Consider the one-period macro model from class. The representative consumer maximizes her utility by choosing consumption and leisure subject to a budget and time constraint, where MRSl,c = w. The representative firm maximizes profits by choosing labor up to the point where MPN = w.

(a) Suppose that the representative consumer’s utility function takes the following form: U(C, l) = 1/4 ln C + 3/4 ln l, which is subject to the usual budget constraint. Suppose the following parameter values: w = 9, h = 30, π = 10, T = 100. Write down the consumer’s maximization problem and solve. Your answer should include the Lagrangian associated with the problem and the first-order conditions. To solve this you must find the optimal level of (i) consumption, (ii) leisure and (iii) labor in equilibrium.

(b) Now suppose that the only factor of production is labor, and the representative firm’s production function is Y = zNd . Given this production function, solve for MPN . Then write down and solve the representative firm’s profit maximization problem and prove that MPN = w in equilibrium.