1. Mr. C enjoys commodities x and y according to the utility function

U(x, y) = .3ln x + .7lny

a. Derive the Marshallian Demand Functions for x and y.

b. Compute the indirect utility function.

c. Compute the expenditure function.

d. Compute the Compensated Demand Functions for x and y.

e. Derive the Slustky equation for a change in the price of x.

2. Consider the CES production function y = [k^ρ + l^ρ]^1/ρ.

a. Show that the Marginal Product of and the Marginal Product of l are functions of output (y).

b. Compute the Marginal Rate of Technical Substitution between k and l.

c. Compute the elasticity of substitution.

3. Suppose the production function of a firm is

y = 2l⁵

Where l is labor. Assume that the firm is a price-taker in the output market and the price of labor is w.

a. Derive the factor demand function for l.
b. Derive the supply function.
c. Derive the profit function.

4. Suppose that the firm's production function is

y = k⁵l²⁵m^.25

Where k is capital, l is labor, and m materials. Assume that capital is constant at level K‾ and its price is r. The prices of labor and material are w and u respectively.

a. Derive the conditional factor demands for l and m.

b. Derive the cost function.

c. Show that the conditional factor demands are homogeneous of degree 0 in input prices.

d. Show that the cost function is homogeneous of degree 1 in input prices.