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ECON 5110 Class Notes Endogenous Fluctuations 1 Introduction In this section I present an overview of businessecycle models that are driven by extrinsic or nonefundamental uncertainty Typical businesscycle models are driven by intrinsic or fundamental uncertainty ie uncere tainty related to preferences technology endowments In these models intrinsic uncertainty is often assumed to arise from technology government spending or monetary shocks Extrinsic uncertainty on the other hand is anything that is unrelated to economic fundamentals Keynes7 animalespirits hypothesis was an early application of the idea that the business cycle might be driven by extrinsic uncertainty The research program advocated in Farmer s Macroeconomics of SelfeFul lling Prophecies which I rely on heavily in this section can be considered a modern formalization of Keynes7 animalespirits hypothesis 2 Classi cation of BusinessCycle Models Most modern businessecycle models can be written as a system of nonlinear expectational difference equa tions 21 Etf lwhh 1 where y is a vector of endogenous variables eg prices output and at is a vector exogenous or predetere mined variables eg government spending money supply technology shocks The expectation operator Et is typically assumed to represent rational expectations conditioned upon all known information at time t and earlier The rational expectations error 62411 le 7 Ew1 obeys E4411 0 To solve these types of models one generally requires that 1 be linearized 3h bEnyt1 cxt 2 We divide the system 2 into two classes depending on the value of b 2 1 Regular Case The regular case is de ned by bl lt 1 Most models of the businessecycle fall into the regular class These models have unique rational expectation equilibria or in other words a single convergent path to the steady state equilibrium Prominent examples include the RBC model and Taylor s overlapping wage model 211 An Example Cagan s In ation Model Cagan s in ation model begins with a simple money demand function mt Pt O Etpt1 Pt where a gt 0 mt is the log of the nominal money balances exogenous and pt is the log of the price level endogenous Notice that this money demand function ignores income and interest rates which is reasonable in times of high in ation Rearranging into the form of 2 gives Pt bEtpt1 077 where b a1 a and c 11 a Because 0 lt b lt 1 this is an example of a regular model 212 General Analysis A Special Case Now consider a special case of the more general model 2 where act ac and EtyHl yHl The model is lit byt1 CHE i yHl bilytibilcx 4 Since we are discussing the regular case we know that If1 gt 1 and there exists only one noneexplosive solution 213 General Analysis A Less Special Case We now relax the assumption of a nonstochastic act and perfect foresight Let expectations be rational and at be a stationary stochastic process Begin with M bEtyt1 CM and substitute for yHl 2h bEtlet1yt2 cxn1l CHE Using the law of iterated expectations we get 21 b2EtM2 Cf bcEtlel Further substitutions produce 371 2h bsEtst 220 bZCEtxti Letting s A 00 and assuming EtyHS does not explode too fast 00 bZE v F M 0220 xt2 t because b lt 1 We Will refer to Ft as the fundamental solution It is useful to verify that Ft is indeed a solution by substituting it back into 2 3h bEtyn1 can C 20 biEtti 11E C 20 biEt1t1i CM C 20 bi1Ett1i can 0 20 biEtacHi This veri es that F is a solution but is it the only one 214 Bubbles Next we look for solutions of the form yt F By Substitution into 2 gives F B bEF1 EtBt1 095 Since Ft bEtFHl Capt this implies that in order for yt F By to be a solution we require B bEth 5 B that satisfy equation 5 are often referred to as rational or speculative bubbles Examples of Bubbles Here are three types of bubbles l Deterministic EvereEXpanding Bubble B Irlii H where b lt l and BO is given This bubble satis es equation N Stochastic Bubble B b lBt1z where z gt 0 is an iid stochastic process satisfying EtzHl 1 This bubble satis es equation 03 Periodically Popping Bubble B BH 7 Bbir1 with probability 77 Bbl 7 7T 1 with probability 1 7 7T This bubble satis es equation Can We Rule Rational Bubbles Out397 Sometimes we can and sometimes we can t For example bubbles cannot be ruled out on the price of an intrinsically worthless asset that goes on into perpetuity especially one for which fundamentals are hard to pin down However bubbles can be ruled out on assets with known nite maturity or if their value becomes so large that they are not consistent with the notion of nite resources Note that all bubbles are explosive B bEtBt1 bEtlet1Bt2l bZEtBt2 B bsEtBHS EtBHS 1243 Every rational bubble therefore satis es limsnooEtBHs 00 if Bi gt 0 Since we are analyzing economies comprised of a xed number of in nitely lived agents rational bubbles can be ruled out Agents purchase the asset at a price higher than indicated by fundamentals only with the expectation of selling it for a gain in the future Without new entrants into the economy it cannot be an equilibrium for everyone to do this In this sense the bubble is similar to a Ponzi game 2 2 Irregular Case The irregular case is de ned by lbl gt 1 These models allow for multiple nonexplosive solutions that is multiple equilibria As you will see this class of model can support the notion of animal spirits or selfeful lling prophecies 221 General Analysis A Special Case As before consider the special case of perfect foresight and nonstochastic at The system can be written as equation 4 yt1 bilyt 1171090 where lb ll lt 1 Therefore this linear difference equation is stable and there exist an in nite number of equilibrium paths that converge on the steady state 17 cap1 7 b 222 General Analysis Complete Class of Solutions To analyze the complete class of solutions7 de ne 77t1 yHl 7 EtyHl where the de nition of rational expectations gives EmHl 0 Substitution into 2 and letting at at we have 11 byn1 77t1 can 7 yt1 71171096 bilyt 77t139 6 Equation 6 is a stationary rst7order autoregressive AR1 process where 77t1 N iid0a Notes a W has many names 7 extrinsic noise 7 nonfundamental noise 7 sunspot 7 sel ful lling prophecy 7 animal spirits The B bubbles in the regular case can be thought of as explosive sunspots There are two sources of indeterminacy 7 yo and m a m can be any variable coordinating expectations and matters only because people believe it does Jevons 1884 thought sunspots actually were affecting the economy 223 Other Types of Multiplicity There are other types of multiple equilibrium that do not fall into the regular or irregular class 1 Regular cycles For some 31 fy1 act it may be possible to generate7 say7 regular two7cycles or three7cycles 2 Multiple steady states Other 31 fy1 1 may produce multiple steady states with the possibility that sunspots may jump you from one steady state to another Models of this type could be used to look at coordination failures 3 Chaos Still other 31 fy1 1 may exhibit chaotic behavior where small changes in the initial conditions may produce a rich variation in dynamic behavior 23 Overlapping Generations 0G Example Economists have long recognized that dynamic rational expectations models may exhibit multiple equilibria We will discuss below how this may occur in an in nitely lived representative agent RA model However7 the best known examples come from overlapping generations 0G models Here s one example Framework one good a constant stock of money7 M twoeperiod lives a no production a constant population 7 half young7 half old endowment 7 51 0 a P is the money price of the good The objective for agents is to choose consumption in period one and two 01 62 1 to maximize V01t WC2t1 7 subject to M d M 01 61 an 02t1 P Pt1 Putting the constraints together gives C2 1 51 CLO t Pt1 where PtPH1 is the gross rate of return on holding money Substituting 8 into 7 gives P V01t W 1 Culp tl n The rsteorder condition is 3M 801 P B v 7 W 7 01 Pt 51 01 RH which after rearranging gives P P t W 62t139 9 t1 VClt Although V and W in 9 do not have explicit functional forms we can imagine solving implicitly for rsteperiod consumption demand P 01 C1617 P t1 Plugging this back into the rsteperiod constraint gives 61 C1617 P 10 1 Men which is a demand function for real money balances The derivative of money demand with respect to 51 will be unambiguously nonnegative ie extra endowment in period one will not cause the agent to want less consumption in period two The derivative with respect to the rate of return on money however could be positive or negative Imagine a small decrease in PHl If the substitution effect dominates then a higher PtPH1 will induce the agent to substitute consumption tomorrow for consumption today and therefore demand more money ie L2 gt 0 If the income effect dominates then a higher PtPH1 increases wealth and periodeone consumption demand ie L2 lt 0 Next we linearize 10 since it is likely to be nonlinear m Pt 751 041 Etpt1 where lowercase letters represent proporitional deviations from the steady state This can be rearranged into our standard form as Pt 1 bEtpt1 11 where b al a and a m 7 7el1 Q There are two cases 1 Regular case In the regular case lbl lt 1 and the substitution effect dominates the income effect This means that a gt 705 The unique solution which can be found by repeated substitutions into to the righthand side of 11 is Pt 1ib Irregular case In the irregular case 1111 gt 1 and the income effect dominates the substitution effect This means that a lt 705 The full set of solutions can be found by substituting marl pt1 7 Etle into 11 and rearranging Pt ab l bilptil 77 where 77 is a sunspot An interesting special case occurs when 71 lt a lt 705 producing the soecalled 7 cobweb model 231 Parametric Example Let the utility functions be CBS 179 CM 7 1 179 179 C2t1 1 5 W6 1 79 W32t1 VC1t 01f W C2t1 5027 Substitution into 9 with some light algebra gives P 1 61 7 19 5 1 7 01 Pt1 Let s look at the three cases 1 to 03 lrregular case lneBetween case Regular case If 0 lt 9 lt 1 then 01 falls when PtPHl increases In other words when today s price increases relative to tomorrow you consume less today save more The substitution effect dominates 1f 9 gt 1 then 01 rises when PtPHl increases In other words when tomorrow s price decreases relative to today you consume more today because you are wealthier The income effect dominates 1f 9 1 then utility is log in consumption and 01 does not depend on relative prices The substitution and income effects cancel one another out 3 Multiple Equilibria Based on Increasing Returns IR While OG models can exhibit multiple equilibria they are dif cult to implement empirically Rather Farmer uses the standard RA framework of the RBC model but adds increasing returns IR to scale The problem with incorporating lR into the neoclassical model is that it is inconsistent with competitive behavior However it is possible to reconcile the neoclassical model with IR by modifying the model appropriately Here are two approaches 31 Externalities Approach Let the representative agent be indexed by 239 on the unit interval 0 l The im agent chooses consumption and labor In to maximize 1 oo X U Zs Et tislogcis 7 2 s 1 7 X subject to Cu km km 5 21m and 21m Atstk tla lim where A is an aggregate production externality organizational synergies given by 1 9 A kgy z1mdi i0 Recognizing that all agents are identical allows us to write aggregate output as 1 1 y yidz39 Atskfv zi1 mdz 20 20 1 stAt knmtzmkmdi 20 1 9 1 5 kzzltvtzit1mdi szzltwigt1mdi 20 20 1 19 s kg7 z1mdi i0 St kWme Stk Ytlt where p m1 9 V 1 7 m1 9 and p V gt 1 Hence we have constant returns to scale at the private level and IR at society s level When making private decisions agents will take A as given and each factor will be paid its private marginal product which in aggregate will exhaust all of national income There is no inconsistency between competitive markets and IR at the social level 32 Monopolistic Competition Approach Under the monopolistic competition approach begin by assuming that each agent produces a distinct inter mediate good with an IR production technology The intermediate goods are aggregated in a competitive sector to form a nal good using the following technology y mallA 12 3 21 FinalGoods Producers Each nalegoods producer then chooses 31239 for 239 E 0 1 to maximize 7139 Ptyt Pityitdi 13 239 subject to 12 Substituting 12 into 13 produces 1A 7139 Pt ipityitdi 2 2 Taking the derivative with respect to y setting it equal to zero letting p 1 be the numeraire and rearranging gives the demands for intermediate goods y A71 it i 14 Pt gt Therefore if A 1 nal output is a simple sum of all intermediate goods ie they are perfect substitutes pit 1 and intermediateegoods producers are price takers If A lt 1 the yit s are imperfect substitutes and there is some market power in the intermediate sector 322 IntermediateGoods Producers lntermediateegoods producers7 taking 14 as given7 choose lit and kit to maximize y A71 39t 7W yit wtlit Ttkit 15 M where yit 5mm with p V gt 1 Substitution of 16 into 15 gives 16 7W yi s lmlmw wtlit Tth where we assume that My V S l The rsteorder conditions give the factor demands 7 7 ALyitpitk1 wt Wynpitl l 3 23 Consumers Consumers are assumed to choose Ci and lit to maximize t s 17X V 7 7 V 7 23 U 7 E mm loge 1 7 X subject to the standard budget constraint The consumers7 Euler equations are 1 E 1 7 6 n1 and Cit Cit1 7 X w 7 owl 324 Combine Sectors Now we impose symmetry across all consumers and rms so that in equilibrium kit kit E kt lit ljt E It and pi pj 17 where the last equality comes from imposing zero pro ts 713 0 in the competitive nalegoods sector This produces 1 1 En 176Auy1k 1 Ct Ct1 Aug ctltl c 33 Overall Increasing Returns Model We now have two economic environments that generate increasing returns at the social level 7 aggregate production externalities and monopolistic competition Although the underlying structural parameters have different interpretations they can be Written in one common framework kHl y 1 7 Dist 7 ct capital accumulation y stk ytlt production technology my ctltl c consumptioneleisure tradeoff 1 1 Et 1 7 5 myHl consumption tradeoff Ct Ct1 kt1 s sfilvt technology shock Where p V gt 1 The parameters are related according to Externalities approach a m private share in capital n private share in labor a m n 1 a p m1 9 I V 17m19 Monopolistic competition approach a m ML n AV m n lt 1 so there are positive pro ts 34 Empirical Evidence for Increasing Returns A welleknown puzzle in the macro literature is that of procyclical productivity in the US data Since detrended output is more volatile than detrended hours worked7 it implies that average labor productivity gtm increases when 31 increases Using this as background7 we can list the following pieces of evidence in favor of IR Increasing returns can solve the procyclical productivity puzzle but of course so can the RBC model a The Solow residual in the RBC model is correlated with things it shouldn t be eg7 military expendie tures Econometric estimates of the marginal product of labor in larger structural models often exceed one a A simple regression of detrended output on detrended hours has a slope greater than one 35 Comparing the RBC and IR Models In either the case of the RBC or the IR economy7 the linearized system can be written as 7 fw1 A vt1 t J HAl R 17 A A wt1 5t 5t1 where le is an expectational error Where the models differ is in the nature of J To help compare the RBC and IR models7 consider the following de nitions a n3 number of forward stable roots n1 number of predetermined variables a n2 number of free variables I nn1n2 A model is regular if m n2 A model is irregular if n3 lt n2 351 RBC Economy The standard RBC economy With m p and n V has one forwardestable root n3 1 and from equation 17 above7 consumption is the only free variable n2 1 Therefore7 the RBC economy is a regular economy exhibiting saddleepath stability Note that s and k are predetermined variables associated With initial conditions so and kg 3 52 IR Economy The IR economy7 on the other hand7 has no forward stable roots nS 0 Therefore7 m lt ng and it is an irregular economy As a result7 the economy Will display expectational indeterminacy so that sunspots may in uence the equilibrium path of the economy 353 Empirical Comparison of the RBC and IR Economies Farmer compares the RBC and IR economies by calibrating each economy Each economy is speci ed to have a single shock process 7 technology in the RBC economy and selfeful lling beliefs sunspots in the IR economy See Farmer7 Table 71 for details The shock processes are given a standard deviation so that the volatility of arti cial output matches that in the US data Relative Volatilities The relative volatilities of the RBC and IR economies are similar in their ability to match the US data7 although IR consumption tends to be smoother and IR investment more volatile than the counterparts from the RBC economy See Farmer7 Table 72 Impulse Response thctions Farmer points out that for the US economy7 the dynamical system 17 is best represented With complex roots so that the economy exhibits cyclical dynamics The typical RBC economy has only real roots7 While the IR economy tends to have complex roots Therefore7 the IR economy is better able to match the apparent cyclical dynamics in the US economy See Farmer7 Figure 74 In nite Horizon Economics and Representative Agents 87 K The RA Model Ct C K A The 0G Model Kt Ct C Figure 51 gt I Comparative dynamics of RA and 0G Models HP ltcned log f GN 391 1950 quarter 1 1990 quarter HP ltcred log of GNP The slope of this line is 14 HP ltcred log Figure 71 GNP and hours in the United States The R model The RA model Output V 39 39 39 39 39 39 Productivity 1 shock h 1 Hours Figure 72 Two alternative explanations for procyclical pmt lucuvity of external effects that should not be present if the neoclassical model i correct Roughly speaking they nd that a regression of industry output or industry inputs and aggregate output comes up with a positive Cicllicien on aggregate output in most manufacturing industries A nal indirect plea of evidence against the real business cycle model comes from the proper ties of the Solow residual which is supposed to he an unbiased estimate 0 the production disturbance Recall that this residual which we discussed ll 39l nble 72 Rchliv Volatilitics in he RA and IR Models Table 71 Variables us Dan RA Model IR model The Parameters of Pmduclion GNP 173 176 174 n 1 m 11 R001 1 R0012 Consumption 086 051 0439 1213c Mode I 054 064 036 036 093 106 Inves39mcnl 778 53973 3939 m Modcl 058 07 121 023 04 1o7111 107 Olli Hours 15 134 144 Producxivily 0111 051 041 Impulse rcsponscs in US data n a I 0 HQ 025 lnvcslmcnl 020 ulpul 015 Hours I g 029 Consumpuon C 010 005 X z W m r 7005 quotIn I I r I r I I I I I I 7 o 3 6 9 12 15 18 21 24 Quarters Impulse responses in an RA economy rm39s mm lnvcsu39nent 03039 Outpu V074quot Hours 018 Consumption 012 mm w 006 Am 0 3 1s 9 I 15 I8 21 24 Qunncrs Impulse msponscs in an IR economy 050 1 lnvcslmem 025 I quot p quot Consumption 00 025 I 397quot I I I r I I 0 3 6 9 12 IS 18 21 24 Quancm Figure 74 Impulse responses in Lhe dam compared with two models ECON 5110 Class Notes Sticky Information Versus Sticky Prices 1 Introduction Mankiw and Reis 2002 present a new model based on the idea that information disseminates slowly through the population Current New Keynesian models typically rely on either sticky prices wages in order for aggregate demand movements to have impacts on both output and in ation These models are the workhorses of modern macro The standard models however make several predictions that are hard to square with the facts 1 Announced credible disin ations can cause booms Ball 1994 2 In ation is not very persistent Fuhrer and Moore 1995 3 lmpulse responses from monetary policy shocks decay monotonically The primary problem with the standard models is that although the price level or wages are sticky in ation can freely adjust to economic conditions In Mankiw and Reis7 stickyeinformation model however some rms compute optimal prices based on current information while the remaining rms set prices based on old information This places restrictions on the adjustment of in ation In this sense the paper is a mixture of Calvo s 1972 random adjustment model and Lucas7 imperfect information model As you will see the stickyeinformation model successfully addresses the three issues above 2 Sticky Price versus Sticky Information Models 21 StickyPrice Model The New Keynesian Phillips Curve Each period a fraction A of rms adjust price and 1 7 A of the rms keep price xed The exibleeprice rms maximize pro ts to get a pricing equation of the form P Pt 04 1 where 172 is the desired price p is the overall price level and y is the output gap All variables are measured in logs The adjustment price 1 is thus determined by 00 x A ZFOG 7 AJEp 2 The nal equation speci es that the overall price level pt is an average of all prices in the economy 00 p AZFOQ 7 WM 3 which given the random adjustment becomes a weighted average of prices set in the past Combining equations 1 through and de ning 7T p 7 p1 gives the standard New Keynesian Phillips curve 7139 521 Et7TH17 4 where aAZl 7 A 22 StickyInformation Model Now assume instead that rms gather information and recompute optimal prices in a Calvo manner ie each period a fraction A recompute optimal prices The rm s optimal price is still given by If a rm has not recomputed optimal prices for j periods it charges the price Etijpg 5 The overall price level is j 139 Pt Zj01 A xt39 6 Putting together 1 5 and 6 gives 00 Pt A ZFOG 0ij Pt aytl 7 The stickyeinformation Phillips curve is then given by 77 AM A 200 7 A39E1m my 8 where A31 is output growth The main difference between 4 and 8 is a The stickyeprice New Keynesian Phillips curve depends on current expectations of future economic conditions a The stickyeinformation New Keynesian Phillips curve depends on past expectations of current economic conditions 3 In ation and Output Dynamics To close the model Mankiw and Reis add an aggregate demand AD equation mt Pt M 9 where m can be interpreted as the money supply They also consider a third backwardelooking model 713 5 71371 10 which can be interpreted as a version of the stickyeprice model with naive expectations E7T1 7131 Now consider three separate experiments 31 Experiment 1 Sudden Permanent Decrease in AD Consider an unanticipated permanent 10 decline in m See Figure 1 for the impulse responses Below is a summary of the dynamics All three models forwardelooking sticky price backwardelooking sticky price and sticky information predict sudden recessions All three models predict the recession is over after 4 years a The backwardelooking model predicts a cyclical response to output a The stickyeprice model predicts a sudden disin ation with a fairly quick return to price stability The stickyeinformation model predicts a delayed disin ation with more in ation inertia This is due to a relatively small calibrated value of a 01 32 Experiment 2 Sudden Permanent Disin ation In this experiment7 there is a sudden decrease in the money growth rate Am from 10 per year to 0 per year See Figure 2 for the impulse responses Below is a summary of the dynamics I The stickyeprice model predicts a painless disin ation In ation jumps immediately from 7T 10 to 7T 0 but output stays constant Although prices are sticky7 in ation is free to jump and is not inertial The disin ation occurs gradually in the stickyeinformation model because rms are still basing pricing decisions on outdated information The economy falls into a recession with the trough occurring about 15 years after the announcement This matches much of the empirical evidence suggesting that the effects of monetary policy occur with a lag The backwardelooking model displays similar behavior to the stickyeinformation model7 however the disin ation and recession occur more gradually 33 Experiment 3 Anticipated Permanent Disin ation In this experiment7 the decrease in the money growth rate from Experiment 2 is announced two years in advance and is fully credible See Figure 3 for the impulse responses Below is a summary of the dynamics The backwardelooking model has the same dynamics as in Experiment 2 a In the forwardelooking rational expectations stickyeprice model7 we get the result that agents begin decreasing in ation prior to the money growth slowdown7 increasing real money balances7 and creating a boom This is an unappealing result a In the stickyeinformation model7 the preeannounced policy does not generate a disin ation until money growth falls Consequently7 there is no boom However7 the preeannouncement does impact agents7 future plans and generates a quicker and more painless disin ation 34 In ation Persistence Fuhrer and Moore 1995 claim that the standard sticky price models of Phelps 1978 and Taylor 1980 deliver too little in ation persistence They present a new contracting scheme whereby agents are concerned with relative gal wages that ts the data better1 However for realistic AR1 money growth processes Mankiw and Reis show that all three models backwardelooking stickyeprice and sticky information actually deliver reasonable amounts of in ation persistence see Table 1 Critics could argue however that much of the in ation persistence in these models comes from the exogenously speci ed persistence in money growth rather than through endogenous propagation mechanism 4 Acceleration Phenomenon Table 2 shows correlation between detrended real GDP and changes in in ation The US data show a clear positive relationship 7 times of high economic activity are associated with rising in ation The three models make very different predictions about the acceleration phenomenon The backwardelooking model automatically builds in a very strong positive correlation The Phillips curve for this model equation 10 is 7T i 71371 5 which implies a correlation equal to 1 a The stickyeprice model does not satisfy the acceleration principle This is seen clearly in Figures 2 and 3 a The stickyeinformation model on the other hand does a good job of replicating the acceleration phenomenon found in US data This can also be seen clearly in Figures 2 and 3 5 Responses to the Skeptics Critics of the stickyeinformation model are most likely to ask quotWhy would agents ever use outdated information in forming forecasts of the futurequot Here are Mankiw and Reis7 responses A recent study by Zbaracki et a1 2000 shows that the most important costs associated with changing prices are quotmanagerial and customer costsquot associated with information gathering decisionemaking negotiation and communication 1Recently Holden and Driscoll 2005 have shown that Fuhrer and Moore s 1995 in ation persistence results disappear if agents care about current relative real wages rather than past relative real wages Another recent study by Carroll 2001 shows that expectations of the general public respond in a lagged fashion to the expectations of professional forecasters MankiW 1985 shows that When rms With monopoly power do not update prices or information it can impose large costs on the macroeconomy7 even though the loss to rms may be very small ECON 5110 Class Notes Overview of New Keynesian Economics 1 Introduction The primary distinction between Keynesian and classical macroeconomics is the exibility of prices and wages In classical models eg RBC model Lucas supply model etc prices and wages are completely exible so that labor and goods markets continually clear In Keynesian models prices andor wages are temporarily in exible so that in response to outside shocks eg changes in scal or monetary policy quantities adjust New Keynesian economics refers to the retooling of traditional Keynesian models to be consistent with microeconomic fundamentals 2 Traditional ISLMAS Model The traditional ISeLM model describes the aggregate demand AD side of the economy The analysis is static and not grounded in microfundamentals 21 Aggregate Demand An ISLM Framework 211 IS Curve The IS relationship describes all combinations of interest rates 7 and output Y that generate equilibrium in the goods market The building blocks are C CY 7 T Consumption Function I 7 Investment Demand G G Government Spending T T Taxes NX W Net Exports Y C I G NX Goods Market Equilibrium Therefore the IS curve can be written as YCY7TIrGNX 1 where there is an inverse relationship between 7 and Y A higher interest rate reduces investment demand which requires a reduction in output to restore equilibrium between output and expenditures Total diff ferentiation of 1 with some algebra produces various multipliers For example the government spending multiplier is dY 1 E lt1 0 where Cy dCdY is the marginal propensity to consume Because 0 lt Cy lt 1 the government spending multiplier is greater than one 212 LM Curve The LM relationship describes all combinations of interest rates 7 and output Y that generate equilibrium in the money market The building blocks are MPCl LY 7 Money demand MPS MP Money Supply MPCl MPS Money Market Equilibrium Therefore the LM curve can be written as M P LY 7 where there is a positive relationship between 7 and Y A higher interest rate reduces the demand for real money balances so that income must rise in order to bring demand back to the level that equates it with the xed money supply 213 AD Curve The AD curve depicts combinations of P and Y that simultaneously clear the goods and money markets The AD curve is found by using 7 to combine the IS curve YQYi H GNX and the LM curve MP LY r The AD curve slopes down because a higher P reduces the supply of real money balances and thus Y decreases and 7 increases to restore equilibrium in the money and goods markets Assuming the LM curve is not horizontal the government spending multiplier is smaller when incorporating the money market because the increase spending raises interest rates and ends up crowding out some private investment The lSeLM relationship involves three endogenous variables 7 P and Y and only two equations Therefore we need aggregate supply AS to determine the general equilibrium 2 2 Aggregate Supply In the short run there are various theories as to why the aggregate supply curve P on the vertical axis and Y on the horizontal axis has an upward slope We will explore several of these theories in later papers In the long run the AS curve is vertical such that output is determined by technology and the factors of production The AS curve takes the form YYMP7V where the natural level of output Y is given by the production function the capital stock and the labor force Z Y F K Z Since a gt 0 when the price level P is higher than expected PE output is above its natural rate Conversely when P 3 gt P output is below its natural rate Finally when P9 P output is at its natural rate ie Y Y The lSeLMeAS equations uniquely determine the equilibrium levels of 7 P and Y for given values of the exogenous variables The advantage of the traditional ISeLMeAS framework is its simplicity It can be a useful tool for analyze ing7 for example7 how changes in government policy will impact the macroeconomy However7 the traditional ISeLMeAS model lacks microfundamentals7 dynamics and a wellespeci ed expectations mechanism 3 Microfoundations of the Canonical New Keynesian Model Dynamic New Keynesian NK models are typically represented by three equations an IS curve7 a Phillips curve and a monetary policy rule 301 IS curve The IS curve in NK models is similar to the one in traditional Keynesian models except it incorporates expected future output The IS curve takes the form 9 S lit Et77t1l Ett1 V 2 where the variables are de ned as n gt E output a 7 E potential or trend output act yti 73 E output gap E nominal interest rate measured as deviation from longerun level a N a 713 E in ation measured as deviation from longerun level rt 239 7 E7T1 E real interest rate vi E demand shock The output gap is negatively related to the real interest rate because higher returns induce agents to save more today ie7 the intertemporal substitution effect7 which reduces current output demand The output gap is positively related to the expected future output gap because agents wish to smooth their consumption over time Higher expected output next period will raise their desire to consume today and thus current output demand increases Microeconomic Derivation Equation 2 can be derived from the consumers7 problem Consider the standard RBC framework where the representative agent maximizes E 22 W0 7 arlcilf subject to the usual constraints The Euler equation for consumption is a En Q12 1 7 6 mm 3 where we have used the fact that the real interest rate is 8 7 z 7 E 7T t 81 t t t1 Linearize to get t Et t1 ozE Hl 4 where hats A over variables indicate proportional deviations from steady state and a i TO39 Next we substitute in the resource constraint 3 E Q which ignores investment and trade This produces in Etl wl t1l aE n or rearranged to give lit OtEt H Et t1 77t where 77 iEtAng Finally noting that g act 7 n a 750 and V 77 we get the IS curve in equation For a more detailed derivation of the IS curve see Yun s 1996 Journal of Monetary Economics article 302 Phillips Curve The NK Phillips curve 713 Aft 7Et77t1 5 also differs from its traditional form Although in ation is positively related to the output gap or inversely related to unemployment it also depends on expected future in ation Et t1 and a costepush shock pt 39 39 D 39 U 7 Lucas 39 l E L 39 E Model One microeconomic derivation of 5 is Lucas7 1972 model of imperfect information As you learned last semester the Lucas model assumes that producers have limited information about the overall price level and attempt to infer whether changes in the price of their output is an overall price level increase or an increase in relative prices As Lucas shows the rational inference is that it is a little of both causing rms to increase production This leads to the following aggregate supply curve Qt q 770 Etpt1 6m which after rstedi ferencing can be written in the form of equation 39 39 D 39 U 7 quot C L39 with Price Staggering Following Yun 1996 there are two sectors 7 an intermediate and a nal goods sector The nal goods sector is perfectly competitive and produces the nal good Y according to the technology 00 nn71 y y 139 n71n di t M t where is the intermediate good Given prices and B for the intermediate and nal goods the implied demand for intermediate goods is yin 13335777 The intermediate goods sector is monopolistically competitive They produce the intermediate good using a CobbeDouglas production function FKL by hiring labor L and renting capital K in competitive input markets Market power in the intermediate goods sector implies that factor payments will be distorted according to W MotFmKth R MCtFKKt7Lt where marginal cost MCt is independent of the level of output given the assumption of constant returns to scale Finally to introduce a nominal rigidity ie price stickiness it is assumed that prices are set in a staggered manner Fischer 1977 Each period lN of the rms set their price for the next N periods The remaining fraction of rms 1 7 1N leave their prices xed Aggregation of this type of staggered7price scheme is tedious A simpler but similar scheme is credited to Calvo 1983 In each period each rm has probability 1 7 V of changing price and probability V of leaving price xed As a result the probability of changing price is independent of time and the average duration of price stickiness is 11 7 V1 As Yun 1996 shows this leads to the linearized aggregate Phillips curve 713 mct 7Et77t1 M where m0 is real marginal cost Real marginal cost through appropriate restrictions on preferences technology and the labor market can then be linearly related to the output gap to arrive at the familiar Phillips curve 303 Monetary Policy Rule There are various ways to represent monetary policy Traditional lS7LM models assumed that the money supply was exogenous In accordance with the times early work by John Taylor eg 1979 AER paper to be presented assumed that the policy instrument was the money stock mt which was assumed to follow an accommodative form like mt 151 where w is the aggregate wage level and ab is a parameter measuring the degree of accommodation of aggregate demand to wage changes More recently central banks around the world have begun to use interest rates as opposed to the supply of money as policy instruments In more current work Taylor has shown that a simple policy rule such as it i 9xEtxt1 97rEt7Tt1 appears to do a good job of tting the recent dynamics of the US federal funds rate The money supply under these so7called Taylor rules is assumed to adjust as necessary to reach the target nominal interest 1If intermediate producers are free to change price in every period V 0 they will choose prices that lead to a price index for the nal good that is a constant markup over marginal cost P NMCM where N gt 1 rate7 it A third alternative is to specify a central bank objective such as minimization of 39 2 2 Et 220 zl wa39 7Ttil by choosing a target interest rate it Which Will produce an optimal policy rule 304 Summary In sum7 the dynamic New Keynesian model can be represented by three equations act Split Et7Tt1l Ett1 V IS curve 7B At 7En7Tt1 M Phillips curve it i 91Ett1 gwEtTrt17 Interest rate rule Where the endogenous variables are 1 7T and it Furthermore7 because three equations above are derived from 39 39 L t the A t 50 A777979 are tied back to the economic primitives ECON 5110 Class Notes Coordination Failures 1 Introduction We have discussed three different models of the business cycle 1 Businessecycle models driven by supplyeside factors including imperfect information eg RBC and Lucas7 model 2 Businesscycle models driven by sunspots eg Farmer s 1R model 3 Businessecycle models driven by wageprice rigidities eg Taylor s overlapping contracts model menu cost model One could argue that there is a fourth Class that does not rely on any of these mechanisms One example is Cooper and John s 1988 game theoretic model Where strategic complementarities lead to coordination failures Another example is Diamonds 1982 model of coordination failures that arise due to search and trading externalities 2 Diamond s 1982 Search Model 21 Island Metaphor Diamond uses an island metaphor to describe the basic idea of the model Islanders are employed by searching for palm trees With coconuts All trees have nuts but they vary in their height There is a taboo against eating the nuts you have picked and so you must trade a If many are employed picking nuts ie production is attractive lf feW are employed picking nuts ie production is unattractive Trading externality my employment affects others7 likelihood of making a trade a Multiple equilibria 7 If everyone thinks few are picking pessimistic it is best not to pick 7 If everyone thinks picking is high optimistic it is best to pick Expectations can be selfeful lling 22 Basic Model All individuals are identical riskeneutral and maximize expected lifetime utility given by EM EiZ157N1Ut1l by choosing consumption and the cost of production lnstantaneous utility is given by Uy7c Production opportunities follow a Poisson process and arrive at exogenous rate 1 Recall the Poisson distribution is 670 pac at Each opportunity produces y and costs 0 With cumulative probability distribution Costs are seen T quot quot cannot consume their own before the production decision and A 39 is 39 t output and cannot produce until all inventories are exhausted There are two types of individuals Unemployed Individuals with 0 units looking for production opportunities Employed Individuals With y units looking for a trade The arrival of a trading partner also follows a Poisson process With arrival rate 115 Where e is the fraction employed It is assumed that b e gt 0 The law of motion for aggregate employment is a1 7 eGc 7 be Where 0 is the cost ceiling Note that as expected the steadyestate employment rate 0 rises With cquot This can be found by setting 0 and totally differentiating the resulting equation to get de a1 7 eG c gt 0 dcquot 15 eb e aGc Note also that ddcquot2 lt 07 When evaluated at 0 23 Individual Choice Begin by denoting We3 and Wu as the discounted expected lifetime utility from employment and unemploy7 ment7 respectively Standard dynamic programming arguments give TW 3 by 7 W93 Wu 0 TWu 1 We3 7 Wu 7 cdGc 0 These equations are in the form quotinterest rate X asset expected contemporaneous bene t expected capital gainquot Similar conditions can be found in Shapiro and Stiglitz s 1984 AER e iciencywage pape Unemployed persons accept production if We 7 Wu gt cquot This produces an expression for cquot by a fog ch W 7 W C e u 7 b aGc Differentiating With respect to 5 gives dc i b yr b aGc 7 by af0 ch b 7 My 7 C gt 0 de TbaGc2 TbaGc 39 Note also that pd1amp2 lt 0 24 SteadyState Equilibrium Figure 1 shows three possible steady7state rational expectation equilibria 7 zero7 low and high employment The economy Will always be on the c 06 curve but Will only be on the 0 curve in the steady state The zero and high employment steady states are stable The low employment steady state is unstable 2 5 Static Model To highlight the possible inef ciencies associated With the trading externality7 consider a simple static version of the model Agents produce output With cost function 0 and take the probability of making a sale y as given However7 in aggregate7 p depends positively on y as more agents Will be trading Expected utility follows In making private decisions agents will fail to recognize that by increasing their production it increases the likelihood that others can successfully make trades Therefore the decentralized equilibrium will result in too little production A social planner however will internalize the trading externality and choose y according to My yp y My which will lead to higher production By subsidizing the cost of production via lumpesum taxes the decentralized economy can realize the socially ef cient level of production 26 Policy in the Dynamic Model Consider an increase in 0 via a production cost subsidy Diamond shows that 8Welfare 0 80 gt so that as in the static example private decisions in the dynamic model without the subsidy result in an equilibrium with too little production The intuition is as follows lntuition By increasing the maximum cost at which individuals are willing to produce more people will produce trade will be easier which will in turn induce people to produce more That is the production subsidy partially offsets the trading externality 27 Conclusion Diamond s model is a highly abstract model of coordination failures However the basic concept applies to large and complex modern economies where there is not a Walrasian auctioneer to make sure all mutually bene cial trades are exhausted Search is still an important feature of our economy In the presence of search costs7 an unsubsidized7 decentralized economy is likely to underproduce due to trading externalities This is one possible explanation of business cycles7 Whereby shocks bounce us between inef cient equilibria In this type of environment7 reaching the socially ef cient equilibrium Will almost surely require government intervention to induce agents to internalize the trading externality and increase production ECON 5110 Class Notes RBC Theory Addressing Shortcomings 1 Introduction In this section we examine several welleknown shortcomings and modi cations of RBC theory The topics while far from exhaustive give a general sense of the direction that RBC research has taken since the seminal works of Kydland and Prescott 1982 and Long and Plosser 1983 2 Labor Market 21 Hansen 1985 Labor Indivisibilities Student presentation 22 Christiano and Eichenbaum 1992 Government Spending Student presentation 23 Aadland 2001 HighFrequency Real Business Cycles This paper attempts to explain the same two laboremarket puzzles addressed in Hansen 1985 and CE 1992 volatility of hours worked relative to output and a correlation of real wage and hours worked While those papers added new features ie labor indivisibilities and government spending to the base line RBC theory this paper maintains the baseline theory and simply changes the decision interval Nearly all RBC models assume that agents make decisions at a quarterly frequency ie four times per year This assumption is made so that the arti cial data from the model can be easily compared with the US quarterly data However agents likely make decisions more frequently than four times per year In this paper I allow agents to make decisions more frequently eg once per week temporally aggregate the arti cial data up to the quarterly level and then reassess the performance of the RBC model 231 Benchmark Model A representative agent is assumed to choose weekly consumption ct and hours worked m streams to maximize j E 2H 9 1 7 a5 logmj IogN 7 mm subject to the resource constraint mama 2 c km 7 1 7 319 and the law of motion for technological change at at W71 eXPW sat 232 HighFrequency Calibration The weekly model is calibrated by adjusting the standard quarterly parameter values using ad hoc transfore mation rules Denoting the weekly parameters with asterisks7 for example 5 5113 09926113 09994 n N N13 136913 1053 a 515 qb 2 3 See Table 1 for the full set of calibrated parameters 233 Simulation and Temporal Aggregation The calibrated RBC model is then solved and used to simulate weekly arti cial data The weekly data are then aggregated over time up to the quarterly frequency using the same procedures employed by the US data collection agencies eg7 Bureau of Economic Analysis7 Bureau of Labor Statistics A particularly important fact is that the household and establishment surveys administered by the BLS use the quotcalendar week that contains the 12th day of the monthquot as the reference period for the entire month The temporal aggregation operators are BTAL1LL2quot39L12 for ow variables such as output and consumption and 13 BSSL gm L5 L for ow variables such as total hours worked where L is the lag operator satisfying LjX Xj The aggregation operators are more complex when the variables are measured in logarithms and rst differences 234 Comparison of the Basic and Aggregate Covariances Since RBC theorists focus on the secondemoment properties of the data ie standard deviations and cross correlations it will be useful to relate the aggregate quarterly and basic weekly covariances The general relationship is Fy0 Ww nymW 1 where Fy0 is the zerotheorder covariance between aggregate ac and aggregate y a was is the vector describing the temporal aggregation of at a my is the vector describing the temporal aggregation of y ryw0 is the matrix of autocovariances between basic as and basic 3 For example consider threeeperiod aggregation n 3 of a ow variable as with BTAL 1 L L2 and endeofeperiod sampling of a stock variable y with BSSL 1 Using equation 1 and assuming the variables are measured in growth rates we get 1 Frym L4 3773 6772 7771 670 3 71 72 Continuing with this example imagine that at is output and y is the real wage The RBC model predicts a large positive contemporaneous covariance say 70 09 and weaker cross covariances say yj 7025 for j 74 73 7 2 71 12 So even if the RBC theory were the true model we would observe after aggregation I W0 721025 609 005 in the US data and approximately RENO 09 from the standard quarterly RBC model Therefore we would mistakenly reject the RBC theory 23 5 Results For the complete set of results see Table 2 on page 285 Here is a summary of the main results Statistic US Data Baseline RBC Model Home Production Model Quarterly Weekly Quarterly Weekly stdy 096 096 096 096 096 stdn 094 030 038 067 088 corrwn 7035 097 060 012 7030 M where indicates failure to reject that the statistic is equal to that in the US data The home production model improves the performance of laboremarket uctuations because homeeproduction technology shocks generate additional substitution between market and home activities which also weakens the correlation between market wages and hours worked 236 Conclusions A strong assumption in RBC models as well as other models of the business cycle is that agents make decisions once per quarter In reality agents in our economy make economic decisions on a much more frequent basis This paper modi es a standard RBC model to allow agents to make decisions on a weekly basis With careful treatment of the time aggregation and sampling properties of actual US data the weekly RBC model comes closer to resolving a couple of welleknown laboremarket anomalies In fact the weekly RBC model with household 1 quot is 39 39 quot 39 quot 39 39 from the US economy along these two dimensions 3 Propagation 31 Cogley and Nason 1995 Output Dynamics Cogley and Nason 1995 ask whether standard RBC models can match the stylized facts about output dynamics Two welleknown facts about US output are 1 GNP growth is positively autocorrelated over short horizons and 2 GNP has a strong humpeshaped trendereverting component See Figure 1 page 494 311 Baseline RBC Model Cogley and Nason use the CE 1992 RBC model as their baseline case A representative agent maximizes 139 En ZFO 5 10gctj YN ntjl subject to the resource constraint 1h k atntllig 2 C kt1 1 576 and two shocks The technology shock follows at M71 eXPW Eat and the government spending shock follows 9 X ll exp gt where g exp1 7 Mg and g gtat The model is calibrated as follows Parameter 7 9 6 p g p am 0399 Value 103 025 00037 0344 0021 0004 0177 096 00097 00113 Using Monte Carlo simulations 1000 arti cial samples are generated each with a length of 140 quarters 312 Autocorrelation thction ACF Results Using the generalized Q statistic with an asymptotic chiesquare distribution QACF e 7 c V1e 7 c where E is the vector of US autocorrelations c 23220 Ci and Va 23300239 7 cci 7 c Cogley and Nason test whether the CE RBC and the US output growth autocorrelations are different CN soundly reject the null that they are the same ACFs see Table 1 and Figure In fact the autocorrelation function in the CE RBC model is very nearly zero at all horizons 313 Impulse Response thction IRF Results Using the same generalized Q test CN also reject that transitory and permanent lRFs from the CE RBC model are equal to those in the US data see Table 1 and Figure In particular the CE RBC model produces a damped hump7less response to transitory shocks 3 14 Propagation In an earlier paper in Economics Letters Cogley and Nason 1993 show that output in the CE RBC model can be decomposed into its permanent and transitory parts yt yp ZITU where ZIP2 109 gm 016 The rst part of each expression refers to the shock dynamics and the second the propagation dynamics Clearly there is no propagation mechanism for the transitory part ie dampening scalar 016 and very little with the permanent part since 1 7 087L and 1 7 094L nearly cancel out Overall this implies that output in the CE RBC model inherits the dynamics of the exogenous shock processes with little additional dynamics coming from within the model This can also be seen in the ACFs because CE RBC output growth and the growth in total factor productivity are each white noise 315 Employment Lags and Labor Adjustment Costs Some progress on output dynamics can be made by incorporating employment lags eg labor7hoarding model of Burnside et al 1993 in which rms choose the number of workers before observing the state of the economy and then vary work effort afterwards and adjustment costs to changing employment Both models do a better job of matching US output dynamics however they still require implausibly large transitory government spending shocks to match the magnitude of output dynamics 3 16 Conclusions The main contribution of Cogley and Nason 1995 is to document that standard RBC models are incapable of matching the output dynamics of US GNP This is primarily due to the absence of any endogenous propagation mechanism in the models Therefore7 output in the model basically inherits the dynamic properties of the exogenous shock processes 4 International 41 Backus Kehoe and Kydland 1992 International Real Business Cycles Student presentation ECON 5110 Class Notes RBC Theory Basic Model Calibration Solution and Simulation 1 Introduction In this section 1 present the details of a basic real business cycle RBC model I rely heavily on Prescott s 1986 seminal paper and the material in Farmer Chapter 4 of Romer also provides a nice summary 2 Real Business Cycle Theory 21 Prescott 1986 quotTheory Ahead of Business Cycle Measurementquot Prescott presents the following puzzle Why do industrial market economies display recurrent large uctuations in output and employment over relatively short time periodswhen the associated movements in labor s marginal product are small The surprising answer according to Prescott is that there is no puzzle 7 this is exactly quotwhat standard economic theory predicts RBC theory explains business cycles by incorporating stochastic technology shocks into a standard Neoe classical growth Ramsey model Agents in the economy vary both intra and interetemporally their consumption and leisure choices in response to technology shocks that alter the returns to working This endogenous variation in consumption and leisure creates business cycles 2 2 The Model Begin by assuming that a representative agent maximizes her discounted future utility which is given by t 7 7 J 7 E0 2H 5 1 lt15 loge log1 mm by choosing ch n 0 subject to the following constraints a at c S stktlien resource constraint kHl 1 7 516 act capital accumulation I 5H1 sf expe1 technology shock process Et1 N iid0a392 driving shock kg and so given 2 21 FirstOrder Conditions To see the rsteorder conditions Write out the objective function starting at period t With the constraints substituted in J 7 1 7 qgt1og7kt1 7 1 7 6m stktkenf qgt1og1 7 710 E15 1 lt15 logl fktn 1 6kt1 St1knllignf1l 151ng1 nt1 The rsteorder conditions are found by taking the derivatives With respect to kHl or equivalently 0 and n and setting them equal to zero Begin With capital 8 8kt1 L CH1 1 6 1 95t1k91nf1l 0 1 71 7a E1 1 7 a5 Which after rearranging gives 1quot5EM176179y 1 2 Ct Ct1 kt1 The intuition behind 2 is straightforward To maximize future utility the agent must equate the utility of an extra unit of consumption today 3373 and the discounted expected utility of foregone consumption 3Ut1 3Ct1 3yt1 3kt1 39 7 39 39 39 act 78 ght a for each perlod t 7 0oo The rsteorder condltlon W1th respect to tomorrow labor is 887 1 w testkkenf l agt1 1 0 3 Which after rearranging gives 1 7 c 9illbm39 4 The intuition behind 4 is that the extra utility from consuming the fruits of labor giggiigifi must equal to the disutility of labor 7273 Finally combining the resource constraint and capital accumulation equation gives km 7 1 7 6m 7 c stkfenf 5 The equilibrium is therefore a nonlinear expectational system of four difference equations 7 equations 2 4 5 and the law of motion for s 7 in four variables st kcn 23 Solving the System There are many different solution methods for nonlinear rationaleexpectations difference equations such as those above A good reference is the book Frontiers of Business Cycle Research Which is a collection of papers edited by Thomas Cooley Below I present the method outlined by Farmer Which is a variation of the procedure of Blanchard and Kahn 1980 in Econometrica 231 Steady State The steady state is de ned as the values all variables would converge to in the absence of any technology shocks The steadyestate value of a variable is denoted by the lack of a time subscript The system of steadyestate equations are 1 l a5 1 7 a5 1 7 n c a 7 lemme 8 k 1 7 6k 7 c 18479 9 Given values for the structural parameters 45945 the system can be solved for the steadyestate values 31 k c and n 232 Linearizing the Transition Dynamics For pedagogical purposes assume n 1 for all periods so that equations 4 and 7 are not relevant The rst problem in solving for the transitional dynamics is that equations 2 5 and the law of motion for technology shocks are nonlinear Standard solution techniques for rational expectations models require a linear system Consider taking a rsteorder Taylor series expansion to the system around the steady state For an arbitrary equation as fy 27 the linearization With variables Written in proportional deviations from their steady states Will take the form M v yytiy ZZt Z if 7 i f nun y h m it aygt 1th 7h Where m is an approximation error and hats over variables denote proportional deviations from the steady state The linearized versions of 2 5 and the law of motion for technology shocks are E Et t1 12E31 0L3En t1 10 EH1 1141 15 ae t 11 t1 9 t1 12 Where the coef cients are de ned as a2 91 7 9159 a3 7 1 7 9159 a4176179k 9 a5 159 a a6 7ck 233 Standard Form The next step is to Write the system in standard matrix form7 Where we de ne the conditional expectation of EH1 to be Et EH1 411 way where wf1 is an expectational error that satis es Etwt rl 0 Similarly for kHl and 11 The system in matrix form is t1 1 0 0 E 1 a2 13 411 0 1 a2 a3 a A A wt1 a6a4a5k010kt10000 will 00p t 0013 71000 s wt1 or Written more compactly as Y AYt1 BVn1 13 A A A I k I Where Yt Ct7kt75t 7 VH1 t17wta17wt17w 1 7 0 0 p 0 0 1 and 7 1 1 0 0 0 1 a2 13 B a6 a4 a5 0 0 0 0 0 0 p 7 1 0 0 0 234 Diagonalization The dynamic properties of the model depend critically on the matrix A It Will be convenient to calculate the roots or eigenvalues of the matrix A7 ie7 the As that solve the following matrix equation A 7AIY 0 The interesting solution to this problem requires that A 7 AI be singular7 or in other words7 1A 7 All 0 Solving this equation produces the eigenvalues A1 A2 An where n is the number of rows or columns of A Eigenvalues less than one in absolute value ie inside the unit circle are called forward stable roots The associated eigenvectors satisfy AYU Aili for 239 1n For the RBC example above stacking these equations together produces A1 0 0 A 30 y2 y3y1 y2 y3 0 2 0 0 0 A3 or AQ QA where Q is the matrix of stacked eigenvectors and A is the diagonal matrix with the eigenvalues along the main diagonal Diagonalization of A involves writing it in the form A QAQ l We will use this representation of A to write the system as a set of independent equations 235 Transformation Begin by taking expectations of 13 conditional on time t information which produces Y AEthH Using the diagonalization of A above we get Y QAQilEtYHJ i Qilyt AEtQillVHJ If we let Z Z Q lYt then we can write the matrix system as three independent equations Zn MEMMH ZAEZ1 22 Mam 14 Zan ASEtZ3 1 236 Law of Iterated Expectations Equations 14 hold for all t 0 00 Using this fact we can substitute for Z on the righthand side of 14 to get Z AEtlAEt1ZtZl AZEZ2 where we used the law of iterated expectations for the last equality Repeated substitutions produce Z ATEZ1T For the n3 forwardestable roots7 if we let T A 00 and impose the condition that the lianoo EZT does not explode too fast7 then we have zit 0 for z 1ns 237 Impose Constraint The RBC model7 written in this form7 will possess one forward stable root ie7 n3 1 Using this fact and the relationship Z Q lYt7 we now have an additional restriction on the model t 1111 112 15 where the q coef cients are associated with the rst ie7 forwardestable row of Q l 238 Vector Autoregression and the Policy thctions Finally7 we can substitute 15 into 11 to obtain 344 7 141 115 116011115 112 14 aeqnm a5 aeq12 7 1211 1223 Along with 127 kg and so we can describe the evolution of the state variables in vector autoregression VAR form as 344 171 b2 1 t1 EH1 0 p 5 1 C and the evolution of the remaining variables are determined through the following quotpolicy functions Emljmit H 2 4 Calibration Now back to Prescott Rather than attempt to estimate the structural parameters via econometrics7 Prescott calibrates the model by choosing parameter values that are consistent with longerun historical averages and microeconomic evidence Prescott chooses the following values a 9 064 labor s share of national income I 6 0025 corresponding to about 10 depreciation per annum qb 23 productive time to non7market activities a p 1 random walk technology in logs I 099 corresponds to a subjective discount rate of 4 per annum a39 0763 standard deviation of technology shocks from Solow residuals 25 Simulation Once the model has been calibrated and solved for its VAR form it is possible to feed in counterfactual technology shocks similar to those experienced in recent US history and simulate arti cial data on variables such as output consumption investment and total hours worked The properties of the arti cial data can then be contrasted with the actual US data Before discussing the results of such an exercise we need to address the issue of trend7cycle decomposition 251 HodriCkPrescott Filter In order to remove the long7run growth trend component from the US data Prescott employs a exible detrending procedure known as the Hodrick7Prescott HP lter The HP lter chooses the trend 739 or equivalently the cycle Y 7 n to minimize 211m 7 n A 21mg 7 n 7 n 7 my where A is the smoothing parameter Higher values of A result in a smoother trend For instance a A 0 results in 739 Y and there are no cycles a A A 00 results in 739 a bi or a linear trend Prescott chooses A 1600 which causes the HP lter to focus on cycles with periodicity of 8 years or less 252 Simulation Results for Postwar US and RBC Economies Prescott presents his ndings for the HP7 ltered US economy between 1947 and 1982 The focus is on standard deviations of and comovements between various macro time series Here is a sampling of the results reported in Figure 2 and Table 1 US Stylized Facts Hours worked and output are highly correlated over the business cycle Hours worked and output are almost equally volatile The standard deviation of consumption is less than the standard deviation of output a The standard deviation of investment is much greater than the standard deviation of output a The average product of labor YL is weakly procyclical Table 2 presents the average statistics for 20 simulations of the RBC economy M a The standard deviation of output is approximately 45ths of that in the US economy Consumption is less volatile than output lnvestment is much more volatile that output Hours worked are only about oneehalf as volatile as US hours worked a The average product of labor is strongly procyclical The rst three RBC facts led Prescott to claim that quotgiven people s ability and willingness to intrateme porally and intertemporally substitute consumption and leisure it would be puzzling if the economy did not display these large uctuations The last two facts are failures for the model and have been a major source of further research among RBC theorists 26 Modi cations Prescott discusses two possible modi cations to the quotplain vanilla RBC model to increase the volatility of hours worked 7 adding a distributed lag of leisure and allowing for a varying workweek of capital 261 Distributed Lag of Leisure Kydland and Prescott modify the utility function as follows 1 2 14 20 ailwil 10g0t 10gz0 ailtii where l 1 7 n is leisure 20 04239 1 a0 05 and the remaining 04 decline geometrically at rate 09 This preference structure captures the notion of quotfatiguequot that is high work effort in the recent past increases the marginal utility of leisure today To see this note that But 2 05 37 0az In other words7 low levels of leisure in the past increase the marginal utility of leisure today and hence the degree of intertemporal substitution of labor 262 Varying Workweek for Capital Allowing the workweek of capital to vary with the workweek of labor also increases the intertemporal sube stitution of labor The typical chain of events in an RBC model is 8 positive technology shock s i T 87 i T m Th so that hours worked increase in response to a technology shock that raises the returns to labor With a varying utilization of capital7 we get the additional path 8 8 positive technology shock s i T i i T n i T k i T i i T m am am where higher capital utilization in response to more hours worked further increases the marginal product of labor and labor market activity 263 Results By adding these two modi cations7 the standard deviation of hours worked in the modi ed RBC economy is nearly 70 the standard deviation of output as opposed to 50 in the basic RBC model Furthermore7 the standard deviation of output in the modi ed RBC economy is now approximately equal to that in the US economy 3 Economic Mechanisms In this section7 we look deeper into the economic mechanisms that generate businessecycle uctuations A useful starting point is to ask the following two questions 1 Why is consumption less and investment more volatile than output 2 Why are hours worked so volatile over the business cycle Keynes would answer the rst by saying rms have volatile expectations animal spirits and answer the second by saying there exists a failure of effective demand working through sticky wages The neoclassical answers are much different and are treated in order below 31 Consumption Smoothing To see how consumption smoothing emerges in the neoclassical model consider a simple twoeperiod model where agents maximize lifetime utility 1777 1 C1 C2 17n 17 7 ucl 02 7 n by choosing rsteperiod consumption cl and secondeperiod consumption 02 subject to the budget constraint 02 212 1 Ry1 7 cl where R is the rate of return on savings The Lagrangian function is 5 6177 5617 M312 62 1 RZI1 01 1 77 1 77 and the rsteorder conditions are 7 cn71R0 7 ogquot 7A0 y2621Ry1C10 Rearranging the equations we get Cl 1 177 i 7 16 02 1 B To see how consumption smoothing works consider a temporary increase in income 311 increases but 312 remains constant Because neither 311 or yg show up in equation 16 the household will not change its relative consumption bundle so that the ratio of cl to 02 remains constant ie the household smooths consumption This story is a little misleading for RBC theory however as income changes are driven by technology shocks which are typically permanent in nature Permanent changes in income would raise 311 and yg by equal amounts which again does not directly affect the household s relative consumption bundle However with a permanent technology shock the return to savings 7 7 79 9 RiMPkis179kt n will increase We can look at three cases 1 A household with a very low willingness to intertemporally substitute consumption 7 A 00 will choose to keep cl 02 regardless of the change in R 2 A household 10 with a very high willingness to intertemporally substitute consumption 7 gt 0 will reduce cl increase savings 311 7 cl and reap the bene ts of the higher R by increasing 02 A household with 7 1 log utility and offsetting substitution and income effects will not alter savings decisions in response to R but 0102 will fall as households earn a higher return on their savings Thus the neoclassical model predicts consumption smoothing and as a consequence high investment volatility 32 Intertemporal Substitution of Labor ln explaining laboremarket uctuations the neoclassical model does not rely on any market failures or nominal frictions but rather on the ability and willingness of households to substitute labor across time Consider again a twoeperiod model with perfect foresight Households maximize lifetime utility written in Lagrangian form n1Jr n1Jr mm c ln61 7 1 1n02 7 2 w1n1 22 51 2 1 1 1R by choosing consumption 0 and labor in both periods given the wage rates ml and 1122 Rearranging the rsteorder conditions produces 3 50 R0 n2 102 where a39 is the intertemporal elasticity of substitution of labor Similar to the consumptionesmoothing story above a permanent technology shock that does not change the relative return to labor wlwg will not directly cause an intertemporal substitution of labor However a positive technology shock will raise R and thus induce households to supply more labor in period one and reap the consumption bene ts of the higher returns on savings Thus the neoclassical model naturally predicts businessecycle uctuations in hours worked in response to technology shocks 4 Conclusions For the conclusion l ll focus on two items the title of Prescott s paper and the general arguments for and against RBC theory 41 Prescott s Title Why does Prescott title his paper quotTheory Ahead of Businessecycle Measurement To answer the question we need to de ne the labor elasticity of output within the model which is Blny Blnn 9 11 and calibrated to be 9 064 Now in the actual US and RBC arti cial detrended data7 if we were to run a regression of the following form in 97h 5t we would most certainly get an upwardly biased estimate of 9 because of the positive correlation between technology shocks 5 and hours worked For the US economy7 the estimate is 9 11 and for the basic RBC model it is 9 19 Prescott interprets the large magnitude of the US estimate as quotstrongly supporting the importance of technology shocks in accounting for business cycle uctuations Furthermore7 he argues that the gap between the US estimate and the RBC estimate is due to measurement errors in US output and states that quotthis deviation could very well disappear if the economic variables were measured more in conformity with theory Hence the title 42 Pros and Cons of RBC Theory Here are some of the standard arguments made for and against RBC theory Pros Simple neoclassical growth model without any money or nominal frictions can explain the comovements between and variation in several key macro variables There is no room for countercyclical policy 7 uctuations are optimal responses to changes in relative prices Focus on microeconomic fundamentals makes the analysis less ad hoc and allows for welfare analysis Avoids Lucas critique of macroeconometric policy I Dif cult to believe that the standard deviation of technology is nearly 1 per quarter a The Solow residual has been shown to be correlated with things that it shouldn t be if it were a pure measure of technology shocks The intertemporal substitution of labor is small in micro studies lmplausible that technological regress can explain major recessions Empirical studies typically show that money is noneneutral7 but would be neutral in a standard RBC model Calibrated models are not subject to formal hypothesis testing The propagation mechanism of the RBC model is weak ECON 5110 Class Notes Staggered Wages 1 Introduction A popular method for generating persistence and wageprice stickiness in current macro models is through staggered contracts The earliest work in this area is credited to Stanley Fischer 1977 and John Taylor 1979 1980 1 will focus on the Taylor s 1979 AER article Taylor s model is important because it presents a framework where aggregate demand disturbances can have real effects that are spread out over long periods of time ie display persistence without sacri cing rational expectations It also produces a familiar Phillips curve tradeoff 2 Taylor s 1979 Model 21 Staggered Wage Setting Begin by assuming that rms and workers negotiate labor contracts that specify xed nominal wages for two periods ie one year Contracts are staggered Half are set in January and half in July The wage setting equation is 13 b hii dit1 Ybt d t1 6t 1 where a at is the log of the contract wage in period t I b d and y are positive parameters with b d 1 a y is log excess demand ie log output gap a hat over a variable represents rational expectations based on time t 7 1 information Firms unions and workers care about relative wages Therefore when setting wages in the beginning of period t which will be in effect through periods t and t 1 agents care about wages set in period t 7 1 known with certainty and wages to be set in period t 1 unknown in period t Agents also care about laboremarket conditions throughout periods t and t 1 as measured by yt 2 2 Money Demand Money demand is taken from the quantity equation mtytwtivt where mt is the log of money demand a w is the log of the aggregate wage level vi is a velocity shock The aggregate wage level is assumed to follow w 0535 11 23 Money Supply Money supply is given by a simple policy rule Assume that the monetary authorities set the money supply according to 4 mt gwt where g is indicates the degree of accommodation to changes in the aggregate wage level If g 07 then the central bank does not accommodate wage changes lfg 17 then they accommodate them oneeforeone 2 4 Aggregate Demand By equating equations 2 and 47 we get the aggregate demand curve 31 wt vt gwt OF M wt l vt where 1 g 25 Solving for the Reduced Form The model currently has three equations in three endogenous variables th7 act and yt The system can be collapsed down to a single linear expectational difference equation by substituting equations 5 and into equation This produces it 175671 156771 Yb505it 5571 170 d 505it1 5 17771 En it 7 biH dim 7 057mi 7 057b iH 7 057431 7 0539yd a c 0 7 b 7 05ryb i1 7057b 7 05743 7 1m d 7 05 yd fc1 0 7 biH 7 Ci 7 dim 6 where C 7 1 05 7 1 7 0573 Equation 6 is of the same form we have analyzed throughout the semester 7 a linear dynamic rational expectations difference equation We have solved this type of model using repeated substitutions and using Farmer s eigenvalue method Another solution technique is that of undetermined coef cients see Romer section 66 or Blanchard and Fischer appendix 5A First7 we guess the form of the solution based on experience with coef cients to be determined act 041371 6 7 Using equation 7 to form expectations 3 and fowl we equate coef cients to nd ac70274d17d05 2d Using equations 7 and 37 we get the reduced form for aggregate wage dynamics wt awt105eet1 00 V 7 05 Zi0azet7iet717i and the reduced form for output dynamics 2h Olytil 05 t t71vt O Util oo 705 220 azei 714 39Ut 26 Interpreting the Results 261 Phillips Curve Tradeoff Output and wage dynamics are governed by the parameter a Lower values of a lead to less persistence ie higher stability in aggregate wages Note also that g and a are positively related Therefore the central bank can generate more stability in aggregate wages by making 9 small However this comes at an expense Lower values of 9 imply higher values of and thus a atter aggregate demand curve This means that shocks to contract wages will lead to more output volatility The tradeoff exists in reverse if the central bank is more accommodative ie makes 9 large The Phillips curve tradeoff can be summarized as follows Accommodative policy 9 high low 7 low variation in y and high variation in w a Not accommodative policy 9 low high a high variation in y and low variation in w 262 Degree of ForwardLooking Behavior It is interesting to see how wage and output dynamics depend on the degree of forwardelooking behavior As expected wage persistence as measured by a is decreasing in d This makes intuitive sense Consider the limiting case of d 1 where agents only look forward to next period s wage In this case a 0 and a shock to the contract wage only lasts as long as the contract period On the other hand when d 0 and agents only look backwards a is very nearly one and wages are very persistent In sum Only forwardelooking behavior I 0 d 1 a 0 7 low persistence in w and y a Only backwardelooking behavior I 1 d 0 a 2 1 7 high persistence in w and 31 263 HumpShaped Impulse Response Mnctions A welleknown feature of US output is that it displays a humpeshaped impulse response that peaks somewhere in the neighborhood of four quarters For example if we let I d 05 y 02 g 05 and v 0 for all t we get the following equation for output dynamics 31 06351 7 1256 61 which is an ARMA11 process Assuming yo 60 07 61 71 and e 0 for all t gt 17 we get the following sequence for output period j w 1 025 2 041 3 026 4 017 Therefore7 a onetime shock to the contract wage produces a humpeshaped response for output that peaks at four quarters ie7 one year after the shock 3 Menu Costs Student presentation ECON 5110 Class Notes Shapiro and Stiglitz 1984 Equilibrium Unemployment as a Worker Discipline Device 1 Introduction to Ef ciency Wages This is the seminal paper in the area of efficiency wages The model can be used to explain involuntary unemployment above market wages shirking on the job The important idea is that aboveemarket wages exist as an optimal7 equilibrium response to imperfect monitoring and the incentive to shirk 2 Model 2 1 Workers There are N identical workers with utility function U w 7 e7 1 where w is the wage and e is effort There are two effort choices 5 0 or e gt 0 When unemployed7 a worker receives unemployment bene ts if and e 0 When employed7 a woker receives wage w and e gt 0 An employed worker may be exogenously separated from his job with probability 1 Workers maximizes lifetime utility that is discounted at rate 7 22 Effort Decision Workers can choose 5 0 to shirk or e gt 0 to not Shirk If e gt 07 the worker gets paid w a lfe0 7 and he does not get caught shirking With probability 1 7 q7 he gets paid w 7 and he gets caught shirking With probability q7 he gets red and receives 1D The rsteorder condition for a shirker is rv57wltbqgtltvu avg 2 While for a noneshirker it is TVENw7ebVu7VEN 3 Where V5 VEN and Vu are the discounted7 expected lifetime utility for a shirker7 noneshirker and unemployed person7 respectively Solving for V5 and V gives bqVu VS w 4 E Tbq and 7 bV VNW 5 E 71 Workers Will shirk unless V v gt V5 The noshirking constraint NSC is therefore erVuTbqeqw 6 The NSC can also be Written as AVE 7 Vu gt 67 7 Which implies that unless V gt Vu and there is a penalty involved With shirking7 everyone would shirk The critical wage is increasing in effort 5 a lifetime utility of unemployment Vu probability of not being caught 1 7 q a discount rate 7 exogenous quit rate 2 3 Employers There are M identical rms With production function where L is effective labor Assume that so that full employment is ef cient F Ngte 9 Firms offer workers a compensation package w if 12 Where 117 is the legal minimum set by law Assuming for simplicity that if 07 the aggregate labor demand curve is 2 4 Market Equilibrium w F L The rsteorder condition for unemployment is Where a is the jobeaquisition rate TV 117 aVE Vu Solving 5 and 11 simultaneously gives wiearu7b V TE abr TV wieawbr abr Substituting 13 into the NSC gives wZiDeeabTqu7 We can remove a by noting that in the steady state7 we must have bL aN 7 L 10 11 12 13 14 15 Solving for a and substituting into 14 gives the NSC w2weeqburu7 16 where u N 7 LN is the unemployment rate The market equilibrium is found by equating NSC with the demand for labor to give F L1Deeqbur 17 as shown in Figure 2 Unemployment is involuntary but lowering the wage is not an option because workers cannot make credible promises not to shirk 25 Comparative Statics Using Figure 2 we can consider the effects of changes in a the quit rate 1 a the monitoring intensity q a the level of unemployment compensation a technology shocks ie7 shifts in F N 2 6 Welfare Analysis The decentralized unemployment equilibrium above is subeoptimal The central planner s problem with 1D 0 is to choose w and L to maximize w 7 eL 18 subject to the NSC wZeeqburu7 19 and the resource constraint wL g 20 The optimum occurs where the average product of labor APL is set equal to the NSC This results in less unemployment than the decentralized equilibrium and a higher wage The centralized Pareto optimum could be reached by a tax on pro ts that can be used to nance a wage subsidy See Figure 4 The decentralized equilibrium is inef cient because a each rm sees the private cost of hiring another worker as w when the social cost is 5 higher too few workers a each rm fails to recognize that hiring another worker reduces u and therefore makes it more attractive to shirk hire too many workers The former effect outweighs the latter so that equilibrium unemployment is too high in this model 3 Extensions Endogenous Monitoring Risk Aversion Endogenous Turnover 4 Alternative Enforcement Mechanisms Performance Bonds Other Costs of Dismissal Heterogeneous Workers 5 Summary This paper presents a model where unemployment through aboveemarket wages serves as a worker discipline device However7 the decentralized equilibrium is not generally ef cient 7 rms don t recognize how their level of monitoring and wages impact other rms Government intervention in the form of a subsidy on wages can improve ef ciency ECON 5110 Class Notes Real Business Cycle Theory Solow and Ramsey Growth Models 1 Introduction The Solow and Ramsey growth models are the backbone of modern business cycle theory In fact7 real business cycle RBC theory7 as initiated by Long and Plosser 1983 and Kydland and Prescott 19827 is simply a Ramsey neoclassical growth model with stochastic technology shocks Modern New Keynesian models also share many of the features of the Solow and Ramsey growth models 2 Solow Growth Model 21 The Basics The Solow growth model begins with a constant returns to scale CRS production function where the terms are de ned as YO 7 output KO 7 capital stock AU 7 knowledge LO 7 labor When A and L enter 1 multiplicatively7 technological progress is said to be laboreaugmenting7 which implies that the KY ratio will be constant in the steady state It will be convenient to use the CRS property to rewrite the production function in its intensive form by multiplying all terms by lAL 1 s y fk 2 where y YAL and k KAL are output and capital per unit of effective labor In addition to being CRS7 the production function is assumed to satisfy f0 07 f k gt 07 f k lt 0 and the lnada conditions limkdo f k 00 and limkdoo f k 0 It will be convenient to work with the CobbeDouglas production function which in its intensive form is 22 Dynamics The model is set in continuous time Labor L and knowledge A are assumed to be exogenous and grow at exponential rates according to L0 nLt Lt emL0 Au gAtsAte9 A0 By assumption a fraction 5 of output is devoted toward savings and hence investment and a fraction 1 7 s of output is devoted toward consumption Therefore 5 is the exogenous saving rate Capital evolves according to I39m 5w 7 6Kt 4 where 0 S 6 S 1 is the depreciation rate Writing 4 in its intensive form gives I39m ska 7 n g mm 5 The evolutions for yt kta and Ct l 7 syt can be found easily from 2 3 Steady State In the steady state capital per unit of effective labor is constant ie kt 0 From 5 this implies that total savings per unit of effective labor skta must equal breakeeven investment n g 6kt It is common to denote this steadyestate level of capital as kquot If kt gt kquot then because capital exhibits diminishing marginal returns savings will not be sufficient to replace the capital lost to depreciation labor growth and technological progress As a result the capital stock will fall lt 0 and return to kquot The opposite is true if kt lt kquot This implies that the system is stable See Figure 12 on page 15 in Romer 24 Balanced Growth Path For any initial value of 160 the economy will eventually settle down to the steadyestate level 16 Since 16 KAL this implies that in the steady state the actual capital stock not per unit of effective labor will be growing at rate n 9 Furthermore since AL also grows at rate n9 this implies that Y will grow at rate n 9 We often use output per person YL as a measure of standards of living across time and countries YL will be growing at rate 9 along the balanced growth path 25 Conclusions and Shortcomings I begin by highlighting the major ndings of the Solow model 1 For any initial 160 the economy will converge to a balanced growth path where YL will grow at the exogenous rate of technological progress 9 There is a unique saving rate 59 which maximizes consumption per worker CL This often referred N to as the goldenerule saving rate and is given by the condition f k n g 6 3 The model predicts conditional convergence That is once you control for differences in savings rates depreciation and labor growth rates poorer countries with smaller capital stocks should grow faster and eventually catch up with richer countries The empirical evidence supporting conditional converge across countries is mixed at best 4 Growth accounting can be used to decompose output growth into the parts coming from capital ac cumulation and technological progress Total differentiation of along with some minor algebra gives YO K0 LU mamliaml t where RU is often referred to as the Solow residual or total factor productivity There are several welledocumented shortcomings of the Solow model 1 Steadyestate growth in YL is entirely exogenous In response a new endogenous growth paradigm led by Paul Romer has emerged 2 The micro fundamentals of household and rm decisions are assumed away The Ramsey model below addresses this shortcoming 3 The Solow model7 for reasonable parameterizations7 is unable to explain the vast differences in living standards across countries or across time It requires unrealistically large differences either across time or countries in capitalelabor ratios andor technology 3 Ralnsey Growth Model The Ramsey model extends the Solow model to allow for explicitly optimal behavior by rms and households 31 The Basics Begin by assuming the following There are a large number of identical households with each member supplying one unit of labor a For simplicity7 the initial amount of labor is set at unity L0 1 and there is no labor growth n 0 Households own the rms Each rm hires labor and rent capital in competitive input markets Each rm sells its output in a competitive output market a Each rm has access to the CRS production function with AU growing at rate 9 and A0 1 There is no depreciation of capital ie7 6 0 32 Household Behavior Households are assumed to be in nitely lived and maximize a discounted stream of future utility given by 00 e ptuCtdt t0 by choosing Ct the control variable at each point in time The instantaneous utility function uCt is assumed to be in the constant elasticity of substitution CES class 02 7 1 C t ul l 1 7 9 where 9 iCu u gt 0 is called the coefficient of relative risk aversion Larger 9s indicate more curvature in the utility function and less willingness to substitute consumption intertemporally a39 19 is commonly referred to as the intertemporal elasticity of substitution As 9 A 0 the utility function becomes linear in consumption and a39 A 00 The households7 constraints are given by the flow budget constraint W 100 WNW 00 where at is the sole asset wt is the wage rate and TO is the asset rate of return ii the no Ponziegame condition lim eimna 2 0 tgtoo RU 10 Tsds and iii 20 given Solving this continuousetime dynamic problem involves using calculus of variations no derivations will be provided at this time Begin by writing down the present value Hamiltonian He pt 179 7 i mm mm 7 can 79 where Mt the costate variable is the presentevalue shadow price of income The rsteorder conditions for maximization are 8H 0 s w 0er 6 w 788 1 Am 4mm 7 and the transversality condition is Combining 6 and 7 we get Equation 8 is known as the Euler equation for consumption 33 Firm Behavior Firm behavior is simpler Firms choose labor and capital to maximize pro ts per period Which Written in its intensive form is 1W WWW WW wie g l Since rms take TO and wt as given they Will rent capital up to the point that its marginal product equals rental rate or T0 f ki 9 This Will result in zero economic pro ts if labor is also paid its marginal product or 1W WWW kif39kil 34 Equilibrium Dynamics and Welfare Equilibrium dynamics for this economy are given by the capital accumulation equation 8 and Letting Ct egtc and at egtk along With the appropriate substitutions the equilibrium for this economy reduces to Q monkey cm 9 7 km ma 7 cm 7 gm along With the transversality condition and 60 given The dynamic properties of this economy have been studied extensively Figure 24 on page 58 of Homer depicts the dynamics in a phase diagram The primary conclusions are 1 There exists a unique 0 k combination that produces steadyestate c39 0 growth This is given by point E in Figure 24 of Homer 2 For a given 160 there is a unique 00 that will result in a nonedivergent path to the steady state This path is known as the saddle path and this general property is known as saddleepath stability 3 Since markets are competitive and there are no externalities7 the rst welfare theorem of economics states that this competitive equilibrium is Pareto optimal ie7 no agent can be made better without making another worse off This is also the same outcome that would be reached by a benevolent social planner that treated all agents equally 4 The steadyestate level of consumption per worker in the Ramsey model sometimes referred to as the modi ed goldenerule level is less than the goldenerule level derived from the Solow model This happens because impatient optimizing agents are willing to trade off a permanently lower level of future consumption fo a higher level of consumption today Am Iquot 7 k FIGURE 24 The behavior of l and k for various inilial ECON 5110 Class Notes Learning 1 Introduction This section relies heavily on the material in George Evans and Seppo Honkapohja s book Learning and r1 A i m 7m 1 A u i r of future economic variables play an important role in macroeconomic theory Examples include the permanenteincome lifecycle consumption hypothesis monetary policy and asset pricing models The evolution of expectations in macroeconomics can be classi ed as follows Naive expectations Under this mechanism expectations of a future variable yHl are given by gal 21t Adaptive expectations An example of adaptive expectations AE is 11511 215 My 7 y where the parameter A governs how current expectations adjust to the previous period s forecasting errors AE were commonly used in Keynesian models that dominated the macro landscape in the 1960s and 1970s For example the expectationseaugmented Phillips curve often employed AE In terms of policy AE imply that policymakers can continually adjust policy instruments such as government spending or the money supply to manipulate macro aggregates Rational expectations The rational expectations RE revolution in macroeconomics began in the mid 1970s with the research of Robert Lucas and Thomas Sargent It has dominated macroeconomic theory ever since RE assumes economic agents are very sophisticated They form expectations of future variables according to 11211 Eyt1lnt where E is the mathematical expectation operator and t is the information set containing all informa tion dated at time t and earlier RE assumes agents know the structure of the economy and all relevant parameter values In terms of policy RE imply that policymakers are no longer able systematically manipulate macro aggregates 7 agents understand policymakers7 incentives to do so and adjust their behavior accordingly a Learning Learning in macroeconomics is a reaction to the strong assumptions made with RE In particular it seems unreasonable to assume that economic agents know the relevant parameter values with certainty when even the best econometricians must themselves estimate the parameters Learning generally assumes that while agents are able to gure out the reducedeform equations governing the economy they must continually update their estimates of the parameters An interesting question is whether through the learning process agents can grope their way toward the rational expectations equilibrium Learning is also a useful tool to choose between multiple REE 7 only those that are stable under learning would be expected to be observed 2 Learning Techniques 21 The Setup Begin by considering a structural macroeconomic model similar to the one discussed in the previous set of lecture notes in a blEtilyt bZEtilyHJ can 1 9h 1771 6t where E l is some arbitrary expectations mechanism and E le 0 The REE solution takes the form in EboJF igEtil 77t 2 where 77 06 Assume now that agents do not know ojbl but are able to gure the structure of equation Agents instead specify a perceived law of motion PLM 3h 150 15171 77t where 0 bl are the agents estimates of 350 The actual law of motion ALM is found by substituting the forecasts for y and yHl from the PLM into the structural model 1 y a bll o 1t71l b2l o 1Pt71l 0 which after rearranging gives 21 a 150071 52l 151071 9172 CPlxtil 77 4 This implicitly de nes a mapping from the PLM to the ALM T qbo aqgt0b1b2 151 1511 1 9 07 The relevant questions are whether the parameters in equation 4 converge under reasonable learning rules discussed below and if so do they converge to the REE 3504351 2 2 LeastSquares Learning Let qbtil 0 t71q51 t71 be the estimate of ab O 1 at time t7 l and let 2 lxi The leastesquares estimates are lt15o 71 71 lb 2332871471 ZiQizielyil 5 M71 Note that the least squares estimates can also be written in a recursive manner as 15 t71t71R1zt71ytiqbiilztil 6 Rt Rt71i71zt712 1Rt717 which is known as recursive least squares RLS Here f1 is referred to as the gain y 7 q57121 is the most recent forecast error and R is the moment matrix for 27 Substituting these estimates back into 4 we have M 1 07151 b2ll llt151n71bl 1752 C lhil 77 or alternatively 2h Tleliztil 77t39 7 Substituting 7 into 6 gives 15 1571171Rflzt71T t71Ii iillzt71 770 8 Rt Rt71i71zt712 1Rt717 9 which is a recursive stochastic system Showing convergence of this recursive least squares system is complicated see chapter 6 of Evans and Honkapohja and by no means obvious Under learning economic variables depend on agents7 econometric forecasts of a system which in turn depends on their forecasts This type of learning environment can lead to either divergence from or convergence to REE Fortunately the concept of expectational stability Estability can be used to establish convergence or lack thereof 23 Expectational Stability Before presenting the conditions necessary for Eistability rst note that the REE solution 350 351 is a xed point of the mapping ab We will show this explicitly in an example below We say the REE is Eistable if the REE is locally asymptotically stable under the differential equation 1 150 T 150 7 150 10 d T 151 151 151 where 739 denotes arti cial time In other words an REE is Estable if small deviations from an REE under a perceived law of motion and a given learning rule gradually return back to the REE Using the framework above we would look for the conditions under which the equations d 0 aq50b1b27 0aq50b1b271 d 151011952CP 1 151Pb271cp generate stability in a neighborhood of the REE Assuming that 0 S p S 1 a suf cient condition for Eistability is 111 112 lt 1 3 Economic Applications 31 Cobweb Model Consider a competitive market for a single good The demand for the good is given by d 040 i 04117 u and since there is a production lag supply depends on expected price 5t 50 lEttlpt V where u and V are mutually uncorrelated meanezero whiteenoise shocks Assuming markets clear ie d 5 then we have the reducedeform equation Pt a bEZAPt 77 where a a0 7 0Oq b i lOq lt 0 and 77 is meanezero white noise 311 Naive Expectations Under naive expectations Efilp p1 we have Pt aJFthil 77 11 There are two cases 1 lrregular case If the supply curve is steeper than the demand curve ie 0 gt b gt 71 then equation 11 is a stationary stochastic process the equilibrium is indeterminate and the xedepoint p libfla is a quotsinkquot Note This appears to be inconsistent with the results in the previous set of notes but notice that equation 11 is written in its backwardelooking as opposed to forwardelooking form 2 Regular case Conversely if the demand curve is steeper than the supply curve ie b lt 71 the unique fundamental equilibrium is p l 7 b 1a 77 a noisy steadyestate 312 Rational Expectations Under rational expectations E5 11 E 117 we have Pt a l bEtilpt 77 12 Begin by taking expectations of both sides of 12 conditional on information at time t7 17 which gives E1p l 7 b 1a Substituting back into 127 produces the unique REE Pt 1 7 b 1am lt3577r 13 313 Stability of the REE Under LS Learning Assume that agents do not know 35 in equation l37 but instead use the following PLM to forecast prices Pt lt15 77 Plugging the forecasts back into equation 127 gives the ALM Pt 1 blt15 77t The condition for Estability and hence convergence under leastesquares learning is that 35 be locally7 asymptotically stable under 1amp5 ETW ab a b1 14 Therefore7 if b lt l the REE is stable under learning Since I lt 07 the REE is indeed stable under learning For example7 in the unlikely event that supply sloped down more steeply than demand ie7 l lt 0 and ll gt lall then the REE solution would be unstable under learning Finally7 note that 35 l 7 b 1a can easily be seen as the unique xed point of 14 32 Lucas Aggregate Supply Model Lucas7 aggregate supply function is Qt q 77Pt Eti h 5t 15 where qt is aggregate output pt is the price level 7Tq gt 0 and 6 is meanezero white noise Aggregate demand is derived from the quantity equation mt 39Ut Pt Qt 16 where v is a velocity shock and m is the money supply which is white noise around a constant mean m m m pt 17 All variables are measured in logarithms Some simple algebra produces the reduced form 17 abE 1p77t 18 where m i 1 a7 177 71739r and 77 1 lt u 7 e 1 7T m t 1 Since equation 18 is in the same form as the Cobweb equation it has the same condition for stability under learning blt1gt7rlt17r This condition is satis ed so that the REE from the Lucas supply model is always stable under learning 33 Ramsey Model 331 Framework Consider a discreteetime version of the Ramsey growth model which abstracts from population growth technology shocks and depreciation Labor supply N is normalized to one The representative agent maximizes E 20 t ll 7710t14fi0 subject to C Km w 1 7Kt Firms7 given the CRS production function fK Kg maximize pro ts given by KO 7 th wt This produces the standard Euler equations Tt flag wt fag thKt Plugging these into the consumer s problem and assuming perfect foresight gives CH1 C 1aKKf iCta 11 KH1 Ktthct The Ramsey model has a unique equilibrium7 involving a saddle path that converges to a nonestochastic steady state C In other words7 for a given K07 there is a unique choice of Co that Will put the economy on a convergent path to the steady state All other choices for Co Will lead to divergent paths that violate some nonenegativity constraint or transversality condition 332 Learning Now let s introduce some uncertainty and learning Given the knifeeedge nature of the equilibrium7 it is an open question as to Whether the economy Will converge to the rational expectations equilibrium When agents start With nonerational expectations and use some sort of adaptive learning Begin by linearizing the system A A Ct ME Ct1 WE kt1 IACHJ bl rbzf 19 20 The REE takes the form JAW1 ligt t1 7 35275 Now assume that agents use the following PLM in forecasting 19 1511 t1 15275 Substituting agents7 forecasts of EH1 ancl IAcHl into 19 gives the ALM 5H1 151012 a1 2llj t1 19 111 lull The ALM can also be Written as tj l 1ta2 a1 2 ll T1lt151n Q52 ict j lbl 2n bZl T2 1w 1520 The law of motion for the parameters uncler aclaptive learning is 151 1 71 7ti t71j t71 1 71l 1 71 7tlT1 1 71l 152 152th Jr hlUAWj il 152t71l 2 71 7tlT2 nal Convergence to the REE can be veri ed computationally Evans and Honkapohja state that the convergence to 351652 is rapid for parameter values 09 a 03 a39 05 and constantegain learning Because MM lt 1 this also implies that CK converge to OJ So the Ramsey REE appears to be stable under learning
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