1 Answers to Final Exam, ECN 200E, Spring

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1 1 Answers o Final Exam, ECN 200E, Spring A good answer would include he following elemens: The equiy premium puzzle demonsraed ha wih sandard (i.e ime separable and consan relaive risk aversion) preferences, he consumpion based capial asse pricing model is no consisen wih he differences in average reurn beween socks and bonds. The reason was ha he covariance beween agens marginal uiliy of consumpion and he reurn on equiy was oo low. Since he reurn on equiy is fairly volaile, he fac for he low covariance is ha consumpion is very smooh and, herefore, wih sandard preferences and moderae levels of risk aversion, marginal uiliy is relaively smooh. One can increase he volailiy of MU by increasing risk aversion - bu hen one runs ino he risk free rae puzzle. Increasing he risk aversion parameer causes he risk free rae o increase. This can be seen from he ineremporal equaion for bonds: (1 + r ) 1 = βe cg γ +1 (1) where I have assumed cg denoes one-period consumpion growh and, for convenience, I have assumed ha his is i.i.d. I is clear ha, since consumpion growh is on average greaer han one, increasing relaive risk aversion (i.e. γ) will resul in a greaer risk free rae. The reason hese puzzles are of ineres o macroeconomiss is ha i shows dramaically ha our models of ineremporal choice are missing somehing when i comes o capuring risk preferences. This is imporan since i cass doub on Lucas s exercise abou he cos of business cycles. He found ha agens would pay a very small amoun o perfecly insure heir consumpion agains he volailiy implied by business cycles. Lucas used per-capia consumpion which obviously ignores he differenial impac of business cycles bu he did use he same kinds of preferences examined in he original equiy premium puzzle paper by Mehra and Presco. Since ha paper demonsraed ha hese preferences severely undersae he way invesors perceive he risk associaed wih equiy, i is no surprising ha Lucas found business cycle risk (of consumpion) o be quie small. Somehing along hese lines would be sufficien. 2. The definiion of equilibrium need no be as complee (noe ha he consisency condiion beween individual and aggregae policy funcions is he assumpion of raional expecaions) bu i should have he funcions noed and make a disincion beween he individual and aggregae sae variables. Also, he social planner problem should be saed - he equilibrium behavior should be described. Here are he deails: 1

2 For he individual, he maximizaion problem is: ( v (z 1, ζ ) = max (c,h,z ) c 1 γ 1 γ + A (1 h )+βe v z, ζ +1 ) (2) subjec o : w h + z 1 (π + q )=c + q z where z denoes equiy holdings in period and π = ζ h α w h denoes profis. Noe he value funcion, v (z 1, ζ ), is a funcion of he individual s beginning of period equiy holdings and he curren echnology shock. Prices are also par of he sae vecor, bu since hese will be funcions of he aggregae sae (ζ ), I suppress hese in he sae vecor. The associaed necessary condiions are (using λ o denoe he Lagrange muliplier for he budge consrain): c : c γ λ =0 h : A + λ w =0 " # V z, ζ z : βe +1 λ q =0 z Or, combining erms and using he envelope heorem: Ac γ = w c γ q = βe c γ +1 (π +1 + q +1 ) (3) These have he sandard inerpreaions. The firms maximizaion problem is saic and is simply: max (ζ h α w h ) h which leads o he necessary condiion: αζ h α 1 = w (4) b. Definiion of a recursive compeiive equilibrium A recursive compeiive equilibrium can be defined by a se of funcions: a value funcion ha defines he household s problem (2), a wage funcion, w (ζ ), an equiy price funcion, q (ζ ), a se of decision rules for households, c (z 1, ζ ),h(z 1, ζ ), and a corresponding se of aggregae per capia decision rules, C (ζ ),H(ζ ). These funcions mus saisfy: a. The household s problem (as noed above for he value funcion.) b. The necessary condiion for households: i.e. q = q (ζ )ineq. (3)when evaluaed a marke clearing quaniies. c. The necessary condiion for firms: i.e. w = w (ζ ) in eq.(4). d. The consisency of individual and aggregae decisions: c (1, ζ )=C (ζ ),h(1, ζ )=H (ζ ). (ha is, marke clearing requires z = 1. 2

3 e. The aggregae resource consrain: C (ζ )=ζ [H (ζ )] α c. Social planner problem Since here is no capial, he social planner problem is simply o maximize uiliy each period given by: " # c 1 γ max c,h 1 γ + A (1 h ) The necessary condiion is: subjec o c = ζ h α Ac γ = αζ h α 1 (5) This is clearly he same as in he compeiive economy when he household s and firm s necessary condiions are combined. Noe ha equiy has no relevance for he social planner problem since i does no influence equilibrium allocaions. d. Characerizaion of equilibrium Using he necessary condiion, eq.(5) and he resource consrain implies: h = Γζ 1 γ 1 α(1 γ) where Γ = 1 α 1 α(1 γ). Given he resricions on (α, γ) hisesablishesha A labor is an increasing funcion of he echnology shock. The wage rae is given by eq.(4) so ha, in equilibrium: w = αγ (α 1) ζ γ 1 α(1 γ) so ha wages are also increasing in he echnology shock. Equiy prices are given by q = βe c γ +1 (π +1 + q +1 ) c γ In he numeraor, all variables are funcions of nex period s echnology shock. Given he assumpion of i.i.d. shocks, his implies he numeraor is a consan. Since consumpion is an increasing funcion of he echnology shock, hen equiy prices will be as well. 3. a. I was looking for a discussion abou he use of he Second Welfare Theorem so ha he compeiive equilibrium prices are cerainly presen in he behavior of RBC models. Hence, Summers criique is misplaced. b. Many sudens discussed he behavior of he labor marke in RBC models and he inconsisency wih he daa - his was fine bu did no answer he quesion. The ypical RBC model does no have srong propagaion mechanisms so he dynamic properies are primarily inheried from he exogenous 3

4 echnology shock. See he aricle by Cogley & Nason, he aricle by Sadler, and he discussion in Romer s ex. 4. This problem was clearly oo hard: nobody se up he budge consrain correcly in his problem. Here is wha he problem saed: In he asse marke, agens receive he reurns from capial (i.e. he revenue from he sale of oupu in las period s goods marke and he sale of undepreciaed capial), he reurns from bonds, money no spen in las period s goods marke, and he lump-sum moneary ransfer and use his o buy new capial, bonds and money. Nex agens visi he goods marke where money is used o finance consumpion (i.e. consumpion is subjec o he cash-in-advance consrain). This corresponds o he following budge and CIA consrains: 1 f (k 1 )+ k 1 (1 δ)+b 1 N 1 + b 1 R 1 +(M 1 1 c 1 )+T (6) = k + B + b + M M c (7) The dynamic programming problem is à U (c )+βv (s +1 )! P 1f (k 1) V (s )=max +λ + k 1 (1 δ)+ B 1N 1 + b 1 R 1 + (M 1 1 c 1 ) + T k B b ³ M M +γ c (8) where he sae vecor is s = (M 1,c 1,k 1,B 1,b 1 ). Noe ha consumpion las period affecs how much money is brough ino period hence ha is why i is a sae variable, along wih M 1. Taking derivaives and applying he envelope heorem yields he following firs order condiions: c : U 0 = γ + βλ (9) P k : λ = βλ +1 f 0 (k )+1 δ +1 (10) M : λ = γ + βλ +1 (11) +1 B : λ = βλ (12) b : λ = βλ +1 R (13) Firs combining eqs. (11) and (9) yields: λ = U 0 This is he answer o (b) - he reason is ha wealh of any kind can be convered ino money in he asse marke which precedes he goods marke. N 4

5 Using his in eq. yields: (10) and solving for seady-sae (imposing +1 = 1 1+µ ) β 1 = f k 0 1+µ +1 δ This is he answer o par (c) - money is no superneural in his economy since he inflaion ax affecsherevenuesfromcapial. (Whyishisdifferen from Sockman s model?) Finally, manipulaing he las wo necessary condiions yields he Fisher relaionship: R = N +1 5