Market forces meet behavioral biases: cost misallocation and irrational pricing.

([a - [p.sup.*]/b])[([p.sup.*] - c]/1 + ([N.sup.*] - 1)[theta]] = F. (13)

It is straightforward to verify that [N.sup.*] > [N.sup.0].

6. Discussion

* Experimental results. As noted in the Introduction, there is strong evidence from the psychology literature that individual decision makers commit the sunk cost fallacy. A well-cited example is a study by Arkes and Blumer (1991) which found that the sunk costs incurred by season ticket holders for a university theater company apparently influenced attendance decisions.

Experimental economists have also studied sunk cost biases, but the evidence from this work is more mixed. Economists' experiments on the sunk cost fallacy differ from those in the psychology literature in that the latter typically do not focus on decision making in market settings in which competition might be expected to create strong pressures for decision makers to "get it right." By contrast, economists' experiments on sunk costs generally place subjects in the context of some sort of market rivalry. For example, Phillip, Battalio & Kogut (1991) study bidding in first-price sealed bid auctions in which subjects paid a sunk admissions fee in order to participate. They found that 95% of the subjects correctly ignored sunk costs in their bidding behavior. Summarizing a variety of experiments in competitive market settings in which sunk costs might effect market outcomes, Smith (2000) writes: "These results showed no evidence of market failure due to the sunk cost fallacy."

The finding that subjects in competitive market experiments do not succumb to the sunk cost fallacy is consistent with the prediction of our theory that it would run counter to the self-interest of a price-taking firm, or a price-setting firm in a homogeneous product oligopoly, to base decisions on sunk costs. However, a more direct test of the implications of our theory would be to explore pricing behavior in a differentiated product oligopoly market in which firms incur sunk costs. Unfortunately, most of the existing experimental investigations of price competition in a differentiated product oligopoly (e.g., Dolbear et al., 1968) are not relevant to our theory. This is because these experiments typically provided subjects with a table showing profit as a function of own price and the prices of rivals. Participants were simply asked to choose a price, and they did not have to figure out for themselves which costs might be relevant for pricing decisions.

An important exception to this approach is a recent paper by Offerman and Potters (2006). In this study, subjects assumed the role of decision makers in a differentiated product duopoly, and they were given information on firms' demand curves and marginal costs. Subjects were placed in one of three treatments: one in which rights to operate in the market were auctioned to the two highest bidders; one in which entry rights were allocated randomly to two subjects who were required to pay an upfront fee (23); and one in which entry rights were allocated randomly without payment of a fee. Offerman and Potters find that when an entry fee is charged (either via a fixed fee or by means of an auction), prices are significantly higher than the undistorted (symmetric) Nash equilibrium price, but when no entry fee is charged, prices were close to the undistorted equilibrium price. By contrast, in a monopoly market, prices tend to equal the profit-maximizing monopoly price, irrespective of the level or form of the sunk entry fee.

Our theoretical model gives exactly the same predictions. Our theory implies that a monopolist would have no incentive to succumb to the sunk cost fallacy. In a duopoly with differentiated products but with no sunk costs, firms might ultimately benefit from distorting their relevant costs, but because they lack a compelling justification for distortionary behavior, our theory would predict that they would end up charging the undistorted Nash equilibrium price. When sunk costs are present, firms have the opportunity to distort relevant costs upward, and given the demand curves and marginal costs in the Offerman and Potters environment, it makes sense to do so. Indeed, in this case, the fit between our model and Offerman and Potters' quantitative results is remarkably tight. In the calculations in Table 1 in the previous section, we chose parameters so that when [theta] = 0.80 and N = 2, we have an economic environment that is identical to that in Offerman and Potters's duopoly experiments. (24) As shown in Table 1, the equilibrium price in the two-stage game predicted by our model is 78.18, about 30% higher than the undistorted Nash equilibrium price of 60. In Offerman and Potters's fixed entry fee and auction treatments, the average prices were 75.65 and 72.73, respectively. (25)

Offerman and Potters interpret their results as suggesting that entry fees generate tacit collusion. In fact, their explanation itself relies on a bounded rationality assumption as they study finite repetitions of an oligopoly game which has a unique subgame perfect equilibrium. Moreover, their explanation predicts that tacit collusion should also occur in a Bertrand oligopoly with homogeneous products. This does not coincide with our predictions and hence provides a way of testing our theory against theirs.

In conclusion, we note that Offerman and Potters's experiment is not a formal test of our model. Our model requires separate pricing and costing decisions and a specific timing structure in which adjustments in these decisions occur at different rates. Nevertheless, models should ideally provide robust insights that hold beyond the narrow formalism on which they are based. It is in this sense that we view their results as suggestive of the plausibility of the core intuition of our article.

[] Cournot competition and behavioral biases. The results for Bertrand competition with product differentiation do not transfer to Cournot competition. In contrast to the Bertrand model, firms in the Cournot quantity competition model benefit from becoming more aggressive by producing higher quantities. To develop intuition, assume there are just two firms and firm 1 reduces its output as it suffers from a sunk cost bias. Learning will take place via quantity adjustments, so firm 2 will eventually increase its output and firm 1's economic profit will go down. Thus, in contrast to a price-setting industry, the sunk cost bias is not reinforced in the cost adjustments phase and our model suggests that the sunk cost bias should disappear in a Cournot setting. This provides another refutable implication of our theory that can be tested using experimental or field evidence.

In fact, our model suggests that managers may reinforce a bias to overlook relevant costs in a Cournot setting. In an experimental study, Phillip, Battalio, and Kogut (1991) found evidence that subjects treat opportunity costs as different from direct monetary outlays. In practice, a manager whose firm has purchased an input under a long-term contract may treat the cost of that input as sunk when, in fact, it is variable if the input can be resold in the marketplace. A similar bias can arise if the contract price of the input is less than the current market price, and the manager computes marginal cost based on the historical, rather than current, market price.

[] Alternative explanations: noisy observation of cost, delegation, and evolution. Cost and price distortions may arise as a result of firms observing their total cost but not its split between fixed, sunk, and variable components. In this case, rational firms' estimates of relevant costs are likely to include non-zero errors similar to the distortions introduced in this article. In this case, a plausible scenario is that firms' errors have zero mean, in which case no systematic distortion in relevant costs, prices, or profits would appear.

It also is instructive to compare our model with models of strategic delegation (Fershtman and Judd, 1987) and strategic commitments in general. Strategic commitments matter only to the extent that they are observable and irreversible. By contrast, firms' internal accounting methodologies and their confusions about relevant costs are neither observable nor irreversible. Firms in our model have myopic incentives and arrive at their decisions through a process of naive learning. They do not actively fine-tune their incentive schemes in anticipation of more favorable second-stage equilibrium outcomes. Our model has testable implications that separates it from the delegation framework. First, the delegation model would predict higher prices regardless of the presence of sunk cost, a prediction that can be experimentally tested. Second, the logic of delegation has no bite when prices are set by the firms' owners, as in experiments where each firm is a single subject (so there is literally no delegation). In our model, whether the owner of the firm is the same as the agent setting prices is irrelevant.