Structural screens in stochastic markets.

I. Introduction

Antitrust policy in the U.S. has evolved into a sophisticated process that takes into account a variety of relevant factors, including potential entry and other dynamic or conduct-oriented considerations, in accordance with modern economic theory. However, simple structural tests are still used as an initial screen to identify those merger cases requiring more detailed analysis. This emphasis on market concentration, rooted in the structure-conduct-performance paradigm of Mason and Bain, is spelled out in written guidelines of the U.S. Department of Justice (DOJ) and has been reiterated in recent public statements by DOJ officials.(1) Moreover, a resurgence of academic support for structure-based antitrust policy has been evident in some recent literature [1].

This structural emphasis means at a minimum that relatively more DOJ resources are deployed in the scrutiny of large mergers in concentrated markets, compared with other mergers. Such scrutiny can also consume resources from the merging firms. Moreover, even apart from involvement in a merger, a firm may be subject to closer scrutiny and a higher probability of sanctions or challenges as an increasing function of its market share. For example, a firm with a large market share may face a higher probability of being charged with collusive pricing or predatory pricing (the "lightning rod" effect). The policy of awarding treble damages makes a firm's expected costs of such charges non-negligible under any positive probability of being found liable.

Since there are social costs to this sort of scrutiny, the optimal policy would attempt to minimize unnecessary scrutiny and focus as much of the scrutiny as possible on those firms most likely to be in violation of the law. The purpose of the DOJ's initial structural screen is precisely such optimization. This paper explores the theoretical implications of uncertainty in demand and cost on the efficacy of the structural screen in attaining that purpose.

A long-standing literature maintains that competitive behavior results in more volatile market shares than would be observed in a collusive market [4; 5; 8; 9; 11; 12]. In a market of similar firms, this hypothesis would imply that competition induces a mean-preserving spread in the distribution over time of each firm's market shares, in the sense of Rothschild and Stiglitz. This, in turn, would imply a greater probability that a firm's market share will exceed a given high threshold in any given period, and hence a greater probability that a competitive firm might fail the DOJ's structural screen and draw closer scrutiny with its attendant costs than would a collusive firm. Thus, stochastic effects might tend to undermine the validity of a structure-based antitrust policy or the efficacy of structural screens as currently applied.

We examine this argument in a simple framework of duopoly with linear demand and cost. This simple framework is chosen because it allows explicit solutions and, consequently, sharp results. At the same time, the model is formulated in terms of general patterns of symmetric conduct and general statistical distributions of the random components, with some additional results derived for the special case in which the stochastic factors vary independently across firms. The parameterization of conduct permits the model to characterize outcomes for firms exhibiting any degree of collusion from the perfectly competitive or contestable extreme to joint monopoly, whether sustained by explicit collusion or by an implicit dynamic enforcement mechanism as in Friedman [7]. In either extreme, of course, it is well known that pricing and output levels are independent of structure; in such cases, a structural antitrust policy would have no impact on social welfare apart from possible deadweight losses or the stochastic effects analyzed here. For certain intermediate patterns of conduct, such as Cournot, it is well known that structure is monotonically related to equilibrium pricing and output levels; in these cases, stochastic effects could still distort the impact of a structural screen or antitrust policy.

Stochastic marginal costs, demand slopes, and demand intercepts are analyzed separately. We assume in each case that firms are able to observe the realizations of the stochastic variables before making their production decisions. For quantity-choosing risk-neutral firms, this is the only sequence by which a stochastic mechanism could affect market shares, and is therefore the only sequence relevant to our analysis of structural screens. Regardless of which variable is stochastic, the sign of the effect of a structural screen is shown to depend on parameter values. For homogeneous goods and independently distributed marginal costs, we show that stochastic shifts in marginal cost render the likelihood of failing the structural screen a decreasing function of the degree of market power, consistent with the argument outlined above; this result undermines the validity of structure-based antitrust.

When products are differentiated, the probability of failing the structural screen is in general a function of the degree of differentiation. In the special case of independently distributed stochastic marginal costs, the probability of failing the structural screen is an increasing function of the degree of substitutability of the products of the two firms, when products are gross substitutes (equivalently, when firms compete in the same product market) and the threshold market share is sufficiently high. The outcome of structure-based antitrust in this case is again detrimental, since for a given pattern of conduct the degree of competition between two firms will be higher if their products are close substitutes. Thus, a structure-based antitrust policy will selectively attack those firms that are more direct rivals, constituting a second way in which stochastic costs subvert the validity of the policy.

When the demand slope is stochastic, the probability of failing the structural screen is an increasing function of the degree of substitutability of products, for a sufficiently high threshold. This result, which does not require statistical independence of firm shocks, likewise undermines the validity of a structure-based antitrust policy. In other cases, the probability of failing the structural screen is an increasing function of the degree of competition between firms. Similar results are derived when the demand intercept varies across firms. All such cases further undermine the validity of a structure-based antitrust policy when products are differentiated. When products are homogeneous, random shifts in demand would affect both firms equally and have no effect on the incidence of a structural policy.

The next section presents the homogeneous case with stochastic marginal cost, while section III introduces product differentiation. Section IV examines stochastic demand in the differentiated case, and section V concludes.

II. The Homogeneous Case with Stochastic Costs

The calculations that follow require an explicit solution for each firm's equilibrium output level. Rather than rely on strictly numerical methods, we adopt a simple linear duopoly framework to permit a closed-form analytical solution. Hence, the results are illustrative rather than exhaustive.

In this section we focus on the relationship between the pattern of firm conduct or pricing and the effect of random marginal costs on the firm's likelihood of incurring punitive regulatory costs in the context of a structure-based antitrust policy. To do this, we abstract temporarily from issues of product differentiation and stochastic demand; we return to these issues in subsequent sections. Pure cost shocks could arise either from exogenous changes in input prices or from stochastic process innovation, for example. The possibility of asymmetric cost shocks across firms is needed to produce nondegenerate market share effects. Although input price changes might be expected to have symmetrical effects on two firms operating in the same input markets, different choices of technology by each firm could result in unequal input mixes and, consequently, unequal marginal cost shocks from equal changes in the input price vector. Likewise, if firms negotiate multi-period contracts with their suppliers, then differences in the timing of contracts may lead to variations in input prices across firms, even within a single input market; such effects are not explicitly modeled in the static framework used here, but may underlie some of the assumed interfirm asymmetries in practice.

Consider a homogeneous duopoly with firms i and j, each facing stochastic marginal costs and able to adjust its output level after observing the realized marginal cost. The profit for firm i is given as:

[[Pi].sub.i] = [x.sub.i](a - b[x.sub.i] - b[x.sub.j] - [c.sub.i]) (1)

implying the first-order condition for profit maximization:

0 = a - (2 + r)b[x.sub.i] - b[x.sub.j] - [c.sub.i] (2)

where [c.sub.i] is a positive random variable bounded above such that the equilibrium output is always positive (i.e., [c.sub.i] [less than] a) and r is the conjectural variation, which parameterizes the oligopoly solution concept (equivalently, the equilibrium markup or degree of competition) [15, 19; 14, 245; 3, 1027]. This representation of the firm has also been called the "quasi-Cournot" model [13]. Perfectly competitive conduct implies r = -1, Cournot equilibrium is given by r = 0, and joint monopoly is given in the symmetric duopoly by r = + 1 [6]. In the special case of homogeneous products with constant marginal cost, r = -1 for the Bertrand equilibrium as well [2, 937]. The second-order condition for profit maximization is satisfied since -b(3 + r) [less than] 0. Throughout the analysis, we are only interested in the case in which both firms may earn nonnegative profits at positive output levels, and accordingly assume that the demand intercept a is greater than both [c.sub.i] and [c.sub.j].

To focus on the impact of purely stochastic influences rather than of systematic differences across firms, we assume that r is identical for each firm so that both firms exercise the same degree of market power. If we allowed r to vary across firms, the equilibrium conditions would imply that the firm with the smaller (i.e., more competitive) value of r would produce the larger output quantity, contrary to the assumptions underlying a structure-based antitrust policy.

Combining (2) with the corresponding condition for firm j implies:

[x.sub.i] = [a - [c.sub.j] - (2 + r)(a - [c.sub.i])]/{b[1 - [(2 + r).sup.2]]} (3)

[x.sub.i]/[x.sub.j] = [(1 + r)a + [c.sub.j] - (2 + r)[c.sub.i]]/[(1 + r)a + [c.sub.i] - (2 + r)[c.sub.j]]. (4)

The assumed upper bounds on [c.sub.i] and [c.sub.j] ensure that both the numerator and denominator are positive in (4).

Equations (1) through (4) provide a framework within which to model the effects of concern. As noted in the introduction, the regulatory emphasis on market structure will tend to make a firm's likelihood of incurring punitive regulatory costs (which may be some combination of direct costs and opportunity costs, as when an intended merger is prohibited) an increasing function of its market share. The simplest way to model this phenomenon is to postulate a fixed threshold [Theta] [element of] (0, 1) such that the firm fails the structural screen if its market share exceeds that threshold; this approach is also consistent with U.S. horizontal merger guidelines, which specify structural thresholds above which closer scrutiny is performed and the merger may be challenged. The market share of firm i will exceed [Theta] if and only if [x.sub.i]/[x.sub.j] exceeds z [equivalent to] [Theta](1 - [Theta]). The probability of failing the structural screen is thus 1 - [F.sub.z](z) where Z [equivalent to] [x.sub.i]/[x.sub.j], and this probability changes with the degree of market power as -[Delta][F.sub.z](z)/[Delta]r.

Both the probability of failing the screen and the sensitivity of that probability to the conduct parameter can be interpreted either in a time-series sense for an individual firm or industry, or in a cross-sectional sense in comparing different markets or industries. The relevant calculations can be carried out without addressing the further question of whether market power increases the stability over time of each firm's market share, as assumed in the literature cited above.

To find [F.sub.z](z), note first that Z [less than or equal to] z (or firm i passes the screen) if and only if [c.sub.i] [greater than or equal to] H where

H = {a(1 - z)(1 + r) + [c.sub.j][1 + z(2 + r)]}/(z + 2 + r), (5)

substituting from (4). If either r = -1 or z = 1, then H = [c.sub.j] so that a low-cost firm ([c.sub.i] [less than] [c.sub.j]) would always fail the structural screen - clearly an undesirable outcome. If r [greater than] -1, then H [greater than] [c.sub.j] for all z [less than] 1 since a [greater than] [c.sub.j]; however, we need not worry about this case since we may assume z [greater than] 1 for duopoly. If r [greater than] -1 and z [greater than] 1, then H can take either sign depending on the relative magnitudes of a and [c.sub.j].

In general, where f([c.sub.i], [c.sub.j]) denotes the joint marginal density functions of marginal costs for firms i and j, we obtain:

[F.sub.z](z) = [integral of] [[integral of] f([c.sub.i], [c.sub.j])d[c.sub.i] between limits [infinity] and H] between limits [infinity] and -[infinity] d[c.sub.j] (6)

from which:

-[Delta][F.sub.z](z)/[Delta]r = (1 - [z.sup.2]) [integral of] (a - [c.sub.j])f(H, [c.sub.j])d[c.sub.j]/[(z + 2 + r).sup.2] (7)

which has the opposite sign as the integral expression since we may assume z [greater than] 1 for duopoly. Since a [greater than] [c.sub.j] in the relevant case considered throughout, and f (H, [c.sub.j]) is always nonnegative and is strictly positive for a range of values, the integrand is everywhere nonnegative and is strictly positive over a range of values so that the integral itself is positive. Then -[Delta][F.sub.z](z)/[Delta]r [less than] 0, so a duopolist with more collusive conduct and pricing is less likely to fail the structural screen merely from the impact of random cost changes. This result undermines the validity of a structural screen or, more generally, of a structure-based antitrust policy.

Note that the conduct parameter r enters equation (7) only in the denominator, in such a way that a smaller (more competitive) value of r increases the sensitivity of [F.sub.z](z) to r. That is, not only does a structural screen exhibit an anticompetitive bias when -[Delta][F.sub.z](z)/[Delta]r [less than] 0, but that bias is worse when firms' conduct is the most competitive. Even if the opposite case could arise, in which -[Delta][F.sub.z](z)/[Delta]r [greater than] 0 so that the structural screen statistically reveals market power, that statistical test would be less sensitive for high (collusive) levels of r.

III. Stochastic Costs in a Differentiated Duopoly

In this section we relax the assumption of homogeneity and examine the probability that a firm's market share exceeds a given threshold, as a function of the degree of substitutability or product differentiation. The inverse demand for firm i becomes:

[P.sub.i] = [a.sub.i] - [b.sub.i][x.sub.i] - [Gamma][x.sub.j] (8)

and similarly for firm j. Here the parameter [Gamma] indexes the degree of substitutability of goods i and j, where [Gamma] [greater than] 0 for substitutes, [Gamma] = 0 if the two goods are independent in demand, and [Gamma] [less than] 0 for complements. Analogous first-order conditions to (2) can be derived for each firm and solved jointly for each firm's output, as in (3), to obtain:

[x.sub.i] = [([a.sub.i] - [c.sub.i])(2[b.sub.j] + [Gamma]r) - [Gamma]([a.sub.j] - [c.sub.j])]/[(2[b.sub.i] + [Gamma]r)(2[b.sub.j] + [Gamma]r) - [[Gamma].sup.2]] (9)

and similarly for [x.sub.j], implying:

[x.sub.i]/[x.sub.j] = [([a.sub.i] - [c.sub.i])(2[b.sub.j] + [Gamma]r) - [Gamma]([a.sub.j] - [c.sub.j])]/[([a.sub.j] - [c.sub.j])(2[b.sub.j] + [Gamma]r) - [Gamma]([a.sub.i] - [c.sub.i])]. (10)

Proceeding as in the homogeneous case, note that [x.sub.i]/[x.sub.j] [less than or equal to] z if and only if

[c.sub.i]([Gamma]z + [Gamma]r + 2[b.sub.j]) [greater than or equal to] [a.sub.i]([Gamma]z + [Gamma]r + 2[b.sub.j]) - (2[b.sub.i]z + [Gamma]rz + [Gamma])([a.sub.j] - [c.sub.j]) (11)

which takes two cases depending on the sign of [Gamma]z + [Gamma]r + 2[b.sub.j]. This latter term is positive if [x.sub.i] and [x.sub.j] are substitutes, which is the only policy-relevant case since antitrust is concerned with rivals in the same product market (that is, where 0 [less than] [Gamma] [less than or equal to] [b.sub.i]). Alternatively, [Gamma]z + [Gamma]r + 2[b.sub.j] is positive if the threshold [Theta] is sufficiently low (in particular, if [Theta] [less than] (2[b.sub.j] + [Gamma]r)/(2[b.sub.j] + [Gamma]r - [Gamma]), implying a small value of z) even if [x.sub.i] and [x.sub.j] are complements ([Gamma] [less than] 0). We therefore analyze only the case where [Gamma]z + [Gamma]r + 2[b.sub.j] [greater than] 0. Then the required condition to obtain [x.sub.i]/[x.sub.j] [less than or equal to] z is [c.sub.i] [greater than or equal to] H where H is now:

H = [a.sub.i] - (2[b.sub.i]z + [Gamma]rz + [Gamma])([a.sub.j] - [c.sub.j])/[Gamma]z + [Gamma]r + 2[b.sub.j]). (12)

The probability that the market share of firm i exceeds the threshold [Theta] is given by:

1 - [F.sub.z](z) = 1 - [integral of] [[integral of] f([c.sub.i], [c.sub.j]) d[c.sub.i] between limits [infinity] and H] between limits [infinity] and -[infinity] d[c.sub.j]. (13)

The impact of [Gamma] on the probability of failing the structural screen is given by:

-[Delta][F.sub.z](z)/[Delta][Gamma] = 2[[b.sub.i](rz + [z.sup.2]) - [b.sub.j](rz + 1)] [integral of] ([a.sub.j] - [c.sub.j])f(H, [c.sub.j]) between limits [infinity] and -[infinity] d[c.sub.j]/[([Gamma]z + [Gamma]r + 2[b.sub.j]).sup.2]. (14)

The expression [b.sub.i] (rz + [z.sup.2]) - [b.sub.j] (rz + 1) is positive for sufficiently high thresholds. For example, if [b.sub.i] = [b.sub.j], the condition is [Theta] [greater than] 1/2, which we expect to characterize any structural screen applied to duopoly. In the Cournot equilibrium with r = 0, the condition is [Theta] [greater than] [([b.sub.j]/[b.sub.i]).sup.1/2]/[1 + [([b.sub.j]/[b.sub.i]).sup.1/2]]. In such cases, equation (14) takes the sign of the integral, which is positive since [a.sub.j] [greater than] [c.sub.j] in all cases considered.

Thus, the result of this section is that a structure-based antitrust policy will generally be biased by the degree of product differentiation when marginal costs are stochastic. If (1) goods are substitutes, (2) the demand intercept is sufficiently high that equilibrium output levels are positive, and (3) the threshold of the structural screen is sufficiently high, then firms are more likely to incur punitive regulatory costs at random if their products are close substitutes. This finding again undermines the validity of a structure-based antitrust policy, since larger values of [Gamma] correspond to more homogeneous products and, for a given pattern of conduct, more direct competition between firms.

IV. Stochastic Demand

Thus far we have assumed that only costs are stochastic, when in fact demand may equally well have a random component. When products are homogeneous, any demand shift must affect both firms equally, and hence will have no impact on the probability that one firm's market share will exceed a given threshold. Only in the differentiated case may we observe firm-specific demand shifts.

Consider first the case in which the slope of each firm's demand curve is stochastic. Then the condition [x.sub.i]/[x.sub.j] [less than or equal to] z is equivalent to [b.sub.i] [greater than or equal to] H where H is now:

H = [(2[b.sub.j] + [Gamma]r + [Gamma]z)([a.sub.i] - [c.sub.i])/([a.sub.j] - [c.sub.j]) - [Gamma](1 + rz)]/2z. (15)

We wish to examine the effect of changes in [Gamma] alone as well as of changes in r alone. The probability that the market share of firm i exceeds [Theta] is:

1 - [F.sub.z](z) = 1 - [integral of] [[integral of] f([b.sub.i], [b.sub.j]) d[b.sub.i] between limits [infinity] and H] between limits [infinity] and -[infinity] d[b.sub.j] (16)

which responds to [Gamma] as:

- [Delta][F.sub.z](z)/[Delta][Gamma] = [(r/z + 1)([a.sub.i] - [c.sub.i])/([a.sub.j] - [c.sub.j]) - 1/z - r] [integral of] f(H, [b.sub.j]) between limits [infinity] and -[infinity] d[b.sub.j]/2. (17)

and to r as:

- [Delta][F.sub.z](z)/[Delta]r = {[Gamma]([a.sub.i] - [c.sub.i])/[2z([a.sub.j] - [c.sub.j])] - [Gamma]/2} [integral of] f(H, [b.sub.j]) between limits [infinity] and -[infinity] d[b.sub.j]. (18)

Equation (17) takes the sign of the term in square brackets, which is ambiguous but nonzero in general. In the symmetric case where [a.sub.i] - [c.sub.i] = [a.sub.j] - [c.sub.j], the term in brackets reduces to (z - 1)(1 - r)/(2z) and is unambiguously positive for all r [less than] 1 (that is, for all patterns of conduct short of perfect collusion) since z [greater than] 1 as noted above. Here, the probability of failing the structural screen is an increasing function of the degree of substitutability. More generally, this positive sign arises whenever [Theta] [greater than] [([a.sub.j] - [c.sub.j]) - r([a.sub.i] - [c.sub.i])]/[([a.sub.i] - [c.sub.i] + [a.sub.j] - [c.sub.j])(1 - r)]. In all such cases the effect would undermine the validity of a structure-based antitrust policy, just as in the case of stochastic cost analyzed in the previous section. The minimum threshold to achieve this result simplifies in the Cournot case to [Theta] [greater than] 1/[1 + ([a.sub.i] - [c.sub.i])/([a.sub.j] - [c.sub.j])].

Equation (18) takes the sign of the term in braces, which is also ambiguous in general. In the symmetric case where [a.sub.i] - [c.sub.i] = [ a.sub.j] - [c.sub.j], equation (18) takes the sign of ([Gamma]/2)(1/z - 1) which, since we assume z [greater than] 1 for duopoly, has the opposite sign as [Gamma]. Then, in the policy-relevant case of substitute products ([Gamma] [greater than] 0), equation (18) is negative, further undermining the validity of a structural screen. This result further holds whenever the gap between the demand intercept and marginal cost is not too dissimilar across firms, and will also hold for the firm with the smaller such gap even if the gaps are widely dissimilar. Interestingly, equation (18) is independent of the pattern of conduct, reflecting the linearity of equation (15) in r.

Turning to the effect of a stochastic demand intercept, we note that Z [less than or equal to] z if and only if [a.sub.i] (2[b.sub.j] + [Gamma]r + [Gamma]z) [less than or equal to] (2[b.sub.i]z + [Gamma]rz + [Gamma]) ([a.sub.j] - [c.sub.j]) + [c.sub.i] (2[b.sub.j] + [Gamma]r + [Gamma]z), which takes two cases depending on the sign of 2[b.sub.j] + [Gamma]r + [Gamma]z, just as in the stochastic cost case treated above. As before, a positive sign characterizes the more relevant case. Then [x.sub.i]/[x.sub.j] [less than or equal to] z if and only if [a.sub.i] [less than or equal to] H where:

H = [c.sub.i] + ([a.sub.j] - [c.sub.j])(2[b.sub.i]z + [Gamma]rz + [Gamma])/(2[b.sub.j] + [Gamma]r + [Gamma]z). (19)

The probability of failing the structural screen is given by:

1 - [F.sub.z](z) = 1 - [integral of] [[integral of] f([a.sub.i], [a.sub.j]) d[a.sub.i] between limits H and -[infinity]] between limits [infinity] and -[infinity] d[a.sub.j] (20)

and this probability varies with product differentiation according to:

-[Delta][F.sub.z](z)/[Delta][Gamma] = 2[[b.sub.i]z(r + z) - [b.sub.j](rz + 1)] [integral of] ([a.sub.j] - [c.sub.j])f(H, [a.sub.j]) between limits [infinity] and -[infinity] d[a.sub.j]/[(2[b.sub.j] + [Gamma]r + [Gamma]z).sup.2], (21)

which is analogous to equation (14), reflecting the symmetric roles of [a.sub.i] and [c.sub.i] in equation (10). Although [Delta]H/[Delta][Gamma] takes the opposite sign for equation (19) as for equation (12), reflecting the opposite signs of [a.sub.i] and [c.sub.i] in equation (10), note that the limits of integration must be interchanged in equation (20) versus equation (13). This exchange introduces a sign change in going from [Delta]H/[Delta][Gamma] to [Delta][F.sub.z](z)/[Delta]G that is present in one case but not in the other, and this difference cancels out the sign change in the two expressions for [Delta]H/[Delta][Gamma]. More simply but less generally, it is easy to see that an increase in the variance of [a.sub.i] has exactly the same effect on the variance of [x.sub.i]/[x.sub.j] in equation (10) as does an equal increase in the variance of [c.sub.i].

For sufficiently high thresholds, [b.sub.i]z(r + z) - [b.sub.j](rz + 1) is positive. Then equation (21) takes the sign of the integral which, since [a.sub.j] [greater than] [c.sub.j], is also positive. The probability of failing the structural screen is thus an increasing function of the degree of product substitutability for sufficiently high thresholds. This result further undermines the validity of a structure-based antitrust policy.

Equation (20) varies with conduct according to:

-[Delta][F.sub.z](z)/[Delta]r = [Gamma][2z([b.sub.i] - [b.sub.j]) + [Gamma](1 - [z.sup.2])] [integral of] ([a.sub.j] - [c.sub.j])f(H, [a.sub.j]) between limits [infinity] and -[infinity] d[a.sub.j]/[(2[b.sub.j] + [Gamma]r + [Gamma]z).sup.2] (22)

which again has an ambiguous sign in general. When [a.sub.j] [greater than] [c.sub.j] and products are gross substitutes ([Gamma] [greater than] 0), then equation (22) takes the sign of the term in square brackets. For [b.sub.j] [greater than or equal to] [b.sub.i], or when [b.sub.j] is not too much smaller than [b.sub.i] (i.e., when [b.sub.j] [greater than or equal to] [b.sub.i] + [Gamma](1 - [z.sup.2])/(2z)), that sign will be negative, implying that more competitive firms are more likely to fail the structural screen - a result that again undermines the structural approach to antitrust. Equivalently, this result holds for sufficiently high thresholds - in particular, when [Theta] [greater than] {[b.sub.i] - [b.sub.j] - [Gamma] + [[[[Gamma].sup.2] + [([b.sub.i] - [b.sub.j]).sup.2]].sup.1/2]}/[2([b.sub.i] - [b.sub.j])]. As in equation (7), conduct enters equation (22) only in the denominator, such that the bias is stronger for more competitive conduct (smaller values of r).

In relating these results to real markets, it is useful to understand the composition of typical shocks. For example, demographic shifts, possibly made firm-specific by location effects not modeled here, could alter the demand intercepts without substantially changing the slopes. Exogenous preference shocks could alter either slopes or intercepts, as could wealth effects over a business cycle. Some factors influencing demand, such as the business cycle, would likely affect costs as well. If demand shocks result from product innovation, simultaneous changes in a, b, [Gamma], and c would all be likely. Similarly, demand changes resulting from advertising or other marketing efforts would tend to involve increases in a and c (where marginal cost is defined inclusive of marketing costs) but decreases in b and [Gamma]. The relative magnitudes of these changes, along with other parameters, would determine the overall impact of structural screens; this is an important empirical question that has not yet been comprehensively addressed.

V. Conclusion

Within the illustrative framework of a stochastic duopoly facing linear costs and demand, this paper has analyzed the impact of conduct and product differentiation on the probability that a given firm's market share will exceed an arbitrary threshold. Each of three stochastic components was analyzed first in terms of an arbitrary density function and then, to derive additional explicit results, when the stochastic variation is independent across firms. The model was sufficiently general to allow for arbitrary symmetric patterns of conduct. The results were interpreted in terms of the stochastic bias inherent in a purely structure-based antitrust policy or in structural screens as currently applied by the U.S. Department of Justice.

It was shown that this bias is nonzero in general, but has an ambiguous sign except in special cases. Either cost uncertainty or demand uncertainty can undermine the validity of a structure-based antitrust policy in some cases while reinforcing it in others, depending on parameter values. However, in a variety of plausible cases the impact of stochastic shocks to any of the components analyzed was shown to be in the direction of undermining the structural approach to antitrust, by making more competitive firms more likely to fail a structural screen. In several instances a higher market share threshold, corresponding to a relatively lenient antitrust policy, made this perverse outcome more likely. In at least two cases, the bias was more severe at more competitive levels of conduct.

In particular, therefore, structural screens or a structure-based antitrust policy can penalize either competitive conduct among firms or close substitutability among products. These distortions introduce direct misallocation of resources. In addition, if conduct and product differentiation have endogenous components, then structural screens or a structure-based antitrust policy can create incentives for firms to shift to less competitive conduct and less substitutable products, in the presence of stochastic market elements.

As traditional antitrust theory and enforcement have not focused on the possibility of stochastic effects, the results here indicate that further attention is warranted in this area; they also re-emphasize the importance of considering additional economic information besides structure alone. Moreover, additional empirical research on the decomposition of typical stochastic shocks into cost and demand components, along with the timing of shocks relative to firms' production decisions, could assist the interpretation of the relevant theory as it applies to real markets. Finally, to the extent that an antitrust policy or agency has committed to a particular role for structural screens, analysis of the sort introduced here might permit the development of auxiliary regulatory policies or procedures that could mitigate or reverse the detrimental interaction between stochastic variation and structural screens.

1. The Deputy Assistant Attorney General for the Antitrust Section of the U.S. Department of Justice has stated [10]: "It is important for you to understand that we use a two-step process at the Justice Department. Step No. 1 is the screening process. With 2,000 applications coming in every year, we obviously don't have the resources to investigate every one. The screen can be summarized by a general rule of thumb. It is: 1,800, 200, 20. We won't look at the investigation stage unless a merger involves a large market with an HHI - a Herfindahl-Hirschmann Index (a common measure of market concentration) - of over 1,800, where the change in the HHI is over 200...."

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