International macroe.. - Free

4.1. THE BARTER ECONOMY 109The foreign agent has the same utility function and faces the analogousproblem to maximizesubject to⎛∞XE t⎝j=0β j u(c ∗ xt+j ,c∗ yt+j ) ⎞⎠ , (4.9)c ∗ xt + q tc ∗ y t+ e t ωxt ∗ + e∗ t ω∗ y t= ωxt−1 ∗ (x t + e t )+ωyt−1 ∗ (q ty t + e ∗ t ). (4.10)The analogous set of Euler equations for the foreign individual arec ∗ yt : q tu 1 (c ∗ xt ,c∗ yt )=u 2(c ∗ xt ,c∗ yt ), (4.11)ωxt ∗ : e tu 1 (c ∗ xt ,c∗ yt )=βE t[u 1 (c ∗ xt+1 ,c∗ yt+1 )(x t+1 + e t+1 )], (4.12)ωyt ∗ : e ∗ t u 1 (c ∗ xt,c ∗ yt) =βE t [u 1 (c ∗ xt+1,c ∗ yt+1)(q t+1 y t+1 + e ∗ t+1)].(4.13)A set of four adding up constraints on outstanding equity shares andthe exhaustion of output in home and foreign consumption completethe speciÞcation of the barter modelω xt + ωxt ∗ = 1, (4.14)ω yt + ωyt ∗ = 1, (4.15)c xt + c ∗ xt = x t , (4.16)c yt + c ∗ yt = y t . (4.17)Digression on the social optimum. You can solve the model by grindingout the equilibrium, but the complete markets and competitive settingmakes available a ‘backdoor’ solution strategy of solving the problemconfronting a Þctitious social planner. The stochastic dynamic bartereconomy can conceptually be reformulated in terms of a static competitivegeneral equilibrium model—the properties of which are well known.The reformulation goes like this.We want to narrow the deÞnition of a ‘good’ so that it is deÞnedprecisely by its characteristics (whether it is an x−good or a y−good),the date of its delivery (t), and the state of the world when it is delivered(x t ,y t ). Suppose that there are only two possible values for x t (y t )in