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12. Consider a consumer who chooses consumption levels in two consecutive time periods to maximize utility. The consumer's utility function is U(c1,c2) : clcz, where

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12. Consider a consumer who chooses consumption levels in two consecutive time periods to maximize utility. The consumer's utility function is U(c1,c2) : clcz, where ct denotes consumption in period t, income in each period (3),) is exogenously given, and the consumer can borrow or save in period 1 at the exogenouslv given interest rate r. Consumption in period 2 (c2) will be equal to period 2 income (3/2} plus principal and interest from any first period saving. Thus, the consumer's intertemporal budget constraint is c2 = y2 + (1 + r)(y1 - c1). a. Show that the intertemporal budget constraint can be written in the form: g(cl,c'2) = k, by setting g(cl,cz) = C2 + (1 + r)c1 and k = 312 + (1 + r)y1. b. Write down the Lagrangian function for the consumer's constrained utility maximization problem, and differentiate it with respect to C1, C2, and A. c. Solve the first order conditions for the utility maximizing quantities of first and second period consumption (01*, CE), expressed as functions their exogenous determinants: y1, y;, and r. d. Use the Hessian matrix of the Lagrangian function's second derivatives to verify that utility is maximized at (Ci, cg). e. Derive def/6321 and aci/ar and determine their signs. f. Given that saying in this model is s E yl Cl, what does this model predict will happen to saving when first period income rises? When interest rates rise? Explain

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