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4.1. THE BARTER ECONOMY 111subject to (4.16) and (4.17). The Euler equations for this problem areφu 1 (c xt ,c yt ) = (1− φ)u 1 (c ∗ xt,c ∗ yt), (4.19)φu 2 (c xt ,c yt ) = (1− φ)u 2 (c ∗ xt ,c∗ yt ). (4.20)(4.19) and (4.20) are the optimal or efficient risk-sharing conditions.Risk-sharing is efficient when consumption is allocated so that themarginal utility of the home individual is proportional, and thereforeperfectly correlated, to the marginal utility of the foreign individual.Because individuals enjoy consuming both goods and the utility functionis concave, it is optimal for the planner to split the available x andy between the home and foreign individuals according to the relativeimportance of the individuals to the planner.The weight φ can be interpreted as a measure of the size of the homecountry in the market version of the world economy. Since we assumedat the outset that agents have equal wealth, we will let both agents beequally important to the planner and set φ =1/2. Then the Paretooptimal allocation is to split the available output of x and y equallyc xt = c ∗ xt = x t2 , and c yt = c ∗ yt = y t2 .Having determined the optimal quantities, to get the market solutionwe look for the competitive equilibrium that supports this Pareto optimum.The market equilibrium. If agents owned only their own country’s Þrms,individuals would be exposed to idiosyncratic country-speciÞc riskthatthey would prefer to avoid. The risk facing the home agent is that thehome Þrm experiences a bad year with low output of x when the foreignÞrm experiences a good year with high output of y. One way to insureagainst this risk is to hold a diversiÞed portfolio of assets.AdiversiÞcation plan that perfectly insures against country-speciÞcrisk and which replicates the social optimum is for each agent to holdstock in half of each country’s output. 2 The stock portfolio that achieves2 Agents cannot insure against world-wide macroeconomic risk (simultaneouslylow x t and y t ).