1Problem 1: The Consumption-Leisure Framework (30 points). In this question, you will use the basic (one period) consumption-leisure framework to consider some labor market issues. Suppose the representative consumer has the following utility function over consumption and labor,( , )lnu c ncAn=−, where, as usual, cdenotes consumption and ndenotes the number of hours of labor the individual chooses to work. The constant A > 0is outside the control of the individual. usual, ln( )⋅is the natural log function.) Suppose the budget constraint (expressed in real, rather than in nominal, terms) the individual faces is (1)ctw n=−⋅⋅, where tis the labor tax rate, wis the realhourly wage rate, and the number of hours the individual works. Recall that in one week there are 168 hours, hence n + l = 168 must always be true. a.(4 points) Construct the Lagrangian for the consumer’s utility maximization problem, defining any new notation you need to include. (As nis
b.(4 points) Based on the Lagrangian in part a, compute the representative consumer’s first-order conditions with respect to consumption and with respect to labor. Clearly present the important steps and logic of your analysis.