale fluctuations in employment such as occurred in the Great Depression Keynes an 1 swers this question in the negative Keynes retains y fn and f n wp but does away with the postulate that households deter mine labor supply by maximizing Uy 11 He introduces as an axiom fixed money wages This allows money supply changes to affect real variables The three variables y n and p are then de termined by 139 There is also a methodological criticism of relying on changes in U and f to explain business cycles Unless we have a theory which restricts the changes in U andf we are able to observe it is doubtful if the theory is testable If we allow arbitrary changes in U and f we can explain 7 or rationalize ex post 7 almost any observations What then would count as a good test of the theory How can we ever know whether the the ory is right or how good it is 22 f39n wOp 18 y ll 19 MV py with M and w0 given as exogenous variables To determine the effect of changes in the exoge nous variables M and w0 on the endogenous variables y n and p we totally differentiate this sys tem of equations w0 1 z 0 0 f 2 dy p dwO P dn 21 1 f 0 d 0 0 dM P p 0 y 0 v which we can write in matrix notation as A dy B dx 22 where y endogenous variables x exogenous variables The determinant of matrix A is given by detA f quoty f wOp lt 0 23 and the solutions for the effects of the exogenous variables on the endogenous variables are giv en by Cramer s rule 0 l f W dyOP P2 f y0lt0 24 dw detA O f 0 f yp f w 00 y 23 dn dw dp dw detA 1 f 0 1w 0 2 pp2 y 0lt0 detA 1 0 0 fypfw p0y 1 0 f I 1 p fp gt0 f yp f wO p 0 0 and similarly we get the effect of M on the endogenous variables as dy m E dM dp m 0 l W 1 0 f 2 vf w0 gt0 detA 0 f 0 fIyp2fW0p v 0 y W0 0 0 F in detA 1 0 0 fIyp2fW0p P V y 1 ff 3 Lgt0 detA y 0 p 0 V f yp fw 25 26 27 28 29 so that a money supply shock produces a pro cyclical movement in output employment and the nominal price of output p If in the longer run wO adjusts upward to the higher p real output and employment will fall and p rise further From the original equations we can see that an adjust ment of p and w proportional to a change in M will lead to the same values for y and n Short run fixity but long run exibility of the nominal wage appears to offer the promise of providing an 24 explanation of short run non neutrality but long run neutrality of money supply changes 0 As an initial working hypothesis the assumption of a fixed nominal wage appears to hold some promise However Keynes wanted to include investment in his model while he also wanted to explain the determination of the interest rate To discuss the Keynesain model in any further depth we clearly need to move beyond the simple static general equilibirum model presented above In particular one cannot discuss the determination of interest rates without introducing intertemporal choices into the model 6 An Empirical Application of the simple general Barro notes that a common feature of economic development is that hours of work tend to de cline early in the development process whereas in later stages of development hours of work tend to be relatively constant Although this is a feature of an equilibrium economy as a result of the theorem we can discuss it using the Pareto optimum of the representative agent economy Although it is jumping ahead of the story to some extent we associate development with an increase in output for a given level of labor input that is increases in labor productivity Graph ically development will be associated with an upward movement of the production function However the production function will continue to pass through the origin since zero labor input will yield zero output regardless of the level of development Thus the production function will also rotate around the origin as productivity of labor increases The development process is thus represented by the sequence of production functions in Figure 9 Then if at low levels of development the tangency between the representative agent s indiffer ence curve and the representative firm s production function is far to the right hours of work will initially be quite large More to the point increases in development will be associated more with an upward shift of the production function and less with a rotation of it The income effect will be strong and the substitution effect weak Labor supply will decline 25 0 At higher levels of development the tangency between the representative agent s indifference curve and the production function will be further to the left Here additional growth in produc tivity will produce more of a rotation of the production function The substitution effect will be larger and the resulting increase in labor supply can largely counteract the negative impact on labor supply of the income effect so decreases in hours of work largely disappear growth in labor productivity V I1 Figure 9 Effect of growth on the production function Figure 10 Labor supply total hours as the economy grows 26 0 The indifference curves in Figure 10 have also been drawn to reflect the idea that as market con sumption increases non market activity becomes relatively more valuable so that long hours of work bring greater disutility This effect also contributes to the decline in hours of work as de velopment proceeds Nevertheless when the substitution effect becomes large actual hours of work may not change much as development proceeds 7 Allocation over Time and under Uncertainty While this static equilibrium theory might appear limited in scope its applicability can be ex tended by reinterpreting the commodities in the general algebraic n dimensional version 1 For intertemporal resource allocation we can call consumption at different times different goods Prices then become futures prices though we will give an alternative interpretation to them later in the course 2 Uncertainty can be introduced by thinking about the future as a list of possible states of nature which might prevail tomorrow taken as known by everyone and agreed upon Define a set of new commodities by distinguishing which state has occurred The same good in two different states becomes two different commodities in the model Prices for these different commodities are called contingent prices Note that preferences between the various contingent commodities will be affected by the probabilities of occurrence expected of the various possible states con cerned The equilibrium prices will also in general depend on the probabilities of occurrence of different states This reinterpretation of the model widens its scope we can talk about changing economies and economies subject to random but predictable shocks The model should also warn us that there is no easy way to distinguish equilibrium from disequilibrium when the equilibrium we are talk ing about can involve such abstract commodities We can ask whether a particular model of the equilibrium is a good description of reality but its difficult to say whether a particular situation is or is not a competitive equilibrium In particular equilibrium does not mean unchanging in this model In modern microeconomics equilibrium resource allocations and prices are dy 27 namic changing over time and stochastic depend on random variables that determine various states of the world 8 T woperiod ConsumptionSaving Model 0 Before we look in later chapters at a more general intertemporal and stochastic equilibrium mod el it is useful to consider a simple two period model 0 Suppose a consumer has income y1 in period 1 and y2 in period 2 Let the interest rate on one period savings or borrowings be r with y1 y2 and r all known with certainty The consumer chooses c1 c2 to maXUc1 c2 30 Cl CZ subject to the intertemporal budget constraint Y2 l H 31 02 c 1 1 r yl The budget constraint 31 equates the discounted present value of consumption to the discount ed present value of income2 0 We can represent the consumer s maximization problem as in Figure 11 0 In a simple special case the intertemporal utility function is additively separable between con sumption today and consumption tomorrow and can be written as U0102 U01 BU02 32 239If we let B the quantity of one period bonds held at the end of the first period then B yl 7 c1 Consumpi tion in the second period will then equal second period income plus the bonds from the first period plus the interest earnings on those bonds c2 yz B Br yz Y1 C11r This equation can be rearranged to give 31 28 Here 5 lt 1 is the time discount factor and p defined by 11p B is the rate of time discounting period 2 y11ry2 U 01102 slope 41H yl y1 1L period 1 I Figure 11 The two period consumptionsaving model 0 The slope of an indifference curve is found by differentiating uo Ucl02 to get 3U 3U dcz TOETCZE 33 that is dcz 3U 3U U1 a a a U 34 In the additively separable case 34 becomes dc2 U cl 7 35 dc1 BU 02 29 so the slope is 15 along the 45 line where 01 02 as illustrated in Figure 12 slope NB Figure 12 Indifference curves for an additiver separable utility function 9 Algebraic Solution for the ConsumptionSaving Decision 0 Now we want to examine the maximizing choices of the consumer using algebra The consumer chooses 01 and 02 to maximize intertemporal utility 30 subject to the intertemporal budget con straint 31 Transform the problem into an unconditional maximization Given cl consumption in period 2 02 is given by 02 Y2 Vi Ci1r 36 after rearranging the constraint 31 Second period consumption equals second period income plus first period savings and the return on those savings Then the consumer chooses 01 to max Ucly2y1 cl1r 37 1 0 The first order necessary conditions FONC for a maximum are U1 U21r0 38 30 with second order sufficient conditions SOSC DU11 U12l1 U2111 U22l1 2lt0 that is U11 2U121r U221r2 lt 0 which we shall assume to hold Observe that the first order condition 38 amounts to requiring that at the optimum the slope of an indifference curve equal the slope of the budget constraint This is what Figure 11 implied period 2 ll Case 2 c1 decreases Case 1 c1 increases YZ r r r 7 r r r 7 n period 1 Figure 13 A change in income in period 1 0 Now consider the effect of a change in y1 keeping y2 and r constant Graphically the effect can be illustrated as in Figure 13 A change in income in either period leads to an income effect which can either increase or decrease c1 depending upon whether or not c1 is a normal good Algebraically we differentiate the FONC 38 that gives the maximizing c1 as an implicit junc lion of the parameters of the problem with respect to y1 31 a D U121 r U221 r2 0 41 By1 that is 3c1 1 rU12 U221 r a D 42 which can be positive or negative 3c For an additively separable utility function U12 2 0 and 71 gt 0 3y1 In what follows we shall assume that neither c1 nor c2 is inferior That is we shall assume that 3c U is such that 1 gt i gt 0 In light of the FONC 38 this is implied by the following restrictions 1 on the utility function U U12 1rU22 Ulz U iU22gt0 43 and U1 Ull 1rU12 Ull U 2U12lt0 44 Henceforth we shall assume U satisfies 43 and 44 Now consider the effect of a change in r holding y1 and y2 constant Figure 14 graphs the effect 32 of an increase in r y1 Figure 14 An increase in the interest rate 0 From Figure 14 we conclude that there will be a substitution effect tending to decrease c1 and an income effect that depends on whether the individual is a net borrower or lender in the first period If households on average are net lenders to the rest of the economy firms and govern ment the income effect for most households will be positive Algebraically we again differentiate the FONC 38 with respect to r 3c Dair1U12y1 c1 U2 1 rU22y1 c1 0 45 thatis 3c U y c U 1r U ail 2 1 1 22 12 46 r D and 3c1 Tylgt0gtU221r U12lt0 47 33 so that 3c 1 0 Brlt 3c 1 y1 c1gt 0 gt air can be pos1t1ve or negative while y1 c1lt 0 gt 0 In theory an increase in interest rates need not necessarily decrease average current consump tion or equivalently for fixed current income increase average savings although this is a common assumption Implicitly the assumption is that substitution effects dominate income ef fects 10 Combined labor supply and saving We now allow for variable labor supply in the two periods but specialize the intertemporal util ity function to be additively separable Thus the utility of the representative consumer worker household becomes UCly 111 BU02 n2 0 Suppose the household can work in either period and can save or borrow at the interest rate r The household also receives non labor income from the firm in each period Now let 11 and mi denote the real profits and real wages in period i The intertemporal budget constraint then be comes c 152 02n2 lt151coln1 c1 2 1r 1r Now that we have four choice variables we cannot use the budget constraint to eliminate one of the variables and thus obtain a simple calculus problem involving maximization of a function of one variable We need to consider constrained maximization Define the Lagrangian mction for the representative household as 34 152 cozn2 c2 L Uc1 n1 BUc2 n2 9 11 coln1 c1 1 H where the Lagrange multiplier 9 on the budget constraint written as an inequality that is 2 0 is positive We shall shortly that the multiplier has the interpretation in this case of the marginal value to the household of an increase in period 1 real income 0 The constrained maximum of the original function is now found by maximizing the Lagrangian function with respect to the choice variables c1 c2 n1 and n2 and minimizing it with respect to the multiplier 9 0 The first order conditions are 33 9 48 Ci 5 Ii 49 3731 9 co1 50 53 2r 51 9 11coln11wr2nZ cl lc 0920 1 1n11wr2I1220110 r 52 0 Note first that if the marginal utility of consumption is strictly increasing no satiation then 48 or 49 imply 9 gt 0 and then 52 implies that the budget constraint must hold with equality We shall assume that this is the case We then have five equations 48 51 and the budget con straint as an equality to solve for the five endogenous variables c1 c2 n1 n2 and 9 35 0 From the household s perspective r col 602 11 and 152 are exogenous variables A change in any one of these variables will generally change all of the endogenous variables We can write the solutions to the first order conditions for c1 c2 n1 n2 and 9 as functions of the exogenous variables col 602 11 and 12 0 It is useful to consider a change in 11 in particular This will generally change each of the endog enous variables Differentiating the Lagrangian function with respect to 151 we can write dL BL ac1 BL 802 BL 8111 BL 8112 BL The first order conditions for the endogenous variables imply however that the partial deriva tive of the Lagrangian function with respect to each endogenous variable is zero so the effect of 151 on L is given simply by the partial derivative This result is an example of the envelope theo rem In our case the envelope theorem implies that 9 dLd rtl so that the Lagrange multiplier 9 has the interpretation of the marginal value of real income in period 1 as suggested above Note that the first order conditions 48 and 49 again imply BU BU B1rai02 Tel which can be interpreted as an intertemporal arbitrage condition Specifically forgoing a unit of consumption in period 1 costs in utility terms BUBcl Saving the released resources allows 1r units of consumption in period 2 The marginal value of each unit of increased consump tion in period 2 is BUBc2 in period 2 utility but this has to be discounted by B to yield a corre sponding period 1 utility value 0 Equations 50 and 51 can similarly be solved to yield 36 0 In general a change in the real interest rate for example will also change labor supply This is known as intertemporal substitution in labor supply Households are concerned not only about the trade off between market and non market activity but also about when they supply labor time to the market For example labor time might be bunched into workdays with non market time collected into a block at the weekend In order to explain such bunching however the above simple specification of preferences is inadequate Perhaps a better approach is to introduce a technology for producing utility from non market activity It is quite feasible that such a tech nology is likely to have a fixed cost component so a certain minimal amount of time is required to make non market activity worthwhile We can use this simple model to discuss some basic issues regarding fiscal policy Introduce government spending and taxes For simplicity we ignore issues such as public goods and ex ternalities and assume that the government spending is worth to households what it costs to sup ply Let the real spending in period i be g and now assume that utility is U01 g1 111 BU02 g2 112 Assume initially that taxes Ti in period i are lump sum Then the budget constraint of the house hold assuming non satiation so it holds with quality becomes 12 2 02n2 TEl T1 601111 2 c 1 1r 1r The household again chooses ci and ni but now takes as given g and Ti The Lagrangian and first order conditions now become 37 n2 12 cozn2 c2 L Uc1 g1 n1 5Uc2 g2 n2 KPH 131 coln1 1 c 1r 1 1r c1 g1n1 9 53 302 ggyng Ii 54 3731c1 g1 n1 2 01 55 B E2 112 i 56 c11 rTrl rlcoln11r 2n2 57 0 In this two period world the government also faces a budget constraint If expenditure exceeds tax revenue in period 1 the government has to sell bonds to finance the short fall These must then be redeemed in period 2 together with interest by raising revenue in excess of period 2 ex penditure g1 111r Tzg2 58 0 Now let 6 62 E1 and 2 solve the original problem and consider c1 61 g1 c2 62 g2 n1 2 E1 and n2 2 32 as candidate solutions to the problem involving government The govern ment budget constraint 58 ensures that the latter satisfy the household budget constraint 57 Furthermore since 61 62 E1 and 32 solve the original problem the candidate values will solve the first order conditions 53 56 for the revised problem If U is such that the solution to the original problem is unique the candidate solution will also uniquely solve the new problem 38 0 This result is known as Ricardian equivalence In this model it makes no difference whether the government finances expenditure with current taxes or through selling bonds Actually in the above model we have more The government has no effect on resource allocation at all Private agents simply offset whatever actions the government takes One reason for this strong neutrality result is that we have ignored public goods and externalities Another is that we have assumed taxes are lump sum 0 We can think of two types of distorting taxes with different bases consumption and income taxes In the case of a consumption tax at rate Ti per unit of consumption in period i the house hold budget constraint changes to c21 12 12 cozn2 c1T 1 con 1 1 1r 1 11 1r In order to consume ci the household has to relinquish real income ci1 ci The key difference from the previous model is that the taxes now enter the first order conditions Zclglnl A1 Cl 59 502 2 112 9 60 31M g1n1 tc01 61 302 g2 112 b 02r 62 Now taxation must alter the equilibrium We can no longer obtain the same solution as in the model without government Furthermore if the Pareto optimum is unique the new equilibrium 39 must yield a level of welfare that is inferior to the original equilibrium which corresponded to the Pareto optimum Thus the new equilibrium will be inefficient and the taxes are distorting Similarly in the case of an income tax the household must pay a tax at the rate 51 on labor in come earned in period 1 and a tax at rate 52 on labor and interest income earned in period 2 However if the household borrows interest payments are not tax deductible unless they are used to buy real estate and are payments on a mortgage We assume for this discussion that any borrowing is a pure consumption loan such as a credit card debt The budget constraint now becomes c2 11 colnl1 31 c11 r1 52 1l392 602n21 52 63 when the household saves in period 1 and c1 1r1 colnl1 rl1 r 1r2 602n21 52 when the household borrows in period 1 For the remainder of this discussion assume the house hold saves in the first period which arguably is more consistent with the representative agent assumption and the fact that firms and usually also the government are net borrowers To en sure that the multiplier has the same interpretation re arrange the budget constraint 63 as c1 c2 co 12 602n21 12 l rl1r1 121 Tl quot1 1n11r1 521 Tl The first order conditions for a constrained maximum analogous to 59 62 now become 3U 871101 l39glyn1 1 Tl 64 3U 9 367 gm 65 40 BU aTllc1 g1 n1 2 01 66 M020 12 1r1 521 Cl 67 3 2 g2 112 2 2 Again we get a distortion between consumption and leisure Now however there is an additional distortion to saving The income tax reduces the incentive to save relative to an otherwise equiv alent consumption tax We also conclude as with the consumption tax that the income tax pro duces an inefficient allocation 0 Over the doors to the Internal Revenue Service are inscribed the words of Justice Oliver Wendell Holmes Taxes are the price we pay for a civilized society Suppose public goods such as a good legal system are indeed a necessary component of a civilized society What is the differ ence between a tax used to pay for a public good and a price charged for a private good In particular why do taxes distort behavior and produce inefficiency while prices charged for a good do not 11 Multiple Period Budget Constraint with Oneperiod Loans 0 Return to the case where we do not distinguish labor and non labor income and we do not exam ine the labor supply decision Now suppose however that the representative consumer will live for n periods earning a real income y1 y2 yn in each of those periods Let the consumptions for the n periods be c1 c2 cn First period real savings will be yl c1 giving a total second period income y2 y1 c11r Second period savings therefore are y2 c2 y1 c11r Third period real income will then be y3 y2 c21r yl c11r2 Continuing in this way the total real income available for consumption in the nth period will be cn yn yn71 n11r y1 c11rn391 68 41 0 Equation 68 can be rearranged so that it says the present value of consumption has to equal the present value of real income H c H y t t 69 1rquot1 1rquot1 1 1 0 If we let the number of periods increase without bound and if the sequences of real income and consumption do not grow faster than the rate of interest so the resulting infinite sums are finite the present value budget constraint becomes N ct N yt Z 70 171 2 171 111r 111r The infinite period additively separable intertemporal utility function can be written 2 Bquot 1Uct 71 tl where again 5 is the time discount factor 42 STOCHASTIC DIFFERENCE EQUATIONS 1 Cycles in Second Order Difference Equations Ordinary difference equations of low order can generate cycles but in general these cycles are not appropriate as models of the sort of uctuations we nd in economic variables We can il lustrate the type of cycles we get from ordinary difference equations by studying the second or der linear difference equation Xt2 aXtH bXt 1 Xt 0 is the only stationary solution to this difference equation but let us look at the evolution of Xt over time To do this we look at the corresponding characteristic equation M a b 0 2 which has roots x int Aa2 4b 3 IfD a2 4b gt 0 we get two real roots while ifD a2 4b lt 0 we get a pair of compleX roots We consider the solution to the difference equation in each of the three possible outcomes for D aDa24bgt0 Then Xt AM BM 4 where kl and M are the roots of the characteristic equation and A and B are two as yet undeter mined constants We can verify this by substituting into the original difference equation We want to verify 99 AM2BM2 aAM1Bk 1bAM Bkg 5 We can rewrite the expression to check as Al klziakl7bBk k 7akzib 0 6 This must hold since kl and M are roots of the characteristic equation Now examine the solution for Xt There are no cycles here Iflkll or lkgl gt 1 the values oth will diverge to ice Otherwise Xt will converge to 0 exponentially b D a2 4b lt 0 which of course requires b lt 0 The roots will be complex conjugates so let them be kl reie and M re39ie Then1 Xt Arteiet Brteiet for constants A and B Since Xt is real for all t A and B must be complex conjugates For exam ple X1 Areie Bre39ie rABcos9 irABsin9 7 is real Therefore AB must be real and AB purely complex so that if A xiy we must have B x iy Let A aei l and B ae39i l Then we can write Xt artei9t e39KGVr 1 2art cos9t 1 8 Here Xt will cycle In fact we can get almost any continuous function of time by taking sums of sine and cosine curves this is the idea behind Fourier representation theory If lrl gt 1 the ampli tude of the oscillations will explode exponentially If lrl lt l the amplitude will decay to zero We need lrl 1 that is a function of the parameters has to be exactly equal to 1 if we are to get a cycle of the same amplitude continuing forever cDa24b0 139That this is a solution to the original difference equation can be verified as above 100 Here the solution to the difference equation is a t Xt A Bt 9 Again we can verify this by substitution XM fax 1 ebXt szm Ba 2 7a A Bt17bA Ba 10 and observing that the terms in braces can be factored as A Bt 4 a24b 11 In this case Xt will follow a linear trend multiplied by an exploding or decaying exponential de pending on whether a gt 2 or a lt 2 2 0th er Deterministic Difference Equations Higher order linear difference equations will not give us any different behaVior of the solution Xt Again the solutions can generally except for the knifeedge case of equal roots be written as a sum of exponentials in the roots which will all be either real or complex conjugate pairs These will therefore also either explode or decay exponentially or be sums of oscillations with exploding or decaying amplitudes Are the cycles we get from these linear difference equations reasonable as a model of a business cycle The cycles will tend to be too regular unless we have a large number of periodic functions added together In that case we might try to find a more parsimonious representation which fits the data as well Nonlinear difference equation models of a second and higher order can yield more interesting 101 behavior For example there has recently been interest in the socalled chaotic models which produce pseudorandom data from a deterministic model However chaotic models are still open to the inprinciple objection that we can use a model that has been fit to the data to get a perfect prediction of the future This runs counter to our intuition that there are many inherently unfore castable events which in uence the evolution of the economy Nonlinear chaotic models also have the property that a small perturbation in the initial conditions can lead to a very large change in the values of the endogenous variables Nevertheless chaotic models have spawned a large literature on nonlinear stochastic time series models and there is some promise that these nonlinear stochastic models might provide a better description of many economic time series more accurate predictions in a model involving fewer parameters than the linear stochastic models we shall concentrate on in this course In particular there is a growing literature on models where the parameters depend on the regime the economy is in and where the switch from one regime to another depends on the values taken by the endog enous variables Some econometricians have also argued that asymmetries in business cycle be havior require nonlinearities in the model 3 Stochastic Difference Equations As an alternative to deterministic models most macroeconomists have focussed on exogenous stochastic disturbances as the source of business cycles The large econometric models of econ omies used by most central banks Treasury or Finance departments and private forecasters con sist of a set of simultaneous stochastic difference equations 4 T h e rst order equation Consider the first order linear stochastic difference equation Xt1 CC pXt 8H1 12 102 where St is distributed normally with mean 0 and variance 0392 we write 8 N N00392 and we as sume 8 is serially uncorrelated that is that E8t8ts 0 Vs 0 Now observe that X10LpX081 X2ocpocpX08182oc1pp2X0p8182 13 Guess that Xt oc1 p p2 pt1 th0 pt39121 pt39222 p8t1 at 14 It is easy to verify this solves the stochastic difference equation Xt will be a random variable since the 8 s are random variables In fact X will be normal since a linear combination of normal random variables is a normal random variable The mean and variance are sufficient statistics for a normal random variable For Xt we get EXt E Ht 061ppZpt391 tho 15 varxt EXtExt2 E o 62192p4p2ltt1gt 16 and Xt Nut lt53 17 We can get ordinary difference equations in M and 039 from the original stochastic difference equation Ht10 PHtH0X0 18 531 pzcst2 0392 0390 19 These ordinary difference equations have the same solutions as we just found for u and 0392 The 103 stationary solutions to these difference equations will be 7 CL iii 17p 20 2 62 6 21 17p The difference equations will be stable that is M gt it and 039 gt 6392 if lpl lt 1 If the system has been going for a long time we don t have to worry about the initial conditions We use the limiting or stationary distribution Y N N 6 2 to approximate the distribution of Xt 5 Autocorrelation Now look at the joint distribution for Xt and XHk for k gt 0 Since both Xt and XHk are normal the joint distribution will again be normal with mean vector and covariance matrix as sufficient statistics We want to calculate 039ng E covXtXtk EXtutXtkutk E8tp8t1pt391818tkp8tk1ptk3918122 Now use the fact that E818j 0 unless i j where E818i 0392 to conclude Ot k 02pt1ptk1pt2ptk2ppk1pk 02pk1p2p3p4p2t1 pkOtZ 23 The covariance is a measure of association of two variables If p gt 0 the covariance is positive and successive values of X tend to move in the same direction If p lt 0 adjacent values tend to be negatively related but values two periods apart are positively related on average If l pl lt l the degree of these associations tend to die out over time We conclude that the joint distribution of Xt and Xtk is 104 X 52 6 t N N l l39t t t t k 24 Xtk wk O39LHk 012 Note that we also have a limiting or stationary distribution for this joint distribution of Xt and XHk if lpl lt 1 We let t gt 00 while keeping k constant The limiting covariance stationary joint distribution will again be normally distributed with mean vector El 25 and covariance matrix pk62 62 62 pkOZ 26 Observe that we can also obtain the limiting or stationary covariance as follows Write the sto chastic difference equation as Xt Ht PXt1 Ht1 t 8t 27 Multiply through by Xtk utk and take expectations to get EXt HtXtk Htk PEXt1 Ht1Xtk Htk E9tXtk Htk 28 Now observe that Xtk utk involves 8 s from periods tk and earlier so that the second expec tation on the RHS is 0 Therefore we get O39ttk Pot1tk 29 Now let t gt co and use 7k for the limiting or stationary covariance between two X39s k periods apart Then the above ordinary difference equation for 7 reduces to 105 Yk PYkl 30 Conclude Yk PkYo pk z 31 6 Second Order Stochastic Difference Equation Now look at a second order stochastic difference equation Xt aXH bXt2 8t 32 Assume St is a white noise process that is St N N00392 and E8t8ts 0 for s7 0 Now take expectations of the stochastic difference equation to get Ht aHt1 f but2 33 This is a second order ordinary difference equation in u and as we saw above this has the solu tion excluding again the knifeedge case of equal roots ut AM BM 34 where kl and M are the roots of the characteristic equation If a and b are such that WM and lkgl lt 1 then X will have a limiting or stationary distribution as t gt 00 Then the stationary distribu tion for X will have mean zero and if we multiply through by Xtk and take expectations the co variances of the stationary distribution for X will satisfy Yk aYk1 f bYk2 35 with 70 21270 b270 2ab71 52 36 106 and Y1 3Y0 bYl 37 The autocovariances of the stationary distribution will satisfy an ordinary second order differ ence equation with two initial conditions The covariogram of the stationary distribution or the graph of the autocovariance as a function of the lag period k will therefore follow either an ex ploding or damped exponential trend or will display oscillations Similarly the covariogram of an nth order stochastic difference equation will obey an nth order ordinary difference equation and therefore will be a function of exponentials and cosines 7 Stochastic Difference Equations and Business Cycles Now observe that a regular cycle in the covariogram of a time series is very different to a cycle in the series itself A cycle in the covariogram says that Xt will tend to covary in a cyclical fash ion with lagged values of itself If Xt instead solved a deterministic difference equation then Xt would follow an exact and perfectly predictable cyclical pattern Using this analysis we might say a variable possesses a cycle of a given frequency if its co variogram displays damped oscillations of that frequency This will be true if the corresponding characteristic equation has complex roots with 9 equal to that frequency If we follow the NBER chronology for minor business cycles the cycle would be a business cycle if it had a periodicity of from two to four years In addition to studying the autocovariances of each series we can study the covariance between one variable X and current and lagged values of another variable Y A feature of the business cycle is that we would expect to find a high degree of correlation between aggregate variables at lags corresponding to the business cycle frequencies One can argue that econometric models have been a good tool for modeling business cycle uctuations because they give us a set of si multaneous stochastic difference equations The simultaneity will introduce the correlation be 107 tween the different aggregate variables and it will be of the right frequency if the dynamics of the model is chosen appropriately 8 The Lag Operator Another way of arriving at the limiting or stationary distribution is to use the lag operator For any sequence of real numbers Zt which is a function of time we define the lag operator L by LZt Zt l The first order stochastic difference equation can then be written XtOLpLXt8t 38 39 The advantage of the lag operator is that we can manipulate it as an algebraic symbol2 Then we can write the difference equation as CC f at Divide through by lpL to get CL 1 Xt lipL lipLgt Now for lpl lt l we expand the rational fraction in L to 1 i piLi 1 7 pL 1 0 where we interpret Li as the operator which lags i periods Li zt ZH 239Functional analysis is needed to prove this is legitimate 108 40 41 42 43 For the constant term 0L we have LiOL 0L since 0L does not depend on t Thus we get as the so lution to the stochastic difference equation 0L Xt 1 E Platii 7 p I 0 1 9 The second order equation and the Lag Operator 44 We could also obtain the autocovariances in the second order equation by expressing Xt as a function of the 8 s using the lag operator 1aLbL2Xt at 01 17vlL172LXt 8t DiVide through by the polynomial in L and let m m 1 7 ML Conclude that c and d have to satisfy the equation 1 clML dlle and therefore lcdand0c7 2d7 1 that is 45 46 47 48 49 50 Therefore we get 1 Xt klMMkietrkzkaem lt51 We can also use this expression to derive the autocovariances of the limiting or stationary distri bution for Xt and they will be the same as found above 110 UNEMPLOYMENT AND OTHER FEATURES OF LABOR MARKETS 1 U n employment Thus far we have focused on explaining uctuations in employment In the postKeynesian era economists have focused more attention on unemployment An individual is recorded as unem ployed if they do not have a job and are currently looking for work These terms have a specific definition that varies from country to country Thus the unemployment statistics of different countries are often not comparable The basic idea however is that an unemployed individual is someone who wants to work but cannot find a suitable job at the wages and with the condition they are willing to accept Even when the economy is in a business cycle upturn we do not observe a zero level of unem ployment Friedman referred to the level of unemployment consistent with a constant rate of wage and price in ation and constant in ationary expectations as the natural rate W D s unemployed Minimum wages or mandatory disequilibrium award wages will result in structural unemploy ment for the least skilled workers by both contracting the demand for unskilled labor and tem porarily at least increasing the supply of workers willing to work at the legal minimum wage Similarly if trade unions or labor court judges are more effective at raising real wages in some 134 industries than they are in others there will be queues for the plum union jobs and these will be measured as unemployment Equilibrium explanations of the natural rate of unemployment and variations around that natural rate also often rely on incomplete information Relative demand or technology shifts lead to a loss of employment at some rms but a demand for labor at others Workers who lose employ ment will in general find it desirable to search before accepting a new job Measured unemploy ment levels typically correspond to the number of people actively looking for a job Labor economists have elaborated on these models in various ways to obtain a better description of labor markets The models have been tested using quite sophisticated econometric techniques to cope with the nonlinearities and have performed reasonably well when tested on microeco nomic data 2 A Model 0fJ0b Search Assume a worker can either search or work but not both Assume he allocates his time between search and work to maximize the expected present value of his earnings stream Let his discount factor be Assume that each time he searches he draws a wage independently from the distri bution w If he searches we assume he earns nothing this period and if the current job offer is not accepted this period it cannot be recalled at a later date Let Vw expected present value of a worker with wage offer w who behaves optimally If he works at the wage w he gets wBwB2w wlB 1 If he doesn39t work he gets 0 this period draws w from I and behaves optimally from there So 135 g ifhe works VW co 2 BIVw39 w39dw39 if he searches 0 Since the individual has a choice between two alternatives he maximizes by choosing the biggest S0 Vw maX iv 1 B ngwwwvdw39 3 Observe that the second term is independent of w since the current wage w gives the individual no information about likely future wage offers under our assumptions Let it be A W 17B BEVW39 A l 2 We conclude that 136 A if wsw Vw i if wzw 4 where w is known as the reservation wage A wage offer greater than or equal to w is accepted while an offer less than w is refused and the worker chooses to search for another period To solve the problem we need to specify values for A and w To do so note that at w A BVw 0 A wlB so that w A1B 39gt ltw39dw39 BJA ltw39gtdw39 Bj ltwvdw 0 Let I be the distribution function corresponding to I That is 13w Prw W j w39dw39 0 Then equation 6 for A can be written Hence A A mom rB jw Mwwa B m lt176gtltIABltIgtltmgtlW ltWgtdW 5 6 7 8 9 and 7 B m w A1 5 176 WJw wdw 10 W We can do various comparative static exercises with this expression for the reservation wage w by differentiating with respect to the parameters of interest Note that we can rewrite the expression 10 for the reservation wage 56m v mm 5w ij39mw39mw39 11 so that B m w 176Jw7w wdw 12 W The individual can work at w or attempt to get another job at the differential w w by searching The right hand side of 12 equals the discounted expected gain from searching and this must equal the reservation wage w at the margin One implication of the model is that the availability of unemployment bene ts will tend to ex tend the period of search by reducing search costs below the foregone wage It has been argued for example that the introduction of unemployment benefits into the UK in the 1920 s was one factor contributing to the historically high unemployment in that country at that time If current wage offers convey information about future wage offers we get more complicated dynamics The reservation wage will in general change over time This will also happen if the horizon of the individual is finite 138 We have taken I as xed in the above analysis Why does this distribution persist over time One answer is that it takes time and other resources to arbitrage across markets Furthermore if relative demands or production technologies are continually changing the opportunities for arbi trage are also changing An alternative interpretation of I as the productivity of the match be tween workers and rms also would result in a distribution of l which persists The productivity of different matches seems to be an important element in search models This has been pursued in the work of Jovanovic and others The assumption that the worker either works or searches was imposed on the model In practice search occurs on the job Taking a wage cut and searching for a better job while working seems to dominate the above strategy unless the marginal value of leisure andor unemployment com pensation is high One way to derive a dichotomous model endogenously might be to suppose employers take the individual39s previous wage as a signal of his ability This would impose a cost to taking a wage cut on the current job before searching It might also be more expensive to go to interviews and ll out and le applications if still working That is search costs might be low er if you39re unemployed Finally taking a wage cut and staying employed while searching may not be a feasible option if the rm has some lumpiness in production so that at lower levels of output the additional workers are unpro table at any wage acceptable to the worker The work by Hansen and others has studied the effect of xed costs either for the rm or the worker on employment relationships The xed costs could include for example the fixed costs of getting to work and getting started With such xed costs it is not optimal to reduce work hours of each employee when desired output falls Reducing the number of employees is preferable The em ployees laid off would be offered unemployment bene ts and for the period the bene ts last will search for a new job 3 Implicit Contract Theory and Involuntary Unemployment Another strand of the literature has focused on explicit or implicit wage contracts which are a means for firms and workers to share risks In effect the rms insure workers but in return have 139 the right to lay off workers when they face an unfavorable demand for their output A conse quence of the insurance is that real wages and employment are less variable than they would be in a spot labor market and one can explain ex post involuntary unemployment and the availabil ity of private unemployment insurance The basic hypothesis underlying the contract models is that rms are less risk averse than work ers A justification for this assumption might be that human capital is nonmarketable and there fore allows for less risk sharing for workers than can be achieved in capital markets for firms39 assets1 Assume a firm has the production function fn with f39 gt 0 f lt 0 and n the level of employment of homogeneous labor Assume the firm is competitive and able to sell all it wants in period t at the real price pe 9 12 representing two states of the world Assume p1 gt p2 Assume the firm and its workers share a common view of the probabilities of the two states emerging and denote these E1 and 72 Denote by we the real wage paid by the firm in state 9 The f1rm39s profits in state 9 are then given by 13 Pe ne 39 Wene Assume all workers are identical and possess an indirect utility function which is concave in the real wage and hours of employment VgWe L g1gt05g11lt05g2205g220 14 Suppose hours of work are fixed institutionally this might be derived by assuming preferences or technology is convex such as in the fixed costs model examined by Hansen Assume that if 139 To quote Azariadis Employers possess substantial information about the status of each employee control a supervisory apparatus which monitors current job performance and enjoy superior access to financial mar kets It is sensible therefore for firms to underwrite the insurance policies their employees are unable to place directly and in effect to become in part financial intermediaries shifting risk from owners of human capital to owners of financial capital 140 the worker works he has leisure L0 while if not he has leisure L1 gt L0 De ne U as the utility of the real wage when working Uwe a gwe L0 with U gt 0 U lt 0 15 and let r be the pecuniary value of the extra leisure or other nonmarket activity when not work ing that is r is what he would have to be paid to make him indifferent between working and not working g0 L1 g0 Lo Ur 16 The rm will employ ne workers in state 9 with n1 2 n2 Workers are aware of this and assume workers are laid off at random in state 2 so there is a probability of employment of n2n1 in state 2 Assume the worker maximizes expected utility Ill 7 n2 H v 71Uw1 Tun 2 Uw2 72 17 1 H1 The rm can hire as many workers as it wants subject only to the constraint that its jobs offer workers a level of expected utility v at least as great as the market determined level of expected utility workers can get from other rms v The maximization problem for the rm is max 7E1p1fn17w1n1 n2p2fn27w2n2 18 WI WZ nl n2 subjectto n2 n1 112 v 7E1Uw1TE2n Uw2rc2 n Ur l9 1 1 141 and n1 2 112 20 De ning the Lagrangian with 7 the Lagrange multiplier for the constraint 19 and M the Lagrange multiplier for the constraint 20 and setting the appropriate derivatives to zero yields the set of rst order conditions II II Tclplf n17 Tclwl iknzn iUw2 Mal jun 711 21 l l 1 1 7E2p2f n2 7 7E2w2 anII Uw24m2n Ur 11 22 l l Tclnl m1 U w1 0 23 n2 7 72112 kznzn U w2 0 24 1 together with the constraint l9 and the KuhnTucker condition n1 n2 0 25 From 23 and 24 we get n1 XU w1 and n1 XU W2 so that w1 W2 w and the real wage should be independent of 9 Now consider employment variation over the two states If n1 gt n2 then M 0 and the remaining rst order conditions become 142 n2Uw n2Ur n1p1fn17n1w7n2m 752m 0 UW Ur p2fn27wmim 0 n17n2Ur II v TE1UW n2 2Uw 72 n1 n1 Then from 27 l imlUWPUUH E HWJ pzf n2 7w The concavity of U and U gt 0 imply l w r gt w r so that 29 implies 1329012 lt r 26 27 28 29 30 31 If workers derive no utility from additional leisure when unemployed r 0 and f39n2 gt 0 yields a contradiction Therefore if r 0 we must have n1 n2 and the workers bear no risk 7the risk neutral rm provides complete insurance On the other hand if r is large enough that p2f39n2 lt r we can have n1 gt ng and there will be layoffs in state 2 Observe that l w r is the risk premium workers have to be paid to accept the chance of unem ployment in state 2 When this risk premium plus the value of marginal product p2f39n2 falls short of w the wage workers have to be paid in state 2 everyone is better off on average if fewer than n1 work in state 2 This reduction in employment is achieved by temporary layoffs rather than wage cuts 143 Returning to the case n1 gt n2 26 implies 7E2nzUltwgtiUltrgt1 W 32 plf n1 W Tcln1 U w so the real wage the rm pays in both periods is between the values of the marginal physical product of labor in the two periods The situation is illustrated in the diagram below Here the values with superscript 0 denote the outcomes we would observe in a spot market In a spot labor market we would have real wages 0 7 Wwn Gil 33 r if92 and levels of employment n n1 if 9 1 n3 if e 2 34 where 0 0 0 w p1fn1 and p2f n2 r 35 Thus in the case n1 gt n2 we have n2 gt n In the contract market the value of the marginal product of labor is equated to the marginal private cost of leisure w l w r lt r so the level of employ ment is higher in the low demand state The rm insures the workers by providing less wage vari ability and less employment variability than would occur in a spot market Of course the workers pay a premium for this insurance in the form of wages below the value of their marginal product in the state of high demand 144 0 n2 n2 n1nO We would have measured unemployment of n1 n2 in state 2 in the contract model Notice how ever that the measured unemployment would in fact be higher if the market were a spot market In that case measured unemployment in state 2 would be n1 n8 Also observe that if the spot market does not have unemployment in state 2 neither does the contract market The key implication of this model is that some in exibility of real wages across different states of demand might well be consistent with a competitive model The mere observation of such real wage in exibility does not necessarily imply we cannot obtain an explanation consistent with individual maximizing behavior mediated by competitive markets The model does not howev er explain high employment variability 7 the variation in employment is less in the contract model than it would be in a spot labor market Now we can ask whether the unemployment in state 2 is involuntary Given state 2 unemploy ment is ex post involuntary individuals would rather work at the wage w than collect the lower sum r which represents the value of nonmarket activity On the other hand the contract itself is freely arrived at and unemployment resulting from it is ex ante voluntary Furthermore to pre 145 I I I 4 I serve some of39 y F J even in the ex post sense one needs to ap peal to employer risk aversion moral hazard or some capital market imperfection which would prevent rms offering workers an actuarially fair insurance policy For suppose rms can pay workers 2 for not working in state 2 Then if unemployed in state 2 workers get utility gz L1 Look at the constrained maximum problem for the rm with this new choice variable and gz L1 substituted for Ur and pro ts decreased by n22n1 n2 As before the rst order con ditions for the choice of w1 and w2 imply w1 w2 and n1 7 U w Differentiating the Lagrangian with respect to z we obtain n1 4th so we conclude gZ U w The marginal value of unemployment compensation in state 2 must equal the marginal value of wages in state 1 Now consider the particular utility function gw L hw BL B gt 0 h gt 0 h lt 0 36 so real wages and leisure are perfect substitutes Then Uw hw 3L0 37 and r BL0 BL1 or r BL1 L0 38 Also since gz L1 hz BLl hz r BLO a Uzr 39 gZ U zr 40 2 and in this case gZ U w gt U w U zr gt w z r With unemployment compensation allowed the workers bear no risks trading them all to the rm 239 Looking at the first order conditions for n1 and n2 it can again be shown that a necessary condition for the firm to set n1 gt n2 is pzf39n2 lt r 146 For more general utility functions 2 will not equal w r Nevertheless in general the rm will choose n1 gt In and pay some unemployment compensation At the margin we must have gZ U w so the workers welfare will be independent of employment status Workers are in sured at actuarially fair terms and all unemployment is purely voluntary 4 The Ef ciency Wage Model The efficiency wage hypothesis has also been used to explain relatively in exible real wages This theory assumes that while the employer cannot observe the level of effort he believes that by increasing wages he will encourage his workers to supply greater effort The aim of these models is to explain why an excess supply of labor might not reduce real wag es While a reduction in wages would reduce costs the productivity of the work force might fall so much that real profits decline when wages are lower Calvo and Wellisz Lazear and Moore and Malcomson and others have used the idea that employees can vary their level of effort per haps in a way that is only partially detectable by the firm to explain the hierarchical structure of wages and ageeamings profiles within firms Lazear and Moore claim that the desire to provide employees with incentives not to shirk may be more important than onthejob training as a source of increasing earnings with job tenure The efficiency wage model can also be used to explain the apparent ability of wages to adjust to prescribed award or minimum wages As we noted above the standard model of the labor market implies that minimum wages above the marketclearing level should increase unemployment The high minimum wage reduces the demand for labor The reduction in employment raises the marginal product to equal the new higher real wage Until workers become discouraged from their inability to find employment at the high minimum wage the legally prescribed minimum also increases the supply of individuals willing to work The queue of individuals searching for a job is measured as increased unemployment 147 According to the standard analysis the costs of a minimum wage include both the net value of the lost employment opportunities and the net value of the lost output from firms There may also be losses associated with the increased unemployment as individuals spend more time and other resources searching for the limited jobs available at the legal minimum wage Losses in current employment may also have future costs as individuals who are denied valuable work experience suffer a reduction in future productivity Time out of the work force can also produce a deterio ration in previously acquired work skills Some individuals displaced from the sectors covered by a minimum wage take jobs in uncovered sectors at a reduced wage rate The opportunity to nd work in occupations that aren t covered by the minimum wage lessens the adverse impact of the law on overall levels of employment Some of the effects of the minimum wage are how ever spread from the covered to the uncovered sectors as workers in the latter sectors suffer de clines in their real wages In addition since the marginal product of labor is higher in the covered than the uncovered sectors the value of output could be increased by transferring labor back to the covered sectors Some of the efficiency costs of the minimum wage will therefore take the form of an inefficient allocation of labor across the different sectors of the economy However the standard analysis cannot explain why so many employers appear to be able to pay the legally prescribed minimum wage and other award wages without greatly reducing em ployment In the standard analysis it is only through a reduction in employment that the margin al product of workers can be increased to equal the legally prescribed minimum real wage The following figure illustrates the distribution of wages following the imposition and enforce ment of a minimum wage as predicted by the standard theory Any individual whose marginal 3 product after the minimum is imposed is less than the minimum would lose his job The reduc tion in employment following the imposition of a minimum wage generally raises the marginal 339 Some individuals with a marginal product less than the legal minimum wage before the minimum is im posed will keep their jobs and be paid the minimum However we would expect the demand curves for a nar rowly defined skill category of labor to be quite elastic A legal minimum real wage of wo would therefore eliminate most jobs with a marginal product less than wo before the imposition of the minimum wage 148 products of labor and shifts the wage distribution to the right particularly for low wages where most of the employment changes are concentrated Thus the postminimum distribution is not merely a truncated version of the preminimum distribution Nevertheless we would expect the postminimum distribution to be a truncated version of a smooth distribution proportion unemployed legal marginal product minimum amp real wage In practice the imposition and effective enforcement of a minimum wage leads to a large num ber of workers receiving the minimum as illustrated in the following figure Furthermore a change in the minimum wage shifts the mass of workers receiving the minimum 149 proportion legal marginal pgduct minimum amp real wage Similarly the distribution of wages in many European economies has noticeable peaks at various award wage levels Many workers in different industries or parts of the country receive exactly the same wage4 In a deregulated labor market the wages paid for similar jobs in different in dustries or different locations would be related since the employers are drawing on a common labor pool However the marginal products of workers in a particular job classification are likely to vary from one industry to the next Even in the same industry different employers use differ ent technologies so the marginal products of workers nominally doing the same job are likely to differ Finally similarly classified jobs in different industries or locations are likely to have dif ferent nonpecuniary characteristics This would also lead to wage variations as workers and firms competed on the attractiveness of the overall employment opportunity and not just wage rates In summary in a free and competitive labor market we would expect to see a smooth dis tribution of wages within a particular job category rather than the high degree of uniformity char acteristic of European wage distributions 439 As with the simple minimum however the award wage is nonbinding in some cases and the firms make overaward payments 150 It can be argued that just as other price controls distort the quality of market goods and services minimum wages distort the quality of labor services exchanged in the market place The dis tortion in quality following the imposition of a price control lessens its adverse effect on the quantity of trade The quality of labor services can be identified with the amount of effort or the net output per hour of labor input Many firms can pay legally prescribed wages without reducing employment by adjusting their operations to increase the amount of effort per hour of labor input For example consider a firm operating an assembly line When a minimum wage that exceeds the current productivity of workers is imposed the firm may be able to afford the wage by making the assembly line run faster This will make the workers worse off and may also raise costs by for example increasing wear on the equipment or the proportion of defective products Yet it may enable the firm to stay in business without having to lay off workers or see them resign to find more attractive alternative jobs Other changes in working conditions might also affect output For example the employer can increase supervision decrease the number or length of work breaks reduce the amount of socializing on the job cut fringe benefits or con trol losses from breakage or pilfering Output also might be increased if less of the labor time of employees is used to maintain the cleanliness or safety of the work place A firm also might be able to increase current production by reducing onthejob training This will however reduce productivity growth and therefore future wages and profits More generally we assume that the employer can to some extent observe and control the effort net output per hour of labor input of his employees Such modifications to technology or work practices may require additional maintenance addi tional supervisory staff extra equipment or other expenditures by the employer Employees also are likely to bear some costs The work environment may become less pleasant and rewarding or employees may be forced to expend more energy per hour of work Employees also can change their productivity in subtle ways that are not easily observed or con trolled by the employer For example they can alter the care they take with their job or the num 151 ber or quality of suggestions they make for improvements to the production process Employees who have worked for the same employer for some time acquire skills and abilities that are particularly useful to that employer but are of limited value in alternative jobs The costs of acquiring such rm speci c skills along with signi cant search costs make it expensive for employees to change jobs Search and training costs also make it expensive for rms to hire new employees Firms and workers who have formed a productive match thus often have a surplus to share between them even in a reasonably competitive labor market An employee is better off in his current job than he would be in the next best alternative The rm is better off with its ex isting employees than it would be with the next best alternative employees If an employee be lieves he is being cheated in the sense that he is not getting a fair share of the surplus arising from a productive match he can reduce his effort level To encourage greater productivity an employer therefore needs to share some of the rents with employees This is the key idea be hind the ef ciency wage hypothesis The employer has both a carrot and a stick to in uence the effort level of employees Employee productivity depends on both the real wage and the ef fort level enforced by the employer So that we can represent the analysis using simple diagrams we assume the hours of work of each employee is xed For a given number of labor hours increases in effort enable the employ er to pay a higher wage while maintaining pro ts This gives an isopro t locus in the follow ing gure where combinations of w and 6 keep pro ts constant As effort increases the marginal product of effort decreases and the marginal cost of enforcing even higher effort levels increases Hence as w increases larger increases in effort are required to compensate for equal increases in w Beyond some level of effort further increases in 6 reduce the level of w the rm can pay The locus in the gure therefore becomes negatively sloped for e gt We also assume workers are worse off when they are forced to supply more effort but are better off with a higher real wage This leads to a set of indifference curves that trace out all combinations of w and e yielding quot f quot Since the mp1 J dislikes being forcedto sup aparticular level of of the mp1 J 152 ply effort he must be compensated by higher wages if he is to remain equally satis ed as e in creases Hence the indifference curves are positively sloped in we space The concavity of these indifference curves follows from the assumption of decreasing marginal utility of wages and increasing marginal disutility of effort As 6 increases larger increases in w are required to compensate for a given increase in e effort A isoprofit locus e r r 7 r r r r 7 r r r r 7 r r r r 7 r r r r 7 r r indifference curves w real wage In an unconstrained market equilibrium the chosen wage and effort level will maximize the worker s utility for a given level of firm profits as at w 8 The particular levels of firm profits and worker utility at the equilibrium the split of the rents between the employer and the employ ee depend on the tradeoff between enforced effort and worker surplus as alternative means of increasing productivity Now suppose the firm and the worker are forced to negotiate in the context of a minimum wage law specifying that the worker must be paid a real wage wo per hour The firm can pay the high er minimum wage by increasing the amount of effort of the employee but the deal between the employer and employee is no longer efficient It does not make either the firm or the worker as 153 well off as they could be given the technology If the exogenously imposed minimum is too high there may be no level of e that would enable the employer to pay wo and earn a pro t The minimum wage would then produce unemploy ment as in the standard analysis5 Alternatively the employer might have to increase 6 so much that the worker is worse off than his reservation level of utility U0 Unemployment would again increase as workers leave their current job to search for an alternative they believe will yield higher utility For some unskilled workers the relevant alternative might be long term un employment and accompanying bene ts or unemployment insurance payments The indiffer ence curve corresponding to utility level U0 intersects the zero pro t locus for the rm at wmax Any exogenously imposed real wage between w and wmax would not lead to unemployment or ef ciency losses as usually measured effort TEgtIlt unconstrainef X wage we v 539 In the short run the firm only needs to cover its operating costs to remain in business It may be willing to earn less than a competitive rate of return on its capital stock for a short period of time so long as it expects to be able to earn more than the competitive return on its capital at some time in the future The firm may be reluctant to lay off employees to cover a temporary decline in profits when it has invested in a relationship with those employees 154 The model can also explain why minimum wages are more likely to reduce the employment of younger workers These workers have accumulated fewer rmspeci c skills and therefore are probably earning fewer rents from their current job than more established workers There would be less of a difference between the utility level and U0 for these workers Wages could not be increased as much above w before utility declined below U0 To allow for changes in hours of work and the number of employees in addition to effort levels we need an algebraic formulation of the model The additional freedom the firm has to vary hours of work and the number of employees would enable it to adjust to an even larger range of exogenously imposed real wages without going out of business In so far as the firm reduces the number of employees however a change in the minimum wage will produce a fall in employ ment We might identify this fall in employment with layoffs ratherthan quits when U falls to U0 or bankruptcies when firm profits fall to zero The key difference between this more general model and the simple model with fixed employment and hours however is that a reduc tion in total hours worked is now an alternative to an increase in e as a method of raising the mar ginal product of labor to equal wo If the increase in the real wage together with the change in hours worked by each employee raises utility and if 6 does not increase too much then the utility of those individuals who keep their jobs can rise We can restore the proposition from the stan dard analysis that the imposition of the minimum wage may be favored by a union representing unskilled workers when it raises the utility of those who keep their jobs This is only likely to happen however in those cases where the minimum wage reduces total hours worked and therefore imposes high efficiency losses as conventionally measured 155 CASH IN ADVANCE MODEL OF MONEY DEMAND In previous chapters we studied the overlapping generations model as a model of the productive role played by money In that model we have individuals living for two generations with utility function Uc c39 There is a quantity M of money in circulation Money enables the efficient allocation of resources across generations We have the following graphical representation of the equilibriums with and without money without money The key aspect of the model is that the trading period is finite but the desired trades cover an infinite number of periods One can view this as an allegory for the more general idea that money helps overcome the double coincidence of wants inherent in a barter economy That is in a barter economy people will in general buy from and sell to different individuals It will only be coincidental that the person I want to buy from will simultaneously want to buy what I have to sell Money helps overcome this problem because of its universal acceptability as a medium of exchange In the overlapping generations model people would like to do deals with as yet unborn generations These deals cannot be done in the overlapping generations context Money facilitates the trading because the current young generation will accept money in exchange for output because they believe future generations will in turn accept the money in exchange for output Money enables the trades to be made because it is believed to be universally acceptable as a means of payment for output 0 One might criticize the models on the grounds that we have abstracted from the family as an institution to implement inter generational transfers and then noticed its absence It does not seem that inter generational transfer of consumption is likely to be a large part of the role money plays in the economy If we treat the intergenerational transfer part of the model as allegorical rather than literally we are left to wonder how good an allegory it is A feature of the overlapping generations model is that the introduction of money takes the economy from an inefficient equilibrium to a Pareto optimum The movement of the economy to a Pareto optimum with the introduction of money perhaps makes money seem more advantageous than it really is As an alternative we shall look at the cash in advance model It too has problems as a model of the role of money in the economy but at least it gives us a different perspective on some of the issues of interest We consider a dynamic model where people trade over discrete periods but there is a cash in advance constraint goods in period t must be bought with money in hand at the beginning of period t That is we have consumers with preferences EOZB Uc 1 0 with U gt 0 U lt 0 and Ua 00 as caO Let exogenous income each period be y Let M be the money demand in period t and assume purchases cl can only be acquired using money on hand at the beginning of the trading period plcl S M 2 Goods are assumed non storable but money can be held over from one period to the next Maximum cash balances available in period t1 will then be given by income from the current period plus any cash left over from current period trading Mt1 S ply Mt ptct 0 How might we motivate the cash constraint One justification is to suppose the single consumption good in fact comes in n quotcolorsquot Assume each color costs the same to produce and the market is competitive Therefore the different colors will sell at the same prices Let c11c21cm be the consumption vector and let preferences be H H Cit UH w1th 011 gt 0 and 2 011 1 11 11 Since the different colors all sell for the same price optimal consumption will be c 0 011 where ct cit 1 This is in a sense what we are thinking of when we look at a one good economy Index numbers enable us to move from a many good economy to a one good economy Now if we suppose there are some fixed set up costs for producing goods of a given color there will be specialization of producers We might assume each worker specializes in the production of one color each day but can shift colors from day to day we assume producers shift around so money balances are uniformly redistributed from one day to the next To motivate a demand for money introduce a parameter x x1x2x drawn from a distribution Fx which indicates what a particular individual wants to buy on a given day Then individuals will in general wish to buy a bundle of goods different in color to the single color good they are selling and there will be a problem of the quotdouble coincidence of wantsquot There is a continuum of traders so economy wide demand doesn39t change even though its distribution across individuals does Imagine each household has two people one works and the other shops for goods of a combination of colors depending on x One day the worker gets currency in exchange for output and the next day the shopper spends it on goods Imagine the same economy without money Shoppers would have to be monitored to make sure they don39t get more than y units of goods in total Each time a purchase is made that information would need to be passed to all the other stores There will be n 1 messages to be transmitted each stop of the n shoppers The total number of messages will increase with n2 The use of money works perfectly to economize on these information costs The cash in advance model implements the idea originally proposed by Clower that money facilitates exchange in an economy where goods can be exchanged for money and money can be exchanged for goods but goods cannot be directly exchanged for goods Townsend Models of Money with Spatially Separated Agents in Models of Monetary Economies edited by Kareken and Wallace 1980 examines various models where agents are spatially separated and money arises endogenously as a means of overcoming the problem of the double coincidence of wants under barter As in the simple cash in advance model the equilibriums in Townsend s models are generally non optimal He also shows however that equilibriums in his models do not correspond to equilibriums in simple cash in advance models where the constraint is imposed exogenously Nevertheless he does show that the requirements that agents use money to purchase consumption goods and that money balances must be non negative imposes Clower type constraints on consumption and money balances The main difference between Townsend39s model and the simple cash in advance model is that Townsend allows agents to consume their own output without using money balances whereas the simple cash in advance model requires that cash be used to finance all consumption Kiyotaki and Wright J PE 1989 analyze a general equilibrium matching model to show how different commodities can emerge as media of exchange depending on their intrinsic properties as well as extrinsic beliefs They distinguish commodities by their storability and show that storability influences the suitability of a commodity as a medium of exchange They also show however that the likelihood that an object will serve as a medium of exchange critically depends on agents beliefs that it will Kiyotaki and Wright identify a commodity as fiat money when it is accepted in trade not to be consumed or used in production but to be used to facilitate further trade A commodity money is a commodity consumed by some agents but it also is accepted by at least one agent not to be consumed or used in production 0 A money in advance constraint applies to the use of either fiat or commodity monies but it again is derived endogenously from the assumed structure of the economy rather than imposed exogenously Further in the Kiyotaki and Wright model money is not necessary for financing all consumption In 4 particular if two agents who meet happen not to have a double coincidence of wants problem then they can exchange without having to use money Also in contrast to the cash in advance model fiat money would not be used in their model if everyone believes it will be unacceptable to others 0 These models certainly illustrate some of the limitations of the cash in advance model However the Townsend and Kiyotaki and Wright models and other models like them have their own limitations In particular the trading structures in these models are very artificial and stylized This makes them good for illustrating some of the key issues However the properties of the models seem to be very dependent on the detailed assumptions about the trading structures who trades with whom who likes to consume what how do people meet the costs involved in using different media of exchange and so on Since the assumed trading structures bear little resemblance to actual economies the models may not be very good for answering many questions about the behavior of actual monetary economies Modifications to make the models more closely resemble actual trading patterns would greatly complicate them perhaps to the point where we cannot solve them or derive any general or testable propositions from them 0 In the remainder of these notes we focus on the simple cash in advance model In a subsequent set of notes we discuss a model related to the Townsend model with a more explicit trading structure Returning to the formal analysis we wish to maximize 1 subject to the sequences of constraints 2 and 3 Since U gt0 and Ua 00 as caO the budget constraint 3 will always hold with equality For simplicity we also initially consider a model where income is constant every period Then in equilibrium the chosen level of consumption must satisfy cl yVt If we take y as a parameter like 5 the only state variable will be M Furthermore if the per capita money supply is also constant we will assume that the equilibrium prices are constant and there will no longer be any random elements in the optimization and we can drop the expectations operator from preferences In addition with prices constant choosing nominal money balances is equivalent to choosing real money balances Therefore define and re write the cash in advance 2 and budget 3 constraints in real terms as c S m 4 and ml1 S y m cl 5 0 Let 7 be the multiplier on the cash in advance constraint 4 for period t Bellman s equation for the constrained optimization problem can then be written VmtmcaXUctBVmtcy m c 6 Observe that we can look at the right hand side of 6 as an indirect utility function involving c and real balances m but we have motivated the inclusion of real balances The first order condition for the choice of c is Ucl BVmt1 9 1 1mt 0 0 9 2 0 mt 2 ct 7 Now we want to apply the envelope theorem If there were no inequality constraint or at an interior solution to the constrained problem where the constraint is irrelevant the first order condition 7 would become Uct BVmt1 and the envelope theorem would imply Vmt BVmt1 Uct On the other hand if the constraint is binding cl m and 6 becomes Vmt Umt Uct Hence we conclude that in either case Wm U c 8 Updating 8 one period we find V m1 U c1 9 and substituting 9 into the first order condition 7 we obtain Ucl I3Uctl t In equilibrium however we must have cl c 1 y and 10 becomes 9 1 BU ygt0 11 which implies that the cash in advance constraint must hold with equality The velocity of circulation then is fixed at unity and results solely from the exogenous requirement that cash is needed to make purchases 0 Many other models with a cash in advance constraint also deliver a binding constraint This does not lead to a very satisfying theory of money demand It is not necessary however that the constraint bind For example if we add unanticipated shocks to income or desired consumption or introduce heterogeneity between consumers we can get a non trivial model of money demand 0 Consider a consumer with preferences which involve a taste shock On some days the consumer gets more utility from consumption than on other days This is meant to reflect the fact that opportunities to make purchases arise randomly There will be a precautionary demand for money balances in order to take advantage of unusual consumption opportunities 0 Specifically assume the representative consumer now has preferences 2 stage g 5 et s 6 91 F 0 with 9 independent over persons and time We assume U is bounded twice continuously differentiable with Uc gt 0 Ucc lt 0 Use gt 0 U9 gt 0 and 7 l9in 1UCc9 00 for all c while limUcc9 0 is implied by the assumptions U is monotonic bounded and concave With the distribution of F fixed across periods and a large number of consumers aggregate demand will not vary from period to period and will equal its mean value in every period We are going to postulate a constant money supply so it is natural given the constancy of aggregate demand to once again look for an equilibrium with nominal prices constant The constraints on the household can then once again be written in real terms as mtl mtY39 01 c s m with m given 0 The lowest end of period balance in real terms is y Also if the constraint cl 5 mt never binds the individual must be holding too much money More could be consumed without loss of future utility so the policy could not be maximizing Therefore we conclude the lower bound y on money holdings must be obtained with positive probability There is no apparent upper bound to the possible holdings of money that could be accumulated We assume the individual does not know 9 when m is chosen but does know it when c is chosen For notational convenience we drop the t subscripts and use a prime C to denote period t1 Bellman s equation for this problem can then be written vme manxUc9BJVm 9 dF9 12 subject to the constraints c s m m s m y c c m 2 0 The set we are maximizing over is as shaded in the diagram below 8 Since an increase in m expands the feasible region V must be at least as good with a larger m Hence V is increasing in m Thus the objective function we are maximizing is increasing with respect to both c and m Therefore at the optimum c and m39 the budget constraint must hold with equality c m m y 13 Using 13 and again taking 9 as the multiplier on the cash in advance constraint 12 becomes Vm9 maxUc9BJVmy c9 dF9 Km c 14 0 We can use the concavity of U in c and the convexity of the feasible region to prove that if V on the right side of 12 is concave in m then V on the left side will be concave in m1 In particular since the 1 Let c0 mo39 be the maximizing choice for m0 0 and c1 ml39 the maximizing choice for m1 0 and consider mx 0 with mx kmo 179 m1 Use the convexity of the constraint set to show MD 179 c1 kmo39 179 m139 is feasible Then since V is the maximized value over the feasible region Vmx02 U7ncD 1 7nc10BJVmx y 7ncn 1 7c1039dF039 UWD 1 hc 0BJVhmn y cD1 hml y cl039dF039 2hUcD01 7 Uc107LBJVmU y cu039dF039l 7LBJVm1 y cl039dF039 Since c0 mo39 is maximizing for m0 0 and c1 ml39 is maximizing for m1 0 the budget constraints m0y7c0 mo39 m1y7c1 ml39 9 functional equation is a contraction mapping the function V that is a fixed point can be obtained from an initial function that is concave in m It follows that the fixed point V must be concave in m m m Since we are maximizing a concave function over a convex set the maximizing c and m39 will be unique for each m and 9 Thus we can write the maximizing c and m as functions of m and 9 c cm9 m39 gm9 These so called policy functions will in fact be continuous Again we can show that the value function V is continuously differentiable with respect to m and Vmm9 Uccm99 In the interior case a this again follows from the first order condition for c and the envelope theorem When the cash in advance constraint is binding we have c m and vme Umeijye39dFe39 are satisfied Hence Vmx0 ZAUCD0BJVmD39039dF0391 LUcl0BIVm139039dF039 xvmne1 xvmle39 10 so that Vmm9 Ucm9 Uccm99 by differentiation Finally since c is continuous and Uc is continuous Vm will be continuous as a function of m Now we want to carry out some comparative static exercises In an interior solution the first order condition for consumption c will be Ucc9 BJVmmy c9 dF9 0 15 If we could differentiate V one more time we could find the effect of m on c as follows Ultcegt BImedF9 cm me ijdFlte39gt 0 that is cmm9BJV dl UCCBImedF and cmm9 E 0 1 for me Ucc lt 0 But V may not be twice differentiable As an alternative one can look at a sequence of policy functions corresponding to a sequence of value functions Each of the policy functions in the sequence will be differentiable with m gt m gt 0 s cm9 cm9 lt m m and this property will carry over in the limit In effect from the point of view of getting the derivative of c we do not mind if Vm has a kink because we are integrating it Now look at m39 gm9 m y cm9 Then gmm9 1 cmm9 We can represent these results graphically as follows A r A f 550119 3 x j x y c x i E 39 cm0 i 45 i r n In El In For low values of m the cash constraint will be binding so that c m and m39 y As m increases c will increase one for one until m reaches m The level of money balances m is determined by the intersection of c with the 45 line where c is the level of c that would have been chosen had the cash constraint not been binding Until m reaches m gm9 y For increases in m beyond m both c and m39 increase but less than proportionately with the increase in m 0 Note that m will be a function of 9 From the first order condition we can see that consumption increases as 9 increases Therefore cm9 shifts up and gm9 shifts down by the same amount as 9 increases 0 For each individual we get a stochastic difference equation governing money holdings mtl amber Suppose we begin with a distribution of real balances across consumers lquotm and suppose consumers believe prices will remain constant when they solve their maximization problem For total momey supply M the equilibrium nominal price will solve M v gm9 d lquotm dF6 Observe that we can use the budget constraints of individual consumers to conclude that if the money market equilibrium is achieved by a particular price then goods markets will also be in equilibrium at that price y cm9 d Iquotm dF9 We want the equilibrium p to match the p assumed by consumers in solving their maximization problems Thus for the assumption that prices will remain constant to be consistent with long run rational expectations we need the distribution 1quot to replicate itself This does not mean we need each individual to hold the same money from one period to the next Rather declines in money balances from one consumer must be matched by increases from another so that the overall distribution of real balances is replicated For Am39 m9 gm9 s m the distribution of real balances next period given the distribution 1quot this period will be determined by lquotm39 Ild mdF9 The equilibrium we are looking for is the long run rational expectations equilibrium This equilibrium concept makes most sense in an economy where consumers have experienced many changes in the money supply and have learned the relationship between money changes and changes in the equilibrium price level As an alternative one might try to model a learning process by postulating consumers react to a difference between the assumed price level and the actual outcome by updating their expectations We could then look for a convergence of the actual price level to an equilibrium value Definition An equilibrium in this model is a distribution function Iquot a value function V policy functions c and g and number p satisfying the market clearing and stationary distribution conditions with c and g defined as the policy functions corresponding to V the value function for the representative consumer s maximization problem assuming the consumer takes the equilibrium price p as given 0 To prove an equilibrium distribution of money balances exists we need to show that the functional equation Iquotm d l mdF6 Am39 where Am m9 gm9 s m39 has a fixed point in the space of probability distributions One cannot apply a contraction mapping theorem 0 Lucas Equilibrium in a Pure Currency Economy discusses how theorems on ergodic sets1 can be used to prove 1quot exists The basic idea is that Iquot will settle down to an equilibrium distribution if money holdings have a tendency to fall into a range of values and once they get into that set money holdings will stay in that set Also we don39t want the set of equilibrium values to split up into smaller subsets with money holdings jumping between these subsets and never settling down To show the distribution settles down we also want the ergodic set E to be noncyclic That is we want to show E C E 14E gt 0 x e E gt pxE gt 0 Once we get into any proper subset of E the probability of staying in that set must be non zero If this were not the case we could imagine quotcyclesquot continuing forever whereby we continually enter and then immediately exit some subset of E Putting together the conditions for E to be ergodic and non cyclic all proper subsets of E have to be such that the probability of staying in those sets once you get into them is strictly between 0 and 1 For a proof that these conditions suffice to guarantee the distribution settles down to a stationary distribution Lucas refers the reader to the book by Doob on Stochastic Processes Now apply these conditions to our money demand model Consider intervals on the m axis in the following diagram Suppose we can find m such that m 2 y and gm9 lt m for all m gt m and all 9 1 Intuitively for E to be ergodic we want it to be true that once we arrive in the set E the probability of staying there is 1 and there is no proper subset of E with size of sets measured according to the measure function u with this property More explicitly for a transition probability pxA Prx39e Alx and a measure 14 an ergodic set is a set E such that IKE 1 xe EpxE1 i E39 c Ewith 14E39 lt uE such thatx e E39 pxE39 1 and V1gt gt0 Elcgt 0 such that Vx 14A 6 z pXA17 The final condition says that for sets A which are small by the measure 14 the probability of getting to those sets is bounded away from 1 14 Then when consumers hold money balances in excess of m money balances next period will be lower This will continue until money balances fall below m Also because of the budget constraint for the individual consumer money balances cannot fall below y m39 quot gm 9 A gone i y i K i i I 45 i E gt m 0 Hence we can consider E ym as an ergodic set where m solves gm Q m That is m if it exists is the level of balances such that the consumer with m and low demand Q chooses to hold m at the end of the period If m gt y then the consumer must choose to hold over some balances from consuming this period when initial balances are m and 9 Q That is c s m must be slack and the FONC for an interior maximum must be satisfied with cy Ucy B fvmurw39 dFe39 Thus we want to know if the latter equation can be satisfied for m gt y m lt 00 The right hand side of the equation is a monotonic decreasing function of m and Vm Uccm 9 939 2 0 so we have just two possibilities These are illustrated in the following diagram and are denoted as case 1 or case 2 We know case 2 is possible because that is what we found in the certainty case where m y in every period and the consumer carries over the same balances each period If 9 jumps around enough then we can get case 1 where m gt y vamh n e dF Ucy 9 V In either case 1 or case 2 E ym will be an ergodic set with the degenerate case y m a possibility Also E will not have cyclic subsets No matter where you start on the interval ym EIG such that gm9 crosses the 45 line at that m with positive probability Once we have determined that the stationary distribution 1quot exists the equilibrium price level will be determined from m 2 fm d lquotm y Velocity of circulation will now depend on preferences and how noisy the economy is that is the shape of F9 in addition to institutional factors The demand for money in this economy is intimately linked with the uncertainty about future consumption prospects Money provides consumers with a form of insurance We can ask what the equilibrium 1quot looks like For example we can look at two economies with different y s and ask what happens to the distribution of money holdings Alternatively we can look at individuals with different y s in the one economy To do the latter we need to modify the set up of the problem Suppose each agent draws a constant income y from a distribution Altyy where y f y dAltny We can then go back to the maximum problem facingOthe individual consumers and consider functions Vm9y cm9y and gm9y and we can get a solution for each individual39s income y as 16 above This will lead to a stochastic difference equation mm gmt9uy which will have a stationary distribution Iquotmy such that if the individual has the constant income y lli mbal rml S m lquotmy We no longer interpret Iquot as an economy wide distribution but the mathematics is the same Given the individual problem was solved on the assumption that prices are constant for consistency we now want to know if the equilibrium p will be constant That is will may p 1 J md lquotmyd1yy yield a constant p as the equilibrium solution We look at the function r rY I md f mayhy which can be thought of as an Engel curve for average money holdings We obtained the stationary distribution 1quot as the limiting distribution solving a functional equation defined on the set of probability distributions We need convergence in distribution implies convergence in mean Lemma Suppose we have an operator S taking probability distributions into probability distributions Hkl Silk IdeGs e Ifyquot 814 and for all measurable A limS u0A uA for all M0 law then for all continuous bounded f0 l lgEIf0mS u0dm Jf0mu dm 16 For a proof see a measure theory text such as Halmos Next observe that in our case the operator S is defined from 17 Iquotm d l mdF6 Am39 where F9 is the distribution of the utility shock 9 Therefore If ms lu0 dm Hf gmes u0 dmdFe Define a sequence of functions ftm f1m I gm9dF9 Now observe that I f0ltmgtsuodm I I f0gltmegtuodmdFltegt I u j fgltmegtdFltegtdm I f1muodm and conclude that we can write Ifmuodm if ltmgtsmdm 0 Hence we can replace 16 by mmmmdm Ifomudm 17 This is true for all initial 40 Therefore the functions ftm must converge to a constant function That is for all m we must have lim m If 11 du am Now we can apply this to derive a comparative statics result Lemma Suppose gm9y is a non decreasing function of m and y Then IE0 hy Imd may is a non decreasing function of y Proof Choose f0myEm Then f1may If gm9yaydF9 and the proof is by induction on the sequence ft It is clear that f0 is non decreasing in In and y Suppose ftIny is non decreasing in In and y Then using g is non decreasing in In and y it follows le is also non decreasing in In and y Then lim my Imd l Iny Qua is non decreasing with respect to y SOLOWSWAN GROWTH MODEL 1 Why study Economic Growth The following graph shows the growth of US real GNP over the second half of the 20th century Speci cally it is a plot of the log seasonally adjusted real output against time 9 5 7 log seasonally adjusted real GDP 7 l 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 The graph also includes a cubic trend line The linear term is 00434 estimated standard error of 00011 This suggests and average annual real growth rate of around 4 Both the quadratic es timated value 000026 and cubic estimated value 173E06 terms are signi cantly different from zero at conventional significance levels tstatistics of 589 and 345 Thus there is evi dence consistent with the hypothesis that economic growth trended down in the 1970s and 1980s relative to the 1960s and 1990s An alternative way of allowing for lower trend growth rates in the 1970 s and 1980 s is to fit a piecewise linear time trend with a break in the slope in the early 1970 s and another in the early 1990s 39 The next few sets of notes focus on models of the longer run growth process in a market econo my We are interested in these models for several reasons 1 There can be quite large differences between the average economic growth rates of different countries for a considerable period of time e g compare the postWWII experiences of the UK US and Japan and for the same country over time We would like to be able to explain why trend rates of growth vary over time 40000 i Japan 35000 United Kingdom United States 30000 25000 20000 15000 Real GDP per capita PPP adjusted 10000 5000 1975 197719791981 1983 1985 198719891991 1993 1995 19971999 20012003 2 Such differences in economic growth rates seem to have important implications for economic welfare 3 The effects of growth rates is arguably more important than the effects of business cycles 4 Most market economies showed a downward trend in growth rates in the 1970s and 1980s and we would like to understand the reasons for this What if anything can be done to obtain the relatively good growth rates ofmost ofthe 1960s and 1990s 5 Growth models form the basis of an important class of models of the business cycle and the understanding of business cycles is a major focus of the course 6 The techniques used in analyzing growth models also have application in other models of 40 business cycles but arise in a simpler context when considering growth In particular these mod els allow us to discuss representative agent and the use of the fundamental welfare theorems in a dynamic context 2 The Aim of the Analysis We would like a theory of growth to explain why the economy has grown in the industrial era For most of human existence it has not been typical to encounter growth in access to material goods or services supplied by others More speci cally we would like our theory to explain why this rate has been around 34 pa over this period At the same time we would like the theory to account for the broad outlines of other features of aggregate economic performance such as the change in real wages the growth in employment and variations in the rate of in a tion An issue we discuss again later is whether essentially the same model we use to describe growth can also account for the evident variations in growth decade by decade or whether an alternative model is required to account for the short run uctuations in growth rates year by year or quarter by quarter 3 Factor Supply and Growth The most basic economic model of growth focuses on the growth in population resulting in an increase in the labor force on the one hand and saving leading to an increase in the capital stock on the other hand Later we shall also discuss the role of resources or physical and biological endowments in the growth process One way of interpreting capital and labor in this anal ysis is to see capital as any factor of production that can be augmented through market activity and labor as any factor of production with an exogenous rate of growth that cannot be changed by using market outputs For explaining growth in per capita output we focus on capital accumulation as a means of in 41 creasing worker productivity or relieving resource scarcity In turn security of property rights and a stable and effective legal and law enforcement system appears to be an important factor in producing capital accumulation In The Wealth of Nations Adam Smith identi ed changes in the organization of production as being central to economic growth In particular he argued that the incentive for any one individ ual to specialize depended on the degree of specialization present elsewhere in the economy and that specialization aided productivity presumably as a result of learning by doing A problem with specialization is that it makes one more vulnerable to disruptions of the economy Hence political and social instability might be a factor limiting growth in many countries We will not discuss the issue of specialization because it introduces additional mathematical complexities into the model In practice however it is likely to be a significant factor behind economic growth More recently economists have also focused on human capital accumulation either through ed ucation or learning by doing and technological progress as an important factors in increasing worker productivity We shall look at some simple models of the growth process which focus on the capital accumu lation or saving decision We ignore some of the above issues relating to property rights and specialization A good model of economic development would need to take these issues into ac count 4 Saving and Investment Suppose output at time t can be either consumed or invested saved Investment will have the benefit of increasing the output available for consumption in the future but the cost of reducing current consumption Assume labor supply does not depend on the real wage so that labor supply grows along with 42 the total r r 39 quot 39 J J quotJ or 39J with respect to output Later we shall exam r ine a model with variable labor supply In the simplest model we also assume that a xed fraction of income is saved although later we shall examine the consequences of determining the level of saving through maximizing behav ior We shall also use the maximizing model as a basis for thinking about equilibrium market behavior We assume aggregate output can be represented by the output of a representative rm Yt FKtNt which has constant returns to scale Here K is the current aggregate capital stock and Nt the cur rent aggregate labor supply We assume F 0 if either K 0 or N 0 we also assume FK FN gt 0 FKKlt0FNNlt0andFKNgt0 Because of the inelastic labor supply assumption the labor force is proportional to the popula tion We therefore assume the labor supply is growing exogenously at the population growth rate kgt0 Nt11 7V Nt We assume capital depreciates at the rate 5 gt 0 and denote by It the level of gross investment in period t The evolution of the capital stock is then described by the equation Kt11395Kt1t In turn gross investment is determined by the level of savings It SYt This completes the description of the basic assumptions of the model Now let us see what they 43 imply First use the fact that F is homogeneous of degree 1 to conclude Yt Nt FKtNta 1 E fXt Note that f FK gt 0 f FKK lt 0 and f0 0 or in terms ofa picture x Xt Using the equations governing capital accumulation and labor force growth we get Xt1 KtHNt SYt t 1 5Ktl1 MNtl Sl Yt Nt 1 5l Xt which can be written in terms of f as Xt4r1S14r 7vlfXt1 51 70 Xt E gXt 1 This relationship is presented in the graph below Since 0 lt s lt l and 1 gt 1 the rst expression on the RHS ofl S1MfXt is a fraction of fX The second expression 1511Xt is a straight lie through the origin with a positive slope 1511 The function gX is the sum ofthese two expressions 44 gX 5 Difference Equations Equation 1 is a first order nonlinear homogeneous difference equation in Xt We solve it by looking at the function g which is the sum of a positive multiple of f and a linear function of Xt as graphed above Suppose Xt starts out at X0 Xt1 gX XI The value ofX in period 1 will exceed X0 because g lies above the 450 line at X0 As illustrat ed X will approach X monotonically with smaller and smaller steps each period Similarly if X0 starts out above X X will decline back to X in the long run 45 At X X sl7 fX 151k X or rearranging terms M8 Xquot s fX 2 The LHS of 2 represents the replacement of K necessary to make provision for depreciation and population growth The RHS represents savings per capita The above diagram implicitly assumes g390 gt 1 Observe that 175 s k f 0gt1iff 0gt k T 5 Klinog X 11 3 so the productivity of capital when the current stock of it is zero has to be sufficiently large if growth is to take off The above diagram also is drawn on the assumption that g eventually crosses the 450 line This 7 will follow if lim g X lt 1 that is lim f X lt 5 For example iffis bounded we will x a as x a as have lim f X 0 A common constant returns to scale production function where this con x a co dition need not hold is the constant elasticity of substitution function FKN AK1e BNlee with e gt 0 where fX AB Xlee In particular note that f X AB xlef39lx19391 ABX 1e1 gt A as x gt oo For e 14 A 2 and B 1 this function is graphed below 46 fX f X The values 0 and X for X solve gX X and are known as fixed points of g If the initial value of X X0 X then Xt X for all t Similarly if X0 0 then Xt 0 for all t The values 0 and X for Xt are therefore also known as stationary points of the difference equation More complicated f functions will produce more stationary points as can be seen from the following diagram 47 xt1 39 39 Xt Some f functions such as the one produced by a CBS production function with a large value of A may produce no stationary points 7 a point to which we shall return below 6 Comparative Statics in a Dynamic Model We can do some comparative dynamics by asking what happens to X and the path of X as the parameters in the model change given X X in the long run equilibrium the capital stock and hence output will grow at the rate 7 Thus part of the reason for the relatively high growth rates of Japan in the 1960 s was that the opportunity to move large number of workers out of agriculture into manufacturing through the use of more modern farming techniques greatly increased the labor force growth rate in manufacturing and allowed output to expand rapidly in that sector of the economy A similar phenomenon is happening in China and India today changes in the saving rate s do not alter the long run growth rate of output Increases in the sav ing rate s will however make X larger and Xt larger at each t output per head will therefore also be larger in the long run equilibrium of an economy with a higher savings rate 48 Xt higher 5 Algebraically we can use calculus to find 1 69 fX sf Xd that is dX fX XfX 3 d5 157sf X sfX7Xf X But since F is homogeneous of degree 1 FuK uN u FK N for u gt 0 4 Differentiate 4 with respect to u to get KFK NFN FK N 5 The result 5 is known Euler39s theorem Then we have f X FK F NFNK f FND or rearranging 49 fXt 39 Xt f Xt FN gt 0 6 Hence from 3 and 6 we conclude that an increase in the savings rate will raise the long sta tionary per capita capital stock what happens to consumption as s changes If s 0 we end up at X 0 so that consumption 0 in the long run equilibrium If s l we also have zero consumption 7 The Golden Rule Presumably some s 0 lt s lt l is best Suppose we choose s and therefore X to maximize consumption at X This is equivalent to choosing s to II18kXli sfX subject to sfX X 5X Note that given the constraint choosing s is equivalent to choosing X The constraint implies l sfX fX 7 5X so the maximizing X will be given by f Xk5 7 This leads to the socalled golden rule savings rate s Xf XfX elasticity off 8 8 Factor Payments What happens to factor payments along the equilibrium path The rental on capital is given by FK f X and since we have assumed f lt 0 rental rates decrease along the equilibrium path until X is reached The real wage will be given by FN Using the assumption that F is homoge neous of degree one we showed above that FN real wage fXt th39Xt 9 50 Then the derivative of the real wage with respect to Xt is f Xt f X th Xt th quotX0 10 and since f lt 0 the real wage is increasing along the equilibrium path 9 Performance of the Model How well does this model do empirically At a trivial level we can associate developed economies with economies having a more pro ductive technology or a higher level of output F from a given level of inputs These economies also have a higher level of capital per capita X particularly if we include human capital in K Perhaps as economies mature their economic growth rate slows down as predicted by the model There is a large empirical literature on the socalled convergence hypothesis which postulates that per capita GNP growth rates are negatively related to the initial level of per capita GNP so that economies tend to converge over time in terms of per capita GNP levels It has been claimed that this hypothesis is consistent with data from the states of the US regions of Western Europe as well as across many samples of similar countries convergence clubs On a recent research project we estimated an equation for per capita GDP growth using World Bank data from 173 countries the graph of Japan the UK and the US above came from that da ta Although the maximum sample size for any one country was 52 years there was an average of 376 years of data for each country The estimated equation was 5 01 33913Ait714r 321lnYit71 336 1t71ln3 1t71 a4ln it71 IHYUStq where the terms ci are country specific constants The parameter estimates nd their correspond ing estimated standard errors were Parameter Estimate Std error a1 09362 00886 a2 09431 01178 a3 761930 07009 a4 700152 00030 The negative coefficient on the difference between country i GDP per capita and US GDP per capita implies that per capita growth rates of other countries will tend to converge toward those of the United States over time The positive coefficient on the inverse log level of per capita GDP lln yit implies that growth rates will tend to diminish as per capita GDP increases Further more the negative coefficient on the interaction term implies that growth rates will tend to be come more persistent as the economy matures The estimated model sits uncomfortably with the conclusion that growth in per capita terms will cease in the long run The estimates imply growth will slow down over time but it will be a very long time before growth ceases It may not even cease since the level of GDP where that hap pens is way beyond the values included in the estimation sample It might also be argued that some high growth countries have a higher marginal propensity to save s leading to higher growth rates at the same level of X and eventually a higher level of X even though the model predicts that all economies regardless of their level of per capita savings s would end up with no long run growth in per capita terms The effect of different savings rates in a model with international capital flow is however a more complicated issue and is left to a homework problem At a more quantitative level if we can describe aggregate production with the function Y FKN then we will have differentiating with respect to time and dividing through by Y KFK NFNA K N Y Y Y sKK sNN where A s denote percentage rates of change and the si are factor shares in aggregate income The growth rate of output is the weighted growth rate of the inputs with weights given by the relevant factor shares Under constant returns to scale the weights sum to l and can be obtained from the National Accounts For the US over a long period we nd the annual growth rate of output is about 35 labor s share about 075 capital39s share about 025 the net growth of the capital stock about 25 and the change in total hours worked about 125 The weighted sum is about 12 the recorded average annual growth rate We can call the difference between the two numbers exogenous technological change but this is not useful unless we have a theory which leads to some alternative way of understanding tech nological change Recent papers have attempted to model technological change by looking at RampD races and models of emulation of technological developments by other firms Since tech nological innovation can be emulated firms have a reduced incentive to invest in RampD Patents are one way of coping with this extemality but at the cost of allowing monopoly production for a period of time Another problem with patents is that they encourage wasteful RampD which merely attempts to duplicate existing patents but which has no or little social value1 Other recent models of the growth process have asserted that govemmentprovided public goods are an omitted variable from the aggregate production function and may account for some of the missing factors of production In particular some authors have claimed that at least part of the slowdown in growth in the US in the 1970 s and early 1980 s can be explained by a fall in gov ernment investment in infrastructure This claim has been supported not only with time series 139 An alternative method of increasing RampD is to subsidize it directly However it is questionable whether government bureaucrats have sufficient information to choose the best technologies to develop 7 or whether politicians would let them choose the best as determined by some objective criterion rather than for exam ple those investment projects located in the most marginal electorates Using patents and privately funded RampD the firms at least have an incentive to invest in research which is likely to benefit consumers 53 evidence on aggregate government spending on capital projects but also by splitting such expen diture into components for administrative buildings courts hospitals defence equipment and transport public utility and communications infrastructure Only the transport public utility and communications components have been shown to be signi cantly positively correlated with eco nomic growth rates Furthermore changes in government consumption expenditures such as transfer payments have been shown to have zero or negative correlations with private sector growth Other recent papers have also related crosssectional variations in growth rates in the US to differences in infrastructure spending by State and local governments There are also some papers claiming to have found contrary results A particularly difficult issue to answer empiri cally is the cause and effect relationship between government infrastructure spending and eco nomic growth 7 does additional spending in rapidly growing regions cause or result from the high rate of private sector growth Attempts to extend these US results to other economies have found less significant positive results than were found for the US However broad international evidence including less developed economies in the sample would seem to be consistent with the notion that inadequate provision of public infrastructure including an independent legal sys tem guaranteeing private property rights is one reason for underdevelopment in many nations We might be able to adjust for the quality of N by measuring schooling and other investment in human capital We might also attempt to adjust the physical capital stock for quality by measur ing RampD input into its production Some authors have argued that the return to investment in education depends on the level of education of other individuals in the society As the level of education rises individuals find it more profitable to spend more time in human capital invest ment This introduces changes that fundamentally alter the properties of the model In fact these models are one type of socalled endogenous growth model 10 Endogenous growth The key aim of the endogenous growth models is to explain why per capita growth might con tinue inde nitely The simple growth model discussed above has the capital stock and output ul 54 timately growing at the same rate as population so there is no growth in per capita terms in the long run stationary equilibrium From a mathematical perspective the key requirement to ensure continual growth in per capita terms is that the marginal product of capital does not decline to 7 5s as capital is accumulated For example a production function like the CES one discussed above might leave capital suffi ciently productive as the capital labor ratio rises that f X doesn39t decline to equal 7 5s no matter how large X is There will be no fixed point of the first order difference equation and per capita capital X and therefore per capita output and per capita consumption will continue to grow without bound Ljungqvist and Sargent chapter 11 discuss a number of economic mechanisms for obtaining perpetual growth We brie y discuss these within our framework Externality from spillovers In this model the RampD done by some firms is assumed to positively affect the productivity of other firms but in a way that cannot be captured We distinguish the capital under the control of the representative firm Kt from the economywide average physical capital per worker kt Spe cifically we assume that aggregate output is now given by Yt FKtEtNt for F constant returns to scale with an additional equilibrium condition kt KtNt If we proceed as above we find YtNt FKtNtaEt FXtaEt which in equilibrium can also be written tht Xt F11 th1 The difference equation for Xt now becomes KH1 sYt175Kt S 15 X F X k X 11 M NH wt 1 lt t t 1 t and once again in equilibrium equation 11 becomes 7 s 175 7 s 175 XM mflt1xt1 xxt imm T Xt 12 The difference equation 12 passes through the origin If it has a slope gt 1 the economy will grow forever while if the slope lt 1 the economy will decay back to zero capital The necessary and sufficient condition for perpetual growth is then 15 S Sf118gt171 thatis f1gt If this economy does grow its growth rate XtHXt 7 1 is independent of t Xt17 sf1715 11 Xt All factors reproducible Allowing all factors of production to be produced e g by allowing investment to augment the amount of human capital per worker is yet another means of achieving continuous per capita growth While this might seem plausible we should note that with finitelived individuals the human capital has to be of a form that can be passed on to subsequent generations Scientific knowledge might have this cumulative effect on labor productivity Consider the following simple example Suppose aggregate total output can be described by an 56 aggregate production function of the form Y KO NI O 0ltOLlt1 13 t t t where Yt aggregate output Kt aggregate capital services and Nt aggregate labor services Assume that without investment in physical capital capital services would depreciate at the rate 5 per period and that without investment in human capital labor services would grow at the rate 7 per period due to population growth alone with a xed ratio of employees to population and xed labor hours per employee As in the simple Solow Swan growth model assume households invest a constant proportion SI of their income in physical capital accumulation but now assume also that they invest a constant proportion s2 of their income in augmenting their labor productivity through education or train ing Speci cally assume capital and labor services grow according to the difference equations Kt1 175Kt1t0lt8lt1 14 Nt11kNtHt0ltklt1 15 with Its1Yt0lts1lt1 16 and Ht s2Yt 0 lt s2 lt1 17 De ne kt KtNt From the difference equations 14 and 15 for Kt1 and NM we obtain X 7Kt17175Kt1t 1 NH1 1XNtHt Then from 16 and 17 the production function 13 and after dividing through by Nt we get X 7175Xtlet0 7 X 18 H1 M g t Now we can show dXtHdXt gt 0 for all Xt gt 0 7 1 1szxgx175 XSIXta l7175Xt lef Xs2X quot1 7 01 gi 7 1X175170L175s2Xf 17 0Ls1X quot1 7 1ks2Xt0 2 19 EH dXt Also Xt 0 is a stationary point of the difference equation since if we substitute Xt 0 on the right hand side of 18 we obtain Xt 0 Also if we substitute Xt 0 on the right hand side of19 we obtain as Xt gt 0 and using 0L lt 1 dX 7 17M15thialia1552xtl7MXSI 1kocs1 11 7 gt gt alXt X3410 7 SZXE V th a17t2 We conclude that Xt 0 is an unstable stationary point of the difference equation and the phys ical capitalhuman capital ratio will diverge away from Xt 0 Equation 19 also implies dX 11 175X20 170c 178 Xra1koc X Oquot1 41 t Szt 51 t gt0aSXtgtoo dXt 1 MXfM s22 and we conclude that the difference equation has at least one stationary point 58 When Xt is stationary at X the difference equation 18 becomes 175Xs1X X 20 1 1 s2X Equation 20 can be rearranged to get 1 XX s2XO 1 175X SIX or 15X1 0 s17s2X 21 Since 0L lt 1 the left side of 2 1 is monotonic increasing in X while the right side is monotonic decreasing in X In addition the left side equals 0 at X 0 while the right side is s1 gt 0 at X 0 Thus 21 has a unique positive solution for the stationary physical capitalhuman cap ital ratio X gX X1 We have shown that the phase diagram for the first order nonlinear difference equation for Xt is as illustrated in the diagram The function gX starts out above the 450 line and has exactly one fixed point at X Hence Xt will converge to X from any X0 gt 0 59 At the stationary point X the ratio of K to N is fixed so K and N must therefore grow at the same rate The economy is said to be on a balanced growth path With K and N growing at the same rate output Y also grows at that rate Let the growth rate of N and K and Y in the station ary equilibrium be y From the difference equation 15 we obtain N 7 N H t 7 1 t 2 t 22 Nt Nt But from the production function 13 and the investment equation 17 Ht Ssz NE X 23 Nt Nt 52 t In the stationary equilibrium Xt X gt 0 Substituting into 23 and then into 22 we find that the stationary growth rate of N and hence K and Y exceeds the population growth rate 7 y 7 s2X0 24 This economy continues to grow in per capita terms forever The ability to augment labor sup plies through investment in human capital keeps the marginal product of capital from falling as more capital is accumulated so per capita growth does not cease Using the fact that X is an implicit function of the parameters s1 s2 and 7 given in 21 we can differentiate expression 24 for the stationary growth rate to find that y responds positively to an increase s1 s2 or 7 Increases in either saving rate or the population growth rate 7 therefore will increase the long run per capita growth rate of this economy As in the SolowSwan model there is a unique positive stationary state in this economy that is globally stable Unlike the SolowSwan model the economy continues to grow in per capita terms in the stationary state Also unlike the simple model increases in savings rates can have a 60 permanent effect on the growth rate of the economy in this model Finally an increase in the population growth rate actually increases the per capita growth rate in the long run stationary state too This comes about because the new generations are born endowed with the human cap ital we have already invested in The human capital we are thinking of here is therefore like scienti c knowledge or some other disembodied or transferable form rather than being embodied in the nontransferable skills of a current generation Another attempt to explain endogenous growth focuses on increasing returns from specialization in production Ljungqvist and Sargent discuss such a model on pages 2927297 but in an opti mizing framework rather than the simple mechanistic model we are discussing here 11 Limits to Growth Endogenous growth models or models with continuous technological change attempt to explain the empirical phenomenon of continual growth in per capita output over extremely long periods of time Many ecologists and other critics of market economies have argued there are limits to growth implied by the limited availability of natural resources and a limited ability of natural ec osystems to provide essential inputs and accommodate wastes produced by market processes At atechnical level we allow energy minerals and other natural resources to constitute another category of factors of production We might then expect that limitations on the growth in supply of such natural resources would limit economic growth rates That is even if effective labor can be made to grow at rates above the population growth rate 7 so that the marginal product of capital does not decline as KN grows we might still have the marginal product of capital de clining as natural resources become relatively more scarce over time Of course as the relative price of raw materials rises over time firms would also have incentives to exploit resource deposits that are more marginal e g rework old mines or mine the sea bed replace particular resources with substitutes use aluminum instead of copper in electric wires sand fibre optics instead of copper in phone lines etc increase the intensity of use of limited 61 resources e g by switching to hydroponic agriculture to save on scarce land invest in recycling technologies membrane and desalination technology to clean water producing usable energy directly from sunlight and so forth2 Even with regard to energy technologies currently avail able such as solar cells could replace fossil fuels if the latter had a sufficiently high price The supply of solar energy is virtually inexhaustible Nuclear and geothermal resources are also very large Apart from using physical resources more efficiently the productivity of natural biologi cal resources can be augmented through genetic engineering and breeding programs3 As Ljungqvist and Sargent remark the critical technical requirement for perpetual growth is that there must be a core of capital goods that is produced with constant returns technologies and without the direct or indirect use of nonreproducible factors They discuss a simple case where labor is taken as the fixed exogenous resource and the sole capital good is produced without any input of the economy s constant labor endowment Specifically assume that goods output is produced according to Y F KtaNt where 0 S I S l is the fraction of capital employed in producing consumption goods while cap ital goods now are produced entirely with capital Keeping within the spirit of the simple Solow 239Furthermore in practice services are becoming relatively more important over time as economies grow A restaurant meal and a home cooked meal might use the same raw ma terial inputs but the restaurant meal provides greater value because of all the added ser vices 7 the higher quality cooking the atmosphere the new taste experiences and so on In other words as economies grow there is a tendency for the resource intensity of output to fall In fact the continual decline in the relative price of raw materials and agricultural oods suggests that raw materials are becoming less scarce over time 39This is not to say that there is no environmental problem associated with economic ac tivity There usually is but the problem is not one of limited resources per se Rather the problem is one of incomplete property rights Some resources are unowned or communally owned and so are overexploited in a tragedy of the commons Since the resources are unowned they are used without charge and the environmental costs of producing market goods and services are not re ected in their prices The solution is to introduce property rights so use of these resources is re ected in market prices and optimal tradeoffs are made between use and preservation Environmental degradation is a sign of inefficiency resulting from inadequate market structures but is not necessarily evidence of environmentally based limits to growth 62 Swan model we take I as fixed Thus investment is given by It A04 and once again capital accumulation follows Kt1 14319 It Now the difference equation for the capitallabor ratio becomes Xt1 N Kt171 5A1 Kt7175A17 X 7 7 11 t t1 1XNt Thus the per capita capital stock will grow so long as Al gt 57 in which case the growth rate will be A17 75k 1 k Goods output per capita will be given by Yt F Xt 1 f Xt t which will have a growth rate equal to the elasticity of f with respect to X times the growth rate ofX 63 CASSKOOPMANS MODEL AS A COMPETITIVE EQUILIBRIUM 1 The Fundamental Welfare Theorems in a Dynamic Setting In the second chapter we discussed the relationship between competitive equilibria and Pareto optima in a simple two period model In this chapter we illustrate how the argument can be eX tended to a dynamic framework by showing how the CassKoopmans model can be thought of as a competitive equilibrium We shall restrict ourselves to the simplest model with no population growth As in the previous chapter we retain the assumption that labor is inelastically supplied 2 A Competitive Equilibrium Wejoin to the optimum sequences c0 c1 c2 and k1 k2 k3 two sets offutures prices real wages wo w1 w2 and real rental rates on capital u0 u1 u2 such that when rms maximize pro ts taking prices as given and consumers maximize utility also taking prices as given the marketclearing quantities determined are the optimal sequences c0 c1 c2 and k1 k2 k3 More speci cally a set of prices will yield an equilibrium if the capital that households choose to supply to rms at those prices matches the demand for capital chosen by rms at the same prices capital market equilibrium the labor inelastically supplied by households matches the demand for labor chosen by rms at the given prices labor market equilibrium and the output supplied by rms matches the demand for output for consumption and investment from house holds goods market equilibrium An alternative assumption to the futures markets assumption is that we have a sequence of spot markets for nt and kt determining wt and ut However since kt lasts indefinitely into the future consumers cannot make a decision about ct and kt today without knowing all future prices so we must effectively assume they have perfect foresight about future spot prices 73 3 The Representative Firm Maximizatitm Problem For each period t the rms will solve the maximization problem maXF kt nt 7 wtnt 7 utkt nt kt 1 However we imagine the rms solving all these problems simultaneously at time 0 by dealing in futures markets We allow the rms to choose labor demand at t but in equilibrium prices must be such that the rms choose the labor that will be inelastically supplied by households in the per capita quantity n each period 4 The Representative Consumer Maximization Problem The consumers are assumed to rent capital to rms at the real rental rates uo 111 u and sell labor to the rms at the real wage rates wo w1 w2 also determined in futures market trading at time 0 The consumers solve the maximization problem max 2 BtUct ct kt t 0 subject to the constraints Ct kt1 15kttht utkt I 01 2 nt n for all t Observe that the household budget constraints can be rewritten as ct kt1gwtnt1ut5ktt 01 2 74 2 3 4 5 so that we can identify ut 5 E Rt as a rate of return on savings kt that the household takes as given The household chooses consumption ct and savings km to maximize utility subject to a budget constraint that equates consumption and savings to the sum of labor and capital income 5 A Gnessfor the Equilibrium Prices An equilibrium in this economy is a set of pricing functions In order to solve such a problem we have to guess the solution and then verify that our guess works This is analogous to the way we solve other functional equations such as differential equations or the way we integrate Suppose 30 El 52 and El E2 E3 solve the optimum planning problem We guess at FkEt 11 6 Wt FHED 11 7 will be the equilibrium prices to solve the competitive equilibrium 6 Optimization by Representative Agents in Response to these Prices The prices 6 and 7 lead the profitmaximizing firm to choose the inputs E and n in each period t why The Lagrangian for the consumer will be L ZBtUct thtmtn tkt 7 et 7 kt117 5kt 8 with first order conditions for a maximum apart from the transversality condition U Ct qt 9 tht ttht1SPBt qu 0 10 75 ctkt1 vTtn tkt175kt 11 Substitute the above guesses for the equilibrium prices into the rst order conditions 10 and 1 l thtFkEt 11 t tht1 5Bt 1qt1 0 12 ct kw1 Fnlltt nn FkEt nkt 1 7 5kt 13 7 Showing the Prices Result in an Equilibrium Allocation We want to show the equations 12 and l3 describing the maximizing behaVior by households along with our guesses for prices 6 and 7 are solved by the same ct and kt sequences that solve the optimum planning problem Note that if we put ct 2t and kt lg in 13 then by Euler39s theorem we get t12t1 Fl tn175ht 14 and from 9 U39 t qt 15 Now it is easy to see that 15 12 and 14 must hold for St and Ft solutions to the first order necessary conditions apart from the transversality condition to the planning problem L ZBtUCt thtFkty 11 1 5kr Ct kt1 16 That is by the de nition of at and E as the solutions to the planning problem we know they solve the equations 76 U t it 17 thtFkEt ntht1 5 Bti lqtil 0 18 t12H1 Fl tn175ht 19 It is somewhat more dif cult to prove the two problems lead to the same transversality condition and we shall not pursue that issue further in these lectures see an article by Brock and Mirman on the reading list 8 A Trap to Avoid It might be thought that we could merely substitute the Euler equation F69 11 Fn gt IUD Fk gt mfg 20 into the Lagrangian for the competitive consumer39s maximization problem to give us the same Lagrangian as in the planning problem However this is not a valid argument It amounts to as suming each consumer chooses the equilibrium capital stock whereas the idea behind a compet itive equilibrium is that consumers and rms maximize taking prices as given 9 The Intertemporal Arbitrage Condition Note that if we use the rate of return on savings de ned above then the rst order conditions for household maximization 9 and 10 can be written as 5 U Ct1Rt U39Ct 1 21 This equation can be interpreted as an arbitrage condition If the household foregoes one unit of consumption at time tl the cost in utility terms is U ct1 If that unit of output were saved in the form of kt it would yield 1Rt units of output in period t Each of these units of output have 77 value U ct at date t but in terms of units of tl utility each of those time t units are only valued at fraction 78 FINANCIAL INTERMEDIATION 1 Inside versus Outside Money Another criticism of the monetary equilibrium business cycle models of Lucas is that they im plicitly assume all money is outside money In practice much of the money supply is inside It is supplied by financial intermediaries which hold reserves of base money and loans to firms and households as assets against their deposit liabilities The deposit liabilities of banks are almost as universally acceptable as cash as a medium of exchange However changes in the volume of inside money are likely to be associated with changes in the level of loans made by intermediar ies Thus shocks to the demand for inside money might have a direct effect on the level of in vestment and these effects cannot be analyzed with a model that assumes all money is outside money 0 Models that focus on the real effects of uctuations in inside money are also very different from Keynesian or new Keynesian models that emphasize nominal rigidities In models with in side money the nominal price level reflects the value of outside money liabilities of the central bank relative to goods and services The lower liquidity of inside relative to outside money is reflected in the endogenous interest yield on inside money Fluctuations in the average price level result from variations either in the supply of outside money or the demand for outside mon ey as a medium of exchange or as bank reserves However incomplete or lagged adjustment of nominal prices to the excess supply or demand for outside money play no role in linking inside money to real variables in these models 2 Money Aggregates versus Bank Credit 0 There is also a growing body of literature that claims empirical evidence linking monetary dis turbances to business cycles is stronger for the inside components of the money supply than it is for outside money For example Rush has presented evidence that business cycles in the gold standard era were linked to fluctuations in inside money bank liabilities but not correlated with 247 uctuations in the money base Mishkin and Bernanke have presented evidence that the collapse of the banking system played a large role in the Great Depression Most of the massive decline in the US M1 money supply at that time took the form of a decrease in check deposits rather than a decline in base money Although I would not want to make too much of it we can also look at the following evidence using Australian data Recall that we estimated a simple regression link ing GDP fluctuations in Australia to changes in M3 growth rates estimated standard errors in brackets GDPGt 31005 06943 GDPGt71 02104 GDPG174 15883 00581 00575 01378 AM3GPI 00171 Time 00627 00045 Now add changes in the growth rate of the money base to this regression Since I only had money base figures for a shorter period the new regression is based on fewer observations The results were GDPGL 23014 07279 GDPGH 02584 GDPGP4 0074 15480 00752 5 AM3GL71 80512 AMBGtil 00171 Time 01667 007 0326 00045 04 Once the effects of changes in M3 growth have been accounted for changes in money base growth rates have an effect on GDP growth which is not statistically significantly different from zero and which in any case is of the wrong negative sign There have been two strands to the literature focusing on real effects of fluctuations in inside money Some authors such as Bryant and Diamond and Dybvig have concentrated on explaining bank runs as occurred in the US in the Great Depression The basic idea is that bank liabilities are very short term the banks promise to pay cash on demand in exchange for their deposit li abilities However many of the assets of the banks are longer term loans to firms or households 248 that cannot be rapidly liquidated Furthermore if the banks are forced to unload many of their assets in a fire sale as occurred in the Great Depression there is likely to be a large decline in asset values making it difficult for the banks to meet their obligations to exchange cash for their liabilities Normally only a small fraction of a bank39s customers wish to claim their deposits on any given day However if customers come to believe the bank might not be able to meet its obligations they will all queue to get their money out before the bank fails The belief becomes a self fulfilling prophecy As banks call in their loans to meet a bank run firms are likely to fail unless they have access to an alternative source of finance Investment will be adversely affected and the economy is likely to slide into a recession or depression Other authors such as Williamson and Bernanke and Gertler have focused on less extreme epi sodes than bank runs as possible explanations for a correlation between fluctuations in inside money aggregates and real activity They have suggested that normal fluctuations in bank lending could explain normal business cycles In particular they have emphasized the pro cy clical movement in the number of bankruptcies as a key indicator of the role of financial inter mediaries in initiating or propagating business cycles 3 Is Bunk Lending Special The assumption that intermediaries finance all investment is an important feature of the William son and Bernanke and Gertler models since it implies that disintermediation is impossible They argue that intermediaries provide a unique service as a result of an asymmetry of information between lenders and borrowers These asymmetries are also a source of bankruptcy for firms borrowing from intermediaries Williamson s model of bankruptcy can be summarized as follows All loans and investments last for only one period A separate firm undertakes each investment There is a continuum of firms indexed by the productivity of their technology and firms cannot be replicated otherwise all in vestment would be in only the most productive technology Each of the firms has axpost per 249 capita output 8k0 1 6k 1 where 8 is uniformly distributed over 0 80 and is known only to the firm It will always pay the firms to claim 8 equals 0 and to offer their lenders the collateral on the loan 1 6k which will fall short of the agreed upon interest and principal when real loan interest rates are positive In this environment financial intermediaries arise in part as specialists in monitoring firms that can go bankrupt By incurring a monitoring cost C the bank can determine the true ex post produc tivity of the investment In the absence of an intermediary each of the indirect lenders to the firm holders of intermediary liabilities would have to incur the monitoring costs the interme diary is a delegated monitor for each of the depositors There will be a level of 8 say 8 such that if 8 2 8 it will be in the interest of the firm to pay the agreed upon interest and principal to the bank while for 8 lt 8 the borrower defaults In the default state the bank pays the monitoring cost C and receives net income 8k0 1 6k C 2 The state where monitoring occurs is interpreted as bankruptcy Use h to denote the nominal interest rate charged to firms that can go bankrupt Use 7 to denote the actual and expected rate of inflation Firms choose kg and a default level of productivity 8 to maximize expected profits 80 1 h a 0flt8k0 1 6k 1 k0gtd 3 5 subject to the constraints i at 8 the firm is indifferent between declaring bankruptcy and honoring its contractual obli gations 250 sk8 1 6k0 19 4 ii the expected return to the bank minus the cost of providing intermediary services equals the interest payments to depositors Assume a fraction 9 of deposits are held as non interest bearing reserves by the banks and the cost of intermediary services are a fraction x of the level of depos its Then if r the return on deposits the expected nominal return on bank loans will have to be i r x1 p and the expected return on loan contracts will satisfy 1i E0 1 1h CL Z fsk0 1 6kO Cde 60 1 kods 1n 0 5 Use 4 to eliminate h from 3 and 6 The firm then chooses kg and 8 to maximize kg 82k0 gt1 2 gt1 2 gt1 oc 0 28080 s 1 6k0s k0 28 The first order conditions for a maximum of L are lt 1gt k8 lt1 gt J ccil 2 0 8082 1 6ock8quot1lts 11 Eliminate A from equations 8 and 9 and rewrite the constraint 6 to obtain 2DeokO kgsg Cso 8 0 251 5 6 7 8 9 10 kkgltsO s asODk0 11 where we have defined D 6 i 12 a 1 7 0 For given i and hence D equations 10 and 11 can be solved for kg and 8 Equation 4 can then be solved for h Thus the firm side of the model the supply of output and the demand for capital can be solved as an implicit function of the interest rate i 0 Any shocks that affect the demand for deposits from households the costs of intermediation x or p or equilibrium real interest rates will affect the supply of bank loans the number of bank ruptcies and the level of output An important feature of Williamson s model however is that intermediaries finance all investment More generally firms denied bank loans might be able to finance their investment directly from the capital markets or may be displaced by firms that do have access to such finance 4 Household Borrowing Constraints and Bank Lending I have argued that household credit constraints that is restrictions on household borrowing are central to ensuring fluctuations in bank finance have a significant impact on the aggregate CCOHOIIly Banks harness the demand for a medium of exchange to finance investment In effect indirect claims to capital circulate in place of outside money In an economy with only outside money the capital stock merely reflects the desire of consumers to save for future consumption It might be thought therefore that an economy with inside money would support extra capital and pro duce more output There may also be a presumption that shocks to the demand for inside money or changes in bank costs would affect the supply of bank loans and make investment more vul 252 nerable to financial disturbances If the equilibrium capital stock is higher in an economy with inside money then the marginal product of capital and thus the equilibrium real interest rate must be lower In the absence of household credit constraints or restrictions on borrowing intertemporal arbitrage would tie the riskless real interest rate to the household rate of time preference Increased household borrow ing would compensate for increased lending from intermediaries and the equilibrium capital stock would be unaffected Furthermore if households have sufficient access to capital markets to stabilize real interest rates the equilibrium capital stock should also remain largely unaffected by shocks that adverse ly affect the size or efficiency of the intermediary industry A contraction in bank finance would be accompanied by disintermediation or an increase in loans to firms from other sources I also discuss an intertemporal general equilibrium model where an asymmetry of information between intermediaries and firms and the possibility of bankruptcy restricts some firms to bank finance I assume however that households can also lend directly to some firms Furthermore some firms can either borrow from banks or directly from households If households do not face credit constraints the stationary real interest rate on direct loans from households is tied to the household rate of time preference Arbitrage by firms able to choose their source of finance ties bank interest rates to the real interest rate on direct loans Thus the household rate of time pref erence becomes the key determinant of all real interest rates and therefore the marginal product of capital and the capital stock Financial shocks that change the supply of bank loans have few effects on the general equilibrium of the economy Changes in direct investment in firms by households and changes in household borrowing substantially offset changes in the size of the banking sector Disintermediation would become irrelevant only if the economy arrives at a corner where all firms borrowing from banks have no alternative source of finance 0 If households cannot borrow however then I show that the stationary equilibrium real loan in 253 terest rate is below the household rate of time preference The equilibrium capital stock is there fore higher than in an otherwise identical economy with neither inside money nor credit constraints Households willingly hold inside money and loans at such low interest rates in order to finance consumption or self insure against unanticipated uctuations in future liquidity needs Furthermore real interest rates are affected by changes in bank costs or the demand for inside money Since the demand for direct loans to firms is itself related to liquidity needs disinterme diation becomes less effective in insulating firms from changes in the supply of bank loans Changes in real interest rates are accompanied by changes in the equilibrium capital stock out put the number of bankruptcies and the terms of the loans offered to borrowers liable to declare bankruptcy It is informative to consider the way monetary factors and household credit constraints alter the first order conditions for the intertemporal consumption decision of a representative household Throughout this discussion we shall use loans to refer to direct household loans to firms that by pass intermediaries In practice such capital market instruments could be equities instead of corporate bonds In order for inside or outside money to provide liquidity services households either must choose not to or must not be able to freely exchange loans for liquid assets at any time Otherwise households would choose to hold all their wealth as loans They would use cash or check depos its for the briefest interval of time and only because convention demands that direct loans to firms are not acceptable as a medium of exchange We implicitly assume that transaction and information costs limit the acceptability of loans as a medium of exchange and imply that it is optimal for households to exchange loans for liquid assets at discrete intervals In the intervening periods household consumption is subject to a liquidity in advance constraint Inside money yields a positive nominal return while outside money does not If both types of as sets are to be in positive demand then outside money needs to yield additional liquidity services 254 I assume that outside money is more universally acceptable than inside money in the sense that outside money can be used to make any purchase whereas inside money is unacceptable for a subset of purchases The implicit liquidity services of inside and outside money will be reflected in positive Lagrange multipliers from the liquidity constraints These positive Lagrange multi pliers raise the returns on holding outside and inside money to equal the higher real yield on loans They also drive a wedge between the marginal utility of consumption and the marginal value of wealth For a household that is una ected by credit constraints a first order condition of the form 1it 1 EtVWtlVWt 13 will apply to the intertemporal consumption versus savings decision where VW is an indirect utility of wealth function and it is the real return on loans However the marginal utility of wealth will differ from the marginal utility of consumption by a Lagrange multiplier reflecting the li quidity value of the monetary assets required to finance consumption If these liquidity returns are represented by a multiplier u the first order condition becomes 1it 1 Etu39Ct1Mt1u390tMt 14 The marginal utility of consumption axceeds the marginal indirect utility value of wealth by u Hence the level of consumption will be lower than it would be if loans could be used directly to finance consumption The requirement to use lower yielding liquid assets effectively raises the cost of consumption and therefore reduces the amount consumed The first order condition for a credit constrained household would be further modified by a pos itive Lagrange multiplier from the credit constraint Specifically if M is the multiplier for the credit constraint the intertemporal consumption versus savings decision of a credit constrained household would obey 255 1it391 main011 Mt1tluct Mtl 15 The key modification resulting from the presence of liquidity and credit constraints however is that both the values of the liquidity and credit multipliers and the proportion of households that are credit constrained in any period depend on the interest rate it As a result firms effectively face an upward sloping supply curve of investment funds as a function of the interest rate The equilibrium real interest rate and the level of investment become jointly determined by house hold and firm behavior In particular shocks to the marginal productivity of capital or the depre ciation rate of capital alter the equilibrium real interest rate even in a stationary state where per capita consumption is not expected to change from one period to the next In contrast in the stan dard representative agent model real interest rates are determined by household behavior Firms face a horizontal supply curve of investment funds from households at a real interest rate deter mined by the household rate of time preference and anticipated changes in household consump tion 0 In an economy with inside money and household credit constraints there would be some funds available to firms to finance investment even at very low equilibrium real interest rates Inside money with a positive yield is preferable to outside money with a zero rate of return Even when the real yield on inside money is close to zero households will hold some inside money to fi nance purchases that could be made with either type of liquid asset As a result equilibrium be tween the supply and demand for investment funds occurs at a lower real rate of interest than would apply in the standard representative agent economy with neither inside money nor credit constraints Intermediaries would also finance investment in an economy with inside money but no credit constraints However many households would exploit any tendency for real interest rates to de cline by borrowing Such increased borrowing would occur up to the point where households were indifferent between consuming today and consuming tomorrow Intermediaries thus have 256 little impact upon equilibrium real interest rates and investment Increased household borrowing compensates for the increased availability of funds from the intermediaries In the credit constrained economy with inside money the equilibrium real interest rate is very sensitive to shocks affecting the demand for inside money or shocks to the banking sector that affect the supply of loans corresponding to a given demand for inside money Changes in real interest rates are accompanied by changes in the equilibrium capital stock and equilibrium per capita output For example the real effects of shocks to the demand for inside money or bank costs are about 100 to 200 times greater when households face credit constraints The elasticities of the real interest rate on direct household loans to firms upon which the other interest rates are based and the stationary per capita capital stock with respect to key parameters of the model are graphed in the following figures Elasticities of real loan interest rate Elasticities of per capita capital I Unconstrained D Constrained I Unconstrained D Constrained 1DD1 D P DDDDDDD UDDDDPDDDDDDD The key parameters included in the figures are the household time discount factor 5 the coeffi cient of relative risk aversion y the probability of non zero demand for the transaction services of inside money in any given period p the utility value at the same consumption levels of goods which can be bought with inside money relative to goods which can only be bought with cash 6 the reserve ratio for banks p the marginal cost of bank services per unit of deposits x the 257 fixed cost of bankruptcy in units of per capita output C the coefficient of capital in the Cobb Douglas production function 0L and the per period depreciation rate of capital 6 Since cash is universally acceptable as a medium of exchange while inside money is not an in crease in risk aversion Y increases the demand for cash at the expense of inside money In both the credit constrained and the unconstrained economies there is a large fall in the per capita de mand for inside money As a result household risk aversion is a major determinant of equilibri um real interest rates and per capita capital in the credit constrained economy In the unconstrained economy however a large fall in borrowing by less wealthy households and a rise in direct household lending by more wealthy households offset the tendency for real interest rates to rise The increase in net direct lending to firms offsets the decline in bank loans so that per capita capital is virtually unaffected An increase in the probability p of being able to use inside money to fund purchases reduces con sumption uncertainty and therefore the demand for liquid assets The same inventories of cash and inside money finance a higher level of per capita expenditure In the unconstrained econo my households substitute out of liquid assets into direct loans to firms Equilibrium real interest rates fall marginally and there is an extremely small positive effect on the equilibrium per capita capital stock In the credit constrained economy the increase in p reduces average excess liquid assets held by households after goods market trade The demand for direct loans to firms as well as the demand for liquid assets declines Equilibrium interest rates rise and the equilibrium per capita capital stock falls An increase in the value of goods that can be bought with inside money relative to goods that can only be bought with cash 6 raises the demand for inside money In the credit constrained economy the increased demand for inside money dramatically reduces equilibrium real interest rates and produces a large increase in the equilibrium per capita capital stock In the uncon strained economy a large decrease in net direct household loans to firms again offsets the ten dency for real interest rates to fall 258 0 While the elasticities of real interest rates and per capita capital with respect to changes in bank costs x or the bank reserve ratio 9 are not large in either economy they are still about 100 to 200 times larger inthe credit constrained case Again endogenous increases in direct household loans to firms offset the fall in bank loans in the unconstrained economy 0 A striking feature of the figure is that the household rate of time discount 1 5 1 is the only pa rameter to have much off an effect on equilibrium real interest rates when there are no household credit constraints In all other cases endogenous changes in direct household lending to firms offset any tendency for real interest rates to change In the case of changes in CC 6 and C station ary equilibrium real interest rates are completely unaffected in the unconstrained economy By contrast in the credit constrained economy 0L and 6 are key determinants of equilibrium real in terest rates In addition changes in the cost of bankruptcy C affect only those firms at risk of bankruptcy in the unconstrained economy whereas changes in C affect all firms in the credit con strained case 0 A simple partial equilibrium model of the market for loans provides a useful conceptual frame work for viewing the results for the credit constrained economy Changes in CC 6 and C can be thought of as shocks to the demand for loans from firms Since such a shock would produce a movement along the upward sloping supply curve the result is a positively correlated movement in equilibrium real interest rates and the equilibrium quantity of loans On the other hand chang es in 1 5 1 y p 6 p and x can all be thought of as shocks to the supply of loans from house holds and banks A movement of the supply curve along a downward sloping demand curve for loans then produces a negatively correlated change in real interest rates and the equilibrium quantity of real loans It is also noteworthy that most1 of the loan supply shocks also produce endogenous movements in the money supply measured as the sum of inside money and cash balances and average nominal prices that are positively correlated with movements in per capita capital By contrast changes in 0L and 6 produce endogenous movements in the money supply 139 The change in the probability of check goods demand p is the only exception 259 and nominal prices that are negatively correlated with movements in per capita capital There is now considerable evidence particularly from panel data sets that constraints on house hold borrowing can explain deviations of observed household consumption from the levels pre dicted by the permanent income hypothesis Households with relatively high levels of wealth can maintain their consumption when income is temporarily low They effectively self insure by holding assets they can liquidate in time of need Households without much wealth however appear to have limited access to capital markets and therefore tend to display consumption levels that more closely track their current income We have argued that credit constraints in the pres ence of a demand for inside money and a role for financial intermediation can in a similar manner explain many apparently anomalous phenomena in financial markets In particular they offer an explanation for apparently low riskless real rates of interest relatively high risk premiums rel atively large fluctuations in real interest rates in the face of smooth per capita consumption pro files and a large effect on real interest rates of real shocks such as shocks to the marginal productivity of capital They also would appear to account for large effects on real interest rates investment and output of financial sector shocks to the demand for inside money the cost of financial intermediation and the extent or cost of bankruptcy 260 INTRODUCTION TO BUSNESS CYCLES 1 What is a Business Cycle As we saw previously if we plot log real output against time and t a cubic time trend we get a fairly good t but there are signi cant uctuations about the trend line as in the following graph for US GNP 95 7 log seasonally adjusted real GDP 711H 11H 11H HH HH HH HH HH HH HWHWHW 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 It is interesting to note that both the quadratic and cubic terms are signi cantly different from zero at conventional significance levels that is there is evidence consistent with the hypothesis that economic growth trended down in the 1970s and 1980s relative to the 1960s and 1990s An alternative way of allowing for lower trend growth rates in the 1970 s and 1980 s is to t a piece wise linear time trend with a break in the slope in the early 1970 s and another in the early 1990s On the other hand if we regress the annual growth rate of real income against time and time squared neither coefficient is significantly different from zero at conventional significance lev els tstatistic is 106577 for t and 061908 for t2 This re ects a more general issue in time se ries statistics If the underlying process is nonstationary that is the trend rate of growth of the economy is a random process instead of the deterministic trend estimated above deterministic trend lines may appear statistically signi cant when there really is no such trend in the data The simplest random growth rate model makes the trend level of output a random walk with drift lnytalnyt18t 1 so that the growth rate from tl to t which equals a St is stochastic This can be justified on the grounds that growth in F39 J and the capital stock are both likely to have random components while technological change is also likely to be random However we then should fit the output growth to the growth in factor supplies and technological change rather than simply specify a simple trend process such as that given above Many authors essen tially take a weighted moving average of observations from the recent past to define the current trend level of output However we identify the trend we might hope to explain the general trend with growth theory models What explanation is there for the uctuations about trend We would have little hope of getting a theory to explain these uctuations unless they displayed some regularity This was the idea behind the work of Mitchell and his coworkers at the NBER in the 1930 s and 1940 s They studied many economic time series and documented their devia tions about trend in an attempt to identify regularities which might form the basis for a theory of business cycle uctuations Such a theory could then in turn be used to make further testable predictions Why use a large number of individual time series rather than aggregate data 87 i Any one time series is subject to measurement error ii Use of aggregate data already presupposes the components behave in the same cyclical man ner iii Disaggregated data may be more comparable to data from other countries iv Deviations of individual series from the cyclical pattern may be suggestive of possible hy potheses about the source ofbusiness cycles It was believed that business cycles represented an inefficient allocation of resources and should be avoided if possible A theory of uctuations might enable the introduction of reforms which would mitigate their severity if not remove them altogether Koopmans criticized the NBER approach as measurement without theory and effectively end ed that line of research for decades More recently time series analysts have again become in terested in de ning and measuring business cycles However their approach has been somewhat different They have focused on stochastic dszerence equations as a modeling tool 2 Some Regularities in the Behavior 0fP0st WWII US Data We can also look at the cyclical behavior of US time series in the postWWII era using the graphs in the appendix to this chapter You could examine the related graphs and discussion of them in for example Barro s text The particular graphs presented in the appendix are based on data available from the Federal Reserve Bank of St Louis web site httpwwwstlsfrborgfredin dexhtml Implicitly the graphs are based on the stochastic growth model 1 Business cycles are part of the deviations from the average growth rates In Chart 1 the growth rate of real personal consumption and real private investment are graphed along with deviations of annual GDP growth from its average We see that consumption has about the same variability as real GDP while investment varies substantially more The peaks 88 and troughs in both consumption and investment correspond to those in real GDP Chart 2 shows how the different components of consumption vary over the cycle Consumer du rables is the most volatile component of consumption and services consumption the least vola tile The components tend to cycle together Chart 3 plots the growth in the residential and nonresidential components of investment and de viations from trend of real GDP In the early part of the sample residential investment was more volatile but the uctuations in growth rates appear more similar over the last decade There is also some evidence in Chart 3 that peaks and troughs in residential investment tend to lead uc tuations in the other two series Chart 4 shows that business inventories as a proportion of GNP right hand scale also tend to correspond with the cycle in GDP growth rates although there is some hit that they could lag a small amount Inventories are the least volatile component of investment Charts 5 6 and 7 look at variability in real income variables Chart 5 shows that real disposable income growth more or less tracks GDP growth and is of a similar volatility Chart 6 shows that real nonfarm corporate pro ts growth is much more volatile than GDP but again with uctua tions that generally coincide with GDP Comparing Charts 1 3 and 6 real pro ts are more vari able than investment note the scale change Chart 7 suggests that wage and salary growth also is procyclical with a similar volatility to GDP This variable combines movements in employment average hours and real wages It could also re ect changes in the composition of the labor force Chart 815 focus on other labor market variables Chart 8 shows that the ratio of employed peo ple to the civilian population over the age of 16 varies quite closely with GDP with perhaps a slight tendency to lag and has uctuations of a similar magnitude to GDP Chart 9 shows that the growth of average hours in nonagricultural employment is broadly pro 89 cyclical but much less variable than GDP On the other hand the growth in help wanted adver tisements Chart 10 is procyclical and much more variable than GDP The growth in manufacturing employment also is procyclical Chart 11 and of a similar mag nitude to GDP uctuations The growth rate has trended down over time re ecting the fact that manufacturing employment has been declining relative to employment elsewhere in the econo my A shift from manufacturing to services could explain this but so could a faster rate of labor productivity growth in manufacturing than in other sectors of the economy Charts 1215 focus on unemployment as opposed to employment Chart 12 suggests that the un employment rate is a lagged counter cyclical variable although it tends to be somewhat smooth er than GNP growth The median unemployment duration Chart 13 also appears to be a lagged countercyclical variable although it uctuates more than the unemployment rate The countercyclical movements are perhaps clearer in Charts 14 and 15 which graph the growth rates of people who have been unemployed for more than 15 or less than 5 weeks Per haps not surprisingly the growth of longer term unemployed Chart 14 appears to be somewhat more smoothed than the growth in shorter term unemployed people Chart 15 Charts 16 and 17 show the relationship between fiscal variables and uctuations in GNP Chart 16 graphs growth in real government consumption and investment expenditure against devia tions in GDP growth about trend If anything growth in government expenditure appears to be countercyclical However it appears to be procyclical for some cycles Chart 17 suggests that the consolidated government surplus as a percentage of GDP tend to be more procyclical al though perhaps lagging GDP growth somewhat Chart 18 shows that imports and exports uctuate more than GDP If you look closely imports tend to be procyclical Exports were neither pro nor countercyclical early in the sample but also tend to track imports more closely after the move to oating exchange rates 90 Charts 19 and 20 focus on interest rates Chart 19 shows that the 10 year bond rate is smoother than the 3month and 1year T Bill rates the latter two rates tend to move in a very similar fash ion Chart 19 suggests that interest rates became more volatile after the early 1970s This may re ect a change in Federal Reserve Policy away from targeting interest rates toward other objec tives Subsequently interest rates tended to move countercyclically in the 1970s but the corre lation either positive or negative with GDP uctuations appears to have diminished in more recent decades Chart 20 which graphs the spread between the 10 year and 1 year bond rates with GDP growth suggests that this spread tends to lead GDP uctuations Another feature of Chart 19 is the gap between interest rates and GDP growth that opens up in the 1970s and 1980s and then closes again in the 1990s This broad pattern corresponds some what with the in ation rate graphed in Chart 21 Evidently increases in in ation tend to raise nominal interest rates Borrowers and lenders are both concerned about real interest rates not the nominal interest rate If in ation is expected over the life of a loan lenders will demand and borrowers will be willing to pay compensation for the expected erosion in the value of the loan The afterin ation return on a loan or cost of borrowing is the nominal interest rate less the an ticipated rate of in ation Chart 19 also shows that the in ation rate tends to move countercyclically with GDP growth deviations 7 although in some cycles early 1950s early 1980s it is procyclical Different mac roeconomic theories make different predictions about the correlation between in ation and out put real business cycle theories predict a negative correlation between in ation and output new classical theories predict a contemporaneous or short lagged positive correlation between GNP growth and unanticipated in ation and no relation between GNP growth and anticipated in a tion Keynesian models predict a longer lagged positive correlation In sorting the evidence vis avis the last two theories we need to be able to measure anticipated in ation and this is difficult because we don t have good direct data on expectations Charts 21 and 22 also examine money growth rates In the 1960s and 1970s M2 growth tended 91 to be procyclical perhaps with a slight lead over GDP uctuations However the relationship appears to have disappeared entirely in the last decade Chart 22 suggests that M1 and base mon ey growth have remained more procyclical The increasing volatility of M1 and money base growth rates is also very noticeable The large swings in M1 growth from the mid1980s at least partially re ected substitutions into Ml from other components of M2 since they were not matched by similarly sharp changes in overall M2 growth rates The sharp increase in the money base in 2001 re ects the response of the Federal Reserve to the attack on the World Trade Center in New York Chart 23 suggests a stronger relationship between growth in loans and GDP than between the conventional monetary aggregates and GDP However the graph also suggests that the loans variable lags GDP Finally Chart 24 looks at real energy prices and GDP It suggests that there has been a procy clical movement in real energy prices over the last decade or so but the relationship is weak prior to that 92 KEYNES ON MONEY INTEREST RATES amp INVESTMENT 1 Money Interest Rates and Investment 0 We showed previously that as an initial working hypothesis the assumption of a fixed nominal wage appears to hold some promise However Keynes wanted to include investment in his mod el while he also wanted to explain the determination of the interest rate To explain the latter concern we need to discuss some of the debate between economists at the time Keynes was writing about the causes of the Great Depression The prevailing theory of the source of business cycle fluctuations at the time of the Great Depression was that they were primarily the short term consequence of monetary instability However it was also the practice at the time to measure the stance of monetary policy by looking at interest rates Many economists viewed the price of money to be the cost of borrowing on the short term money markets If money was tight in terest rates were high and vice versa This contrasts with the view of the classical quantity theo rists who viewed the inverse of the general price level as the value of money However given that the quantity theorists argued that the connection between money and nominal prices in volved long lags it did not seem very practical to base daily monetary policy on movements in nominal prices Furthermore while it is quite practical to measure the nominal price of particular goods and services on a daily basis it is not possible to continuously monitor the average level of nominal prices Therefore many central banks and in particular the Bank of England focused on movements in interest rates to determine the current stance of monetary policy A problem with such an operating procedure is that nominal interest rates reflect current expectations for the rate of inflation in prices in addition to real interest rates or the current real cost of credit In particular at the height of the Great Depression prices in the United States were de ating at nearly 30 pa and this must surely have reduced nominal interest rates Nevertheless some economists argued that monetary policy could not have caused the Great Depression because nominal interest rates weren39t high enough to suggest money had been very tight This led Key nes to be interested in explaining the short run determination of nominal interest rates and also to look for alternatives to money supply changes as a cause of business cycle uctuations It is quite natural that he should turn to exogenous uctuations in investment as an alternative source of business uctuations He argued that equity markets could be characterized by periods of exogenous swings in investor confidence and that these exogenous movements in equity pric es affected the ability of firms to raise funds to finance investment Keynes talks about invest ment in the stock market as a game of old maid where everyone is concerned with the valuation other investors place on stocks and not the underlying fundamentals In terms of our models of a dynamic economy with capital you might argue Keynes was implicitly suggesting equity prices can deviate from the long run perfect foresight path as investors pay more for stocks only because they expect others to pay yet more for them at a later date Eventually these prices are recognized as being inconsistent with the underlying fundamentals and the period of boom ing investment financed by the rising stock market comes to a crashing halt One apparent difficulty with his model is that if investors sell equities they could be expected to buy bonds This would reduce interest rates making it less costly for firms to raise funds for in vestment via the bond market To close off this channel Keynes emphasized the so called li quidity trap Investors would not ee to bonds but rather to cash and interest rates would not fall far enough to stimulate investment Keynes emphasized that the demand for money was a demand for liquidity and that less liquid assets paying a higher rate of interest were an alternative to holding money This liquidity pref erence theory of the demand for money enabled him to explain how movements in interest rates could affect the demand for money and in the short run lead to deviations between money supply changes and changes in nominal prices 0 The key notion behind Keynes analysis is that an exogenous change in investment or savings requires the real interest rate to change as in the equilibrium intertemporal resource allocation models we have been examining but then the change in interest rates affects the demand for 184 money and therefore the equilibrium average level of nominal prices Stickiness in nominal wag es then prevents a rapid adjustment of the economy to changes in desired intertemporal resource allocation 2 The Textbook ISLM Model 0 The IS LM analysis of Hicks summarized Keynes arguments on the interaction between mon etary and real phenomena in the presence of fixed nominal wages and exogenous shocks to in vestment It is this version of Keynes model which became the standard expository device for the Keynesian model To get an idea of the way the IS LM model works we shall for the moment ignore the labor market even though this was a centerpiece of Keynes39 own analysis We replace the assumption of fixed nominal wages by one of fixed nominal prices In the short run equilibrium in the de mand and supply of goods and in the demand and supply of money jointly determine the two endogenous variables real output and the nominal interest rate 2 real interest rate in the pres ence of fixed prices The demand for goods consists of a demand for consumption and a demand for investment goods Again we shall ignore some of the detail of Keynes analysis by not distinguishing be tween investment goods and consumption goods but merely talk about aggregate output as a homogeneous good Equilibrium in the goods market requires that yci 1 where y the level of output c 2 consumption and i 2 investment Households are assumed to decide on their level of consumption and savings by allocating their current income on the basis of current interest rates 0 Kr Y 2 185 this contrasts with the real business cycle and growth models where consumption depends on wealth rather than current income This issue will be discussed in more detail later Keynes postulates as a fundamental psychological law that the elasticity of consumption with respect to income is less than 1 so that as income increases consumption increases less than pro portionately 3 We shall also assume that substitution effects dominate on average so that increases in r increase savings and so reduce consumption Having determined their level of savings households then allocate those savings between money and interest paying assets on the grounds that i higher income increases the demand for money for transaction purposes as in the classical quantity theory of money ii higher nominal and real with p xed interest rates reduce the demand for money This leads to the money demand function M L 4 p ry with Lr lt 0 and Lygt 0 Investment is determined by firms and depends positively on the current level of output and neg atively on the current nominal rate of interest In addition Keynes argued that the desire to invest was unstable and much governed by whims and herd instincts We can represent this by an exogenous shift parameter 0L in the investment function i gr y 06 5 186 0 The product market equilibrium condition leads to a relationship between y and r called the IS curve y fr y gr y 0c 6 Totally differentiate this equilibrium condition to find Bf dr Bf ngBg i 7 i 7 7 BrdyByBrdy By which can be rearranged to yield Bf Bg 177 E 2 3y 3y 8 dy Bf Bg ii Br Br Then the assumptions on f and g imply the denominator of this derivative is negative Also it is usually assumed that the marginal propensity to consume out of income plus the marginal ef fect of higher output on investment are less than 1 In that case the IS curve will be negatively sloped in the y r plane I ll IS Equilibrium in the supply and demand for money with M and p exogenously given also leads to a relationship between y and r called the LM curve Lr Y MP 9 Totally differentiating this equilibrium condition 9 we find BLdr BL i i 10 Br dyBy sothat dr BL BL 7 i 11 dy By Br which is positively sloped in the y r plane 3 Comparative Statics We can use the IS LM diagram to study the effects of changes in some of the exogenous vari ables Consider first an increase in the money supply M This will shift the LM curve to the right and leave the IS curve unaffected Output y will increase so there will be pro cyclical move ments in M which corresponds to our observations However r moves counter cyclically and there is considerable doubt whether this corresponds with the evidence Our observation of the data suggested r tended to be relatively high as the economy went into a downturn but then tend ed to decline along with output as the recession got under way Similarly low interest rates tend ed to precede periods of rapid output growth but then rise along with GNP at business cycle peaks 0 An increase in the shift variable 0L will move the IS curve to the right and leave the LM curve unaffected The result will be a pro cyclical movement in both y and r which is usually thought to be more consistent with the evidence although as we noted above the correlation between 188 movements in interest rates and output is weak To explain the pro cyclical movement in M we could add the assumption that the monetary authorities accommodate the expansion with higher interest rates following the investment boom the authorities expand the money supply So long as this expansion in the money supply is not too large interest rates can still rise overall In this scenario the causality will run from interest rates to money supply changes rather than money supply being the exogenous driving factor1 0 We can also examine the effects of changes in exogenous variables using algebra Collect the equilibrium conditions together to get y fr y gr y 0c 12 LG Y MP 13 Now totally differentiate 12 and 13 with respect to Ct to get the matrix equation d 1fygyfrgr doc got 14 Ly LI 3 0 doc The determinant is given by A lfygyLr frgrLy lt 0 15 and the solutions for the derivatives by 139This idea of accommodating money supply expansions has difficulty explaining a lead of money supply growth over movements in output but some economists doubt whether this corresponds with the facts The idea that money supply changes are endogenous accommodating movements would also have difficulty ac counting for the evidence of a link between money and output provided by Hume and other economists writ ing before there was a central bank The idea of reverse causation from output movements to changes in the money supply has recently regained popularity in the works of the real business cycle theorists 189 dy gaL I E Tgt0 dr gaLy EL Tgt0 as we noted by examining the IS LM diagram Also we have dc f f EL ydoc rdoc and 18 is greater than zero if Ifyl gt Ifrl and ILrl gt ILyl as we shall assume Also di d 3 y c 1 fyda frdagt0 E 16 17 18 19 so that consumption and investment both move pro cyclically in response to an 0L shock Fur thermore dy dy 1 72 72 71 loclyy ydoc 1doc y y1 fy 1doc y frdoc ny y1 fy 1 c yfy c E gt0 20 21 The final inequality in 21 follows from Keynes fundamental psychological law Thus the investment output ratio moves pro cyclically in response to CL shocks as appears to be the case in practice 190 INTRODUCTION 1 The Subject of Macroeconomics Macroeconomics aims to explain the time series behavior of aggregate economic variables such as the overall gross output of marketed goods and services measured as GDP the average level of prices of that gross output of goods and services measured as the GDP de ator the total level of market employment measured by the number of individuals in the work force the average return to factors of production such as the average level of representative wages or representative interest rates the aggregate level of investment in physical and human capital the net scal de cit or surplus of the government and the manner in which it is nanced the balance of overseas trade the current account balance and the balance of payments the current value of the exchange rate relative to foreign currencies Apart from a scienti c interest in explaining how the world works we are motivated to under stand the behavior of these variables because they are thought to be of considerable practical rel evance the overall level of output of goods and services relative to the population in the economy GDP per capita is thought to be a reasonable proxy for the average level of af uence in the commu nity other things equal people would prefer to live in an economy with a higher level of GDP per capita changes in the average level of prices or in ation is widely believed to reduce the ef ciency with which a market economy operates more particularly high in ation appears to be associ ated with high variability in the in ation rate and this is thought to cause instability in the real economy variations in employment opportunities can adversely affect the welfare of those unable to nd a job at a wage and under the conditions they would accept variations in factor returns can similarly adversely affect the welfare of particular individuals the aggregate level of investment in capital affects the future output of goods and services fu ture factor returns and future levels of employment imbalances in government expenditure and taxation receipts must be financed either by a change in the level of government bonds outstanding or a change in the money supply increases in the former are thought to adversely affect current consumption andor investment while in creases in the latter are thought to increase in ation imbalances in overseas trade can similarly affect future welfare by increasing or decreasing the level of net foreign indebtedness and therefore required future changes in consumption or pro duction balance of payments surpluses or deficits will affect domestic monetary conditions which may in turn have implications for domestic in ation The hope underlying macroeconomics is that if we can understand why the aggregate variables of interest behave the way they do we might be able to implement various policies or reforms to make them behave more satisfactorily This begs the question as to which criteria one should use to evaluate macroeconomic performance 2 The Focus of the Course The focus of this course is on the various models that have been devised in an attempt to un derstand macroeconomic phenomena in a closed economy framework We should first understand what we mean by a model A model of the behavior of aggregate vari ables in the economy is necessarily incomplete as is any other model For example a map is a model The map is inaccurate or incomplete in the sense that it is not an exact replica of the phys ical characteristics of the region it covers It does not include all the trees buildings etc of the real world If it did it would be useless Why use a map to navigate when we could equiva lently look out the window It is precisely because the map abstracts from irrelevant details that it is useful for the speci c purpose of finding one s way around Similarly we want our economic model of the behavior of aggregate economic variables to mimic the observed behavior of those variables We want it to do so with enough detail that we have some con dence that the model will accurately predict the consequences of various relevant events for the behavior of the aggre gates In particular we want our model to guide us as to the likely consequences for aggregate economic variables of various government policies particularly monetary and scal policy The theoretical variables in the model should behave like their empirical counterparts in the way they react to policy changes but the model should not attempt to mirror the economy in all its aspects We shall leave to one side the open economy issues of the balances of trade and payments and the determination of the exchange rate they are discussed in a separate open economy macro course We also shall not discuss policy design or implementation in great detail A complete course could be taught focussing on various aspects of macroeconomic policy design and implementa tion preferably in the context of a discussion of policies with a microeconomic aim such as those to handle extemalities or public goods Much discussion of economic policy in macroeco nomics is incomplete Models with very similar descriptive properties can have very different welfare implications making it very dif cult to draw conclusions about the possibilities for pol icy to improve welfare In addition policy is often discussed with no reference to the way dem ocraticallyelected governments with their bureaucracies and other special interest advocates actually behave or would be expected to behave were they to be entrusted with the task of de signing and implementing macroeconomic policies Nevertheless we want to discuss certain is sues that arise in the consideration of macroeconomic policy and are more or less unique to that class of policies andor that require some understanding of models of aggregate behavior Fur thermore since an underlying motivation for studying macroeconomics is to enable us to say something about the likely effects of different policies we need to discuss macroeconomic pol icies 3 Economic Models What justifies us calling our model an economic one Economics is the study of the allocation of scarce resources to competing ends The goal of the program of research we shall discuss is to explain the behavior of the aggregates discussed above with a model that is based on theories about the way scarce resources are used to satisfy competing ends in a marketbased economy In this sense we want our model to be compatible with microeconomic theory particularly any components of that theory that are both relevant and have survived many attempts at refutation using micro data ie data on quantities and prices in individual markets We shall often use the construct of the representative agent consumer worker firm to derive a model that has some hope of being consistent with microeconomic theory The basic idea is that using a representative agen allows us to abstract from the distributional effects while re taining the idea that demands and supplies are the result of maximizing behavior The claim is that factors such as intersectoral productivity shifts and redistributions of income are not impor tant for explaining aggregate economic phenomena Although it is agreed that distributional or intersectoral shocks affect relative prices and outputs in one sector of the economy relative to another it is a matter of contention whether they are needed to explain aggregate phenomena Strictly speaking to be able to guarantee that our representative agent model is consistent with microeconomic theory we would need to show that our aggregate behavioral equations can be derived from aggregating the behavioral equations of individual agents The theory of aggrega tion will be discussed in your first semester microeconomics course On several occasions in this course we shall note how disparate individual behavior can be aggregated to yield aggregate be havior that could not be adequately modelled using a representative agent model Models with heterogeneous agents are a major area of current research in macroeconomics and will be a focus of the second semester macroeconomics course Even assuming we can adequately represent aggregate behavior using representative agents it is often very difficult if not impossible to solve for an equilibrium We attempt to avoid this problem by invoking the fundamental theorems of welfare economics These state that under certain restrictive conditions there is an equivalence between competitive equilibria and Pareto optima The point of this result for us is that it is often easier to solve for Pareto optima than for the competitive equilibria While it is recognized that the conditions required for the validity of the fundamental welfare theorems are not true in practice for explaining aggregate economic phenomena we hope that the approximation is close enough Nevertheless some of the models we shall consider are based on the premise that deviations from the conditions required for the fundamental welfare theo rems to hold are central to explaining how aggregate economic variables behave Assuming economic efficiency is one characteristic of good economic policy whether or not the observed outcomes are tolerably close to efficient will have implications for the desir ability of interventionist macroeconomic policy Unfortunately this issue infects macroeconom ic discussions with a political avor Often it appears that models are favored more for their implications for the desirability of interventionist policy than for the adequacy of their descrip tion of aggregate economic behavior This course will attempt to focus on the scientific issue of obtaining an adequate description of aggregate economic behavior deemphasizing the im plications of models for the desirability of interventionist macroeconomic policy Recall how ever that the ability of the model to account for the effects of macroeconomic policies will be perhaps the most important factor in judging its adequacy CASSKOOPMANS GROWTH MODEL 1 An Economic Model of Aggregate Savings We now introduce preferences to determine the proportion of income saved and invested and hence the growth of the capital stock Why are we interested in modelling the choice of the sav ings rate as based on preferences There are several reasons i We want to model market economies where the savings are determined by individual choices Although we are interested in market economies we look at the Pareto optimal problem first and then apply the fundamental welfare theorems ii Ultimately we are interested in using our model to make predictions about the effects of var ious exogenous shocks If the variation in an exogenous variable we are considering is within the 1 we could examine the statistical evidence to see what the effect range of historical experience of the shock was in the past However often we are interested in using the model to make out of sample predictions Unless the model is based on some structural theory we cannot have any confidence that the various parameters of the model will remain constant in a new economic en vironment The basic paradigm of an economic theory is that we try to explain choice behavior as the response of a maximizing agent with a relatively stable objective function to changes in the agent39s opportunities If we allowed the objectives of agents to arbitrarily change we can ex plain or rationalize virtually any choice in an expost sense but the model would be of no use for making predictions about the consequences of changes in exogenous variables iii Another reason for basing our aggregate theory on a model of maximizing agents is that we might be able to relate our model to other economic evidence and in this way gain greater con fidence in the accuracy of our basic assumptions In the present case for example we could look at household saving behavior as recorded in crosssectional data sets and see if the same micro economic assumptions on household preferences that enable us to explain the crosssectional ev idence also enable us to explain the aggregate time series evidence We shall return to this point 139And assuming also that the effect of the exogenous variable of interest can be identified in the data sepa rately from the effects of other exogenous variables which happened to be changing at the same time 62 again when we discuss the permanent income hypothesis of consumption iv We observed last lecture that higher savings helped increase future prosperity but higher current savings also reduced current consumption This naturally raised the question of what is the best savings rate So long as we take a democratic approach to answering such questions the criterion for measuring the value of alternative savings rates must depend on individual preferences for consumption today versus consumption tomorrow 2 Saving by a Representative Consumer The way we allow preferences to determine the saving rate st at time t is by introducing a repre sentative consumer with an intertemporal utility function BtUct 1 t 0 with the discount factor B 0 lt B lt 1 We can write 5 llp and interpret p as the rate of time preference and think of it as a natural rate of interest A characteristic of the above preferences is that they are additively separable over time In other words we imagine our representative consumer choosing a bundle of consumption at time peri od t and then reallocating this consumption across goods and services in each period in re sponse to relative prices The tradeoff of consumption today for consumption tomorrow can be considered independently of the tradeoff of consuming different goods within the time period We know that for explaining microeconomic data on certain types of consumption this is likely to be a poor approximation In particular it is likely to work better for explaining consumption of nondurables than the consumption of durables However accounting for nonseparability over time raises additional technical complications Also to avoid comer solutions where ct 0 we assume U 0 0 We retain the same technology we considered last lecture and we also retain the inelastic labor 63 supply assumption Keeping the same model of population growth we can derive the same dif ference equation except that the savings rate is now indexed by t Xt1 St1 fXt f 1517 Xt 2 where again we assume f0 0 lim f X oo fquot gt 0 fquot lt 0 XHO It is more convenient in this model to work in terms of ct rather than st In each period StfXt fXt Gt 3 so the difference equation incorporating our assumptions about technology and labor force growth becomes Xt1 fXt Ctl 11 f 1517 Xt 4 3 The Maximizatitm Problem The maximization problem for the consumer is then to choose sequences c0 c1 c2 and X1 X2 X3 to maximize the objective function 1 subject to the set of constraints 42 De ne a set of Lagrange multipliers Pt tht and write the Lagrangian T L Z BtUct qtf Xt 7 ct 1 7 6xt 7 1 MxH 11 5 t 0 Differentiate L with respect to the ct s Xt39s and qt39s to get the necessary conditions for a maxi mum Bthct 39 Qt 0 t 0 1 2539 T 6 239We allow ct gt fXt so we don39t have to be concerned about any corners for ct We also should constrain XL 2 0 for each t but we ignore this in the formal analysis for the time being 64 Bt lt PXt f 15 39 BHQH 11 0 t 1 T 7 Xm fXt ctl7t 151X X t 0 1 T 8 BTQT XT1 0 9 X0 given 10 Equation 9 comes from considering BTqTXTH gt 0 If that were the case the consumer would be better off consuming more in the nal period and leaving consumption in all earlier periods the same Hence BTqT XTH 0 is a necessary condition for optimality 4 Solving the First Order Conditions From 6 we can invert U to obtain ct gqt Substitute this into the budget constraint 8 to obtain Xm fXt gem11 151k Xt 11 From 7 we obtain B it PXt f 15 it1 11 12 We have reduced the first order conditions to a pair of simultaneous nonlinear difference equa tions in Xt qt with the pair of initial conditions X X0 at t 0 and BTqTXTH 0 5 An Economic Interpretation of the Solution We can interpret these first order conditions as follows i q is the shadow price of capital or shadow value of changes in Xt in current value terms it converts it to present value Thus U ct qt implies that the consumer should equate the mar ginal value of eating a unit of output now to the marginal value of investing it for the future 65 ii From 7 we get f39Xt 1XQt1QtlB 15 13 The right hand side of 13 has the dimensions of an interest rate Writing it in terms of the rate of time preference p and supposing we are in a stationary environment where qt1 qt fxtpmpm5 14 If qt is changing there are also capital gains and losses to take into account If qt is rising the in terest rate is effectively reduced and vice versa 6 Investigating Dynamics with a Phase Diagram To explain the qualitative nature of the solution we use a phase diagram Let T gt co to get a pair of difference equations with one initial condition X0 But we also have the second condition lim BTqTXT1 0 15 T a as as a necessary condition for optimality Equation 15 is the socalled transversality condition at 00 TVCw and provides the second initial condition to give an optimum We have a particular interest in stationary points of the pair of difference equations Xt stationary at 2 and qt stationary at q yields the two nonlinear equations in 2 and q 7 fgt7ltg1 1 757 X 11 171X 16 Bf 7175 1 17 As above rewrite 17 as 66 fgt lt5mppx 18 which is an equation independent of q Let the solution of this be i 2 19 From 16 we get g f0 7 35 7 2 20 which can be solved for q q at 2 gt2 Also observe that 51 U g U fgt7lt757gt7ltl 21 Let X be the solution to f 5XX Then as 2 gt 0 or 2 gt X q gt 00 Further 3 U f78kif ii8k 22 Let be the solution to f39A 5 k This is the socalled golden rule we discussed earlier Then le s lt0 forA0ltTltA 23 gt0forXltltX Finally recall that we let 2 be the solution to f 5 X p pk gt 5 k Then f lt 0 gt 2 lt Thus the golden rule results in a value of X which is too high 67 Putting these results together we get the diagram r Point A is the stationary point of the pair of difference equations Now return to the original difference equations to investigate the dynamics We have f39Xt Qt1Qtl1 t 7 t P t P7 t 5 1 24 IfXgt gt2fxtlt5xp pksothat Qt1Qtl1 7PP70lt0 ktptpk 25 which implies qt1 lt qt or that q is increasing Also gqt fXt 15 Xt HMXH so that 1t U39fXt1 5Xt 1KXt1 26 Then if qt gt U fXt k5Xt U fXt15Xt17Xt1 gt U fXt75th 27 68 and since U lt 0 fXt1 5Xt 17Xt1 lt fXt 75Xt 28 which implies Xt lt XHI or X is increasing There is a unique saddle path such that starting from any given initial X0 El a unique qo on the saddle path so that qt Xt gt q 2 Mathematically qo has to be chosen so that the TVCw holds The requirement that the terminal value of the capital stock gt 0 as t gt 00 will determine a unique initial value of the capital stock consistent with maximization of utility Note that we also have implicit constraints in the above problem that X 2 0 qt 2 0 for all t We did not impose these explicitly to avoid discussion of comers and KuhnTucker conditions Some of the non optimal paths are ruled out because they violate these nonnegativity constraints Others are ruled out because they do not yield lim BTqTXT 1 0 29 T a co 7 Choice of Initial Conditions What do we make of the implication that if qo is not chosen correctly the economy might go down a bad path i Mathematically the model is internally consistent The TVCw delivers a unique path We have set up the mathematics to deliver us a unique optimal solution to the maximization problem ii This is the planning model not on the surface at least a model of a decentralized economy Perhaps taking explicit account of feedback mechanisms can lower the stakes in choosing the right qo at the start However we shall see that the same uniqueness problem arises in a compet itive equilibrium iii The need to get the price of capital correct seems to be a potential source of instability in a market economy Large corrections in the price of capital would appear to provide the potential 69 for exogenous shocks to the economy iv This model is however not suited to examine such issues The aim of a model is to deliver predictions hopefully testable ones If the model says anything can happen then the model is saying nothing there can be no possibility of nding recalcitrant data Thus if we are going to have shifts from one path to another as a feature of our description of a market economy we need to model what causes these shifts 8 Bellman s equation In the analysis above we obtained the first order conditions for maximizing the utility of the rep resentative consumer by defining Lagrange multipliers and setting up a constrained optimization problem with an infinite number of constraints Technically we don t have a theorem that en ables us to write a constrained optimization problem as a Lagrangian when there are an infinite number of constraints We now outline an alternative approach that can be justified more rigorously and which also ca be used when there are shocks The alternative approach uses the Bellman principle of optimal ity This says that in order that the total sum of payoffs over the time interval 0 00 be maxi 3 mized the subtotal of the sum of payoffs over 01 and 1 00 must be maximized Observe that once we know XT in any period 17 the maximization problem is completely speci fied This follows from the infinite planning horizon and the budget constraints Preferences over consumption from remaining periods after 17 are given by x Z Bt TUm 30 tT 339For the Bellman principle to hold the problem needs to be such that the planned choices in the future as a function of possible future information match the actual choices when the future arrives This is a property known as lime consistency The time separable utility function ensures our problem is time consistent 70 while the budget constraints Xt1 fXt ctlk 1511 Xt depend only on the value of XT and the sequence of choice variables ct Define the value function VX as the value of the objective function 30 for the economy with XT X if the planner be haves optimally from period 17 on We do not discuss the conditions under which V can be shown to exist to be differentiable and to be unique but the conditions are satisfied by the current model Some are discussed in the Ap pendix chapter 20 of the Ljungqvist and Sargent text and the issue will be discussed in more detail in the Dynamic Optimization course ECON 523 Assuming that V is welldefined the Bellman principle of optimality implies VXt 03th 1UctBVXt1 31 where the maximization is carried out subject to the constraint X ltfXt7ct175Xt M 32 Since the constraint will hold with equality if U gt 0 we can use 32 to eliminate one of the choice variables in 31 The maximization problem becomes 33 VXt maxUct ct 11 Equation 33 is a special case of the equation 25 discussed in Ljungqvist and Sargent The first order condition for the choice of ct can be written 71 fXt7ct175Xt 7 k 7 0 34 1kU ct7BV 1 Equation 34 gives the optimal policy choice ct as a function of the current state of the system Xt To solve 34 however we need to know the derivative of the unknown function V The Bellman equation approach replaces the problem of solving for an infinite sequence of control variables by solving a functional equation for an unknown function A change in Xt will change the optimal level of ct but again we use the envelope theorem Since c solves 34 the derivative of V with respect to ct is zero The derivative of V with respect to Xt thus is simply the partial derivative f Xt 1 7 5 V X BV Xt1 T 35 Paralleling the development above define the variable qt U39Ct 36 Then we can again invert U to obtain c gq and substituting for c into the budget constraint we obtain X39 fX gnu11 151k X 37 Now observe that the first order condition 34 for the choice of c can be written 1Vqt V39Xt1 38 and assuming that ct1 was also chosen optimally we therefore can lag 38 on period to obtain V Xt 1kqt1 39 72 Substituting 38 and 39 into the envelope result 35 we then obtain the second difference equation 1Xqt1 Bqtf Xt15 40 Equation 40 is also known as the Euler equation for this model see Ljungqvist and Sargent p35 It can also be written as a second order difference equation in X alone Using the definition of qt and the budget constraint 32 as an equality 40 implies 1kU fXt71175Xt7171XXt 41 Bf Xt178U fXt175Xt71XXt1 To ensure a unique solution we need two initial conditions for 41 One is the value of X at t0 The other is the transversality condition TVC By contrast for standard U and f functions the Bellman equation will have a unique solution In a sense the TVC arises because we lose a constant term when we convert from the value function to the derivative form 35 9 Continuous Time Optimal Control The original development of this model by Cass actually sets the model up in continuous time and uses optimal control theory to reach similar conclusions to those above We shall not discuss optimal control theory in any detail in this course but a few remarks are in order The material covered here is further explored in the Dynamic Optimization class ECON 523 The basic optimal control problem can be stated as follows We want to choose a path contin uous function of time for the control variable ut to maximize the value of some integral ob jective function 439These notes are derived from Economic Dynamics An Optimal Control Framework by W A Brock The notes in this chapter are not essential to the remainder of the course which focuses on discrete time mod els 73 T jvxu sds BXT T 42 to where v is the instantaneous payoff function X is the state variable to is the current period and T the time horizon and B is the bequest function giving the value of XT remaining at time T The maximization is carried out subject to some differential constraint equation which relates the change in the state variable to the current value of the state and control variables X fX u t 43 The value of the state variable is also constrained at time to x00 y 44 10 The Hamiltonian and the corresponding rst order conditions The cookbook way of solving this problem involves de ning a Hamiltonian function some what analogous to a Lagrangian by introducing a variable p called the costate variable analo gous to a Lagrange multiplier HPXut VX413 PtfXut 45 The necessary conditions for the control path ut to be optimal then are first the costate equa tions 15 in k 46 x Hp 47 x00 y 48 74 where Hlt is the maximized value of H Also if we de ne T Vy t0 EmaXIVX u sds HXT T 110 t0 subject to X fX u t and Xt0 y we must have the HamiltonianJacobiBellman equation satisfied Vt imaX H 7H um with The transversality conditions VXTT BXTT and MD BXXTT must also be satis ed 11 An Intuitive Explanation 49 50 51 52 The basic idea behind these conditions is as follows We rst use the Bellman principle of op timality to observe that in order that the total sum of payoffs from Xt0 y over to T be maX imized the subtotal of the sum of payoffs from Xt0h over t0hT must be maximized 75 t0h T Vyt0 max I Vxusds I VxusdsBxT T 53 t0 t0h t0h T max I Vxusdsmax I VxusdsBxT T tO t0h t0h max I VxusdsVxt0ht0h t0 Now use the mean value theorem of integral calculus to write to h I Vxusds Vyut0hoh 54 to where oh is a function of order h that satis es Also expand Vxt0ht0h in a Taylor series about xt0 y and t0 Vxt0ht0h Vyt0 Vxyt0 Ax Vtyt0 h oh 55 and note that Ax xt0h xt0 fyut0 h oh Putting these results together we get Vyt0 maXVyauJ0h t Vyto t Vxyto fyu5to h t Vtyto h 0h 56 Subtract Vyt0 from each side of this equation diVide by h and let h tend to zero to get 0 maxVyut0 p fyut0 Vtyt0 57 76 where we have de ned the variable p Vxyt0 Rearrange the equation to nd in maxH 58 um The principle that the optimal control must maximize the Hamiltonian is called the maximum principle It says the control should be chosen to maximize the sum of current instantaneous payoff vyut0 and future instantaneous value PX Pfy 1M0 59 where p is the shadow value of x p Vxyt0 To obtain the costate equations note that x fxut However we can recover this equation via an alternative argument If we differentiate Hlt with respect to p and use the envelope theo rem to drop terms involving changes in the endogenous variable u we get Hpquot fxut x 60 Next differentiate vt Hquot 61 with respect to x to get vtx Hx 62 But vtx ddtVX p The transversality conditions require that at T the value of the problem equal the bequest value and the marginal value of extra x equal the marginal bequest value of additional x If there is an inequality constraint xt 2 0 for all t but B E 0 then the transversality condition pT 77 BXXTT takes the form pTXT 0 which says that nothing of value is left at terminal date T When T is in nite for a large class of problems the condition takes the form Tlim pTXT 0 and is called the transversality condition at 00 When the objective function involves a constant rate of time discounting and the state transition equation does not depend directly on time the above first order conditions simplify Specifically let the objective be T Ie psvxuds e PTBXT 63 to and the state transition equation be X fXu again with initial condition Xt0 y Introduce a change of units by defining Wyt ePtVyt q eptp and h ePtH 64 and the optimality conditions can be written pW iwt mlax hq Xu hq x 65 hqXu vXu qfXu 66 q pq hxquot x hqquot x00 y 67 wxTT BxT and qTBxxT 68 We also have that since p VX q WX 78 When the horizon T gtoo W becomes independent of T so that Wt 0 and the transversality con dition becomes lim e ptqtxt 0 69 ta as 12 The Cass Koopmans Growth Model in Continuous Time Apply the above analysis to our simple one sector growth model without population growth or depreciation of capital We have the instantaneous payoff function v e39Pt Uct 70 and the differential state transition equation k0 fkt ct 71 The Hamiltonian becomes Uc qfk c so that the costate equations can be written 391 Pq qf39k 72 k39 fk c 73 Also we have that the control c must be chosen to maximize the Hamiltonian or U c q 0 74 so that q U c 75 Also since q Wk we have the condition that the marginal utility of consuming must equal the marginal utility of investing 79 Then q 0 yields the equation f39 k p which is independent of q while from the second co state equation k 0 yields fk 7 c gq where g is the inverse of U These are analogous to the equations we studied in the discrete time model and lead to the same phase diagram 1 STOCHASTIC GROWTH AS A MODEL OF A REAL BUSINESS CYCLE A Stochastic Growth Model One attempt to understand business cycles involves modifying the basic growth models to incor porate random shocks The source of these shocks could be for example technological progress in particular industries or shocks to the supply of the factors of production To provide some understanding of how these stochastic growth models work we rst examine a stochastic version of the CassKoopmans model for particular utility and production functions Later we shall look at models which provide better approximations to actual business cycle be havior Suppose consumer preferences are given by x E Z sumo 0ltBlt1 1 t0 and technology by k etkf ict0ltoclt1 2 tl where at is a random variable representing for example random technological progress ran dom uctuations in factor supplies or weather uctuations We assume 8t takes positive values with ln8t independently identically distributed N0 0392 After 8t becomes known output etkf is divided between consumption ct and capital accumulation km We shall show that the consumption savings policy which maximizes 1 subject to 2 is ct l OLB8tktO 3 114 km OLB 8k 4 2 The Bellman Principle and the Value Function We first want to solve this problem using the standard methodology Later we shall show how 1 you can exploit special features of the current problem to obtain the solution more simply Speci cally let Vk080 denote the maximized value of the objective function 1 subject to the constraints 2 given that in time period 0 the capital stock is kg and the shock is 80 That is let Vk080 5 max E0 2 Btlnct 5 at km H where the maximization is carried out subject to the set of constraints 2 and k0 is the initial cap ital stock Here E0 denotes the expected value conditional on information known at time 0 In our problem all shocks capital stocks and consumptions up to and including period 0 are known at time 0 Now note that the original problem looks the same in each time period except that the capital stock and current productivity shock are different The capital stock and current productivity shock are state variables for the problem The Bellman principle of optimality says Vkt8tE max 1nctBEtVkt1y8t1 6 ct kt1 where the maximization is carried out subject to the constraint 2 for time period t Thus Vkt18t1 represents the maximized value of the objective from tl on and if the path c k is maximizing it must maximize over the two subintervals t tl and then tl 00 139A homework problem had you solve the deterministic problem exploiting the special properties of this ex ample 115 3 Guessing a Candidate Value Function The solution method now involves guessing a form for the function V and verifying that the 2 guess works by showing that the first order conditions are satis ed We shall show that V has the form Vko80 A0 A1111ko A211180 7 There is no need to justify this guess either it will work or we will find out it is wrong and hopefully how to modify it to make it right The procedure is analogous to solving integrals in calculus Effectively you have to guess the answer and then show your guess is right by differ entiating to see if you do indeed get back the original function 4 A Motivation for the Guess While the solution procedure does not require us to motivate where the guess came from it might be useful to some students to see a motivation The following manipulations do however in volve a considerable amount of algebra In our case we have already guessed based on our previous analysis of the related deterministic growth problem that the maximizing c and k sequences are given by 3 and 4 Obviously 3 and 4 satisfy the budget constraints they wouldn39t be good guesses for the maximizing choices if they didn t this is a minimal check on their suitability Also if 3 represents the maximizing choices for ct the function V will be given by Vk0 80 E0 2 Bt1n17 dB8tkt0 8 t0 Now use 239We should also check that the second order conditions are satisfied The concavity of In and x will ensure the extremum is a maximum and not a minimum in our case 116 lnl XB8tktO lnl 01B lnet OLln kt to write our guess for V as 1 17 w w Vk0 80 E0HT LB 2Bt1n8t 01 Z Btlnkt 9 10 10 Now use 4 Take logs of 4 to obtain the rst order stochastic difference equation lnkt1 lnOLB ln8t 0L1nkt Conclude that lnkt1 will if our guessed solution is correct have the distribution lnkt 1nxB1xx2xt1 octlnk0 ln8t1 0Lln8t2 oat11n80 Taking expectations at time 0 and using the fact that ln8 is independently distributed with mean 0 E01nkt 1nxB1xx2xt1 octlnk0 oat11n80 t 2 1 10 Substitute 10 into the expression 9 for V to get Vk080 Wmmkowmgow 11 012 Btln0LBl oc012 oct 1 0Ltlnk0 01t 11n80 11 where we have again used the fact that E0 of future ln8 are all 0 Thus we make an intelligent guess for V which takes the form lnliXB ocBlnt 01 1 Vk080 176 176170LBlia lnk0liaBln80 12 117 which we can write as 7 5 Showing the guess is correct Substitute our guess 7 for V into the Bellman equation 6 We want to verify that the following equality is valid Vkt 8t cIILaX lnct BEtA0 Allnkt1 A21n8t1 13 t tl where the maximization on the right hand side is subject to the budget constraint kt1 8tktm ct 14 Now observe that the state variables 8t and kt and the time t choice variables ct km are known at time t In addition ln8t is independently identically distributed so Etln8t1 is 0 Thus under our guess for V the function to be maximized on the right hand side ofl3 can be simpli ed to lnct BA0 BAllnkH1 15 De ne the Lagrangian L lnct BAO BAllnkt 1 M8tkf ictik 1 16 The rst order conditions for a maximum of L are Ct x 17 6A1 x 18 ktl etkt ctkt1 19 118 Substitute 17 and 18 into 19 to nd 1 X15A1 etkf 20 Thus if our guess for V is correct maximizing ct and km will be given by 1 a et 7 1BA18tkt 21 A k Bl k 22 8 tl 1BA1tt Now substitute these solutions for ct and km back into the right hand side of Bellman s equation 13 and we ned to verify that 7 1 a BA1 a Vkt at A0 1 Allnkt 1 A21n8t ln 1 BA18tkt 6 A0 1 A1111 1 BA18tkt is valid for any values of kt and at Expand the terms on the right side of 23 and write it as BA071 BA1ln1 BA1 5A11nBA1 oc1 BA1lnkt 1 5A1lnet 24 and conclude that the functional equation will indeed be valid if the constants A0 A1 and A2 sat isfy A0 BAO 71BA1ln1BA1BAllnBA1 A1 oc1BA1 25 A2 1 BA1 that is the guess will be correct if we choose the constants to be 119 A 1 1416 l A 2 1416 26 Aolt1 B1 7 as ln17 as XML ELEM For the correct value of A1 given in 26 the maximizing ct will be as originally guessed in 3 and the maximizing km will be as originally guessed in 4 This can be veri ed by substituting the value for A1 in 26 into 21 and 22 6 Alternative Approach Speci c to this Problem A more direct approach to this particular problem is possible You can transform the problem and consider choosing a sequence of savings rates kw 1 Xt 1 27 gtkta Using the constraints 2 rewrite l in terms of xt and maximize it directly3 This is much easier than going through the valuation function V but it only works for this problem However this alternative method was probably the way the guesses were originally obtained In fact it is of ten difficult to arrive at guesses for solutions to the functional equation It is possible to numer ically approximate the solution for V for more general functional forms I discuss the numerical approximation of V when Iteach the dynamic optimization course 7 Implications of the Model We want to see what the model implies about the cyclical behavior of output consumption fac tor prices and so on We can then compare the implications with the empirical evidence 339 Do this yourselves for homework 120 If we were to associate the optimal planning solution with a competitive equilibrium the equi librium returns to capital rt will be its marginal product rt detkt l l while the return to the xed factor of production wt will be wt ytirtkt l 7008tkf Given k0 we can use 4 to nd the distribution for In kt lnkt1ln0LB0Llnktln8t Hence lnk1ln0LB0Llnk0ln81 ln k2 10L1n as 12111 k0 In 22 oc In 21 and generally lnkt1oc 0Lt391ln ocBoctln1lt0lnetoclneti1 oct391 ln81 28 29 30 31 32 33 By assumption ln8t is iid N052 so In kt will also be iid normal and since 0L lt 1 ln kt will have a stationary limiting distribution that also is normal with mean Mk E In kt E In kt1 sat isfying anLB lioc uk ln OLB ocuk 0that is uk and a variance satisfying 121 34 E1n kwmz a2 E1n krle E1n 2amp2 35 since In kt and In St are uncorrelated That is 02 52 36 k 1 7 0L2 Now real GNP yt Stkto so In yt 0L ln kt In St and liocL ln yt oc1 0LL ln kt liocL In at 0Lln OLB In 8t1170LLln at OL ln ocB1n at 37 From this we can deduce that 0Lln 1B 7 38 y 1 7 a 2 2 039 5y 1 7 a2 39 covlnyt lnytik aka 40 Next we want to nd the limiting values of covlnyt lnct covlnyt lnrt and covlnyt lnwt First we need to nd the stochastic difference equation In ct ln rt and In wt satisfy From 28 lnrtln0L0Lillnktln 8t 41 so that 17XLln rt 171 In on OL 711n OLB In 8t1170LLln at a 711n Bln at 7 ln 8H 42 122 thatis lnrt7lnBln8t0Lilln8t10Lln8t2 which implies m7mB 20392 2 Gr lOL 1 1 7 E ln8t ln8t lnet1 M y mt 7 liocLGiocLiliochi lion 52 21 2 1 2 2 6 a u a liwo 1a Similarly from 29 In wt ln 171 0c ln kt In St so that 17XLln wt 1711n170c 0Lln OLB 1n 8t1170LLln at lion ln 171 on In 1B In at and hence 1711n170coclnoc5 W l H 70c 2 7 0392 5W7 2 lioc covlnyt lnwt m Finally from 3 ln ct ln 17045 0c ln kt In at so that 123 43 44 45 46 47 48 49 50 liocLln ct 17001n 1706 0Lln OLB 1n 8H 17XLln at 17001n 1416 0Lln OLB In at 51 and hence MC 1711n117 ocBoclnocB 52 70c 2 2 6 00 1712 53 1 1 1 52 54 cov n nc yt t 1412 Comparing these predictions of the model to the empirical evidence on business cycle uctua tions we nd that simply adding shocks to a simple growth model will not result in a good model of the business cycle Output growth does not cycle in this model while consumption and pay ments to labor inelastically supplied are perfectly correlated with output growth Further the model implies that the variance of interest rates could be higher than the variance of output growth consumption and labor income This will be so if 0L lt 05 8 Intertemporal and Intersectoral Correlations There are a number of ways to address the serial correlation problem with the simple stochastic growth model We want the cycles produced by our model to have an autocorrelation pattern matching the pattern observed in measured economic series Speci cally we want the model to match the observed tendency for deviations from trend to persist for several years and then be reversed by a period of deviations of the opposite sign The stochastic version of the simple growth model only gives a oneperiod lag as a result of the capital accumulation equation which makes km depend on kt Kydland and Prescott introduced 124 the notion that it takes time to install capital time to build to explain lags in the evolution of output in response to exogenous shocks This makes the current kt and therefore current output depend on lagged kt several periods into the past The result is a stochastic difference equation for output that has maximum lag length equal to the longest gestation period an investment project To model business cycles however a mechanism is required to achieve a correlation between the random movements of output in dz erent industries We would also like the relative ampli tudes of uctuations in different sectors to correspond to our observations based on the evidence A problem with the simple stochastic growth model is that it is hard to see why productivity shocks would be positively correlated across all industries at once It seems plausible that pro ductivity shocks could be a source of large uctuations in the output of many industries but why would we expect all sectors to experience positive or negative productivity shocks at the same time Perhaps the simplest way to modify a standard growth model to account for correlation between output movements in different sectors of the economy as observed in a typical business cycle is to allow for there to be several industries which can use as inputs the outputs of each of the other industries This can lead to correlated movements in the outputs of each of the industries even though a particular shock to aggregate output might originate in one particular industry It can also introduce some dynamics into the model if outputs from one sector this period are used as inputs in another sector the following period The maximum lag length in output then depends on the number of interrelated sectors in the economy To illustrate this point we shall discuss the model of Long and Plosser JPE 1983 Long and Plosser allow for correlation between output uctuations in different industries by noting that outputs of some industries are used as inputs in other industries We shall show that allowing for these intersectoral in uences also introduces autocorrelation into output uctuations As noted 125 above investment adjustment costs and delivery lags are an alternative way of explaining auto correlation in output uctuations 9 Long and Plosser model Long and Plosser postulate a representative consumer with preferences 0 Z BtUCt Zt 0 lt 5 lt1 55 t 0 where B is a discount factor ct is an le vector of commodity consumption in period t and zt is the amount of leisure or other nonmarket work time consumed in period t The N commodities in the economy can be produced with a constant returns to scale production function y1t1 F1L1tX1t7 1t1 y2t1 F2L2t X2t9 2t 1 l l 56 yNt1 FNLNt XNt7 1N 1 where yim is the total stock of commodity i available at tl Fi is concave and homogeneous of degree 1 Lit is labor allocated to industry i at time t Xit is a lxN vector of inputs from each of the N industries used in the production of commodity i at time t kit is a random shock to the output of commodity i at time t Consumptions cit intermediate inputs Xit and outputs yit available at the beginning of period t 126 satisfy the constraints N 01W 2 int yit 57 j 1 The representative consumer has available a total time of H hours each period which can be al located to leisure or producing each of the N commodities Zt t 2 Lit H 58 To show that this model can lead to random outputs from each industry and the economy as a whole which resemble business cycles Long and Plosser examine a special case They let utility be N UCtZt 10 1nZt Z 11 111 Cit 59 i 1 where Di 2 0 for each i They also take the production functions for each sector ito be N 7 b yim km lLit H X5 60 Jl with the parameters an and bi nonnegative and constant over time and to give homogeneity of degree 1 bi Zj aij 1 Since 7cm is assumed to depend only upon M and not any values of 7 from periods previous to t the current state of the economy or the vector of variables which are sufficient to describe the choices open to the representative consumer can be written st yr kt Let Vst be the value function for the representative consumer or the maximized value of his 127 expected future utility given the current state of the economy So we have Vst cgagiEtlz BS tUcszs 61 s t with the maximization carried out subject to the constraints 56 57 and 58 above The nota tion Et means the expected value is taken with the information about the future values of random variables being that available to the consumer at time t As above we can use the idea that if the consumer is maximizing from t to 00 then the maximiz ing path must also be maximizing from t to tl and then from tl to 00 Hence VSt max UCt Zt t 5 Et VSt1 62 To find the solution to the consumer s problem we guess that V takes the form Vst ZiYilny1t m K 63 with M pi 5 2J yjaji i 123N 64 and WW 5132i Y1 1n7vit1 Et J7 t1 65 and K a constant which depends on the parameters in the utility and production functions but not on the elements of the state vector yt or M Substitute this guess for V into the right hand side of the functional equation 62 and we get VSt max UCta Zt B Et Xi Yi 1nYit1 f J7 t1 Kl 66 128 But from 60 1nYit1 WW1 bilnLit j aij lnxijt Also Um 2t in mm 2i Ln ii in 1nyit Ej XJit 67 68 Now look at the rst order conditions for maximizing the right hand side of 66 with respect to the choice of inputs Lit and XJit DO2t ByibiLit 0 v i quotPicit BYjajiint 0 V iaj From the rst set of these conditions and the constraint 58 we obtain 2 H 0 BZYibi 1 I 7 ByibiH 1t 7 0 BZYibi 1 From the second set of the conditions in 69 we obtain Bya 7 J J1 7 cit 1 int Substitute this into the constraint 57 to obtain 1 BZYjajij it 1Yit J 129 69 70 71 72 73 or using the de nition of Vi N Pi c 74 1t YiYit But then the solution of the rst order condition for Zjit implies that int Bl ajiYiWit 75 These solutions for the optimal values of the endogenous variables may be substituted into the righthand side of the functional equation 62 to obtain Vsi o Ini oH o Bimbo 2iltli1nltlgtiYiYitl B Et 2i viilnkiim bi1nBvibiH o Bimbo 2 ar 1nBviaijvjyjo Jaw K 76 which is ofthe required form since we may write it Vst BK Xiwr ijviaiplnyi BEiiivilnxieiEiiJmigti in mm H o BZmbin lemmavii BiivibilnmvibiHwBimbo BXmZJ arr moverv9 77 To have V of the guessed form we de ne the constant K as the expression within the large braces divided by l B Analogously to our discussion of the simple growth models we can relate this optimal planning problem to a competitive equilibrium by de ning the utilitydenominated prices of commodi ties as the marginal utility values of increasing the supply of commodities available at time t and the utilitydenominated wage rate as the utility value of a marginal increase in leisure consump tion 130 Returning to the solutions for the optimal c39s 2 X39s and L39s above we note the following about the predictions of the model the allocation of the available stock of a commodity is an increasing function of its value in that use the amounts of a commodity allocated to each of its uses is an increasing function of its total supply if the output of some commodity i is unusually high at time t then inputs of commodity i in all its uses will also be high at time t This propagates the shock forward in time and spreads the future effects of the shock across sectors of the economy the allocation of any given commodity does not depend on the contemporaneously available amounts of other commodities given yt none of the allocations made at time t depends on In particular the laborleisure decision is independent of both yt and It is argued that this is a consequence of the particular utility and production function chosen for the closedform solution and not a necessary conse quence of more general models of this type the relative price of commodity i is higher the greater its scarcity relative to other commodities the greater the productivity of a commodity aij the higher its relative price the higher the preference for a commodity 11 the higher its relative price the greater the preference for leisure the higher the utilitydenominated wage The utilityde nominated wage is also higher when the productivity of labor bi is higher The most interesting feature of the model is its prediction of the evolution of outputs over time If we substitute the optimal allocations of inputs Lit and Xth into the production function we get 1nYit1 WW1 bilnBYibiH 0l52mbil EjaijlnBYiaijYjyjtl 78 01 131 Inyim k2iaij1nyjtiit1 79 which is a set of simultaneous first order stochastic difference equations The vector of constants k will in uence the short run movements of outputs as well as their steadystate values In fact the solution to the stochastic difference equation can be written 1nYt1 139 A391k f EH1 f Ait A2Evt1 80 The steadystate value of y is I A391k The inputoutput matrix A aij will have a crucial in uence on the evolution of outputs If E is an iid process the matrix A will completely sum marize the propagation mechanism for the random shocks To get signi cant interaction between the variables the A matrix should display lots of large offdiagonal elements Because of the form of the optimal decision rules for consumption and commodity inputs to production their behavior will mimic the behavior of the output vector Note that if we let I AL 1 I ALdetI AL where M is the transposed matrix of cofac tors of matrix M and detM is its determinant then the deviations of lnym from its long run mean value IA391k can be written IALEt1detI AL or if we denote by y deviations of lny from its mean we have deta Am IAL1 81 Each of the components of yt will have an autoregressive component of the same order and since this order will in general be greater than two the series can easily display a cyclical pattern of autocovariances the autocovariances will solve a deterministic difference equation of order equal to the order of the polynomial in L detI AL Long and Plosser examine a numerical solution for an inputoutput matrix similar to an aggre gated version of the US inputoutput cost shares in 1967 They get agriculture and mining having 132 a higher variance than manufacturing which is not what one would expect for US data In addi tion mining and agriculture shocks do not appear to be highly correlated with business cycle uctuations in the remainder of the economy Their model also seems to imply a relatively much more variable services output than we found in the data The main candidates for sources of business cycles in their model appear to be shocks either to manufacturing or services output Only these sectors have enough interaction with all the remaining sectors to produce the widespread correlated response of output across sectors typ ical of the business cycle Another problem with this model is that it does not give an adequate explanation of the much greater cyclical movement of investment and producer and consumer durables output than ser vices or nondurables consumption These problems might be addressed the same way we shall address them in a single sector model Nevertheless the model does show that the fact that in dustries are linked by the use of intermediate goods and that maximizing consumers and firms will have an incentive to spread out the effects of shocks can account for a considerable amount of the correlation and persistence amongst the movements of economic variables Employment uctuations might be explained by the models discussed in the next set of notes 133 OVERLAPPING GENERATIONS MODELS 1 General Equilibrium Model 0fM0ney In these notes we examine a general equilibrium model of a monetary economy We focus on the Samuelson overlapping generations model since this is probably the most popular model of a monetary economy in the literature In a subsequent set of notes we also look at the cash in advance model The basic idea behind the overlapping generations model is as follows We have a population that is constant from one period to the next although the identity of those individuals who are alive changes from period to period An individual born at time t will live eat in period t and then die in period tl At any time there are N 0year olds and N 1year olds In the simplest model the young are endowed with 1 unit of nonstorable consumption good each and the old are endowed with nothing Let ct consumption of the young people alive at time t and c t consumption of the old people alive at time t and conclude that a person born at time t will consume ctc t1 Let preferences be given by Uctc39t1 The members of each cohort are treated identically so we can discuss representative members Assume the economy lasts forever even though each person has a finite life The set of all feasible allocations in this economy is the set of sequences c0 c39o c1 c l c2 c 2 with ct c t 2 0 and ct c t S 1 Allow the individuals in the economy to trade Since the good cannot be stored the old have nothing to trade and so the young will not trade their 1 unit of output The equilibrium allocation willbe 1 0 1 0 1 0 183 endowment point A 1 ct O l But if all individuals have the same utility function the utilitymaximizing solution will be pre ferred by all and therefore for the whole economy the market equilibrium in this economy will not be efficient The basic reason for the failure of the welfare optimality theorems in this case is that individuals cannot trade with future generations In other words to achieve the maximiz ing pattern of consumption the current young generation would like to alrange trades with the as yet unborn generations whereby i each young generation gives up 1 c to the current old generation and in return ii each old generation is promised I c from the then young generation lI I 0 We can reinterpret this model as an abstract model of a monetary economy with the following modification We have the individuals born in T spend their old age in time zero We can 184 arrange the traders around a circle as in the following diagram 0 TH39 h hl 2 T2 amp3 It makes more sense now to think of the t index as applying to locations or markets rather than generations Individuals in location or market t want to buy from individuals in market or loca tion tl but sell to individuals in market tl The idea that barter is made difficult since individ uals in general want to buy from and sell to different agents is known as the problem of the double coincidence of wants under barter In mathematical terms the reinterpreted overlap ping generations model has the t index calculated in modT arithmetic If we have time we shall show later however that there are some interesting shortrun dynamic consequences of taking the circular trading structure seriously Now introduce money as a costlessly storable commodity with nonzero value Assume con sumption today is transferred to consumption tomorrow using money In other words assume the young will give up some output today in exchange for money if they are confident they can use that money to purchase output tomorrow when they are old This introduces the budget con straints mtd Pt1 ct 1 7 7 d pt1dt1 mtsl mt 2 185 Assume the total money supply per capita is constant at m Assume individuals behave compet itively in the sense that they take prices as given Observe that the choice of money balances in period t taking pm as given is equivalent to choos ing c t The maximization problem becomes max Uct c39Hl subject to mtd ptl ict and pH 1c39H1 mts1 mtd 3 c c39 t tl In equilibrium we must have mt m Eliminating n1d and n1S from the problem we get the bud get constraint ptlct pt c t There is only one constant price solution given by maxUc c39 subject to l c c39 4 c c39 The maximization problem 4 yields the solution c c c39 l c p ml c In this model the introduction of money yields the ef cient outcome Money is perfect at over coming the problem of the double coincidence of wants under barter This result does not obtain in some other models of the microeconomic role of money As a model of the role of money in the economy the overlapping generations model suggests that unbacked currency ie currency that cannot be exchanged for an outside asset such as a precious metal is accepted by an individual in exchange for goods or services because he be lieves someone else will accept it later on an individual who in turn accepts currency because he believes someone else will accept it yet later and so on 2 Expectations and Sunspot Equilibria Now consider solutions apart from the constant price one Form the Lagrangian 186 L U t 39t1 f MIMI Ct Pt1 C39m with first order necessary conditions for a maximum P U1ct t 17 kpt Pt1 P U2 t t 1 00 7VPH1 Pt1 or eliminating 7 from 6 and 7 U1c pt 177 pt U2ct pt 17 0 t Pt1 Pt1 Pt1 The second order sufficient conditions for a maximum are 2 U1172 pt U pt U22lt0 Pt112 Pt1 Now substitute the market clearing conditions m c li andc39 t tl Pt ptl into the first order condition 8 to get U10 m 7 0713 m 0 Pt Pt1 Pt1 Pt Pt1 Equation 1 l is an autonomous first order difference equation in pt Solve this equation for pm 5 6 7 8 9 10 11 gpt One solution to the difference equation will be the stationary solution pm pt 13 from which we can conclude that E is a fixed point of g 187 We also want to examine the nonstationary solutions to the rst order nonlinear difference equation We have from 11 and pt1 gpt that the function g satis es the functional equa tion Differentiate 12 with respect to p and solve for g at the fixed point m U27 TU11 7U21 3 03gt U2 7 5U12 7U22 g 13 Use the second order suf cient conditions for a maximum U11 U12 lt U12 U22 at p to con clude In In U2 U11 U21gtU2 U12 U22gt0 dU21 7 a a In U2gt U12 U22 0 dc Thus ifthe elasticity of U2 is less than 1 for example we can conclude g p gt 1 and p is an unstable equilibrium point Under the assumptions we would expect to be most conducive to stability the stationary equilibrium is unstable The mathematics implies a unique p is consis tent with equilibrium in this perfect foresight world Would it be rational to suppose p i 13 If p lt5 then pt runs into a nonnegativity constraint If p gt pthen the value of money gt 0 But if money has some nonzero utility value gold can be used forjewelry paper money can be burned for warmth or used for art then only p p will be an 188 equilibrium under perfect foresight We can also discuss the outcome predicted by a sequential equilibrium model Suppose people have some arbitrary expectation pf Then the equilibrium for the young generation will be U1179E7ampU2179E 0 14 pt p p Pt p Solve equation 14 for pt hpte Now suppose people use the solution for pt in one period as the expectation for next period Will pt converge and if so what will pt converge to Observe that h is just g391 so that h p lt l and the solution will be a stable equilibrium in precisely the situation we d expect it to be Also if we modify the expectations to be adaptive that is Pte 9P19Pt1 9P19hP1 15 then the model will be stable in the same case Nevertheless the above restriction on equilibria might not hold so the system with sequential equilibria might not be stable In the stochastic case this can lead to sunspot equilibria whereby information irrelevant to the fundamental deter minants of the asset price nevertheless in uence the equilibrium and therefore the asset value 189


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