Problem Set #1

1. Assume that a firm with market power faces an advertising-responsive demand. Let the demand function be

Q = A^(1/2) * (B - P),

where Q is the demand,

A is the number of advertising hours on local television,

B is a constant (the choke price), and

P is the price.

The notation A^(1/2) means A raised to the power (1/2).

Assume that the production cost of the product is proportional to output, with a marginal and average cost of K. Assume that the firm must pay a price of V per unit of advertising.

a) Show that the profit-maximizing price of the product will be independent of the price of advertising and of the amount of advertising chosen by the firm.

b) Show that the profit-maximizing level of sales is inversely proportional to V.

c) Show that the profit-maximizing level of sales will be proportional to the cubic power of the difference between the choke price and the production cost.

2. Consider a monopolistic manufacturer who sells its product through three retailers (indexed by i=1,2,3) located in different ggeographical regions. Each retailer is a profit-maximizing monopolist in its own region, and faces demand function, Pi = A - BiQi. Assume customers from one region cannot buy from other regions.

(a) Suppose that the manufacturer charges retailers a (linear) wholesale price of Ci per unit, where Ci could vary among retailers. Show that in fact, the manufacturer will choose identical wholesale prices for the three retailers. Explain what features of the problem formulation lead to this conclusion.

b) Assume that the manufacturer vertically integrates, buying each of the retailers. Assume that the purchase price for the retail stores is exactly equal to the profit that the retailer would earn otherwise. Then the manufacturer will set retail prices in each region.

Show that this vertical integration will be more profitable for the manufacturer than would be the initial situation, taking into account the cost for buying the retail stores. In addition, show that, as a

result of this vertical integration, the retail price will be decreased in each region.

c) Now assume that, instead of vertically integrating, the manufacturer chooses a new pricing policy:

She will charge a fixed franchise fee, Fi, to each retailer, and a linear price Ci per unit. Compute the optimal retail price in each region. Find the optimal franchise fee. Show that the retail prices in this case will be identical to those of part a). Explain why this is so.

d) Is it more profitable for the manufacturer to set franchise fees or to vertically integrate?

3. Problem 1. Chapter 8. Luenberger.

4. Find a mixed-strategy Nash equilibrium of the following normal-form game.

 
             l              r

	t      2,1          0,2					
	b     1,2          3,0

5. Players 1 and 2 are bargaining over how to split one dollar. Both players simultaneously name shares they would like to have, s1 and s2, where 0 <= s1,s2 <= 1. If s1+s2 <= 1, then both players receive the shares they named; if s1+s2 > 1, then both players receive zero. What are the pure Nash equilibria of this game? Which equilibria would you expect to occur?

6. Problem 12. Chapter 3. Luenberger.

7. Consider the Cournot dupoly model where inverse demand is P(Q) = a - Q, where Q is total output. Firms have asymmetric marginal costs:

c1 for firm 1 and c2 for firm 2.

Fixed cost is zero.

Assume 0 < c1,c2 < a.

What is the Nash equilibrium if c1,c2 < a/2? What if c1 < c2 < a but 2c2 > a + c1?

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