MACROECONOMIC THEORY T J KEHOE ECON 87 SPRING 5 PROBLEM SET # Conder an overlappng generaon economy le ha n queon 5 on problem e n whch conumer lve for perod The uly funcon of he conumer born n perod, =,,, uc (, c, c, c ) = log c βlog c β log c β log c Th conumer endowed wh quane of labor (,,, ) (,,, ) = In addon, here are hree oher conumer, -, -, and Conumer - ha he uly funcon u ( c ) = log c and he endowmen =, Conumer - ha he uly funcon u ( c, c ) = log c β log c and he endowmen (, ) = (, ), Conumer ha he uly funcon u ( c, c, c ) = log c βlog c β log c and endowmen (,, ) = (,, ), The producon funcon f(, ) = A α α, where A = γ A, and capal deprecae a he rae δ per perod, δ There no fa money a) Defne a equenal mare equlbrum for h economy b) Defne a balanced growh pah for h economy Fnd he value of λ > uch ha = λ along he balanced growh pah c) Suppoe ha, afer T = 9 perod, he capal oc grow a he rae λ Tha, = λ, 9,, = Reduce he equlbrum condon n par a o a yem of 8
equaon n he 8 unnown 5 6 7 8, and 9 by conderng he mare clearng condon n perod,,,, 5, 6, 7, and 8 d) Suppoe ha β =, =, (,,, ) = (,, 5, ), A =, γ =, α =, δ = 5 Calculae he balanced growh pah n whch he avng equal o capal oc ha mu be accumulaed for nex perod, λ ˆ (There anoher balanced growh pah n whch rˆ δ = λ, bu o be n h balanced growh pah requre a nonzero amoun of fa money) e) Wre a compuer program o olve he yem of equaon n par c Te he program by verfyng ha, for he rgh choce of, he balanced growh pah an equlbrum Le be equal o 8 percen of her balanced growh pah value Calculae he oluon o he yem of equaon Chec wheher he equlbrum condon n perod 9,,, and are afed f) Now runcae he model a T = Repea par e and compare he reul Ung daa for he Uned Sae or ome oher counry, calbrae he parameer β,, (,,, ), A, γ, α, and δ a) Repea par d, e and f of queon b) Suppoe now ha he economy on a balanced growh pah n whch he populaon growh rae,, a calbraed level In perod conumer realze ha he populaon growh rae wll fall o Tha, (,,, ) = (,,, ) and (,,, ) = (,,, ), =,, Chooe and, >, and calculae he equlbrum, runcang a T = Wha are he predcon of he model ha you can compare he daa for an economy ha ha experenced a lowdown n populaon growh? Conder an economy n whch here are wo dae, =,, no uncerany a =, and uncerany over hree poble ae a = There are wo conumer, =,, wh uly funcon π log c π log c π log c and endowmen ( w, w, w ) Here π > he probably of even, conumpon of he ngle good a even by conumer, and c he w he endowmen of h good The conumer have no endowmen of good nor uly for conumpon a =
a) Defne an Arrow-Debreu equlbrum for h economy b) Defne a Pareo effcen allocaon for h economy c) Prove ha any equlbrum allocaon Pareo effcen d) Suppoe ha π = π = π = /, ( w, w, w ) = (,, ), and Calculae he Arrow-Debreu equlbrum of h economy ( w, w, w ) = (,, ) Conder an economy n whch conumer lve forever In every perod, =,,, one of wo random even occur, = or = A =, he nal ae and a aonary Marov proce gven by a marx wh elemen π = prob( = = ) govern he probably of fuure ae Le π ( ) be he nduced probably drbuon over ae Suppoe ha here are wo conumer, =,, each of whom ha he uly funcon ( ) β π() u( c) S Here S he e of all ae, < β <, ( ) he dae ha ae occur n, and he conumpon of he ngle good n ha ae by conumer Suppoe ha endowmen depend only on he lae even, w = w ( ) There no producon a) Defne an Arrow-Debreu equlbrum for h economy b) Defne a Pareo effcen allocaon for h economy Aumng ha u monooncally ncreang, prove ha an equlbrum allocaon Pareo effcen c c) Defne a equenal mare equlbrum for h economy Carefully ae and prove wo propoon ha relae Arrow-Debreu equlbra o equenal mare equlbra Mae explc any aumpon ha you mae on uly funcon, endowmen, and o on d) Tranlae your defnon of Arrow-Debreu equlbra and equenal mare equlbra o he or of double um noaon ued by Soey, Luca, and Preco e) Suppoe ha u ( c) = log c, w () = w () = 9, and w () = w () = Suppoe oo ha π = π = π, < π < Calculae he Arrow-Debreu equlbrum and he equenal mare equlbrum
5 Conder he opmal growh problem (6) log c = c c = a) Wre down he Euler condon and he ranveraly condon for h problem Calculae he eady ae value of c and b) Wre down he funconal equaon ha defne he value funcon for h problem Gue ha he value funcon ha he form a a log Calculae he value funcon and he polcy funcon Verfy ha he polcy funcon generae a pah for capal ha afe he Euler condon and ranveraly condon n par a c) Le capal ae value for he dcree grd (,, 6, 8, ) Mae he nal gue V ( ) for all, and perform he fr hree ep of he value funcon eraon = d) Perform he value funcon eraon unl V = V ( ) log( ) (6) ( ) ( ) ( ) 5 V V < Repor he value funcon and he polcy funcon ha you oban Compare hee reul wh wha you obaned n par b (Hn: you probably wan o ue a compuer) e) Repea par d for he grd of capal oc (5,,, 995,) Compare your anwer wh hoe of par b and d (Hn: you need o ue a compuer) f) Repea par e for he problem (6) log c = c 5 c = (Here here now no comparon wh an analycal anwer o be made)
6 Conder he dynamc programmng problem whoe value funcon afe he funconal equaon ( θ ) V(, θ) = log c (6) E V( ', ') c ' θ c, ' Here θ a random varable ha ae on he value θ = and θ = 6 a governed by he fr order, aonary Marov proce gven by he marx π π 8 π π = 7 a) Solve h dynamc programmng problem analycally b) Decrbe an economc envronmen for whch he oluon n par a an equlbrum allocaon Defne he equlbrum Calculae he equlbrum c) Le capal ae value for he dcree grd (,, 6, 8, ) Mae he orgnal gue V (, ) θ = for all and θ, and perform he fr hree ep of he value funcon eraon d) Perform he value funcon eraon unl ( ) V = E V (, θ) log( θ ) (6) (, θ') θ (, θ) (, θ) 5, V V < Repor he value funcon and he polcy funcon ha you oban Compare hee reul wh wha you obaned n par b (Hn: you probably wan o ue a compuer) e) Repea par d for he grd of capal oc (5,,, 995,) Compare your anwer wh hoe of par a and d (Hn: you need o ue a compuer) f) Repea par e for he problem ( θ ) V(, θ) = log c (6) E V( ', ') c ' 5 θ c, ' 5