Economcs 8105 Macroeconomc Theory Rectaton 1 Outlne: Conor Ryan September 6th, 2016 Adapted From Anh Thu (Monca) Tran Xuan s Notes Last Updated September 20th, 2016 Dynamc Economc Envronment Arrow-Debreu Equlbrum Sequental Markets Equlbrum Characterzng the Arrow-Debreu Equlbrum 1 Envronment wth Endowments Defnton 1.1. A pure exchange economy s a set of commodtes, {1,...l}, and a set consumers, I = {1,..., n}. Each consumer I has consumpton set X (typcally R l + n a statc envronment), ntal endowment e X, and utlty U : X R. We can express the economc envronment as E = {(e, U ) I }. In ths rectaton, we wll be consderng economes wth the followng characterstcs: Pure exchange economy wth one commodty Dscrete tme t = 0, 1, 2,... Infntely lved consumers, ndexed by I = {1, 2} Each consumer has allocaton c = (c 0, c 1, c 2,...), c t R + Utlty functon: U (c 0, c 1, c 2,...) = βt u (c t), where 0 < β < 1 1
Endowment: e = (e 0, e 1,...) 2 Arrow-Debreu Equlbrum Market Structure: The consumers trade the sngle commodty. Future markets are open n perod 0, durng whch consumers trade contngent clams for all perods. No more tradng occurs. Defnton 2.1. In ths economy, an Arrow-Debreu Equlbrum s an allocaton for HHs: z H, = {c t}, {1, 2} a system of prces: p = {p t } such that (HH) Gven p, {1, 2}, z H, solves (Mkt) For all t, {c t } β t u (c t) p t c t p t e t (λ ) c t 0 (γ t) (Goods Market Clears) I c t = I e t 3 Sequental Markets Equlbrum Market Structure: The agents trade the commodty and one perod bonds. Markets open at the begnnng of each perod, durng whch consumers trade goods and bonds for that perod only. Consumers are constraned by a non-bndng debt lmt. 2
Defnton 3.1. In ths economy, a Sequental Market Equlbrum s an allocaton for HHs: z H, = {(c t, b t)}, I a system of prces: p = {r t } such that (HH) Gven p, I, z H, solves {(c t,b t )} β t u (c t) c 0 + b 1 e 0 c t + b t+1 e t + (1 + r t )b t (µ t) c t 0 (γ t) b t B (Mkt) For all t, (Goods Market Clears) I c t = I e t (Bonds Market Clears) I b t = 0 4 Characterzng the Arrow-Debreu Equlbrum Assumpton 4.1. For all, u s dfferentable. Under ths assumpton, we can wrte the Lagrangan and the Kuhn-Tucker frst order condtons: L = ( β t u (c t) + λ p t e t ) p t c t + γ t c t FOCs: 3
c t] β t u (c t) λ p t + γt = 0 λ ] p t e t p t c t 0 λ 0 ( ) λ p t e t p t c t = 0 γt] c t 0 γt 0 γtc t = 0 Ths s equvalent to c t] β t u (c t) λ p t 0 (= 0 f c t > 0) (1) λ ] p t e t p t c t 0 (= 0 f λ t > 0) (2) Assumpton 4.2. For all, u s strctly ncreasng. Proposton 4.1. Under assumpton 4.2, the budget constrant s bndng. Assumpton 4.3. For all, u satsfes Inada condtons: lm c u (c) = 0 and lm c 0 u (c) =. Proposton 4.2. If lm c 0 u (c) (A4.3), then n equlbrum, c > 0. Thus, under assumptons 4.1 through 4.3, both (1) and (2) bnd wth equalty. Note that proposton 4.2 and the Kuhn-Tucker condtons mply that γ t = 0, t. We can rewrte the FOCs as c t] β t u (c t) = λ p t (1 ) λ ] p t c t = p t e t (2 ) From (1 ) we have the Euler s Equaton, or ntertemporal subttuton of consumpton: u (c t) u (c t+1) = β p t p t+1 4
Queston. Are the FOCs necessary condtons for the mzaton problem? A: Yes. Because the contrants are lnear n c, they satsfy the constrant qualfcaton of Kuhn-Tucker Theorem. Queston. Under what addtonal condtons are the FOCs suffcent for the mzaton problem? A: Utlty s concave. Note that the constrants are lnear, and therefore the constraned set s convex. Also, under assumpton 4.3 the constraned set s open. You can verfy that all the condtons are met to apply Kuhn-Tucker under convexty. (See Sundaram for more detals on the Kuhn-Tucker Theorem.) 5 Envronment wth Producton Dscrete tme t = 0, 1, 2,... Producton economy wth one commodty HHs: Infntely lved n consumers, ndexed by I = {1,..., n} Utlty functon: U ({(c t, l t)} ) = βt u (c t, l t) Consumers nvest x t Consumers have captal stock k t whch deprecates at a rate δ Law of Moton of Captal: k t+1 x t + (1 δ)k t Consumers dvde tme between lesure, l t, and labor, n t Endownment: 1 unt of tme each perod, ntal captal k 0 HHs rent out captal and labor servces to frms, recevng captal and labor ncome. Consumers own a share of frm profts θ such that θ 0, I θ = 1 Frms: only 1 sector producng goods that can ether be consumed or nvested One representatve frm. Fnal good s produced by: y f t = F (k f t, n f t ) Typcal propertes of F are ncreasng, concave, and homogeneous of degree one (constant returns to scale). 5
Defnton 5.1. An Arrow-Debreu Equlbrum s an allocaton for HHs: I, z H, = {(c t, lt, n t, kt, x t)} an allocaton for the frm: z F = {(y f t, k f t, n f t )} a system of prces: p = {(p t, w t, r t )} such that (HH) Gven p, I, z H, solves c t,l t,n t,k t,x t β t u (c t, lt) ] p t c t + x t wt n t + r t kt] + π k t+1 x t + (1 δ)k t, t l t + n t 1, t c t, k t+1, l t, n t 0, t k 0 > 0, gven (Frm) Gven p, z F solves {(y f t,kf t,nf t )} y f t k f t, n f t, y f t ] p t y f t w t n f t r t k f t F (k f t, n f t ), t 0, t (Mkt) For all t, (Goods Market) I c t + x t] = y f t F (k f t, n f t ) (Labor Market) I n t = n f t (Captal Market) I k t = k f t (Profts), π = θ ] p t y f t w t n f t r t k f t 6