Macroeconomic Dynamics ECN 211
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This 14 page Class Notes was uploaded by Sydni D'Amore on Wednesday October 28, 2015. The Class Notes belongs to ECN 211 at Wake Forest University taught by Allin Cottrell in Fall. Since its upload, it has received 49 views. For similar materials see /class/230741/ecn-211-wake-forest-university in Economcs at Wake Forest University.
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Economics 211 Allin Cottrell David Begg s Fullemployment Model The model is given by the following equations YcY7TK6KG Oltclt16gt0 1 Y7a0a1K a0 a1 gt0 2 b1Y7bgr b1 b2gt0 3 ra11T 4 9 5 where Y denotes aggregate real output T real net tax revenue K the real stock of capital G real gov ernment expenditure M the nominal money stock P the price level 7 the nominal interest rate and IT the rate of in ation Equation 1 sets output equal to the sum of consumption 6 Y gross investment and government purchases Gross investment is the sum of net investment K1 and depreciation or replacement investment 6K Equation 2 is a simple linear production function with the role of labor suppressed on the presumption that there is full employment of labor Equation 3 is a standard moneydemand relationship and 4 sets the nominal interest rate equal to the marginal product of capital plus the rate of in ation 1T which is equivalent to saying that the real interest rate equals the marginal product of capital We begin the analysis by deriving the dynamic equation for K using 2 in 1 Ka017ca117c75K7GCT from which we can derive the stationary locus for K setting K 0 K a017c7GcT 6 7 a1 1 7 c This locus is vertical in a space with K on the xaxis and on the yaxis the steadystate value of the stock of capital does not depend on the level of real balances Consider the situation if there is no government sector G T 0 In that case the steadystate value of the stock of capital is positive only if 6 7 a11 7 c gt 0 since a01 7 c is assuredly positive We assume this to be the case On this same assumption the coef cient on K in the K equation is negative That says that if we step off the stationary locus to the right raising K at a given value of then K will go negative shifting us leftward back toward the locus This is therefore a stable locus Now we turn our attention to Using 2 and 4 in 3 we get b1 0 b2 1 bl lK szT For M 7P to equal zero we need to have the price level rising at exactly the same rate as the money stock is expanding ie 1T 6 The stationary locus for real balances is therefore given by APTJ blaro b2a1 bl lK 1726 1The dot notation here indicates the timederivative of a variable ie K E dKdt all m H 0 MP 0 Figure 1 The Saddlepoint Property of the Begg Model In the relevant space this locus has the slope blal gt 0 What about its stability Let us rst note that since the real money stock is the ratio of M to P the proportional growth rate of real balances is the difference between the proportional growth of nominal money stock 6 and the proportional growth rate of prices ie in ation It That is MVP M P Since the proportional growth rate of real balances will always have the same sign as the simple timederivative of real balances we can work in terms of the former From equation 4 1T r 7 a1 and from 3 r Z Y 7 712 Using this information along with equation 2 we get 6 a1 do 04K MP 1172 72 1172 P In this light consider stepping off the stationary locus for real balances in an upward direction raising at given K We start on the locus with M P 0 and raising will then via the positive coef cient 1 b2 generate a positive value of M P That is the motion will be upward away from the stationary locus This locus is unstable We can deduce that this model has the saddlepoint property This is shown in Figure 1 where given the arrows of motion there exists a unique convergent path Within this model what happens if there is an increase in government de cit spending an increase in G without a corresponding increase in T Well the stationary locus for real balances is unaffected it stays where it is But the stationary locus for capital stock is shifted Working from the equation for K given above we see that this locus moves to the left That is the economy will end up with a smaller stock of capital in the steady state This occurs because the use of resources represented by the extra government spending means that less resources are available out of any given income level all MVP 0 Figure 2 The Situation after an Increase in G to maintain the stock of capital The new steady state will also show a lower level of real balances This is because with lower K the level of real output will also be lower Hence the demand for real balances will be less in equilibrium the supply stock of real balances must also be smaller That is the comparison of the new steady state with the old What about the dynamics of transition between the two states Figure 2 shows the situation The new convergent path is indicated as BB it must lie beneath the old one If the economy were to remain for a while at what used to be its steady state the dynamic forces at play would tend to move it further away from the new steady state What must happen in order to initiate movement toward the new steady state The economy must jump onto BB Since the capital stock is not a variable capable of jumping and M is given at any point in time it must be the price level that jumps up in order to reduce real balances and get us onto the new convergent path Clearly for this model to workithat is for the system to be able to converge after a disturbance to G or T or to 6 for that matterithe price level must be free to jump at a point in time This is unrealistic The price level is generally reckoned to be a somewhat sticky variable But the point of this exercise is to familiarize you with the mechanics of analysing this general class of models starting with a relatively simple example The next task will be to analyse a more interesting short run macromodel also with the saddlepoint property Here the jump variable will be one that is indeed capable of jumping namely the foreign exchange rate Exercise I have shown above the analysis of the effects of a change in de cit spending To get a better sense of the model please produce a similar analysis of a the effects of an increase in the rate of growth of money supply 6 and b the effects of a step increase in the economy s capital stock in general the capital stock is not capable of jumping but one could imagine the economy receiving a gift of capital from abroad Economics 211 Allin Cottrell Buiter Miller Model In this exercise we ll analyze a strippeddown version of the model constructed by Willem Buiter and Marcus 1Liller 1981 They used it to examine the effects on the UK economy of North Sea oil production and the tight monetary policy of the early 1980s 1 used it in a 1986 paper to assess a policy proposed by the British Labour Party to tax the earnings of UK residents on their holdings of foreign assets The model highlights the role of jumps in the exchange rate in response to various macro disturbances 1 The model The basic model is set out in the following equations y a7 pbe PPw agtbgt0 1 m kyihrp khgt0 2 P f0 ym fgt0 3 639 r 7 rm 4 The variables are de ned as follows y log of real domestic output endogenous r nominal domestic interest rate endogenous 7 log of domestic price level predetermined m log of domestic money stock predetermined 6 log of nominal exchange rate endogenous rw foreign interest rate exogenous pw foreign price level exogenous Note that the exchange rate is expressed as the domestic currency price of one unit of foreign currencyiso a rise in 6 corresponds to depreciation of the domestic currency The dot notation denotes tim ederivatives x E dxdl note that the timederivative of the log of a variable is the percentage rate of change of the variable eg m is the percentage rate of growth of money supply 639 is the percentage rate at which the exchange rate is changing over time p is the domestic rate of in ation and aw is the foreign rate of in ation Equation 1 says that real output is negatively related to the domestic real rate of interest nominal rate minus in ation but positively related to the degree of trading competitiveness of the domestic economy The rst effect is quite standardiwe can think of the negative link between the real interest rate and GDP operating via investment spending in the usual manner The second effect works as follows A rise in the variable 6 7 p 1 pm can come about in any of three ways a rise in 6 fall in p or rise in 71 Remembering that e is de ned as the log of the domestic currency price of a unit of foreign currency a rise in 6 means a depreciation of the domestic currency making the country s exports cheaper to foreigners A fall in 7 clearly achieves the same effect while a rise in pw also makes domestic output more pricecompetitive with foreign goods In each case we would expect foreigners to buy more of our goods hence raising the level of domestic output Equation 2 is a standard LM schedule with money supply set equal to the demand for money balances which depends positively on output negatively on the interest rate and proportionally on the price level Equation 3 is a Phillips curve of sorts It says that in ation depends on two things the GDP gap and the rate of monetary expansion When GDP is at potential y yquot in ation proceeds at the same rate as the nominal money supply is expanding Finally equation 4 represents the uncovered interest parity condition for the foreign exchange market The basis for this equation is that asset markets will adjust until the returns on holding assets denominated in the various different currencies are equalized e g the return on holding US and British government bonds should be equal We know that the actual rate afinterest is not always the same for assets in different currencies Suppose that bond yields are 2 percent higher in Britain than in the USA Equation 4 says that any such gap between V and rw should be offset by prospective currency movements If US bonds are yielding 2 percent per annum less than British that would be offset by the prospect of the pound Sterling falling against the dollar at an annual rate of 2 percent or in other words the dollar appreciating at that rate relative to Sterling More formally suppose we have V 08 and rw 10 then using 4 we have 08 710 702 which means that the dollar is appreciating at 2 percent per year 2 Analysis of the model The analysis will focus on two variables the real money supply m 7 p and the degree of international price competitiveness e 7 p 71 As a notational convenience we ll refer to these as liquidity L and competitiveness C respectively L2m7p39 CEe7ppw We want to set up a system of two dynamic equations in which L and C percentage rates of change are shown as functions of L C and the exogenous variables in the system Consider rst what the model implies about L and C Using equation 3 Lm7prhifyyrh7fyeygt And using equations 3 and 4 C 7z pwr7rw7fcy7ygt7mpw Sowehave L nyin 5 C r7rw7f01ymi7w 6 Here are our L and C equations the task now is to eliminate the variables y and r from 5 and 6 and reexpress the right hand sides of these equations in terms of L C and the exogenous variables rw 71 m and y Actually if we re doing shortrun analysis and are not interested in changes in y we might as well set this term to zero to simplify matters I have done that in what follows Given that y is in log form this amounts to choosing units such that potential GDP equals 10 The strategy here is straightforward though the actual algebra gets a bit tedious Begin by writing C in place of e 7 p pm in equation 1 and L in place of m 7 p in 2 Solve equation 2 for r and substitute the resulting expression into 1 Then use equation 3 to eliminate p from 1 We are then left with an equation in y C L and m which can be solved for y as below y LC m 7 a a a where a is shorthand for ak 7 hf h The interest rate r gets similar treatment We can use 7 above to eliminate y from equation 2 and then solve for r in terms of L C and m Again I draw a veil over the intervening algebra and present the result r k LC m 8 hot or 01 Now we can use equations 7 and 8 to eliminate y and r from the dynamic equations 5 and 6 It s not too tough to gure out that L EL C m 9 7a 7a 7a 1 b h7k h fia C Tam7rw 17 10 C L id The dynamical system represented by 9 and 10 may also be written in matrix form as L 1 a f b fh L 1 afh o o m c39 70 1 bfh7k C a h a 7a r Pw What can we say about the signs of the coef cients in these equations First of all note that we assumed in writing out equations 1 through 4 that all the individual parameters a b k h f were positive Clearly then any products of these individual param etersisuch as a f I f h and so onimust be positive But what about the term a E ak7hf h andthe term bfh 7k We can t deduce the signs of these from the signs of the individual parameters alone But look back at equation 7 the reducedform expression for real output From that we can see that if an increase in liquidity or competitiveness is to raise rather than lower output then a must be positive We will assume this is the case if it were not the dynamic system would be globally unstable So for is then negative As for bfh 7 k if f is fairly small ie if deviations of actual from potential output have a relatively small effect on in ation then this term will be negative due to the 7k We ll assume this is the case although it wouldn t make a great difference to the analysis to assume otherwise 3 The exercise OK The basic slogging has now been done You have the system of equations 9 and 10 at your disposal along with full information on the signs of the coef cients in those equations You take over 1 Use 9 and 10 to nd the equations representing the L 0 locus and the C 0 locus Using these equations what can you say about the slopes of these loci in a space with L on the horizontal axis and C on the vertical Draw a phase diagram showing the situation 2 Again using 9 and 10 gure out what happens if we move vertically off the C 0 locus What do the arrows ofmotion look like Is the C 0 locus stable 3 Repeat step 2 but this time for a horizontal move off the L 0 locus Is this locus stable 4 Use arrows of motion to show that the system as a whole has the saddlepoint property Show on a diagram what the unique convergent path must look like 5 Suppose now that the monetary authority in this economy believes that in ation is too high and that the rate of monetary expansion rh must be reduced What will a reduction of rh do to the two stationary loci I suggest you tackle this question thus take the equations representing the two stationary loci and solve each one for L Then look at the coef cients on rh in the resulting equations Those coef cients tell you the degree of horizontal shift of the respective loci when rh is changed From the signs of the coef cients you can tell whether the schedules move to the right or the left in your phase diagramibut remember that a monetary slowdown means a negative change in rh 6 Use the system as a whole to determine what the path of the economy will be like on the way from the original steady state to the new one assuming rational expectations Note that e is considered capable of jumping at a point in time but 7 is not What happens to the degree of competitiveness C along the way Referring back to equation 1 and remembering that C E e 7 7 what do you think will happen to real output along the way In plain language what does this model tell you about the effects of a disin ationary policy References Buiter Willem and Marcus Miller 1981 Monetary Policy and International Competitiveness The Problems of Ad justment in W A Eltis and P I N Sinclair eds The Money Supply and the Exchange Rate Oxford Clarendon Press Cottrell Allin 1986 Overseas Investment and Left Policy Proposals Economy and Society vol 15 no 1 pp 1477 6 Economics 211 Allin Cottrell Notes on the math of IS LM Our IS LM model is represented by the following equations in general functional form Y C 1 G 1 CCY7T 0ltC lt1 2 TTY 0ltT lt1 3 I 17 1 lt O 4 LYr LygtOL7 lt0 5 Y FN F gt 0 6 F N F gt 01 lt 0 7 The variables are de ned as follows Y real GDP C real consumption I real investment G real government spending T real tax revenue 7 nominal interest rate M nominal money supply P aggregate price level N employment W nominal wage A local linearization of the model is obtained by total differentiation which yields the following equations dY dCdIdG dC C dY idT 39 dT T dY ill d1 I d iv Yam 7 ffde 1de 17017 v dY F dN vi w 7 gym F dN vii Equations i vii may be represented in vector matrix form Suppose we collect the endogenous variablesidY dC dT d1 017 UP and dNion the left and put the exogenous variablesidG 01M and dWion the right Then the system will look like this 1710710 0 o dy 100 70 1 C 0 0 0 0 dC 00 0 7T39 0 1 0 0 0 0 dT 00 0 dc 0 0 0 171390 0 d1 00 0 01M Ly 0 0 0 L7 Pig 0 dr 00 idWJ 1 0 0 0 0 0 71 VIP 00 0 0 0 0 0 0 gt F dN 00 You may care to check and see if I have this right This system is of the general form Ay BX In principle we could solve for the reduced form of the model which gives us the full effects on all of the endogenous variables of a change in the exogenous variables thus A lAy A lBX y A lBX But of course inverting A would be no mean feat Cramer s rule provides a helping hand here but even for Cramer we don t want to be taking the determinant of an 7 X 7 matrix We need to boil the model down a bit We will have a manageable determinant if we can get it down to 3 X 3 That means eliminating four of the endogenous variables Which are good candidates for elimination Well both C and T are uniquely related to Y we know that if the latter goes up down then so will C and T So let s get rid of these variablesithe explicit inclusion of 01C and dT in the system is not telling us anything we couldn t gure out without them Similarly I is uniquely related negatively to r and so may be eliminated Finally N is uniquely related to Y via the production function so N can go That leaves dY d and VIP as the endogenous variables We could eliminate dP too but it might be handy to keep it in the system The eliminations go as follows First we can use ii iii and iv in i to yield 1 7 C3917 T dY I39dr 016 and vi in vii to get W F 1 Edwi dP F 011 The above two relationships along with v can be put into vector matrix form as 17C 17T 71 0 011 dG Ly L7 air d7 1quot W dW 7 0 F 1P 7 which gives us something to which Cramer s rule is readily applicable Remember the rule states that if you have a system Ay d the solution value for y the jLh element of the vector y of unknowns is given by lAjl y A where A1 is the determinant of the matrix created by substituting the vector d into the jth column of In this case we have nding the determinant by multiplying terms along all the diagonals and giving the products the right signs lAl l 7C 1 7 T LT 70 ltgt lt21 9 070 Now suppose we want to solve the system for dY In that case we substitute the d vector into the rst column of A yielding dc 7139 0 AJ L7 1 7w 0 p z inwhichcase W 1A 7 dGL7ltEgt7O M dW dM W 1 gt E T H gt 7 E 070 W W dM M dW EWG 1727 1 p2 p The solution is therefore 71de 1 1VT 7 13 dY 1 7 017 T017231 7139 gay 7 To obtain the reducedform multiplier with respect to 016 we invoke cateris paribus and set the other exogenous changes 01M and dW to zero Thus we re asking what will be the effect of a small change in government spending on the equilibrium value of output assuming no change in the money supply or the nominal wage The multiplier in question is dG 17C3917T39 ry7 7lg Is it possible to make economic sense of this coef cient Yes I think so Notice rst that if investment spending is totally unresponsive to the interest rate ie I 0 then this coefficient reduces to the simple multiplier with which you should be familiar from intermediate macro namely 4 1 7C3917T39 Now what difference does it make if I lt 0 First L7 is presumed to be negative so I L7 is positive Within the parentheses at the right of the denominator of the multiplier Ly is positive and F is neg ative while of course M P and 13 2 are all positive It follows that the whole term that depends on I is positive That is the responsiveness of investment to the interest rate adds an extra positive com ponent into the denominator of the reducedform multiplier dYdG reducing the overall magnitude of that multiplierior in other words making scal policy less effective Why Due to the crowding out effect The stimulus to GDP afforded by an increase in government spending results in an increase in the demand for money which drives up the rate of interest in turn reducing the level of investment Note the parameters that modulate this effect The greater the value of Ly ie the greater the increase in money demand stemming from an increase in aggregate income the more the crowding out Crowding out is also exacerbated if F is strongly negative that is if the marginal return to labor is sharply diminishing since in that case the increase in GDP stemming from the increase in G must be accompanied by a relatively substantial rise in P to depress WP in line with labor s lower marginal product On the other hand the greater the absolute value of L7 the less the crowding out If the interestresponsiveness of the demand for money is great it takes only a relatively small increase in r to maintain moneymarket equilibrium in face of an increase in demand stemming from a rise in incomes Exercises 1 W Isolate the reduced form coefficient dYdW Ascertain the sign of the effect and give an inter pretation as offered above for dYdG Apply Cramer s rule to nd the solution for VP Write out the reducedform coefficients dPdG dPdM and dPdW and give an account of the economic relationships they represent In place of taking money supply M as an exogenous variable let the monetary authority behave according to the rule M Mr with M gt 0 That is the money supply is adjusted in reponse to changes in the rate of interest in such a way as to stabilize the latter Add this function to the initial seven equations and recalculate the model Derive the reducedform coefficient dYdG and comment on the difference between this and the original version of dYdG discussed above Economics 211 Allin Cottrell Introduction to Linear Dynamic Models 1 A globally stable system Take a system in which the time derivatives of the two variables 26 and y are functions of the levels of both 26 and 3 dx E a26byh 1 013177 dt yicxerdyerk 2 Let the signs of the coef cients be as follows a lt 0 b gt 0 c lt 0 d lt 0 k gt 0 the sign of h is immaterial Consider the locus of points in 26 3 space such that 2396 0 the stationarylocus for 26 It is obtained by setting the lefthand side of 1 to zero which yields x y bb Given the signs of the coef cients the slope of this locus 7a b is positive the higher is the value of 26 the greater must be the value of y to hold 26 constant ie to hold 2396 0 Consider the thought experiment of taking a small step off the stationary locus for 26 in the 26 direction as from A to B in Figure 1 At point A d26dt 0 by construction since we re on the 2396 0 A E Figure 1 Stationary locus for 26 locus At point B 3 remains unchanged but the value of 26 is increased relative to A Look at equation 1 and remember that a lt 0 whenever 26 is increased but 3 is held constant 2396 must fall But this means that 2396 must be negative at B which in turn means that 26 is falling over time and we tend to move back toward the stationary locus The stationary locus for y is derived in the same manner It is represented by 7 c k y d x d and has negative slope icd Repeating the above thought experiment we have the situation shown in Figure 2 Looking at equation 2 and remembering that d lt 0 we can infer that a vertical jump from A to B will make 339 go negative so that 3 falls and we tend to return to the stationary locus Putting two and two together we get the setup shown in Figure 3 This phase diagramias it is callediexhibits global stability The intersection of the two stationary loci represents the equilibrium point for the system where both 26 and 3 remain unchanging The arrows of motion are such that wherever we start outin the phase space we are always drawn inexorably toward the equilibrium y B Figure 2 A Stationary locus for y 339 0 x 3 Figure 3 This system is globally stable x 2 An interpretation IS LM Consider this simple linear IS LM model L 11y 121 P 1 gt 012 lt 0 I i0i17 i0gt0i1lt0 S 5y 0 lt s lt 1 5 The endogenous variables are 3 real output I real investment S real saving L the demand for money and r the rate of interest The exogenous variables are M nominal money stock and P the general price level In equilibrium the model is closed by specifying that I S and L M but suppose that when the system is out of equilibrium the variables 7 and y obey the following dynamic equations 739 kL7M kgt0 6 y wig hgt0 7 This is to say that the interest rate is driven up down by an excess demand supply of money while real output is driven up down by an excess shortfall of investment in relation to saving This seems like a plausible set of dynamics Now let us analyse the disequilibrium dynamics of the system Using 3 in 6 yields 1quot Pklzr Pklly 7 kM 8 while using 4 and S in 7 gives 339 MN 7 hsy Mg 9 Note that equations 8 and 9 are in the same form as the generic equations of motion 1 and 2 above the time derivatives of r and y depend in a linear manner on the levels of r and y The coefficients of the generic equations of motion correspond to those of our disequilibrium dynamic equations for r and y as shown in the following table By inspecting the sign speci cations in equa Generic IS LM a 4 39 Pklz b lt gt Pkll C 4 39 111 01 lt gt 7 hs tions 3 7 we can see that for the IS LM dynamics we have Pklz lt 0 equivalent to a lt 0 Pkll gt 0 b gt 0 hi1 lt 0 c lt 0 and 7h lt 0 d lt 0 This sign pattern corresponds to that assigned in relation to equations 1 and 2 It follows that the IS LM disequilibrium system provides an interpretation or model in the mathematical sense of the term for the globally stable system of section 1 Wherever the system happens to start from in r 3 space it will tend to evolve towards an equilibrium in which 7 and 3 remain unchanging 3 Globally unstable systems Return to equations 1 and 2 and consider the consequences of the following sign pattern for the coef cients a gt 0 b lt 0 c gt 0 d gt 0 If you repeat the reasoning accompanying Figures 1 and 2 do it you will see that if we step off the stationary locus for x in the x direction then according to equation 1 we tend to move further away from this locus and similarly for the y stationary locus The resulting situation is shown in Figure 4 In this case for any starting point other than the equilibrium Figure 4 This system is globally unstable itself the system will tend to diverge progressively from the equilibrium over time Note that for an economic model with this property comparative statics would have little meaning We could determine the change in the equilibrium values of the endogenous variables that results from a change in one or other of the parameters but this would be somewhat pointless as the system will never reach its new equilibrium following such a disturbance 4 The saddlepoint property An interesting situation arises when the sign pattern for the coefficients in 1 and 2 is as follows a gt 0 b lt 0 c lt 0 d lt 0 The signs of a and b are the same as those speci ed in section 3 above which means that the arrows of motion point away from X 0 locus in the x dimension The signs of c and 01 however are the same as those in section 1 so the arrows of motion point toward the 339 0 locus in the y dimension as shown in Figure 2 The resulting system is displayed in Figure 5 From any starting point to the left of X 0 and below 339 0 the resultant motion is unequivocally away from the equilibrium the same goes for starting points that are to the right of X 0 and above 339 0 In the subspace to the left of X 0 and above 339 0 the resultant motionidownward and to the leftican take us toward the equilibrium if the starting point is chosen carefully Similarly for the subspace to the right of 2396 0 and below 339 0 motion upward and to the right will take us in the direction of equilibrium for certain speci c starting points The set of points in the latter two subspaces that Figure 5 This system has the saddlepoint prop erty satisfy this conditioniie if we start from any of them we will head toward equilibriumiis called the convergent locus it is shown as the line AA in Figure 5 The connection with a horse s saddle is clear if you think about it In one dimension namely from the horse s head to its tail a saddle slopes down towards the middle This corresponds to the stable stationary locus 3391 0 in the mathematical example But in another dimension ank to ank the saddle slopes upward toward the middle This corresponds to the unstable stationary locus 56 0 While globally unstable models are basically uninteresting from the point of view of equilibriu analysis we shall see that the saddlepoint system has some quite interesting macroeconomic inter pretations Such systems are of particular interest under the assumption of perfect foresight or its stochastic counterpart rational expectations Suppose we have a twovariable macro system where one of the variables is sticky for some reason or other while the other is free to jump to a new value at a point in time Suppose an existing equilibrium state is disturbed that is the system s equilibrium point is moved to a new location in the phase space Under what condition will the system proceed toward the new equilibrium This will happen only if the system is somehow placed on the unique convergent path corresponding to AA in Figure 5 If one variable is sticky and the other free to jump the jump variable will have to do the job of placing the system on the convergent path We will take a look at two models where this sort of effect is important
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