Question 1. The Consumption-Leisure Framework. In this question, you will use the basic (one period) consumption-leisure framework to consider some labor mar- ket issues. Suppose the representative consumer has the following utility function over consumption and labor: u(c,l) = ln(c) n\"? where, as usual, c denotes consumption and n. denotes the number of hours of labor the consumer chooses to work. The constants A and (,6 are outside the control of the individual, but each is strictly positive. (As usual, in() is the natural log function.) Suppose the budget constraint (expressed in real, rather than in nominal, terms) the individual faces is c = (1 t)wn, where t is the labor tax rate, it) is the real hourly wage rate, and n is the number of hours the individual works. Recall that n +1 = 1 must always be true. a. Set up the Lagrangian function and compute the representative consumer's rst- order conditions with respect to consumption and with respect to labor. b. Based on only the rst-order condition with respect to labor computed in part a, qualitatively sketch in a diagram (with the real wage on the vertical axis and labor on the horizontal axis) the relationship between w and n, and show how changes in t affect the relationship. Briey describe the economics of how you obtained your conclusions. c. Now based on both of the two rst-order conditions computed in part a, construct the consumption-leisure optimality condition. d. Based on both the consumption-leisure optimality condition obtained in part c and on the budget constraint, qualitatively sketch a diagram with the real wage on the vertical axis and labor on the horizontal axis depicting the relationship between to and n. Show how changes in it affect the relationship. e. How do the conclusions in part d compare with those in part b? Are they broadly similar? Are they very different? Is it impossible to compare them? Describe as much as you can about the economics comparing the pair of diagrams. Note: I am not mark- ing this part of the question, we can discuss these implications in class