ECON ENV MNGMNT
ECON ENV MNGMNT ESM 204
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Date Created: 10/22/15
Supplementary Note on the 0G Model Econ 204A Prof Bohn Here are some comments on Romer s exposition and on general OG dynamics Read Romer s Section 29 carefully as it lays out the individual problem In Romer s Section 210 the key equation for the dynamics is 259 or equivalently 267 LogutilityCobbDouglas is a special case The Speed of Convergence section examines convergence in this special case Romer then discusses the general case but without examining convergence This note examines convergence in general as in class but provides more specialized cases as example Slight change to notation Let me follow Romer and use wt for the wage per e iciency unit Let s denote perworker wage income by Wt wLAL wage per AL units of work The individual problem with general utility function Preferences C1t Bquot C2t1 Budget equation for workers C1 a W Budget equation for retirees CZHI 1 rt1 at For a graphical analysis combine the budget equations to obtain the Intertemporal Budget Constraint Wt Clt 39 C2t139 For the algebraic solution there are several approaches set up a Lagrangian with preferences subject to IBC solve IBC for C1 and insert into preferences then maximize with respect to C2 or solve the budget equations for consumption insert into preferences and maximize with respect to assets The latter yields uWt at u1 rHl at as the objective function The rst order condition is u W at 1 rHl u 1 rt1 a0 The solution is a function at aWtr1 of the exogenous current wage and next period s interest rate Its derivatives can be determined by taking total differentials or equivalently apply the Implicit Function Theorem uquotC1th dat Bu39drt11 rt1 uquotC2H1atdrH11 rt1 6151 0 da MquotC1i ldW tl u39d u ai 51 39MquotC2i1 l39drm MquotC1i 1ri1Z39MquotCZL1l Because u lt0 the denominator is positive also 0ltdat1th lt1 The sign of daHl drt1 is gt ambiguous because substitution positive and income effects negative con ict The properties of the savings rate saW follow from the properties of a Note that if utility is homothetic optimal consumption and asset are proportional to W so the savings rate depends only on r and not on W Aggregate dynamics with general production function The capital stock next period equals the savings of the periodt workers Kt1 Lt 39 aWtrtl Lt 39 Wt 39 SWtrtl The labor force next period is LHI Lt 1 n Productivity next period is AHI At 1 g Hence the capital labor ratio is AL LL HEAL SWtrtl SWt Anrtl Wt l kt1 lnlg Note that the productivity index does not drop out unless the savings rate does not depend on the wage That is a steady state in an economy with productivity growth requires homothetic utility To focus on the role of capital let s express the wage as function of capital Wt fktkt 39 f ktaLkt39 fkt7 where aLkWl l alltkt is the labor share in output To conclude the capitallabor ratio in period tl is linked to the periodt capitallabor ratio through ktl msaLkt39 fkt39 Atart139 aLkt39 fkt Note that the return rHl f39kH1 depends on next period s capital Hence equation is an implicit function for km and cannot be interpreted as a solution for km Let s consider some special cases as applicationsexamples a Homothetic power or log utility Then 3H1 srt1 does not depend on the wage so a ktl msohlquot aLkt39 fkt look simpler But it s still an implicit function h Logarithmic utility Then the savings rate is a constant so b kt1m3905L I39fk Here the interest rate drops out which means that the equation provides a solution for next period s capital stock Note that the dynamics are similar to the Solow model but here the young generation s savings rate is constant whereas Solow assumed a constant aggregate savings rate We know that f is increasing and concave but one cannot assert convergence or a positive unique steady state unless one imposes restrictions on the labor share For example there may be multiple steady states if 06Lkt is increasing in k c Homothetic power or log utility and CobbDouglas production Then 06Lkt is constant so a simpli es to c ktl ilirgquot 5rt139fkt This is again an implicit function but the dependence on current capital captured by the known production function d Logarithmic utility and CobbDouglas production Then both savings rate and labor share are constant so 1 ktl 1g39fkt Now next period s capital is directly proportional to current output The mapping has the simple concave shape shown in Romer s Figure 211 Convergence and Steady States One can characterize convergence and steady states by taking the total differential of the mapping either in general or in the special cases Monotone convergence requires dkt 1 dkt 2 0 Local stability requires IdkHl dkt ltl Uniqueness of a steady state is implied if dkHl dkt ltl holds everywhere along the 45degree line because then the mapping cannot cross the 45degree line more than once However even in the easiest logutility and CobbDouglas case local stability is dif cult to verify in equation d for example f is arbitrarily high at low kvalues For this reason it s worth noting that uniqueness also follows if dkm dan u km dki d 111k lt1 holds everywhere along the 45degree line Under this condition the mapping drawn on logarithmic axes cannot cross the 45degree line more than once Or equivalently the elasticity of next period s capital with respect to current capital must be less than one Examples Let s go from special to general One nds dkm 1L v case dk lnlg fkt k 4k ktaii yfw kf39k or L m 7L 06 kltl 61 dki Maw e Na K t The condition in logs is evidently more informative stability and convergence to a unique steady state follows immediately for all ktgt0 Alternatively one may compute the closedform solution for k and evaluate dk 1 dk at k one would nd that the derivative at k also equals the capital share See the endnote below on how to interpret the capital share Case c Here a total differential or appeal to the implicit function theorem is needed From kt srt1 fkt one obtains 1L lnlg a y dktl 1nL1g39Srf39drt1 539 f 39dkt and rt1f39kt1 implies drHl fquotdkt1 so H a a l 1f 1nL1gf39sr39dkt1mmlug S f dkt or k dim ark la k dkl 1 fquotlt1n li1ggtf39srl Note that f lt0 so the denominator is greater than 1 if srgt0 and less than 1 if srlt0 The numerator is bounded as in case d above Thus Sr 20 is suf cient for a unique positive steady state However problems with existence andor multiplicity of solutions may arise if sr is suf ciently negative because the denominator would then approach zero and might become negative dk v v Case b W m39 aLktf kt 05L ktfkt k dig k y y 39 k k or 21 dk1 aLkfk 39 O LU tV 0 L ktmkt K 0 withcf 39 The logarithmic condition reveals that stability problems may arise if the labor share is an increasing function of capital Workers who earn an increasing share of output may save enough that future capital responds more than oneforone to increases in current capital Case a Combining a variable factor share and an interestsensitive savings rate and again using drHl fquotdkt1 one obtains dktl aLf39 Sr 39 fudkHl aLS39 fi39dkt Sf 39 aLi39dkt k dk I 39 k k d Mf Maxltkrgt 1rL 1g Now the elasticity depends on a combination of interestelastic savings capital share and changes in the labor shareithe same elements as above General problem Finally yields a similar eXpression as in case a but with an additional term that re ects the impact of wages through the savings rate through nonzero sw 1 u v v 61km m390 Lf39 Sr 39 f 61km 0 Lf39 5w 39 dWm 0 LS39 f 39de Sf390 L 39dkt Wllt1ggt fosrfquotdkt1l06Lf SW39AHIS3906LS39fl39dktSf3906Ll39dCt kt 11km 1 WWWQ CM dkl 17fn 11L fir DNA0 IACL 1 S SW 1rL 1g Compared to case a swgt0 magni es the response of neXt period s capital stock whereas swlt0 has a stabilizing effect Arrow Debreu Equilibria Recursive Competitive Equilibria Marek Kapicka Econ 204b April 27 2009 Plan for today gt A note on asset pricing gt First and Second Welfare Theorems gt Recursive Competitive Equilibria 1 Arrow Debreu Equilibrium Characterization Asset pricing gt For each state zt there is one Arrow Debreu asset with price determined by U Ctzt t t t M2 3 HZ lzo U Cozo gt Advantage of the A D trading mechanism One can use no arbitrage argument to price any other asset Example risk free bond An asset that delivers 1 unit of consumption in period 1 regardless of the state Example riskless console An asset that pays 1 unit of consumption forever Example stock price An asset that pays dividends dtzt 2 Pareto Optimum gt The social planner solves max 2 tUctztHzt zo Cvkt02ter st ctzt kt1zt lt thkktzt 1 1 7 5ktzt 1 0 given First Welfare Theorem Theorem If c 5 y kd nd are competitive equilibrium allocations then they are Pareto optimal Proof Suppose that a R5 is a feasible allocation that yields higher expected utility Then P gt P otherwise it would be chosen by the households Because P gt P y kd nd was not a profit maximizing allocation for the firm a contradiction ll First Welfare Theorem General Result gt All we need for FWT to hold is local nonsatiation of preferences Second Welfare Theorem Theorem Let c k y k n be a Pareto optimal allocation Then there exist prices q W r such that q W r and c k y k n constitute a competitive equilibrium Proof sketch Find a candidate price system q W r Verify that the candidate price system together with the Pareto optimal allocations constitutes a competitive equilibrium Candidate Prices cm tzi z iiiw WM ZtFnikfquot 2 n dzti Second Welfare Theorem Proof sketch For these prices the first order conditions are satisfied for 6 k y k n Since F U are both strictly concave and differentiable and the Pareto optimal allocation satisfies the transversality condition the first order conditions are sufficient D Second Welfare Theorem General Result 1 Strict concavity and continuity of the utility function concavity of the production set 2 In infinite dimensional spaces Existence of an interior point in the production set gt Differentiability or boundedness of the production function is not required Recursive Competitive Equilibrium gt K economywide capital stock gt Prices of capital and labor can be expressed as a function of z and K only WKz anK1 rKz szK1 gt Assume that the household expects a law of motion for the economywide capital to be K HKz gt k household capital stock gt In equilibrium we will have k K but in a household problem they must allowed to be different Recursive Competitive Equilibrium Household and firm problem gt Household39s value function vk K 2 vk Kz Uc 94 KY 27rz z st ck g rKZ175kWKZ K HKz c 2 0 k 2 O gt Optimal policy functions Ck Kz and hk K 2 gt Firm Maximize profits by setting anK1 WKz szK1 rKz Recursive Competitive Equilibrium Definition Recursive Competitive Equilibrium is given by 1 Value function vk Kz and the optimal policy functions Ck Kz and hk Kz 2 Law of motion for the economywide capital HK z 3 Price functions WKz and rKz such that a Ck Kz hk Kz and vk Kz solve the household problem taking K as given b anK1 WKz and szK1 rKz c Aggregate and individual decision rules are consistent hK Kz HKz d Markets clear CK K z hK K z zFK 1 1 i 6K