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

We assume that [D.sub.n] is differentiable and, wherever [D.sub.n] > 0, satisfies [partial derivative][D.sub.n] / [partial derivative][D.sub.n] [partial derivative][p.sub.n] < 0 and [partial derivative][D.sub.n] / [partial derivative][D.sub.n] [partial derivative][p.sub.n] > 0 for m [not equal to] n. Further, we assume that for any pair of firms n, m, demand is strictly log-supermodular, that is, [[partial derivative].sup.2] log [D.sub..n]([p.sub.n], [p.sub.-n]) / [partial derivative] [p.sub.n] [partial derivative][p.sub.m] > 0. Log-supermodularity is equivalent to the intuitive condition that a firm's demand becomes less price elastic as a rival's price goes up, a property that is satisfied by many common demand systems, including linear, logit, and CES.

Each firm has a constant marginal cost [c.sub.n], with c = ([c.sub.l], ..., [c.sub.N]) [greater than or equal to] 0 denoting the vector of these costs. In addition, each firm has a per-period fixed or sunk cost, denoted [F.sub.n] [greater than or equal to] 0. Our analysis also covers, without any modifications, unsunk flow fixed costs. However, we will use the term sunk cost to avoid redundancy. For example, [F.sub.n] may represent the per-period capital charge for an asset that has no redeployment value. The calculation of [F.sub.n] will depend on the specific accounting standards used by the firm (for instance, how depreciation is computed). We abstract from these issues and take [F.sub.n] as given. In our multiperiod analysis, we also assume that [F.sub.n] is constant over time.

[] Economic versus accounting profits. Firm n's objective measure of success is its economic profit, defined as

[[pi].sup.e.sub.n](p) [equivalent to] ([p.sub.n] - [c.sub.n])[D.sub.n](p) - [F.sub.n].

We model the firm's distorted perception of relevant costs in a way that is both simple and consistent with managerial accounting practices discussed in Section 2.

Introducing the possibility of irrational pricing is delicate, because there is a potentially infinite number of ways a firm can violate the precepts of optimal pricing. Here we introduce irrationality in a tractable and psychologically plausible way that is grounded in common practices. Specifically, we shall assume that a firm may act as though its true marginal cost is inflated by a constant distortion component, [s.sub.n], representing the part of sunk cost (mis)allocated as a per-unit variable cost. We require that

[s.sub.n], [greater than or equal to] 0, and [s.sub.n], = 0 whenever [F.sub.n] = 0.

The second part of this requirement says that firms cannot "make up" sunk costs when there are none. This reflects the idea that a firm must be able to "rationalize" its distortion, that it cannot catch itself in a contradiction. Not doing so would lead to too flagrantly illogical choices that even our irrational firms would be able to identify.

A concrete example in which these assumptions are satisfied is as follows. Imagine that firm n bases its cost allocation on a budgeted quantity [[??].sub.n], (see Section 2). In practice, [[??].sub.n], is based on past realized quantities, and is therefore independent of the firm's current decisions. For now, we take [[??].sub.n] > 0 as exogenously given. In footnote 20, we explain how its choice can be made endogenous after the full learning model is introduced. Given [[??].sub.n], the firm's accounting profit is

[[pi].sub.n](p, [s.sub.n]) [equivalent to] ([p.sub.n] - [c.sub.n] - [s.sub.n]) [D.sub.n](p) - ([F.sub.n] - [s.sub.n] [[??].sub.n), (1)

where ([F.sub.n] - [s.sub.n] [[??].sub.n]) denotes the unallocated sunk cost. The accounting profit function [[pi].sub.n] represents the manager's perception of the relevant payoff function within a given round of short-term price adjustments. Note that in any steady state, budgeted and actual quantities coincide, in which case the difference between economic and accounting profits, [s.sub.n]([D.sub.n].(p) - [[??].sub.n].), disappears. A firm that chooses [s.sub.n]. so that

0 [less than or equal to] [s.sub.n] [less than or equal to] [F.sub.n] / [[??].sub.n]

would satisfy the requirements on [s.sub.n]. stated earlier. Note that we do not require that firms allocate all of their sunk cost. This may be justified in terms of the arbitrariness and flexibility in costing methodologies discussed in Section 2.

[] The static price competition game. Formally, a price competition game [GAMMA](s) with cost distortions s is an N-player game, each with strategy set P and payoffs function given by the accounting profit function defined in equation (1).

In [GAMMA](s), firm n maximizes [[pi].sub.n](*, [s.sub.n]) given a forecast [p.sub.-n], of its competitors's prices, yielding a first-order condition

([p.sub.n] - [c.sub.n] - [s.sub.n]) [partial derivative][D.sub.n](p) / [partial derivative] [p.sub.n] + [D.sub.n](p) = 0, (2)

provided [D.sub.n](p) > 0. This gives rise to firm n's reaction function (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Our assumptions on demand imply that [r.sub.n].(*, [s.sub.n].): [P.sup.N] [right arrow] P is differentiable on the interior of [P.sup.N-1] and strictly increasing:

[for all] n [not equal to] m, [partial derivative][r.sub.n] / [partial derivative][p.sub.m] > 0.

Let r(*, s) : [P.sup.N] [right arrow] [P.sup.N] denote the vector of reaction functions. A (Nash) equilibrium for [GAMMA](s) is a price vector [bar.p] such that [bar.p] = r([bar.p], s). Let [epsilon](s) denote the set of equilibria for [GAMMA](s), which we assume to be compact and contained in the interior of a cube P = [0, [p.sup.+]] [subset] [R.sup.N] for every vector of distortions s.

Given the assumption that demand is log-supermodular, the price-setting game [GAMMA](s) will be supermodular, which implies that a pure strategy equilibrium exists (Milgrom and Roberts, 1990). Our assumptions do not, however, rule out the possibility of multiple equilibria. Note that any equilibrium p such that all firms are active (i.e., [bar.q] >> 0) must satisfy [bar.p] >> c + s.

4. Main results

* Adaptive learning and comparative statics. We first define adaptive pricing processes in the game [GAMMA](s) repeatedly played by myopic players.

Definition 1. An adaptive pricing adjustment for [GAMMA](s) starting at [bar.p] is a sequence of prices {[p.sub.t],} such that [P.sub.0] = [bar.p] and there is a positive integer [gamma] such that for every t, [p.sub.t] satisfies (l3)

r(inf{[p.sub.t-[gamma]], ..., [p.sub.t-1]}, s) [less than or equal to] [p.sub.t] [less than or equal to] r(sup{[p.sub.t-[gamma]], ..., [p.sub.t-1]}, s).

That is, at time t, each firm best responds to some probability distribution on the recent past history of play of their opponents. The class of adaptive pricing adjustment process is quite broad, and includes, as special cases, the Cournot dynamic, (14) a version of fictitious play in which only the past [gamma] rounds of play are taken into account, and sequential best response.

Learning plays two crucial roles in our analysis. First, it is implausible that firms that succumb to the sunk cost fallacy have the sophistication necessary to carry out the complex reasoning justifying Nash equilibrium play. The alternative view, which we take here, is that a Nash equilibrium in the pricing game is the result of a naive adaptive process carried out by boundedly rational players. Second, although a positive cost distortion will shift the equilibrium set upward, (15) equilibrium analysis is consistent with the selection of a lower equilibrium price within the new set. In this case, arbitrary equilibrium selection, rather than fundamentals, ends up driving the persistence of cost distortions. Theorem 1 shows that learning can serve as a foundation for meaningful comparative statics when multiple equilibria may be present. That theorem builds on the following.

Proposition 1. Suppose that [bar.p] [less than or equal to] r([bar.p], s). Then, any price adjustment process beginning at [bar.p] will settle at the lowest equilibrium of [GAMMA](s) higher than [bar.p]. Formally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any adaptive pricing adjustment sequence {[p.sub.t]} for [GAMMA](s) starting at [bar.p].

The proof is essentially that of Theorem 3 in Echenique (2002) specialized to the case where best replies are single-valued. (16) The comparative statics implications of this result is that starting with an equilibrium price [bar.p], if one firm distorts its cost, then any adaptive pricing adjustment must converge to the lowest equilibrium price vector of [GAMMA](s) that is higher than [bar.p]. Note that this result yields an unambiguous prediction on the direction of price changes, but is silent on the effect on economic profits, which is our primary concern. This is dealt with by our first theorem.

* Distortion result.

Theorem 1. There exists [DELTA] > 0 such that given any firm n, vector of distortions [[bar.s].sub.-n] [greater than or equal to] 0, equilibrium price vector [bar.p] [GAMMA] ([[bar.s].sub.-n], [s.sub.n] = 0) with corresponding quantity vector [bar.q] > > 0, if firm n chooses 0 < [[??].sub.n] < [DELTA], then for any adaptive pricing adjustment {[p.sub.t]} for [GAMMA] ([[bar.s].sub.-n], [[??].sub.n]) starting at p there is a time T such that

[[pi].sup.e.sub.n]([p.sub.t]) > [[pi].sup.e.sub.n]([bar.p]) for every t [greater than or equal to] T.

Proofs of this and subsequent results are in the Appendix.