Question

Consider a new firm that is a monopolist in a market with (inverse) demand:

p(y) = 1000y^(−x)

The firm has a cost function c(y) =Cy whereCis a non-negative constant(M C=Cis constant).

1. Show analytically that this demand has a constant elasticity

2. Write out the firm’s profit function in terms of x, y and C.

3. Derive the Necessary First Order Condition for profit to be at a maximum. What is the sufficient Second Order Condition for this point to be a global maximum?

4. For what values of x is the sufficient Second Order Condition satisfied? What would happen to the monopoly price and quantity if this sufficient Second Order Condition was not satisfied? Use this to argue the dangers from monopolies arising in markets where demand is inelastic at high prices, such as food and water.

5. Now suppose x = (1/2). Derive the monopolist’s profit-maximizing price and quantity in terms of the marginal cost, C.

6. If the marginal cost increases by 1 unit, by how much does price rise? Specifically, derive the comparative static: dP M/dC. Use this to discuss the effect of imposing a tax of $ T on each unit sold on the price in the market. Does price rise by more, less, or exactly the amount of the tax?

Answer 1

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