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Dynamic Limit Pricing

There are many different models of a dominant firm's behavior when potential competitors enter gradually in response to prices and predicted profitability. Two well-known models are the lumpy entry model where many fringe firms enter all at once after a lag, and the continuous entry model where fringe firms enter gradually at different times.

In the lumpy entry model, all the potential entrants see an opportunity and start building plants at the same time. The greater the profit opportunity, the faster they build their plants and the sooner they enter.

Because the dominant firm earns lower profits after other firms enter, the dominant firm wants to delay entry as long as the cost of doing so is not excessive. The dominant firm has a choice. It can earn very high profits for a short period of time before competitors flood into the market, or it can earn lower profits for a longer period of time before they enter. The optimal policy, typically, is for the dominant firm to set its price below the short-run monopoly price but above the price that completely deters entry (Kamien and Schwartz 1971, De Bondt 1976). The intuition is that the dominant firm gains from keeping price lower than the short-run monopoly price to deter entry. It trades higher profits tomorrow (due to less entry) against higher profits today (due to setting the price above the limit price today).

Where firms enter gradually and continuously, the optimal policy is to price high initially and then slowly lower the price. If there are no fringe firms initially, the dominant firm starts with the short-run monopoly price, then gradually lowers it. As the number of fringe firms becomes large, the dominant firm lowers the price below the limit price (where the fringe firms' profits are zero). That is, it sets a price so low that fringe firms lose money, so fringe firms gradually leave the industry. As the last of the fringe firms are exiting, the dominant firm raises the price to the limit price and keeps it there, deterring future entry. A key assumption in these dynamic models is the entrants' expectation of future profits. Let us now examine some models in detail.

The best-known model in which fringe firms enter in a continuous flow with a known rate of entry is Gaskins (1971). He examines the pricing behavior of a low-cost dominant firm that takes into account the gradual but continuous entry of competitors in determining its optimal pricing behavior. New fringe firms are assumed to enter at a rate that is proportional to expected profits: The higher the expected profits, the faster they enter. The fringe firms have myopic expectations: They think profits in the future will equal profits today. The dominant firm takes this rate of entry into account in determining the price that maximizes its profits.

Gaskins assumes that fringe firms have constant average and marginal cost, p, and can produce only one unit of output. As a result, as long as the dominant firm picks a price greater than p, fringe firms gradually enter; if it sets a price less than p, fringe firms gradually leave the industry. We call p the "limit price."

Gaskins also assumes that the dominant firm can commit to a particular price adjustment path. That is, in the initial period, the dominant firm specifies its future price p(t) at every moment of time, t. Gaskins shows that the dominant firm's optimal pricing policy, labeled myopic in Figure 11.1, depends on the initial number of fringe firms in the industry. The vertical axis in the figure shows the price set by the dominant firm at time t, p(t), and the horizontal axis shows the number of fringe firms at time t, n(t).

Figure 11.1 Dynamic Limit Pricing

If there are initially very few fringe firms, n1, in the industry, the dominant firm starts with a relatively high price, p1, and then lowers the price gradually to p*, following the myopic line. The arrows on the myopic price-adjustment path indicate the direction of movement.

At p*, the n* fringe firms are exactly breaking even, so they have no further incentive to enter or leave the industry. The dominant firm then continues to price at p* forever, and no further entry occurs. Because the dominant firm's costs are less than p*, it makes positive profits even at p*.

The price p* is a limit price because it is a price that, if maintained forever, prevents further entry (it limits entry). The dominant firm in the Gaskins model may set a price above or below the limit price in the short run, but in the long run it sets a limit price and maintains it.

[The figure shows only the myopic adjustment path to the left of n*. Although not shown in the figure, the straight-line adjustment path continues across the p* line. If the initial number of fringe firms is large, the dominant firm starts with a price below p*. When the price is below p*, fringe firms lose money so some of them leave the industry. As the fringe firms exit, the dominant firm raises its price until it reaches p*, where the number of fringe firms is n*. It then continues to charge p* forever. Thus, in the long run, the price, p*, and the number of fringe firms, n*, are the same, regardless of whether the initial number of fringe firms is large or small.]

It is inefficient for fringe firms to be in such an industry: They produce at higher cost than does the dominant firm. Nonetheless, their presence is socially useful because it keeps the dominant firm's price lower than it would otherwise be, at the limit price. Thus there is a social trade-off between efficiency in production (low costs) and efficiency in pricing.

The assumption that the dominant firm has a cost advantage is crucial. If it has no cost advantage, its market share will ultimately shrink to that of any fringe firm.

In Gaskins' model, a low-cost dominant firm does not drive out the fringe in the long run. That result is surprising because it is inconsistent with the model of instantaneous entry where a low-cost dominant firm with a constant marginal cost drives out the fringe (Chapter 4). With instantaneous entry by the fringe, the dominant firm prices slightly below the fringe's cost of production (any price less than p*) and grabs all the industry's business for itself. This difference is especially surprising because the instantaneous-entry model appears to be a special case of Gaskins' model. If the proportional rate of increase is made very large, entry of fringe firms approaches being instantaneous.

The two models reach different conclusions because of the way fringe firms form expectations about future profits. In the instantaneous-entry model (static model) fringe firms know their present and future profits, which are the same, but in Gaskins' model fringe firms have myopic expectations about profits: They incorrectly assume that future profits will be the same as current ones.

If fringe firms have perfect foresight (they accurately forecast future prices and profits), the dynamic model has the same long-run result as the instantaneous-entry model (Berck and Perloff 1988). [Flaherty (1980) and Judd and Petersen (1986) also present models in which entrants have nonmyopic expectations, but these models differ from the Gaskins model in several other ways as well, which makes comparisons difficult.] The fringe firms can predict their profits perfectly because the dominant firm commits itself to a price path from the first day. As in Gaskin's model, firms enter gradually because it is more expensive to enter quickly than slowly. Fringe firms gradually enter if the expected present value of future profits is positive (even if current profits are negative). They gradually leave if the expected present value of future profits is negative (even if current profits are positive).

As shown by the curve labeled perfect foresight in the figure, with n1 initial fringe firms, the dominant firm initially sets a price p2 (which is higher than in the myopic model) to make extraordinary short-run profits. As fringe firms enter, it lowers its price. Eventually, it lowers its price below p* for a sufficiently long time to drive all the fringe firms out of the industry; whereupon the dominant firm raises its price to the limit price p* to keep them out in the future. Thus, where fringe firms have perfect foresight, the low-cost dominant firm eventually sets price at p* and makes all the sales in the long run as in the instantaneous-entry model.

A dominant firm can achieve this result only by credibly committing itself to a particular price path so that the fringe firms know that it will keep its prices low for a significant period of time. When the dominant firm cannot commit to a price path so that it may revise its pricing policy in the future, it cannot convince rational fringe firms that it will maintain such a low price that they should leave the industry. In the absence of commitment, the dominant firm sets its short-run profit-maximizing price at every moment (Karp 1988, Berck and Perloff 1990). Thus, in the long run, fringe firms with perfect foresight remain in the industry.


Berck, Peter, and Jeffrey M. Perloff. 1988. "The Dynamic Annihilation of a Rational Competitive Fringe by a Low-Cost Dominant Firm." Journal of Economic Dynamics and Control 12:659-78.

------. 1990. "Dynamic Dumping." International Journal of Industrial Organization 8:225-43.

De Bondt, Raymond R. 1976. "Limit Pricing, Uncertain Entry, and the Entry Lag." Econometrica 44:939-46.

Flaherty, M. Therese. 1980. "Dynamic Limit Pricing, Barriers to Entry, and Rational Firms." Journal of Economic Theory 23:160-82.

Gaskins, Darius W., Jr. 1971. "Dynamic Limit Pricing: Optimal Pricing Under Threat of Entry." Journal of Economic Theory 3:306-22.

Judd, Kenneth L., and Bruce C. Petersen. 1986. "Dynamic Limit Pricing and Internal Finance." Journal of Economic Theory 39:368-99.

Kamien, Morton I., and Nancy L. Schwartz. 1971. "Limit Pricing and Uncertain Entry." Econometrica 398 (May):441-54.

Karp, Larry S. 1988. "Consistent Policy Rules and the Benefits of Market Power." Working Paper, Department of Agricultural & Resource Economics, University of California at Berkeley.

© 2000 Dennis W. Carlton and Jeffrey M. Perloff. Reprinted by permission.

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