Economics 246 — Fall 2005International MacroeconomicsProblem Set 1September 27, 2005Due: Thursday, October 13Instructor: Marc-Andreas MuendlerE-mail: [email protected] Productivity Shocks with Initially UnbalancedCurrent AccountsThere are two periods and two countries. Home produces with Y = AF (K)and Foreign with Y∗= A∗F (K∗), where A and A∗are productivity parameters.Both countries’ representative agents have the same period utility so that U1=u(C1) + βu(C2) and U∗1= u(C∗1) + βu(C∗2). Assume period utility u(·) to beisoelastic. International financial market clearing S + S∗= I + I∗(or CA1=−CA∗1) determines the world interest rate r .1. Use the intertemporal optimality conditions for production to show thatan anticipated pro ductivity shock dA2/A2to production in Home changesequilibrium investment and the interest rate in the following wayrdA2A2+µ∂I1∂r¶−1dI1− dr = 0.Derive the according equation for Foreign and dA∗2/A∗2.2. Use the intertemporal optimality conditions for consumption, along withyour results in 1, to show that an anticipated productivity shock dA2/A2to production in Home changes the equilibrium current account level andthe interest rate in the following wayDdA2A2− M dr + V dCA1= 0,for some functions V, D, and M . Derive the according equation for Foreignand dA∗2/A∗2(with V∗, D∗, M∗and using CA1= −CA∗1). Derive the signsof V, V∗, D and D∗.3. Show that anticipated productivity shocks to Home and Foreign have thefollowing effect on the equilibrium interest rate and the Home currentaccount:dr =1MV+M∗V∗µDVdA2A2+D∗V∗dA∗2A∗2¶dCA1= −1VM+V∗M∗µDMdA2A2−D∗M∗dA∗2A∗2¶.14. Suppose Home runs a current account surplus CA1> 0 during period 1.How does an anticipated increase in Home productivity A2during period2 affect the current accounts in period 1? Is the effect unambiguous? Whyor why not?5. Suppose again Home runs a current account surplus CA1> 0 duringperiod 1. How does an anticipated increase in Foreign productivity A∗2during period 2 affect the current accounts in period 1? Is the effectunambiguous? Why or why not?6. Consider anticipated and equal prop ortional increases in Home and For-eign productivity dA2/A2= dA∗2/A∗2. How does this change affect world-wide investment I1+ I∗1during period 1? Does the answer depend on theelasticity of intertemporal substitution? Is the effect unambiguous? Whyor why not?2 Exponential Period UtilityThere are two periods. A country’s representative household has the exponentialperiod utility functionu(C) = −γ exp(−C/γ)with γ ∈ (0, ∞) and maximizes lifetime utility U1= u(C1) + βu(C2) subject toC1+ RC2= Y1+ RY2≡ W,where R ≡ 1/(1 + r) is the price of tomorrow’s consumption in terms of today’sconsumption and W is initial wealth. The value of W depends on R.1. Derive the Euler equation and solve it for C2as a function of C1, R andβ.2. What is the optimal level of C1considering W , R and β as given?3. Differentiate this consumption function of C1with respect to R (differen-tiate W with respect to R too) and show thatdC1dR= −C11 + R+Y21 + R+γ1 + R(1 − ln(β/R))4. Derive the intertemporal elasticity of substitution of the exponential pe-riod utility (−u0(C)/Cu00(C)).5. Use this result to show that the derivative dC1/dR in part 3 can be ex-pressed asdC1dR=σ(C2) C21 + R−C21 + R+Y21 + R.Interpret the three additive terms in this derivative.23 Stochastic Current Account ModelThere are infinitely many periods. A country’s representative household has thelinear-quadratic period utility functionu(C) = C −a02C2with a0∈ (0, ∞) and maximizes lifetime utilityUt= Et"∞Xs=tβs−tu (Cs)#subject toCAs= Bs+1− Bs= rBs+˜Zs− Cs∀s ≥ twhere R ≡ 1/(1 + r) = β and˜Zs(≡˜Ys−˜Gs−˜Is) is random net output.1. Derive the stochastic Euler equations and show that CtsatisfiesCt= rRÃ(1 + r)Bt+∞Xs=tRs−tEt[˜Zs]!2. Show that then CAt≡ Bt+1− Bt=˜Zt− Et[ˆ˜Zt], where the hat denotesthe permanent level of the variable. The permanent levelˆX of a randomvariable˜X is defined asP∞s=tRs−tˆ˜X ≡ EthP∞s=tRs−t˜Xsi.3. Define ∆˜Zs≡˜Zs−˜Zs−1and show that the current account can be rewrit-ten as CAt= −P∞s=t+1Rs−tEth∆˜Zsi.4. Use ∆˜Zt≡˜Zt−˜Zt−1to show that the quantityCAt− (1 + r)CAt−1− ∆˜Ztis stochastically independent of CAsand ∆˜Zsfor all s < t. How is thisfinding related to Hall’s (1978) result that consumption follows a randomwalk?4 Current Account and Terms of TradeIn a small open economy, the representative individual maximizes the lifetimeutility functionUt=∞Xs=tβs−t¡XγsM1−γs¢1−1/σ− 11 − 1/σ,where X is consumption of an exported good and M consumption of an importedgood. The country completely specializes in production of the export go od. The3endowment of this good is constant at Y . The representative individual facesthe fixed world interest rate r = (1−β)/β in terms of the real consumption indexC = XγM1−γ(so a loan of 1 real consumption unit today returns 1 + r realconsumption units tomorrow). There is no investment or government spending.1. Let p bet the price of the exp ort goods in terms of the import good.So, a rise in p is an improvement in the terms of trade. Show that theconsumption-based price index P in terms of imports isP = pγ/γγ(1 − γ)1−γ.2. Show that the home country’s current account identity isBt+1− Bt= rBt+pt(Y − Xt)Pt−MtPt.What is the corresponding intertemporal budget constraint for the repre-sentative consumer?3. Derive the first-order conditions of the consumer’s problem. (Hint: Re-formulate the utility function and budget constraint in terms of real con-sumption C.) What are the optimal paths for X and M?4. Suppose initial expectations are that p remains constant over time. Thereis an unexpected temporary fall in the terms of trade from p to p0< p.What is the effect on the current account CAt= Bt+1− Btfrom part 2?5. Now suppose foreign net wealth B is indexed to the import good M ratherthan to real consumption. Accordingly, let r denote the own-rate of inter-est in imports but assume again that r = (1−β)/β. How does a temporarydrop in the terms of trade from p to p0< p affect the current account now?How do you explain differences, if any, to part