This problem is about the effect of long run repetition on developing a good rep

Question

This problem is about the effect of long run repetition on developing a good reputation. In each period, a manufacturer can choose to produce either a low-quality product at a cost of $8 per unit, or a high-quality product at a cost of $16 per unit (both cannot be produced in one period). A consumer is willing to pay up to $15 for a low-quality product and up to $40 for a high-quality product. The problem is that a consumer cannot tell before purchase whether the product is of low or high quality. If the consumer purchases the product, he learns its quality through usage.

Consider the following game. There is one consumer each period. Each period, the manufacturer (who lives forever) chooses a quality and a price. The consumer observes the price, and decides whether to purchase. A consumer buys as long as the net surplus from the product is nonnegative. The net surplus is the expected value of the product (which is 15 if it is expected to be of low quality and 40 if of high quality), minus the price. The manufacturer’s payoff is the present value of its profit stream, where the profit in any period is zero if the consumer doesn’t buy. If the consumer does buy, the profit is the price charged, less the cost of making the good. The manufacturer’s discount factor is ?.

In each period, the manufacturer decides on the quality and price of the project. After observing the price (but not the quality), the consumer either buys or doesn’t buy. In each period, all past actions, including all past prices, qualities, and purchasing decisions, are common knowledge.

(a) Suppose this interaction takes place in just one period. If the consumer expects the product to be high quality, what is the maximum price at which he would be willing to purchase?

(b) Suppose (again in a 1 period game) that the manufacturer expects the consumer to purchase at the price stated in part (a). Given this, is the manufacturer better off producing high or low quality?

(c) Is there a subgame perfect Nash equilibrium of the 1 period game in which the manufacturer produces at high quality and the consumer purchases the product?

(d) Now suppose this interaction takes place for infinitely many periods. Find a strategy profile that results in the manufacturer’s producing a high-quality product, charging a price of 40, and consumers buying the product every period. Derive conditions for this strategy profile to be a SPNE.

Explanation / Answer

(a) consumer is expecting a high quality product, so he will be willing to pay utmost $40. Anything higher than $40 will result to negative net surplus for him.

(b)  In a 1 period game, if the manufacturer expects the consumer to purchase at the price stated in part (a). Then manufacturer is better off producing a low quality product because it is a 1 period game and he is getting higher payoff by producing low quality good as replacement of high quality good.

(c) when the manufacturer produces high quality product and consumer buys it, then the sumgame perfect equilibrium exists because net surplus of consumer will be non-negative so they will continue to buy.

(d) if this interaction takes place for infinitely many periods. A grim trigger strategy profile that results in the manufacturer’s producing a high-quality product, charging a price of 40, and consumers buying the product every period.

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