Write the monopolists profits as a function of output and

Question

1. Consider a monopolist that faces a linear demand curve and operates with a con-

stant marginal cost of production. Suppose that the monopolist must pay a sales tax on the product. A sales tax is a percentage of the product's price, so that if the sales tax rate is t, the total tax T owed by the monopolist is r percent of total revenue, that is, T = tPQ.

(a) Write the monopolist's profits as a function of output and the exogenous variables.

(b) Find the first- and second-order conditions for a profit maximum.

(c) Solve for the optimal levels of output and price as a function of the exogenous variables.

(d) Find the comparative static effects of a change in the sales tax rate.

(e) Suppose that the government chooses the sales tax rate to maximize tax revenue. Show how to solve for the revenue-maximizing tax rate. Does this relatively simple model yield a solution for the revenue-maximizing tax rate?

2. Suppose that in the model of Section 6.6 the tax on the monopolist is an ad valorem tax (where the tax is applied as a percentage of sales) so that the firm's profit is given by Π(Q)= P(Q)Q0 - - C(Q). Find the effects on equi­librium price and quantity of an increase in the tax.

3. Several demand functions frequently used in applied work arc listed below. For each, show whether or under what conditions demand is homogeneous of degree 0 in prices and income.

(a)  Linear: x₁ = α + β₁p₁ + β₂p₂ + β₃I

(b)  Log-linear: In x₁ = α + β₁,p₁ + β₂p₂, + β₃I

(c)  Log-log: In x₁ = α + β₁lnp₁ + β₂lnp₂, + β₃lnI

(d)  Cobb-Douglas utility: x₁ = αI/βp₁

(e)  Almost ideal demand system: x1 = I/P₁( α + βln(P1/P2) + Υδ(P1/P2)^δ)

4. Suppose that a perfectly competitive firm uses three inputs, L, K, and R; pays input prices of w, r, and v, sells its output at a price of P; and operates with a production function of Q = 3(LX)1/3 + ln R.

(a) Write the expression for the firm's profits. What are the first-order condi. tions? Give an economic interpretation of the first-order conditions.

(b) Check the second-order conditions.

(c) Without explicitly solving for V':

(i)  Find the change in L for a change in r when all other parameters are constant.

(ii) Find the change in L for a change in v when all other parameters are constant.

(d) Solve for L*. Take partial derivatives of V to confirm the results derived in part c.