a. Based on only the first-order condition with respect to labor computed in part a (Based on the given Lagrangian, compute the representative consumer's first-order conditions with respect to consumption and with respect to labor). Qualitatively sketch two things in a diagram with the real wage on the vertical axis and labor on the horizontal axis. First, the general shape of the relation ship between w and n (perfectly vertical, perfectly horizontal, upward-sloping, downward-sloping, or impossible to tell). Second, how changes in / affect the relationship (shift it outward, shift it in inward, or impossible to determine). Briefly describe the economics of how you obtained your conclusions. (Note: In this question you are not to use the first-order condition with respect to consump tion nor any other conditions.) b. Now based on both of the two first-order conditions computed in part a, construct the consumption-leisure optimality condition. Clearly present the important steps and logic of your analysis.
a. Based on only the first-order condition with respect to labor computed in part a (Based on the given Lagrangian, compute the representative consumer's first-order conditions with respect to consumption and with respect to labor). Qualitatively sketch two things in a diagram with the real wage on the vertical axis and labor on the horizontal axis. First, the general shape of the relation ship between w and n (perfectly vertical, perfectly horizontal, upward-sloping, downward-sloping, or impossible to tell). Second, how changes in / affect the relationship (shift it outward, shift it in inward, or impossible to determine).
Briefly describe the economics of how you obtained your conclusions. (Note: In this question you are not to use the first-order condition with respect to consump tion nor any other conditions.)
b. Now based on both of the two first-order conditions computed in part a, construct the consumption-leisure optimality condition. Clearly present the important steps and logic of your analysis.
![4. The consumption–leisure framework. In this question you will use the basic (one-
period) consumption-leisure framework to consider some labor market issues.
Suppose that the representative consumer has the following utility function over con-
sumption and labor:
A
u(c, 1) = Inc -
1+¢
where, as usual, c denotes consumption and n denotes the number of hours of labor
the consumer chooses to work. The constants A and are outside the control of the
individual, but each is strictly positive. (As usual, In) is the natural log function.)
Suppose that the budget constraint (expressed in real, rather than in nominal, terms)
the individual faces is c= (1-t) w n, where t is the labor tax rate, w 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. The Lagrangian for this problem is
A
Inc-n** +2[(1-1)wn- c],
1+0
where å denotes the Lagrange multiplier on the budget constraint.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F441c8085-cc50-463f-bd14-89a7f2b458b6%2Fcd8a175d-f494-45b4-bb56-d5cc3d392c54%2F1bix04n_processed.jpeg&w=3840&q=75)
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