Intermediate Macroeconomics PROBLEM SET-

1. Inada Conditions

Show whether the following utility functions satisfy the Inada Conditions for C and l:

1. U(C, l) = γ log C + (1 - γ)log l, where γ  > 0.

2. U(C, l) = (C^{1 - σ}/1 - σ) + l, where σ > 0.

3. U(C, l) = C^αl^1-α, where 0 < α < 1.

4. U(C, l) = [C^σ-1/σ + l^σ-1/σ]^{σ/1 - σ}, where σ > 0.

2. Consumer's Problem

Consider a representative consumer with utility function U(C, l) = (C^1-σ/1-σ) + (l^1-σ/1-σ). He has h available hours to divide between work and leisure. If he works, he gets an hourly wage of w. Regardless of the amount of time he spends working, he receives an extra income in the form of profits π, because he is the owner of the firm. Also, he has to pay lump-sum taxes to the government, T.

1. Clearly state the problem of this consumer.

2. Characterize the solution to the problem of this consumer, with a system of 2 equations and 2 unknowns.

3. Find expressions for C and l that only depend on the variables taken as given by the consumer, and the parameters h and σ.

4. Suppose there is an increase in taxes. What is the effect on the optimal choice of C and l? Prove your result mathematically and give the intuition/rationale for your answer in no more than 5 lines.

3. Properties of the Production Function

The most commonly used production function in macroeconomics is the Cobb-Douglas, given by:

zF(K, N) = zK^αN^{(1- α)}

Where 0 < α < 1

1. State and prove that this production function satisfies the five properties we saw in class.

4. Firm's Problem

Consider a firm with production function zF(K, N^d) = zK^α(N^d)^{(1- α)}, where 0 < α < 1. The productivity parameter z and the capital stock K are given to the firm, and cannot be changed. The only production factor that the firm can choose is the amount of labor it hires N^d. For each unit of labor, the firm pays a wage w.

1. Clearly state the problem of the firm.

2. Find the condition that characterizes the solution to the firm's problem. In no more than 3 lines, explain the intuition behind your condition.

3. Suppose w = 1; z = 4; α = 1/2; K = 3. Find the optimal level of labor for the firm.

4. Suppose a new invention available today increases the productivity of the firm (z) from 4 to 6. Everything else is unchanged. What is the effect on the optimal amount of labor for the firm? Find the value of the new level of labor, and explain in no more than 3 lines the intuition for your answer.

5. Income Expenditure Identity

In chapter 2 we saw that in every economy, the income expenditure identity holds:

Y = C + I + G + X - M

In our one-period model, total output Y is given by total production of the firm z^F(K, N^d). Taking quations (1)-(7) in slide 15 of chapter 5, show that the income expenditure identity also holds in our model.

(Hint: Remember that in our model there is only one period, so there is no incentive to invest for the future. Also, we are working with a closed economy.)