### Problem of the book assuming that the utility is cobb-dougla

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Exercise 1

Consider the problem of the book assuming that the utility is Cobb-Douglas (U (C, l) = Cαlβ). First formally write down the consumer's problem. Then, answer the questions in the problem. No algebra is necessary to answer the questions, explain your answers using clearly labeled graphs and intuition.

Problem as follows:

Suppose that a consumer can earn a higher wage rate for working overtime. That is, for the first q hours the consumer works, he or she receives a real wage rate of w1, and for hours worked more than q he or she receives w2, where w2 > w1. Suppose that the consumer pays no taxes and receives no nonwage income, and he or she is free to choose hours of work.

1. Draw the consumer's budget constraint and show his or her optimal choice of consumption and leisure.

2. Show that the consumer would never work q hours or anything very close to q hours. Explain the intuition behind this.

3. Determine what happens if the overtime wage rate w2 increases. Explain your results in term of income and substitution effects. You must consider the case of a worker who initially works overtime, and a worker who initially does not work overtime.

Exercise 2

Consider a competitive firm that has a production function given by Y = zF (K,N) where z > 0 and K > 0 are exogenous and faces wage w > 0.

(a) Write down the problem of the firm

(b) Now suppose that Y = N1/2. Calculate the demand function for labor of the firm. Explain the intuition of the algebraic expression you found.

Exercise 3

Suppose a firm has a production function given by F (K;N) = KαNβ.

(a) What are the restrictions on α and β implied by the assumption that the marginal product of capital and labor are positive?

(b) What are the restrictions on α and β implied by the assumption of constant returns to scale?

(c) Prove that the restrictions obtained in parts (a) and (b) imply that the marginal product of labor decreases as the quantity of labor increases.

Exercise 4

Consider a simple one-period, closed-economy model where the representative consumer has utility function U(C,l) = C1/2l1/2 and has h available hours to divide between work and leisure. The representative firm has technology given by Y = zK1/3N2/3. There is a government that sets its expenditure level at a value G.

(a) Define a competitive equilibrium for this economy.

(b) Prove that Walras' Law applies.

(c) Find the competitive equilibrium values ( ^C, ^Ns, ^Nd, ^T, ^w) given parameters h = 20, z = 1, K = 1000, and G = 0.

Exercise 5

Consider the same economy described in Exercise 4.

(a) Set up the equivalent social planner's problem with the choice variables being (C,N).

(b) Using the parameters in Exercise 4, part (c), solve for the Pareto optimal consumption and employment levels, (C*,N*).

(c) Formally state the FirstWelfare theorem relating your solution to part (b) and the competitive equilibrium allocation found in Exercise 4, part (c).

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