Shapiro (1983) "Premiums for High Quality Products as Returns to Reputations" QJE 98

Think about the market where producers can change product quality over time and consumers cannot observe quality prior to purchase. Then, what will happen? To answer this question, Shapiro (1983) develops a model that explores the implications of firm-specific reputations in a perfectly competitive environment. The one of the most interesting results is that in the equilibrium, firms produce higher quality products earn larger premiums. The premiums are needed for the following two reasons;
First, there is a cost to establish reputation and to offset the cost, positive return (=premium) is needed. Without a premium, no firm chooses high quality.
Second, in this market, sellers can always increase profits in the short-run by reducing the quality of their products (="fly-by-night strategy"). To prevent this deviation, positive return on the faithful path (which dominates the short-run return induced by fly-by-night strategy) is needed.
Although, the above point has already been recognized (Klein and Leffler (1981) explored this idea informally), this is the first paper which models reputation under competitive markets.

Tirole (1988) (2.6.2) provides a simplified version of the Shapiro model; one firm, and two types of qualities. He also points out two problems of Shapiro's model, which are the reliance of infinite-horizon time and bootstrap aspects of the equilibria. As is easily seen, only the lowest quality product is provided in each period with finite horizon model (by backward induction). Bootstrap aspects mean that reputation matters only because consumers believe it matters. Indeed, if, for example, consumers believe the firms produce the low quality no matter what the past history, then their expectation would again be fulfilled. In other words, the analysis suggests only that repetition may offer incentives to supply quality, not that it necessarily will.

Note) In the paper, this possibility is excluded since the author poses strong assumption about reputation formation; the expected quality of the firm's product at t is simply the product quality he chooses at t-1, i.e., R(t)=q(t-1). This simple adjustment expectation turns out to be a rational expectation. However, as Tirole mentions, there are other rational expectation equilibria and Shapiro (1983) does not mention them.

Finally, notice that Kreps and Wilson (1982) and Milgrom and Roberts (1982) showed that reputation effects can be obtained even with a finite horizon by introducing asymmetric information about firm's type. Their models also pin down the equilibrium strategy and high quality is necessarily observed.

References

Klein and Leffler (1981) "The Role of Market Forces in Assuring Contractual Performance" JPE, 81
Kreps and Wilson (1982) "Reputation and Imperfect Information" JET, 27
Milgrom and Roberts (1982) "Predation, Reputation, and Entry Deterrence" JET, 27
Tirole (1988) "The Theory of Industrial Organization" MIT Press