A company sells a product which is either of low or high quality where the quality is known to the company but not to the buyer, though the buyer learns the product’s quality after purchasing and consuming it. The product is of high quality with probability q, and low quality with probability 1 - q. It is of value v to the buyer when it is high quality and of zero value when it is low quality. After Nature chooses the product’s quality, it is revealed to the seller but not to the buyer. The seller then chooses an introductory price (the price charged for the buyer’s first purchase) which can be any non- negative number. The buyer observes the introductory price and decides whether to buy. If the buyer chooses not to buy then the payoff to both the seller and the buyer is zero. If the product is purchased at an introductory price p and proves to be low quality then the buyer never buys it again and realizes a payoff of -p. If the buyer purchases it and it proves to be high quality then the the buyer and seller enter into an ongoing relationship that is of value B > 0 to the buyer and S > 0 to the seller. In that event, and given an introductory price is p, the buyer’s payoff is v - p + B (where the buyer realizes net value of v - p from the initial purchase and B from future purchases) and the seller’s payoff is p - c + S where c is the seller’s cost of production so the seller earns profit of p - c on the first purchase and S from future purchases. Assume v > c > 0 so that the high quality is valued more by a buyer than it costs the seller to produce. a) Find the conditions for there to be a separating PBNE in which the buyer buys with positive probability. (Hint: In specifying the buyer’s beliefs in response to a nonequilibrium price being selected, assume the buyer assigns probability 1 to the seller having a low quality product.) b) Derive conditions for there to be a pooling PBNE in which the buyer buys. c) Consider a strategy profile in which both seller types price at p' and the buyer does not buy for all prices. The buyer believes the seller has low quality with probability 1 when price differs from p'. i. Find conditions on p' such that this is a PBNE. ii. Determine whether your answer in (i) satisfies or violates the Intuitive Criterion.
A company sells a product which is either of low or high quality where the quality is known to the company but not to the buyer, though the buyer learns the product’s quality after purchasing and consuming it. The product is of high quality with
a) Find the conditions for there to be a separating PBNE in which the buyer buys with positive probability. (Hint: In specifying the buyer’s beliefs in response to a nonequilibrium price being selected, assume the buyer assigns probability 1 to the seller having a low quality product.)
b) Derive conditions for there to be a pooling PBNE in which the buyer buys.
c) Consider a strategy profile in which both seller types price at p' and the buyer does not buy for all prices. The buyer believes the seller has low quality with probability 1 when price differs from p'.
i. Find conditions on p' such that this is a PBNE.
ii. Determine whether your answer in (i) satisfies or violates the Intuitive Criterion.
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