International macroe.. - Free

136 CHAPTER 4. THE LUCAS MODELProblems1. Risk sharing in the Lucas model [Cole-Obstfeld (1991)]. Let theperiod utility function be u(c x ,c y ) = θ ln c x +(1 − θ)lnc y for thehome agent and u(c ∗ x,c ∗ y)=θ ln c ∗ x +(1 − θ)lnc ∗ y for the foreign agent.Suppose That capital is internationally immobile. The home agentowns all of the x−endowment (φ x = 1), the foreign agent owns allof the y−endowment (φ ∗ y = 1). Show that in the equilibrium underportfolio autarchy, trade in goods alone is sufficient to achieve efficientrisk sharing.2. Consider now the single-good model. Let x t bethehomeendowmentand x ∗ t be the foreign endowment of the same good. The planner’sproblem is to maximizeφ ln c t +(1 − φ)lnc ∗ tsubject to c t + c ∗ t = x t + x ∗ t .Under zero capital mobility, the home agent’s problem is to maximizeln(c t )subjecttoc t = x t . The foreign agent maximizes ln(c ∗ t )subjectto c ∗ t = x∗ t . Show that asset trade is necessary in this case to achieveefficient risk sharing.(100)⇒(101)⇒(102)⇒3. Nontraded goods. Letx and y be traded as in the model of this chapter.In addition, let N be a nonstorable nontraded domestic goodgenerated by an exogenous endowment, and let N ∗ be a nonstorablenontraded foreign good also generated by exogenous endowment. Letthe domestic agent’s utility function be u(c xt ,c yt ,c N )=(C 1−γ )/(1−γ)where C = c θ 1x cθ 2y cθ 3N with θ 1 + θ 2 + θ 3 = 1. The foreign agent has thesame utility function. Show that trade in goods under zero capitalmobility does not achieve efficient risk sharing.4. Derive the exchange rate in the Lucas model under log utility, U(c xt ,c yt )=θ ln(c xt )+(1 − θ)ln(c yt ) and compare it with the solution under constantrelative risk aversion utility.5. Use the high and low growth states and the transition matrix givenin section 4.5 to solve for the price-dividend ratios for equities. Whatdoes the Lucas model have to say about the volatility of stock prices?How does the behavior of equity prices in the monetary economy differfrom the behavior of equity prices in the barter economy?