International macroe.. - Free

110 CHAPTER 4. THE LUCAS MODELeach period–a high value x h (y h )andalowvaluex`(y`). Then thereare 4 possible states of the world (x h ,y h ), (x h ,y`), (x`,y h ), and (x`,y`).‘Good 1’ is x delivered at t = 0 in state 1. ‘Good 2’ is x deliveredat t = 0 in state 2, ‘good 8’ is y delivered at t =1instate4,andso on. In this way, all possible future outcomes are completely spelledout. The reformulation of what constitutes a good corresponds to acomplete system of forward markets. Instead of waiting for nature toreveal itself over time, we can have people meet and contract for allfuture trades today (Domestic agents agree to sell so many units of xto foreign agents at t = 2 if state 3 occurs in exchange for q 2 units of y,and so on.) After trades in future contingencies have been contracted,we allow time to evolve. People in the economy simply fulÞll theircontractual obligations and make no further decisions. The point isthat the dynamic economy has been reformulated as a static generalequilibrium model.Since the solution to the social planner’s problem is a Pareto optimalallocation and you know by the fundamental theorems of welfareeconomics that the Pareto Optimum supports a competitive equilibrium,it follows that the solution to the planner’s problem will alsodescribe the equilibrium for the market economy. 1Wewillletthesocialplannerattachaweightofφ to the homeindividual and 1 − φ to the foreign individual. The planner’s problemis to allocate the x and y endowments optimally between the domesticand foreign individuals each period by maximizingE t ∞ Xj=0β j h φu(c xt+j ,c yt+j )+(1− φ)u(c ∗ xt+j ,c∗ yt+j )i , (4.18)subject to the resource constraints (4.16) and (4.17). Since the goodsare not storable, the planner’s problem reduces to the timeless problemof maximizingφu(c xt ,c yt )+(1− φ)u(c ∗ xt,c ∗ yt),1 Under certain regularity conditions that are satisÞed in the relatively simpleenvironments considered here, the results from welfare economics that we need are,i) A competitive equilibrium yields a Pareto Optimum, and ii) Any Pareto Optimumcan be replicated by a competitive equilibrium.