need HW sulotion

1. (a) Consider a simple game in which two consumers, A and B, are deciding whether to

adopt a new network technology or not. Consumer A has an individual valuation equal

to vA=100, and consumer B has an individual valuation of vB=200. If both consumers

adopt the technology then each consumer receives their own valuation. If at least one

consumer does not adopt the technology, each consumer receives $0. The price of the

technology is $50. Construct the normal form of this game assuming consumers move

simultaneously and choose between two strategies: “Adopt” or “Don’t Adopt.” Solve

for any (pure strategy) Nash equilibria.

(b) Suppose that there are 50 potential consumers in the market for a new technology

that exhibits network effects. There is a uniform distribution of consumers with

individual valuations, v, ranging from $1, $2,…, $50. If consumers’ valuation from

consuming the technology is given by vN, where N is the number of consumers

adopting the technology, graphically and algebraically solve for the three possible

equilibria if price is $600. Which equilibrium is most likely?

2. Consider the market for a software application. There are 100 consumers who value

technical support, which they can only receive if they purchase the application. There are

100 consumers who do not value technical support. Consumers who value technical

support obtain a payoff of 2n – p if they purchase the software and n if they pirate the

software, where n is the total number of users, and p is price. Consumers who do not value

technical support receive payoff of n – p if they purchase the software, and n is they pirate

the software. Suppose the software is costless to both produce and to protect from piracy.

(a) If the software is not protected, so that piracy is an option for consumers, what is the

firm’s profit-maximizing price? Explain.

(b) If the software is protected, so that piracy is not an option for consumers, what is the

firm’s profit-maximizing price? Explain.

3. Suppose there is a uniform distribution of consumers with individual valuations, v,

ranging from 0 to 1, i.e. there is a consumer with v=0.123 and a consumer with

v=0.753 etc. Suppose each consumer’s payoff from purchasing a network good is v +

θN, where N is the fraction of the market that purchase the good. The price of the

technology is p. The total market size is normalized to 1, so that N = 0.5 if half the

market purchases, and N = 1 if everyone purchases.

(a) If v is the valuation of the (marginal) consumer with the lowest valuation that

purchases the product (i.e. those with v < v do not purchase, and those with v ≥ v do

purchase), what is the fraction of the market that purchase the product?

(b) Using your answer from (a), construct an equation to determine the valuation of the

marginal consumer.

(c) Solve for the valuation of the marginal consumer.

(d) Using your answer from (c), construct the profit function for the manufacturer

assuming it faces zero costs. It should be a function of p.

(e) Solve for the profit maximizing price, and derive the firm’s optimal profit.

(f) How does profit depend on the parameter θ? Why?