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

[LAMBDA] = [partial derivative][[bar.p].sub.n]/[partial derivative][s.sub.n] {-[beta] + [[(N - 1).sub.[gamma]] + [(N - 1).sup.2][gamma][xi]] [partial derivative][[bar.p].sub.m]/[partial derivative][s.sub.n]}.

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. because [partial derivative][[bar.p].sub.n]/[partial derivative][s.sub.n] > 0 and [partial derivative][[bar.p].sub.m]/[partial derivative][s.sub.n] = [beta]/2[beta] + [gamma] [gamma]/2[beta]-[(N - 1).sub.[gamma]], [LAMBDA] < 0 if and only if

[(N - 1) + [(N - 1).sup.2][xi]] [[gamma].sup.2]/(2[beta] + [gamma])(2[beta] - (N - 1)[gamma]) < 1. (A6)

Now, straightforward algebra establishes that

[[gamma].sup.2]/(2[beta] + [gamma])(2[beta] - (n - 1)[gamma]) = ([theta]/(N - 1)[theta] + 2(1 - [theta])) ([theta]/(2N - 1)[theta] + 2(1 - [theta])).

Furthermore [xi] < 1, so (N - 1) + [(N - 1).sup.2] [xi] (N - 1) + [(N - 1).sup.2] = N(N - 1). Thus

[(N - 1) + [(N - 1).sup.2][xi]] [[gamma].sup.2]/(2[beta] + [gamma])(2[beta] - (N - 1)[gamma]) <((N - 1)[theta]/(N - 1)[theta] + 2(1 - [theta])) (N[theta]/(2N - 1)[theta] + 2(1 - [theta])).

But N < 2N - 1, because N > 1. Thus

((N - 1)[theta]/(N - 1)[theta] + 2(1 - [theta])) (N[theta]/(2N - 1)[theta] + 2(1 - [theta])) < ((N - 1)[theta]/(N - 1)[theta] + 2(1 - [theta])) (2N - 1)[theta]/(2N - 1)[theta] + 2(1 - [theta])).

Because each term in the above expression is positive but less than one, we immediately have

((N - 1)[theta]/(N - 1)[theta] + 2(1 - [theta])) (2N - 1)[theta]/(2N - 1)[theta] + 2(1 - [theta])) < 1,

which establishes that [LAMBDA] < 0. The equilibrium in [s.sub.n] is thus unique, and because firms are symmetric, the equilibrium distortion will be symmetric across firms.

Proof of Proposition 4.

Proof(i). We begin by noting that [theta] = 0 [??] [gamma] = 0 and [beta] = 1/b. Using this, it is straightforward to verify that [lim.sub.[theta] [right arrow] 0 = [partial derivative][[bar.[pi]].sup.e.sub.n]/[partial derivative][s.sub.n] - [s.sub.n]/2b < 0. Hence, no firm has an incentive to choose a positive distortion. Proof(ii). It is straightforward to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now let [s.sup.*](1) = [lim.sub.[theta] [right arrow]] 1 [s.sup.*] and [p.sup.*](1) = [lim.sub.[theta] [right arrow] 1 [p.sup.*]. Taking limits of each side of(8) and (9), and using the expressions above, we have

[s.sup.*](1) = ([p.sup.*](1) - c)/N

[p.sup.*](1) = c + [s.sup.*](1).

These can only be satisfied if [s.sup.*] (1) = 0 and [p.sup.*] (1) = c.

Proof(iii). It is straightforward to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking limits of(8) and using the above expressions implies that [s.sup.*] ([infinity]) = 0.

Proof of Proposition 5. If [theta] [member of] (0, 1), then [p.sup.0] > c. Furthermore, it is straightforward to show that

(N - 1)[[theta].sup.2]/[(2 - [theta]) + (N - 1)[theta](1 - [theta])][(1 - [theta]) + (N - 1)[theta]] > 0.

which implies that [s.sup.*] > 0. When [theta] [member of] (0, 1),

(2 - [theta]) + (N - 1)[theta]/(2 - [theta]) + (N - 1)[theta](1 - [theta]) > 1.

Thus, [p.sup.*] - c > [p.sup.0] - c, or [p.sup.*] > [p.sup.0].

Proof of Proposition 6. The expression for [p.sup.*] - c can be written as follows:

[p.sup.*] - c = (a - c)(l - [theta])y(N)z(N),

where

y(N) = 1/2 - [theta] + (N - 1)[theta](1 - [theta])

z(N) = 2 - [theta] + (N - 1)[theta]/2 - [theta] + (N - 1)[theta].

Clearly Y(N) is strictly decreasing in N and so too is z(N):

z'(N) = - [[theta].sup.2]/[[2(1 - [theta]) + (N - 1)[theta]].sup.2] < 0.

Hence [p.sup.*] - c is strictly decreasing in N.

Proof of Proposition 7. The monopoly price [p.sup.M] satisfies

[p.sup.M] - c = a - c/2.

From (10), [p.sup.*] - c can be written as

[p.sup.*] - c = (a - c)(1 - [theta])[2 - [theta] + (N - 1)[theta])]/ [2 - [theta] + (N - 1)[theta](1 - [theta])][2(1 - [theta]) + (N - 1)[theta]].

[p.sup.*] < [p.sup.M] if and only if

(1 - [theta])[2 - [theta] + (N - 1)[theta])]/ [2 - [theta] + (N - 1)[theta](1 - [theta])][2(1 - [theta]) + (N - 1)[theta]] < 1/2. (A7)

This can be rewritten as

[2 - [theta] + (N - 1)[theta](1 - [theta])][2(1 - [theta]) + (N - 1)[theta]] > 2(1 - [theta])[2 - [theta] + (N - 1)[theta]].

Now, for N [greater than or equal to] 3, [2(1 - [theta]) + (N - 1)[theta]] [greater than or equal to] 2 because [theta] [member of] (0, 1) and N - 1 [greater than or equal to] 2. Hence, we have

[2 - [theta] + (N - 1)[theta](1 - [theta])][2(1 - [theta]) + (N - 1)[theta]] > 2[2 - [theta] + (N - 1)[theta](1 - [theta])] > 2(1 - [theta])[2 - [theta] + (N - 1)[theta]].

For N = 2, the inequality (A7) boils down to

1 - [theta]/2 - [theta] < 2 - [[theta].sup.2]/4,

which can be shown to hold for [theta] [member of] (0, 1). Hence, (A7) holds for all N [greater than or equal to] 2, establishing [p.sup.*] < [p.sup.M].

Proof of Proposition 8. These results follow directly from expression (11).

We thank Martin Cripps, Ronald Dye, Federico Echenique, Jeff Ely, Claudio Mezzetti, Bill Sandholm, Lars Stole, Beverly Walther, the editor, Joe Harrington, and two referees for helpful comments. We also thank seminar participants at Cambridge, Davis, LSE, NBER, Oxford, UCL, Pompeu Fabra, IESE, UCSD, and the Second Annual Foundations of Business Strategy Conference at Olin for their comments.

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