Question

1. CRRA Utility Function: Constant relative risk aversion, or CRRA, utility function has been extensively used in macroeconomic analysis to represent consumer behavior. It takes the following general form u(x)- where σ is known as the curvature parameter. For the remainder of this question assume that σ>0. Assume that a representative household in a one-period model has the following preferences over consumption and leisure where l is leisure. The budget constraint is (in nominal terms) Pc nominal wage and n +1 1 (a) Plot the indifference curve for the utility level-24 ) Wn, where W is the 20.51-σ-1 for σ 0.5 and σ 20n the same graph. Set the domain of l E [0.4,0.6]. You don't need to upload the excel file, but make sure to label your graph carefully (b) Derive the optimality condition for utility maximization and solve for optimal leisure and consumption in terms of the real wage. For which values of σ does the substitution effect dominate the income effect? For which values does the income effect dominate the substitution effect?] (c) Note that the utility function above cannot be evaluated when σ = 1 . However, there exists a well-defined function in the limit when σ approaches to one. Derive that limiting function. What happens to labor supply curve when the utility function takes this form in the limit? (Hint: Use L'Hôpital's Rule.)

Answer 1

u(c) = c^1 - sigma - 1/1 - sigma + l^1 - sigma - 1/1 - sigma n + 1 = 1 P_c = W_n Budget constraint u^~ = 2(0.5^1 - sigma - 1/1 - sigma) For sigma = 0.5, u^~ = -1.17 sigma = 2, u^~ = -2 For sigma = 0.5, u(c) = c^0.5 - 1/0.5 + l^0.5 - 1/0.5 sigma = 2, u(c) = c^-1 - 1/-1 + l^-1 - 1/-1 We need graph for 1/0.5 (squareroot c + squareroot l - 2) = -1.17 squareroot c + squareroot l = 1.415 equation (i) -(c^-1 + l^-1 - 2) = -2 1/c + 1/l = 4 equation (ii) \r\n (b) Maximize c^1- sigma - 1/1 - sigma + l^1 - sigma - 1/1 - sigma Subject to n + l = 1 P_c . c = W_n n L = c^1 - sigma - 1/1 - sigma + l^1 - sigma - 1/1 - sigma + lambda [P_c . c - w_n (1 - l)] partial differential L/partial differential c = (1 - sigma) c^-sigma/(1 - sigma) + lambda P_c = 0 equation (iii)