Quesion 1: (55 poins) Macroeconomic Theory (ECON 8105) Larry Jones Fall 2016 Miderm Exam Sepember 27h, 2016 Consider an economy in which he represenaive consumer lives forever. There is a good in each period ha can be consumed or saved as capial as well as labor. The consumer's uiliy funcion is β (γ log c + (1 γ) log l ) where 0 < β < 1 and 0 < γ < 1. The consumer is endowed wih 1 uni of ime each period, some of which can be consumed as leisure l, and some of which is supplied as labor n. The consumer is also endowed wih k 0 > 0 unis of capial in period 0. Feasible allocaions saisfy c + k +1 Ak α n 1 α + (1 δ)k where A > 0, 0 < δ < 1, and 0 < α < 1. (a) (5 ps) Wrie down he social planner's problem of maximizing he represenaive consumer's uiliy subjec o feasibiliy condiions. Social planner's problem is max s.. β (γ log c + (1 γ) log l ) c + k +1 Ak α n 1 α c, k +1, l, n 0 l + n 1 k 0 > 0 given + (1 δ)k (b) (10 ps) Wrie down he Euler's equaion for consumpion, he inraemporal marginal raes of subsiuion beween consumpion and labor, and he ransversaliy condiion. Express each equaion in erms of allocaions (i.e. do no include Lagrange mulipliers or prices). Noe ha since uiliy is sricly increasing in consumpion and leisure, we will have n +l = 1, herefore we can subsiue l for 1 n. Le λ be he lagrange muliplier on he feasibiliy consrain. The rs order condiions for he social planner's problem are given by, 1
β γ c = λ (1) β 1 γ = λ A(1 α)k α n α 1 n (2) λ = λ +1 (Aαk+1 α 1 +1 + (1 δ)) (3) To nd he Euler Equaion (ineremporal rae of subsiuion), combine (1) and (3): c +1 c β γ c β +1 γ λ +1 c +1 = λ c +1 c = β λ λ +1 = β[aαk+1 α 1 n1 α +1 + (1 δ) To nd he inraemporal rae of subsiuion beween consumpion and labor, combine (1) and (2): β 1 γ 1 n β γ c = λ A(1 α)kα α n λ (1 γ)c γ(1 n ) = A(1 α)kα n α The ransversaliy condiion saes ha he presen value of he capial sock in he limi mus be equal o 0. There are several equivalen ways o express his condiion. γβ lim [Aαk α 1 c n 1 α lim λ k +1 = 0 lim γβ c k +1 = 0 + (1 δ)k = 0 (c) (5 ps) Dene an Arrow-Debreu equilibrium for his economy. An Arrow-Debreu Equilibrium is an allocaion for he HH: z H = {(c, l, n, k +1 )} an allocaion for he rm: z F = {(y f, kf, nf )} a sysem of prices: p = {(p, w, r )} such ha (HH) Given p, z H solves 2
max c,l,n,k β (γ log c + (1 γ) log l ) s.. p [c + k +1 (1 δ)k (Firm) Given p, z F solves (Mk) For all, [w n + r k l + n 1, c, k +1, l, n 0, max y f,kf,nf s.. k 0 > 0, given [ p y f w n f r k f y f Akα n 1 α, k f, nf, yf (Goods Marke) (Labor Marke) (Capial Marke) 0, c + k +1 = y f + (1 δ)k n = n f k = k f (d) (10 ps) Carefully explain how he objecs in par (c) can be used o consruc a sequenial markes equilibrium. Le (Z, p) form an Arrow Debreu equilibrium as dened in par (c), where Z = (z H, z f ), z H = {(c, l, n, k +1 )}, zf = {(y f, kf, nf )}, and p = {(p, w, r )}. Dene, for all ime, ĉ = c ˆl = l ˆn = n ˆk +1 = k +1 (HH) ŷ f = yf (Firms) ˆk f = kf ˆr b = ˆn f = nf ˆr k = r p ŵ = w p p p +1 1 (Prices) 3
Also, allow me o dene a sequence of bonds {ˆb +1 }, ˆb1 = ŵ 0ˆn 0 + ˆr k 0 ˆk 0 ĉ 0 ˆk 1 + (1 δ)ˆk 0 ˆb+1 = ŵ 0ˆn 0 + ˆr k 0 ˆk 0 ĉ 0 ˆk 1 + (1 δ)ˆk 0 + (1 + ˆr b )ˆb and a large deb limi B such ha for all, ˆb +1 B. Then (Ẑ, ˆp) form a Sequenial Markes equilibrium where Ẑ = (ẑh, ẑ f ), ẑ H = {(ĉ, ˆl, ˆn, ˆk +1, ˆb +1 )}, ẑf = {(ŷ f, ˆk f, ˆnf )}, and ˆp = {(ˆr, b ŵ, ˆr k )}. (e) (15 ps) Dene Pareo Opimaliy in his economy. Sae and prove he Firs Welfare Theorem. Our economy has only one agen, so he deniions and proofs can be modied slighly. An allocaion z is Pareo Opimal if i is feasible, and here exiss no oher feasible allocaion ẑ such ha ( β γ log ĉ + (1 γ) log ˆl ) > β (γ log c + (1 γ) log l ) Firs Welfare Theorem: Le E be a producion economy such ha i, k i 0 > 0 and U i is sricly increasing. If (z, p) is an Arrow-Debreu (compeiive) equilibrium (where z = { (z H,i ) i I, zf } ), hen z is Pareo opimal. Proof. Suppose, by conradicion, ha z is no Pareo opimal. By deniion of Arrow-Debreu equilibrium, he allocaion z is feasibly. Then i mus be ha here exiss anoher feasible allocaion ẑ such ha ( β γ log ĉ + (1 γ) log ˆl ) > Claim p [ĉ + ˆx > [w ˆn + r ˆk Suppose no, i.e. p [ĉ + ˆx [w ˆn + r ˆk + π β (γ log c + (1 γ) log l ) Then we have ha ẑ H saises budge consrain and yields higher uiliy, which conradics z H being par of he Arrow-Debreu equilibrium. Thus, we have p [ĉ + ˆx > [w ˆn + r ˆk Noe ha, since he rm is CRS, and k = k f, n = n f, p y = p y = p Ak α n 1 α = r k + w n p Ak α n 1 α = r k + w n 4
Subsiuing in his condiion, we ge, p [ĉ + ˆx > [ p Aˆk α ˆn 1 α Subiuing in he pro condiion and he rm's problem, we have Noe ha by conradicion hypohesis, ẑ is feasible. ĉ i + ˆx i Aˆk α ˆn 1 α Muliplying boh sides by p and summing across of ime, we have which is a conradicion. p [ĉ + ˆx [ p Aˆk α ˆn 1 α (f) (10 ps) Sae he Second Welfare Theorem and explain is implicaions for his economy. The saemen of he Second Welfare Theorem is in he second week reciaion noes. The uiliy funcion is sricly increasing and concave, so we know ha he Second Welfare Theorem applies o his economy. Furhermore, since here is only one agen in he economy, we know ha here are no feasible redisribuions of endowmens of iniial capial sock. Therefore, i mus be ha he Pareo Opimal soluion se o he social planners problem in par (a) is he same as he se of soluion allocaions o he AD equilibrium dened in par (c). 5
Quesion 2: (45 poins) Consider an innie horizon economy wih an inniely lived represenaive agen. There is a single commodiy ha can be consumed or invesed in capial. Capial fully depreciaes from one period o anoher. The agen supplies 1 uni of labor inelasically. The agen's preferences are of he form β u(c ) where 0 < β < 1. The aggregae resource consrain is c + k +1 = F (k, 1) where c denoes period aggregae consumpion, and k he aggregae capial sock. Assume ha boh u is diereniable, sricly concave, sricly increasing and saises he Inada condiions. Assume ha F is diereniable, sricly concave, sricly increasing, saises he Inada condiions in k, and has consan reurns o scale. (a) (5 ps) Wrie he social planners problem as a dynamic programming problem for capial sock. Le f(k) = F (K, 1). The social planners problem can be wrien recursively as v(k) = max k Γ(k) u(f(k) k ) + βv(k ) Γ(k) = {k R + 0 k f(k)} (b) (10 ps) Sae any properies of he value funcion ha you can infer from characerisics of u and F, and for each propery, explain which characerisics of u and F allow you o do so. Firs noe ha since f saises he Inada condiions, here is a maximum susainable level of capial, and since f and u and boh coninuous, boh funcions are bounded. This allows us o use heorems from Chaper 4.2 of SLP. Since u and f are sricly increasing, We know ha v is sricly increasing. Since u and F are sricly concave, We know ha v is sricly concave. Since u and f are coninuously diereniable and sricly concave, We know ha v is coninuously diereniable. (c) (5 ps) Wrie he rs order condiion and he envelope condiion. u (f(k) k ) = βv (k ) v (k) = u (f(k) k )f (k) (FOC) (ENV) (d) (10 ps) Dene a seady sae for his economy. Prove ha such a seady sae exiss and is unique. Call his seady sae level of capial, k. A seady sae has k = k. Thus, u (f(k) k) = βv (k) u (f(k) k) = βu (f(k) k)f (k) 1 β = f (k) 6
Since lim k 0 f (k) = and lim k f (k) = 0, and f (k) is a coninuous funcion, here mus exis some k such ha f (k ) = 1, by he inermediae value heorem. Moreover, since f is β sricly concave, f ( ) is sricly decreasing. Thus, k mus be unique. (e) (15 ps) Suppose ha policy funcion g k (k) is he soluion he recursive problem dened in par (a). Use he rs order condiion and he envelope condiion o show ha g k (k) is sricly increasing. Suppose for a conradicion ha here for some k 1 > k 2, g k (k 1 ) g k (k 2 ). Then, since f is sricly increasing, Since u is sricly concave, Since v is sricly concave, his implies ha f(k 1 ) g k (k 1 ) > f(k 2 ) g k (k 2 ) u (f(k 1 ) g k (k 1 )) < u (f(k 2 ) g k (k 2 )) βv (g k (k 1 )) < βv (g k (k 2 )) v (g k (k 1 )) < v (g k (k 2 )) g k (k 1 ) > g k (k 2 ) This is a conradicion. Thus, i mus be ha g k (k 1 ) > g k (k 2 ). 7