A buyer wants to buy one unit of a good from an incumbent seller. The buyer's valuation of the good is 1, while the seller's cost of producing it is 1/2. Before the parties trade, a rival seller enters the market and his cost, c, is distributed on the unit interval according to a distribution function with density g (c). The two sellers then simultaneously make price offers and the buyer trades with the seller who offers the lowest price. If the two sellers offer the same price the buyer buys from the seller whose cost is lower. 1. Determine the price that the buyer pays in equilibrium, p, as a function of c. Given p, write the payoffs of the expected payoffs of the buyer and the two sellers. 2. Suppose that the distribution of c is uniform. Show p and the expected payoffs of the parties graphically (put c on the horizon- tal axis and the equilibrium price function on the vertical axis and show the payoffs by pointing out the appropriate areas in the graph). 3. Now suppose that the incumbent seller offers the buyer a contract before the entrant shows up. The contract requires the buyer to pay the incumbent seller the amount m regardless of whether he buys from him or from the entrant, and gives the buyer an option to buy from the incumbent at a price of p (this is equivalent to
A buyer wants to buy one unit of a good from an incumbent seller. The buyer's valuation of the good is 1, while the seller's cost of producing it is 1/2. Before the parties trade, a rival seller enters the market and his cost, c, is distributed on the unit interval according to a distribution function with density g (c). The two sellers then simultaneously make price offers and the buyer trades with the seller who offers the lowest price. If the two sellers offer the same price the buyer buys from the seller whose cost is lower. 1. Determine the price that the buyer pays in equilibrium, p, as a function of c. Given p, write the payoffs of the expected payoffs of the buyer and the two sellers. 2. Suppose that the distribution of c is uniform. Show p and the expected payoffs of the parties graphically (put c on the horizon- tal axis and the equilibrium price function on the vertical axis and show the payoffs by pointing out the appropriate areas in the graph). 3. Now suppose that the incumbent seller offers the buyer a contract before the entrant shows up. The contract requires the buyer to pay the incumbent seller the amount m regardless of whether he buys from him or from the entrant, and gives the buyer an option to buy from the incumbent at a price of p (this is equivalent to
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:A buyer wants to buy one unit of a good from an incumbent seller.
The buyer's valuation of the good is 1, while the seller's cost of producing
it is 1/2. Before the parties trade, a rival seller enters the market and
his cost, c, is distributed on the unit interval according to a distribution
function with density g (c). The two sellers then simultaneously make
price offers and the buyer trades with the seller who offers the lowest
price. If the two sellers offer the same price
the buyer buys from the seller whose cost is lower.
1. Determine the price that the buyer pays in equilibrium, p, as a
function of c. Given p, write the payoffs of the expected payoffs of
the buyer and the two sellers.
2. Suppose that the distribution of c is uniform. Show p and the
expected payoffs of the parties graphically (put c on the horizon-
tal axis and the equilibrium price function on the vertical axis
and show the payoffs by pointing out the appropriate areas in the
graph).
3. Now suppose that the incumbent seller offers the buyer a contract
before the entrant shows up. The contract requires the buyer to
pay the incumbent seller the amount m regardless of whether he
buys from him or from the entrant, and gives the buyer an option
to buy from the incumbent at a price of p (this is equivalent to
Transcribed Image Text:giving the buyer an option to buy at a price m + p and requiring
him to pay liquidated damages of m if he switches to the entrant).
If the buyer rejects the contract things are as in part 1. Given p
and c, what is the price that the buyer will end up paying for the
good? Using your answer, write the expected payoffs of the buyer
and the two sellers as a function of p and m.
4. Explain why the incumbent seller will choose p by maximizing the
sum of his expected payoff, 71, and the buyer's expected payoffs,
UB.
5. Write the first-order condition for p and show that the profit-
maximizing price of the incumbent seller, p**, is such that p** <
1/2. Also show that if g(0) > 0 then p > 0.
6. Explain why the contract is socially inefficient. Is the outcome in
part 1. socially efficient? Explain the intuition for your answer.
7. Compute p** assuming that the distribution of c is uniform, and
show the expected payoffs of the parties and the social loss graphi-
cally (again, put c on the horizontal axis and the equilibrium price
function on the vertical axis).
8. Compute p** under the assumption that G(c) = cº, where a > 0.
How does p** vary with a? Give an intuition for this result.
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