Question

Problem 1 Consider the following two-period utility maximization problem. This utility function belongs to the CRRA (Constant Relative Risk Aversion) class of functions which can be thought of as generalized logarithmic functions. An agent lives for two periods and in both receives some positive income. subject to +6+1 4+1 = 3+1 + (1 + r) ar+1 where a > 0,13 € (0, 1) and r>-1. (a) Rewrite the budget constraints into a single lifetime budget constraint and set up the Lagrangian. (b) Obtain the first order conditions for cand ce+1 Express +1 as a function of (c) Using the lifetime budget constraint obtain the formulas for optimal and C+1 (d) Set a = 1 and verify that the formulas for optimale and ce+1 are identical to the ones we obtained in class for the utility function U = Inc + In Ce+1 (c) Return to expressions obtained in (c). Assume now that ye+1 -0. How does react when interest rater increases? How does it depend on o? How does o impact the relative strength of income and substitution effects? Problem 2 Consider the following two-period model with production: C. max U = In ce + S In ce+1 bet. + subject to a +be+1+ke+1 = ye 6+1 +1 + (1+r) 6+1+(1-5) ket1 91+1=2412 where he+1 denotes bonds, ke+1 denotes physical capital invested in the households firm and 8 € (0.1) stands for capital depreciation rate. (a) Write down the problem in the form of a Lagrangian (b) Find the optimal value of ke+1. (e) Find the optimal values of +1 and Br+1. (d) Calculate the derivative of optimal ke+1 with respect to r. Provide intuition for this result. (e) In equilibrium the rental rate for capital is equal to the marginal product of capital y 1/8ke41 Show that -. . ing+ This can be sily proven by ting the For = the CRRA function becomes logarithmic. U = Ina + L'Hopital's rule to compute the following limit: lime-

Answer 1

As per the HOMEWORKLIB RULESs, the answer to the first four sub-parts of Problem 1 are worked out in the Snapshots attached below.