PROBLEM SET 7 GENERAL EQUILIBRIUM

PROBLEM SET 7 GENERAL EQUILIBRIUM Queston a Defnton: An Arrow-Debreu Compettve Equlbrum s a vector of prces {p t } and allocatons {c t, c 2 t } whch satsfes ( Gven {p t }, c t maxmzes βt ln c t subject to p tc t p twt and c t 0 for all =, 2 (2 Market Clearng: c t + c 2 t = wt + wt 2 b Defnton: An allocaton {c t, c 2 t } s Pareto Effcent f t s feasble and f there s no other feasble allocaton { c t, c 2 t } such that ( u( c t u(c t for both =, 2; and (2 u( c t > u(c t for at least one =, 2 The followng Socal Planner maxmzaton problem yelds the Pareto Effcent Allocatons: max α {c t,c2 t } β t ln c t + ( α β t ln c 2 t, subject to c t + c 2 t wt + wt 2 = w and c t 0, c 2 t 0 The assocated Lagrangan s L = α β t ln c t + ( α β t ln c 2 t λ t [c t + c 2 t w] + γt c t + γt 2 c 2 t, from whch FONCs lead to the allocatons as functons of the weght α: c t (α = c (α = α w and c 2 t (α = c 2 (α = ( α w Date: Nov 3

PROBLEM SET 7 2 c Defnton: An Arrow-Debreu Equlbrum wth transfers s a vector of prces {p t }, allocatons {c t, c 2 t } and transfers (τ, τ 2 whch satsfes ( Gven {p t } and (τ, τ 2, c t maxmzes βt ln c t subject to p tc t p twt + τ and c t 0 for all =, 2 (2 Market Clearng: c t + c 2 t = w t + w 2 t From the household maxmzaton problem we have β t c o c t = p t p 0 Snce consumpton s constant and choosng p 0 = (e p 0 s the numerare, we have p t = β t Usng the allocatons and prces we have the transfers as functons of α: τ (α = α w β β t wt and τ 2 (α = ( α w β Note that w 2 t = w w t whch mples that τ (α + τ 2 (α = 0 β t wt 2 d We need to solve transfers for c t = c 2 t = 3 (note that ths s the same as settng equal weghts to each household type: τ (/2 = 3 β 5β 2t β 2t+ = τ 2 (/2 = 3 β β 2t 2 + β 5β 2t+ = 2 + β e To apply Negesh-Mantel algorthm and fnd the Arrow-Debreu Compettve Equlbrum we set transfers equal to zero: τ (α = α w β β t wt = 0 Solvng for α we have α = β + 5 6( + β Substtutng α nto the allocatons of consumpton c = β + 5 + β and c2 = 5β + + β

PROBLEM SET 7 3 f Defnton: A sequental compettve equlbrum s an allocaton {c t, c 2 t, b t+, b2 t+ } and prces {r t } such that ( Gven {r t }, c t maxmzes βt ln c t subject to c t + b t+ w t + ( + r t b t, c t 0, b 0 s gven and No Ponz Games Condton for all =, 2 (2 Market Clearng: c t + c 2 t = wt + wt 2 and b t + b 2 t = 0 The FONCs for the household s maxmzaton problem yeld the Euler Equaton: c t+ c t = β( + r t+ Note that ths condton holds for both households types such that the growth rate of consumpton for both types s the same Usng ths fact together wth the market clearng condtons we have that the growth rate of consumpton s zero Solvng the sequence of budget constrant recursvely and usng the prces (+r t+ = /β and the No Ponz games condtons, we can wrte the ntertemporal budget constrant as β t c t = β t wt Usng the fluctuatng ncome and the fact that consumpton s constant we fnd the same allocatons as n tem e: c = β + 5 + β and c2 = 5β + + β Fnally we need to solve for bonds Usng the budget constrants we gave whle b odd = b2 odd = 0 b even = 4β + β and b2 even = 4β + β, Queston 2 a Commodty Space: L = {R 2 : s the Eucldean norm} Consumpton Set: X = {x L : x 0, 0 x 2 and l such that l + x 2 = } Producton Set: Y = {y L : y 0, y 2 0, y 2 yγ γ, 0 } b The Socal Planner s problem s max ln c + φ ln( n c,n

PROBLEM SET 7 4 subject to c n γ /γ, c 0, 0 n c Note that, snce the utlty functon s contnuous and strctly concave, and the constrant set s convex, the socal planner s maxmzaton problem has a unque soluton Snce an allocaton s Pareto Effcent f and only f t solves the socal planner s maxmzaton problem, there s a unque Pareto Effcent allocaton d The Household s problem: subject to max ln c + φ ln( n c,n c wn + d, c 0, 0 n, yelds to the followng relatonshp between consumpton and hours worked c = ( nw φ From frms maxmzaton problem we have w = N γ and d = γ γ N γ Usng the market clearng condtons and the budget constrant we fnd the equlbrum allocatons and prces: c = γ ( γ γ, n = γ γ + φ γ + φ, and w = ( γ + φ γ e The socal planner allocaton would be the one found n the prevous tem, however, snce the producton set s not convex anymore (due to ncreasng returns to scale the second welfare theorem does not hold Then we cannot ensure there exsts a vector of prces that wll lead to the same allocaton as the socal planner s allocaton In fact, for any postve prce vector, the optmal allocaton for the frm s to produce nfnty γ a The household problem s Queston 3 ( max {c t,c 2t,b t+ } β t u(c t, c 2t

PROBLEM SET 7 5 ( c t + p 2t c 2t + b t+ w t + ( + t b t ( c t 0, c t 0 ( Gven b 0 (v No Ponz Game Note that solvng the budget constrant recursvely we have the followng ntertemporal budget constrant: ( t (c t + p 2t c + 2t = s s= ( t wt + s The Frms problem s to choose x t to maxmze p 2t Ax t x t s= p t (c t + p 2t c 2t = p t wt Defnton: A Date-0 compettve equlbrum s a sequence of prces {p t, p 2t }, household allocatons {c t, c 2t } and frms allocatons {x t} such that ( Gven prces, household allocatons maxmze the followng problem: max {c t,c 2t } β t u(c t, c 2t ( p t (c t + p 2t c 2t = ( c t 0, c t 0 p t wt For all = A, B (2 Gven prces, frms allocatons maxmze profts (3 Market Clearng: (a Good : c A t + cb t + x t = w A t + wb 2t (b Good 2: c A 2t + cb 2t = Ax t b The Socal Planner s problem s: max α {c A t,ca 2t,cB t,cb 2t } β t u(c A t, c A 2t + ( α β t u(c B t, c B 2t

PROBLEM SET 7 6 ( ( ( c A t + c B t + x t = w A t + w B 2t c A 2t + c B 2t = Ax t c t 0, c t 0 = A, B c The Lagrangan from the Socal Planner s problem s L = α β t [ln c A t + φ ln c A 2t]+( α β t [ln c B t + φ ln c B 2t] λ t [c A t + c B t + (c A 2t + c B 2t/A ]+ [µ t c A t + γ t c A 2t + η t c B t] Gven the log utlty, we can get rd of non-negatvty lagrange multplers Usng the frst order condtons we have the followng allocatons: c t = α + φ ; ca 2t = Aαφ + φ c B t = α + φ ; A( αφ cb 2t = + φ We have that n equlbrum the rato of margnal utlty between two dfferent goods must be equal to the rato of ther prces Then, β t u (c A t, ca 2t u (c A 0, ca 20 = p t = β t and u 2(c A t, ca 2t u (c A t, ca 2t = p 2t = /A, where we used the fact that consumpton s constant over tme We then compute the transfers as a functon of weghts: τ(α A = p t (c A t + p 2t c A 2t wt A = ( α β t + φ + αφ + φ wa t τ(α A = α β β 2, and solve for α when transfers equal zero The result s α = + β a The household problem s Queston 4

PROBLEM SET 7 7 (2 max {c t,c 2t,b t+ } ( β t [ln c t + φ c 2t] c t + p 2t c 2t + b t+ w t + ( + t b t ( c t 0, c t 0 ( Gven b 0 (v No Ponz Game Note that solvng the budget constrant recursvely we have the followng ntertemporal budget constrant: ( t s= + s (c t + p 2t c 2t = ( t s= + s w t The Frms problem s to choose x t to maxmze p 2t Λx t x t p t (c t + p 2t c 2t = p t wt Defnton: A Date-0 compettve equlbrum s a sequence of prces {p t, p 2t }, household allocatons {c t, c 2t } and frms allocatons {x t} such that ( Gven prces, household allocatons maxmze the followng problem: max {c t,c 2t } β t [ln c t + φ c 2t] ( p t (c t + p 2t c 2t = ( c t 0, c t 0 p t wt For all = A, B (2 Gven prces, frms allocatons maxmze profts (3 Market Clearng: (a Good : c A t + cb t + x t = w A t + wb 2t (b Good 2: c A 2t + cb 2t = Λx t b The Socal Planner s problem s:

PROBLEM SET 7 8 max α {c A t,ca 2t,cB t } β t [ln c A t + φ c A 2t] + ( α β t ln c B t ( ( ( c A t + c B t + x t = w A t + w B 2t c A 2t = Λx t c t 0, c t 0 = A, B c The Lagrangan from the Socal Planner s problem s L = α β t [ln c A t + φ c A 2t] + ( α β t ln c B t λ t [c A t + c B t + c A 2t/Λ ]+ [µ t c A t + γ t c A 2t + η t c B t] Gven the log utlty for good and that parameters are such that c 2t > 0, we can get rd of non-negatvty lagrange multplers Usng the frst order condtons we have the followng allocatons: c t = Λφ A c A 2t = Λ αφ A c B t = α αλφ A We have that n equlbrum the rato of margnal utlty between two dfferent goods must be equal to the rato of ther prces Then, p t = β t and p 2t = /Λ, when we use the fact that consumpton s constant over tme We then compute the transfers as a functon of weghts: τ(α A = p t (c A t + p 2t c A 2t wt A = β t ( ( + Λφ A α αλφ A τ(α A = ( + ΛφA α ( βαλφ A β 2, w A t

and solve for α when transfers equal zero The result s α = PROBLEM SET 7 9 + β + β( + Λφ A