International macroe.. - Free

4.1. THE BARTER ECONOMY 107capital inputs. Some people like to think of these Þrms as fruit trees.You can also normalize the number of Þrms in each country to 1. x tis the exogenous domestic output and y t is the exogenous foreign output.The evolution of output is given by x t = g t x t−1 at home and byy t = gt ∗ y t−1 abroad where g t and gt∗ are random gross rates of changethat evolve according to a stochastic process that is known by agents.Each Þrm issues one perfectly divisible share of common stock whichis traded in a competitive stock market. The Þrms pay out all of theiroutput as dividends to shareholders. Dividends form the sole source ofsupport for individuals. We will let x t be the numeraire good and q tbe the price of y t in terms of x t . e t is the ex-dividend market value ofthe domestic Þrm and e ∗ t is the ex-dividend market value of the foreignÞrm.The domestic agent consumes c xt units of the home good, c yt unitsof the foreign good and holds ω xt shares of the domestic Þrm and ω ytshares of the foreign Þrm. Similarly, the foreign agent consumes c ∗ xt,units of the home good, c ∗ yt units of the foreign good and holds ωxt∗shares of the domestic Þrm and ωyt ∗ shares of the foreign Þrm.The domestic agent brings into period t wealth valued atW t = ω xt−1 (x t + e t )+ω yt−1 (q t y t + e ∗ t ), (4.1)where x t + e t and q t y t + e ∗ t arethewith-dividendvalueofthehomeandforeign Þrms. The individual then allocates current wealth towards newshare purchases e t ω xt + e ∗ t ω yt , and consumption c xt + q t c ytW t = e t ω xt + e ∗ t ω y t+ c xt + q t c yt . (4.2)Equating (4.1) to (4.2) gives the consolidated budget constraintc xt + q t c yt + e t ω xt + e ∗ t ω y t= ω xt−1 (x t + e t )+ω yt−1 (q t y t + e ∗ t ). (4.3)Let u(c xt ,c yt ) be current period utility and 0 < β < 1 be the subjectivediscount factor. The domestic agent’s problem then is to choose sequencesof consumption and stock purchases, {c xt+j ,c yt+j, ω xt+j , ω yt+j } ∞ j=0,to maximize expected lifetime utility⎛∞XE t⎝j=0⎞β j u(c xt+j ,c yt+j ) ⎠ , (4.4)