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# 592 Class Note for ECON 570 with Professor Rodriguez-Clare at PSU

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Date Created: 02/06/15

Econ 570 A8 Andres Rodriguez Clare Penn State University September 7 2009 In general one cannot represent the aggregate excess demand function 1131 as the solution ofthe maximization problem of a single representative household But this is possible with quotGorman preferences where the indirect utility function have the form Mp w aha bltpwh In this case can use a representative household with pref erences vltp w M bltpwh where ap E ahpdh and w E whdh Moreover under these preferences there exists a norma tive representative household with N lt 00 and a convex aggregate production possibilities set Y a feasible allo cation that maximizes utility of representative household is Pareto optimal Consider an infinite horizon economy in continuous time and suppose economy admits normative representative household with instantaneous utility Assumption 3 neoclassical preferences Without loss of generality assume that there is a con tinuum of households of measure 1 with each house hold supplying Lt inelastically and LO 1 with L05 2 expnt Utility function is 00 f0 epntuctdt with p gt n Production represented by Y FK L satisfying As sumptions 1 and 2 of Chapter 2 Will use y with k E KL and have R FKKLf k w FLKLfk kf k Capital accumulation together with population growth at n implies 1 fk 6k c k k Law of motion of per capita assets is ar naw C With no government sector have at 2 C75 for all t and 7 R 6 The no Madoff condition takes the form at exp lt 0t7 8 ndsgt Z O A competitive equilibrium of the NGM consists of Ct7 KU 1075 Rt70 st representative household maximizes its utility given KO gt O and taking wtRt0 as given firms maximize profits taking 1005 Rt0 as given all fac tor markets clear More convenient to incorporate equilibrium behaviour of firms inside the definition so that 1005 Rt0 satis fies equations R FKK L W FLK L fk we w Set up Hamiltonian to get necessary conditions and im pose transversality condition lim7ggt00 epntutat 0 where MOE is the costate variable Play with the FOC ofthe maximization ofthe Hamiltonian wrt C75 and the costate equation to get the continuous time con sumption Euler equation 2 1 r cos sultcltt t p where uCC WC is the elasticity of the marginal utility uc From the uc costate equation and the transversality condition we get 5 lim at exp lt 7quot8 ndsgt O t gtoo 0 This is the no Ponzi condition with equality role oftransver sality condition Any pair ktct that satisfies this equation with at and the Euler equation corresponds to a CE The full CE is this combined with market clearing prices which entail t 2 6 Substituting 7quott 6 into the Euler equation we get c39t 1 C15 Memo 190 6 p and into the transversality condition with at yields Iim km exp lt Ala198 5 ndsgt o t gtoo Together with 7305 fkt n 51905 C75 these three equations completely determine the CE How do you go about proving optimality in this case Steady state c39 0 hence p 6 Note that 19 and thus 0 n 6k is independent of function The shape of uc affects the transition but not the steady state Why Compare to kgold defined by fkgold n 6 In the Solow model we had 8k8n lt 0 whereas now 8k8n 0 Why Transitional dynamics are determined by the system of two differential equations 03905 1 m WU 7905 5 P and 7305 fkt n 51905 C75 together with 190 and the boundary condition Iim km exp lt Ala198 5 ndsgt o t gtoo The system exhibits the existence and uniqueness of an equilibrium with saddle path stability Figure 81 Now allow for exogenous constant labor augmenting technological progress so that Yt FKt AtLt with At A0egt We now have y E YL Afk where now k E KAL Is there a balanced growth path The Euler equation is as above 605 1 7 t cos sultcltt p In a BGP we have 7quott 7 and 66 n9 implies that we need cu defined above as the elasticity of the mar ginal utility uc and given by uc u ccuc to be consta nt 5116 is also the Arrow Pratt coefficient of relative risk aversion and also the limit as s gt t of the intertemporal elasticity of subsitution in consumption between times 8 and 75 defined as d 0gc8Ct d ogUC8Ulct aut 8 Integrating 9 u ccuc yields the family of CRRA utility functions as 1 01 LlC leg logo If 9 1 It turns out that the CRRA utility function is the only separable utility function in the family of Gorman prefer ences Canonical model labor augmenting constant exogenous technological change and CRRA preferences 1 0 00 e p ntct 1dt 0 1 0 Let E E cA Then dEZdtgrt p 99 C We also have K AfkL cL 6K K ltALALgtK gt k AL AL2 ALK AL 1 EAfkL cL 6K 2 Z fk3 5 59nk The transversality condition is now lim exp lt 6 n gd8gt O t gtoo Since 7quott 2 R05 6 6 and 5 must be constant along a BGP then 7quott p 99 O and so fTWp599 This equation determines 19 while Efk ng5k Substituting back into the transversality condition we get the condition that p 1 9g ngt0 or p ngtL 9 which is an assumption we must make analogous to p n before This guarantees 7 gt g n Under standard neoclassical assumptions about FK L and CRRA preferences and with p n gt 1 9g then a BGP exists is unique and globally stable Is the rate of conditional convergence higher or lower in the canonical NGM than in the Solow model It depends on the behaviour of the savings rate with kk which in turn depends on 9 and the rest of parameters For reasonable parameters one gets similar rates of condi tional convergence as in the Solow model so endogenous savings do not help the Solow model better match the data See Barro and Sala i Martin39s Economic Growth textbook Formally the dynamic system for Aka is dlnkdt Aka 1 Ek gn6 1 dlnEdt 2 5i Aka l 6p69 Using 239 Iogk and a IogE then we can rewrite this as 2quot Ae10 z 633 2 g n 6 9 a Ae l O Z 5peg The steady state of this system is Ae10 z em Z a2 E g n 6 aAe1O Z a3 E 6 p 99 Taking a first order Taylor expansion ofthis system around the steady state values yields 2quot A1 ae10 z ez z z eza 113 ez Ae10 z aAe1O Z z z Ae10 z a2 a 113 2 am a2 2 z a2 com04 113 1 i gaA ae10 zz z 1 ozagz z Putting this together in matrix notation dzdt am az aZ amoz z z dzcdt 1 04a9 0 113 15 The determinant of the A matrix is negative as long as aaz lt am or which since 04 lt 1 is true as long as 502 lt am or gn6 lt 6p69 p n gt 1 99 The negative sign of the determinant means that the two eigenvalues have opposite signs a result that implies saddle path stability The characteristic equation of the A matrix is 82 am az 39yO where 39y E 1 00am a2 com04 9 The formula for the negative eigenvalue 52 is 252 am a2 ag a22 47 2 The solution for z is z Zgtllt 77bleslt 77b25375 for some constants 1 and 2 Since 51 gt 0 then 1 0 given saddle path stability Then need 2 20 z Thus letting 2 log C75 1 6 675 log 19 6 675 log 190 and since logyt log At alog C75 then log W 1 6 55 log 24 5m logy0 so 5 is the rate of convergence Plugging in the values as before lecture notes 2 chapter 3 with 9 1 get 5 0054

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