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

SMITH, V. "Rational Choice: The Contrast between Economics and Psychology." In Bargaining in Market Experiments, ed. V. Smith, Cambridge UK: Cambridge University Press, 2000.

THALER, R. "Towards a Positive Theory of Consumer Choice." Journal of Economic Behavior and Organization, Vol. 1 (1980), pp. 39-60.

VIVES, X. Oligopoly Pricing. Cambridge, MA: MIT Press, 1999.

(1) To cite one example, Shank and Govindarajan (1989) admonish managers to avoid committing the "Braniff fallacy," the practice (named after the now-defunct Braniff Airlines) of being content to sell seats at prices that merely cover the incremental cost of the seat. They state that "looking at incremental business on an incremental cost basis will, at best, incrementally enhance overall performance. It cannot be done on a big enough scale to make a big impact. If the scale is that large, then an incremental look is not appropriate!" (emphasis in original). Shank and Govindarajan make it clear that they believe that managers should incorporate fixed and sunk costs into pricing decisions, arguing that "business history reveals as many sins by taking an incremental view as by taking the full cost view."

(2) A particularly vivid example is Edgar Bronfman, former owner of Universal Studios, who criticized the movie industry's pricing model because ticket prices do not reflect differences in the sunk production costs of different movies. "He ... observed that consumers paid the same amounts to see a movie that costs $2 million to make as they do for films that cost $200 million to produce. 'This is a pricing model that makes no sense, and I believe the entire industry should and must revisit it.'" Wall Street Journal, April 1, 1998.

(3) An indirect indication of the inertia involved in the selection of costing systems comes from the empirical literature on the adoption of activity-based costing (ABC). ABC is a set of practices for assigning costs to products in a multiproduct firm that is widely believed to be superior to traditional methodologies for allocating common costs. Beginning in the mid-1980s, academics and consultants began touting the virtues of ABC, but available evidence suggests that the adoption of ABC in the mid to late 1990s was not widespread (see, for example, Brown, Booth, and Giacobbe, 2004; Roztocki and Schultz, 2003). Among the factors that have slowed the adoption of ABC systems include the need to make significant new investments in information technology in order to implement an ABC system and the need to obtain acceptance from key managers to support the transition to an ABC system.

(4) In the pricing game, we assume that each firm knows its own demand and the recent history of its rivals' prices.

(5) For example, in a well-known experiment (see Arkes and Ayton, 1999), theater season tickets were sold at three randomly selected prices. Those charged the lower price attended fewer events during the season. Apparently, those who had "sunk" the most money into the season tickets were most motivated to use them.

(6) See Kaplan and Atkinson (1989) for a detailed discussion of full-cost pricing.

(7) This argument may seem strange to economists who might wonder why it is not possible for the firm's managers to price to maximize profits based on standard marginal reasoning and to simply do a "side calculation" to determine whether that profit-maximizing price allows the firm to recover its total costs, including the rental rate on capital.

(8) This emphasis on flexibility and "sticking with what works" is consistent with theoretical models of reinforcement learning.

(9) Sophisticated firms often build flexibility into their budgeting. This flexibility allows the firm to adjust the values of key variables to reflect unexpected shocks (e.g., abrupt increase in the cost of a raw material, labor strikes, and so on).

(10) See Maher, Stickney, and Weil (2004) for a thorough discussion of variances.

(11) We use the following standard vector notation. For any vector x [member of] [R.sup.l], x = ([x.sub.1], ..., [x.sub.l]) and [x.sub.-n] [member of] [R.sup.l-1] denotes the vector ([x.sub.1], ..., [x.sub.n-1], [x.sub.n+]), ..., [x.sub.l]). Also, x [greater than or equal to] 0 means that [x.sub.j] [greater than or equal to] 0 for all j; x > 0 means that [x.sub.j] [greater than or equal to] 0 for all j and x [not equal to] 0; and x > >0 means that [x.sub.j] > 0 for all j.

(12) Our assumptions on demand imply that this is well defined and single valued.

(13) For t < [gamma] we set [p..sub.t-[gamma]] = [p.sub.0].

(14) Under the Cournot dynamic, firms best respond to their opponents' last period price (see Vives, 1999 for instance).

(15) In the sense that the smallest (largest) equilibrium price vector of [GAMMA](s') is larger than the smallest (largest) equilibrium price vector of [GAMMA](s). See Milgrom and Roberts (1990).

(16) As r is a strictly increasing reaction function, the lowest and highest selections from r are equal. Also, the lowest and highest selections from iterated applications of r are also equal. The proof of part 2 of Theorem 3 in Echenique (2002) shows that these selections define the lower and upper bounds of the limit points of any adaptive dynamic. Hence, as r is a best-reply function, this limit is unique. Finally, part 1 of Theorem 3 shows that this limit must be equal to the lowest equilibrium higher than [bar.p].

(17) Most adaptive learning models assume that firms know their own payoff function and the actions taken by their opponents. We are aware of only a few models that can handle the case of unobserved opponents' actions and unknown own-payoff function, all of which rely on players' randomly experimenting with different strategies. Milgrom and Roberts (1991) and Hart and Mas-Colell (2000) describe adaptive procedures that eventually lead to the play of only serially undominated strategies and correlated, equilibrium respectively. Friedman and Mezzetti (2001) propose a procedure that leads to the play of a Nash equilibrium in supermodular games (as well as other classes of games with a property called weak finite improvement property). The problem is that supermodularity of the costing methodology game requires stringent, difficult to interpret conditions on the underlying model. In particular, it does not necessarily follow from the supermodularity of the pricing game.

(18) Thus, formally, we have a double index [{[[tau].sub.t]}.sup.[infinity].sub.[tau],t=1] and where indices are ordered lexicographically: ?? > [[tau].sub.t] if and only if [tau]' > [tau] or [[tau]' = [tau] and t' > t].

(19) The assumption that p(0) is an equilibrium of [GAMMA](s(0)) can be relaxed, but not entirely dropped. For example, Theorem 2.10 (ii) in Vives (1999) shows that this assumption is unnecessary under the Cournot dynamic if one starts with an initial price that is lower (higher) than the lowest (highest) equilibrium price of [GAMMA](s = 0).

(20) With this notation, we can now endogenize the choice of the reference quantity [[??].sub.n] used to determine the per-unit sunk cost [F.sub.n]/[[??].sub.n] that provides an upper bound on [[s].sub.n]. Formally, we set [[??].sub.n]([tau]) = [[q].sub.n] ([tau] - 1) for [tau] > 1 and set [[??].sub.n] (0) > 0 arbitrarily.

(21) See Section 6 for a more detailed discussion.

(22) The parameters for these calculations are as follows: a = 310, b = 350/252, and c = 10.

(23) Unbeknownst to the subjects, the fixed entry fees were set equal to the entry fees generated in the auction treatment.

(24) If we set a = 310, b = 350/252, and c = 10, then when [theta] = 0.80 and N = 2, the firms' demand curves are

[D.sub.1]([p.sub.1], [p.sub.2]) = 124 - 2[p.sub.1] + 1.6[p.sub.2] and [D.sub.2]([p.sub.1], [p.sub.2]) = 124 - 2[p.sub.2] + 1.6[p.sub.1],

exactly as in the Offerman and Potters experiment.

(25) These calculations are based on the data reported in Table 1 in Offerman and Potters.

(26) The special issue of the Journal of Economics Theory, 2001, vol. 97 is devoted to this line of research.

(27) The only other model we are aware of that studies the effect of the sunk cost bias in differentiated Bertrand competition is by Parayre (1995). He considers a two-stage game in a linear environment similar to the special case of symmetric linear demand we study in Section 5 (in his model, no naive learning justification of behavior is considered). Parayre assumes that the sunk cost bias has the implication that firms produce in order to use up whatever capacity is available to them. Compared to the standard Bertrand equilibrium, Parayre's model leads to lower prices and profits. Our model sheds light on the form of distortion that is likely to persist in a competitive environment. Although the form of distortion we propose will be perpetuated in the long run, our model predicts that Parayre's form of distortion must disappear.

(28) The minimum is achieved by the compactness of [B.sub.[alpha]] and ([s.sub.k.sub.-n], 0).

(29) This is the well-known Doob decomposition. See, for instance, Chung, (1974).

(30) E([Y.sub.[tau]+1] | [Y.sub.[tau]]) = E([y.sub.[tau]+1] + [Y.sub.[tau]] | [Y.sub.[tau]]) = E([Y.sub.[tau]+1] | [Y.sub.[tau]]) = E(E([x.sub.[tau]+1] | [Y.sub.[tau]]) | [Y.sub.[tau]]) + [Y.sub.[tau]] = [Y.sub.[tau]].

Nabil Al-Najjar *

Sandeep Baliga *

and

David Besanko *