Dumping with correlated demand.

I. Introduction

The traditional explanation for dumping is monopolistic price discrimination in an international setting. Thus, it is well-known that when frictions such as tariffs or transportation costs separate domestic and foreign markets and home demand is less price elastic, monopoly rents are maximized by charging domestic consumers more than foreign consumers. However, this theory alone cannot explain the paradox that products often appear to be dumped below costs.

Recent literature introduces uncertainty as an explanation for dumping below marginal costs. Notable contribution to this literature include Ethier [4], Davies and McGuinness [2], and Hillman and Katz [5]. In these models, uncertainty is introduced via either the domestic or foreign demand and firms are assumed to determine output before demand is realized. If the realized demand is lower than expected, firms will have over produced and may therefore have an incentive to dump below costs. A similar result has also been obtained by Blair and Cheng [1], BC hereafter, in a model where firms set price rather than output before the realization of demand. A common feature of the above mentioned papers is that they assume domestic and foreign markets to be stochastically independent.

In this paper, we depart from the existing literature by allowing domestic and foreign demand to be correlated. We demonstrate that demand correlation alone can be a basis for dumping, i.e., dumping may arise in the presence of correlated demand when it would not have occurred had the markets been stochastically independent. More interestingly, we are able to show that covariance-based pricing strategy can lead to a new possibility of dumping below costs. The intuition behind this result is that when the covariance between domestic and foreign demand is positive, firms may have an incentive to set price above marginal cost (as well as unit cost since marginal cost is assumed to be constant) in one market and below in the other. This behavior ensures a negative correlation between profits from the two markets and thus dampens the variance of aggregate profits. As our analysis shows, risk aversion may well make this an equilibrium strategy. Analytically, this result is similar to those in finance where risk-averse agents invest in assets with negatively correlated returns to reduce the overall risk of their portfolios.

Given progressive liberalization of trade and capital flows and the consequent increased integration of particularly industrial country markets, positive correlation in demand among these markets is highly likely. Thus, our central result is unlikely to be empirically irrelevant.

II. The Basic Model

Consider a monopolist selling a product in domestic and foreign markets, both of which are subject to random disturbances. It is assumed that the firm sets prices before actual demands are revealed and then produces the quantities necessary to clear the markets. It is further assumed that prices, once determined, will remain rigid for a substantial period.(1)

To focus our attention on the effect of uncertainty, we rule out the standard price discrimination argument by assuming that two markets are identical in the absence of uncertainty [1]. Two demand functions (domestic and foreign labelled by subscripts d and f respectively) are assumed to be

[q.sub.d] = q([p.sub.d])u (1)

[q.sub.f] = q([p.sub.f])v. (2)

where [q.sub.i] and [p.sub.i] are quantity demanded and the price in market i (i = d and f).(2) We postulate that (u, v) has a multivariate distribution with Eu = Ev = 1, [Mathematical Expression Omitted], [Mathematical Expression Omitted], and Cov(u, v) = [[Sigma].sub.uv] = [Rho][[Sigma].sub.u][[Sigma].sub.v], where [Rho](-1 [less than] p [less than] 1) is the correlation coefficient. The existing models assume [[Sigma].sub.u] = [[Sigma].sub.uv] = 0 (i.e., the domestic demand is non-stochastic), thus ruling out interrelation between markets. Our generalization allows us to gain some insights into the effect of covariance components on the behavior of the firm. As shown below, not only the relative volatility between markets but also their interrelation determines whether or not the firm will dump at less than marginal cost.

For simplicity, the cost function is assumed to be linear in output,(3) i.e.,

C = A + cQ (3)

where Q = [q.sub.d] + [q.sub.f]. The profit function of the firm is thus given by

[Pi] = [Pi]([p.sub.d], [p.sub.f], u, v) = [p.sub.d]q([p.sub.d])u + [p.sub.f]q([p.sub.f])v - A - c(q([p.sub.d])u + q([p.sub.f])v)

= q([p.sub.d]) [[p.sub.d] - c]u + q([p.sub.f])[[p.sub.f] - c]v - A. (4)

In the present context, we shall use a quadratic utility function in profit. Since such functions are unique only up to an increasing affine transformation, it involves no loss of generality to write the function as

U([Pi]) = [Pi]([p.sub.d], [p.sub.f], u, v) - (1/2)[R.sub.a][[Pi]([p.sub.d], [p.sub.f], u, v) - [Pi]([p.sub.d], [p.sub.f], 1, 1).sup.2], (5)

where [R.sub.a] [greater than or equal to] 0 is a risk-aversion index.

The firm is assumed to set [p.sub.d] and [p.sub.f] to maximize the expected utility of profit,

[Mathematical Expression Omitted].

The first order conditions are given by

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

III. Dumping below Costs

In this section, we derive conditions under which draping (or reverse dumping) occurs. From equations (7) and (8), we know that there exist multiple solutions.(4) Two cases are considered: (1) a special case where [[Sigma].sub.u] = 0 and (2) a general case where [[Sigma].sub.u] [not equal to] [[Sigma].sub.v] [not equal to] 0.

CASE 1. [[Sigma].sub.u] = 0 (BC's Case).

BC resume that the domestic demand is nonstochastic, i.e., [[Sigma].sub.u] = 0 = [[Sigma].sub.uv]. Equations (7) and (8) imply that either

[([p.sub.d] - c)q[prime]([p.sub.d]) + q([p.sub.d])] = 0 (9)

[([p.sub.f] - c)q[prime]([p.sub.f]) + q([p.sub.f])] = 0, (10)

or

[([p.sub.d] - c)q[prime]([p.sub.d]) + q([p.sub.d])] = 0 (11)

[Mathematical Expression Omitted]

must hold. If [R.sub.a] = 0, conditions (9) and (10) must hold, implying [p.sub.d] = [p.sub.f] by symmetry (no dumping). The result is trivial because uncertainty has no effect on a risk-neutral firm.

When [R.sub.a] [greater than] 0, the solution is determined by conditions (11) and (12). From (11), [p.sub.d] - c = [p.sub.d]/[[Epsilon].sub.d] [greater than] 0 and from (12), [Mathematical Expression Omitted], where [[Epsilon].sub.d] = -q[prime]([p.sub.d])[p.sub.d]/q([p.sub.d]) [greater than] 0. Thus, [Mathematical Expression Omitted] if [R.sub.a] and [[Sigma].sub.v] are sufficiently large (small) and/or if [[Epsilon].sub.d] is relatively small (large). Risk aversion and high market volatility will result in dumping above marginal costs.(5) This leads us to conclude that regardless of risk aversion, dumping below costs cannot occur in our model without the help of covariance components.

CASE 2. [[Sigma].sub.u] [not equal to] [[Sigma].sub.v] [not equal to] 0 (General Case).

In a limiting case where [R.sub.a] = 0, the first-order conditions are given by (9) and (10), implying [p.sub.d] = [p.sup.f]. There is no dumping. For the rest of this section, we assume that [R.sub.a] [greater than] 0. Given this, it is evident from (7) and (8) that the solution can be characterized by the following conditions.(6)

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

provided that [([p.sub.d] - c)q[prime]([p.sub.d]) + q([p.sub.d])] [not equal to] 0 and [([p.sub.f] - c)q[prime]([p.sub.f]) + q([p.sub.f])] [not equal to] 0.

We must note that this new set of the first-order conditions holds when the firm is risk averse and both markets are stochastic. To see this, consider the associated second-order conditions:

[a.sub.11] [less than] 0, [a.sub.22] [less than] 0, [a.sub.11][a.sub.22] - [a.sub.12][a.sub.21] [greater than] 0 (15)

where

[Mathematical Expression Omitted]

[a.sub.12] = -[R.sub.a][[Sigma].sub.uv] [([p.sub.f] - c)q[prime]([p.sub.f]) + q([p.sub.f])][([p.sub.d] - c)q[prime]([p.sub.d]) + q([p.sub.d])]

[Mathematical Expression Omitted].

Notice that [a.sub.11][a.sub.22] - [a.sub.12][a.sub.21] [greater than] 0 is equivalent to [Mathematical Expression Omitted]. Apparently, the second-order conditions are satisfied when (i) [[Sigma].sub.j] [greater than] 0 (j = u, v); and (ii) [Rho] is less than unity in absolute value. Thus, the first-order conditions (13) and (14) require (i) and (ii) above to hold to indicate a utility maximum.

Solving (13) and (14) for ([p.sub.d] - c) and ([p.sub.f] - c) we get

[Mathematical Expression Omitted]

where [Delta] is given by

[Mathematical Expression Omitted].

Since [Rho] is assumed to be less than one in absolute value, [Delta] is strictly positive. Then we get the following pricing equation

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

Relative markup between two markets depends not only on variances but also on the covariance between two random variables. Dumping can thus be viewed as the result of covariance-based pricing. Since the denominators of the above pricing formula are strictly positive, the following statements can be made for [R.sub.a] [greater than] 0 case.

Result (i) If [Mathematical Expression Omitted], then [Mathematical Expression Omitted]. This implies [p.sub.f] [less than] c [less than] [p.sub.d]. The firm dumps below marginal cost.

Result (ii) If [Mathematical Expression Omitted], then [Mathematical Expression Omitted]. This implies [p.sub.f] [greater than] c [greater than] [p.sub.d]. Reverse dumping below marginal cost occurs.

Result (iii) If [Mathematical Expression Omitted], no dumping occurs.

Result (iv) If [Mathematical Expression Omitted], then [Mathematical Expression Omitted]. This implies [p.sub.f] [greater than] c and [p.sub.d] [greater than] c.

If dumping occurs, it must be above marginal cost. From equations (18) and (19), we get

[Mathematical Expression Omitted]

implying ([p.sub.d] - c)q([p.sub.d]) [greater than] ([p.sub.f] - c)q([p.sub.f]) since [Mathematical Expression Omitted] in this case. By the mean-value theorem, ([p.sub.d] - c)q([p.sub.d]) = ([p.sub.f] - d)q([p.sub.f]) + [([p.sub.0] - c)q[prime]([p.sub.0]) + q([p.sub.0])][[p.sub.d] - [p.sub.f]] where [p.sub.0] [element of] [min([p.sub.d], [p.sub.f]), max([p.sub.d], [p.sub.f])]. By substitution, we have [([p.sub.0] -c)q[prime]([p.sub.0])+ q([p.sub.0])][[p.sub.d] - [p.sub.f]] [greater than] 0. Thus, sign[[p.sub.d] - [p.sub.f]] = sign[([p.sub.0] -c)q[prime]([p.sub.0]) + q([p.sub.0])] = sign[-([p.sub.0] - c)[[Epsilon].sub.d]([p.sub.0])/[p.sub.0] + 1]q([p.sub.0]). Thus, [Mathematical Expression Omitted] as [Mathematical Expression Omitted]. That is, dumping ([p.sub.d] [greater than] [p.sub.f] [greater than] c) occurs if the elasticity of demand at [p.sub.0] is smaller than [p.sub.0]/([p.sub.0] - c) and reverse dumping ([p.sub.f] [greater than] [p.sub.d] [greater than] c) occurs otherwise.(7)

Result (v) If [Mathematical Expression Omitted], then [Mathematical Expression Omitted]. Given this, we have [p.sub.f] [greater than] c and [p.sub.d] [greater than] c. Similarly, (20) implies that ([p.sub.d] - c)q([p.sub.d]) [less than] ([p.sub.f] - c)q([p.sub.f]). By the mean-value theorem, we have sign[[p.sub.d] - [p.sub.f]] = - sign[([p.sub.0] - c)q[prime]([p.sub.0]) + q([p.sub.0])] = - sign[-([p.sub.0] - c)[[Epsilon].sub.d]([p.sub.0])/[p.sub.0] + 1]q([p.sub.0]). Thus, [Mathematical Expression Omitted] as [Mathematical Expression Omitted]. That is, dumping ([p.sub.d] [greater than] [p.sub.f] [greater than] c) occurs if the elasticity of demand at [p.sub.0] is larger than [p.sub.0]/([p.sub.0] - c) and reverse dumping ([p.sub.f] [greater than] [p.sub.d] [greater than] c) occurs otherwise.

We have derived conditions under which dumping (or reverse dumping) occurs. Dumping (either below or above marginal cost) hinges not only on the relative volatility but also on the correlation between domestic and foreign markets.

The intuitions behind these results are related to the theory of portfolio choice in financial economics. It is well-known in this theory that if asset returns are stochastically independent, overall risk of a portfolio can be reduced by reducing the risk associated with individual assets, i.e., by including assets with low variance. However, when asset returns are stochastically dependent, the nature of correlation and covariance among assets becomes the more important determinant of overall risk. This is so because, given the variability of individual assets, the overall portfolio risk can be substantially reduced (even to zero) if the asset returns are negatively correlated.(8) This is why negatively correlated assets are so valuable to risk-averse investors.

The implication of the theory of portfolio choice for the present context is easily established. If market demands are stochastically independent, risk-averse firms will seek to minimize the variance of aggregate profits by reducing variance of profits in individual markets. As is well-known, this can be achieved by reducing the markup in each market. When demands are positively correlated however, the firm may set the price below marginal cost in one of the markets in order to convert the positive correlation in demands into negative correlation in profits, thus dampening the variability of aggregate profits. The net benefit of such policy depends on the size of the correlation. Since by dumping below costs the firm's expected profits go down, in order for the utility of aggregate profits to go up, the impact of this must be outweighed by the reduced variance. This is true if the demand correlation is sufficiently high as in Results (i) and (ii). This is the basis for below-cost dumping in our model. When market demands are negatively correlated as in Results (iv) and (v), negative correlation in profits is assured with positive markup in each market. In this case, the firm would have no incentive to price below cost.

Finally, if two markets are identical in all aspects, i.e., [Mathematical Expression Omitted], there is therefore no reason for any price discrimination (see Result (iii)).

IV. Summary

Our analysis extends the price-commitment model of uncertainty and dumping by allowing market demands to be stochastically correlated. We demonstrate that not only can the demand correlation be an exclusive basis for dumping, but a sufficiently strong positive correlation can lead to a new scenario for dumping below costs. The intuition is that in the presence of positive demand correlation, below-cost dumping (or reverse dumping) produces a negative correlation in profits. This serves to dampen the variance of aggregate profits which benefits the risk-averse firm. We also show that when demand correlation is low or negative, dumping can only occur above cost. In these cases, the direction of dumping is determined by relative demand variances and demand elasticites.

We would like to thank an anonymous referee for helpful comments and suggestions.

1. This assumption is not uncommon in the theoretical literature in uncertainty and the behavior of the firm [1]. Clearly, the assumption makes sense in markets where frequent price adjustments are very costly due, for example, to the cost of advertising and production of catalogues, etc..

2. The demand function is assumed to be multiplicative in the random variable. Our results below remain valid when the demands are additive in u or v.

3. This simplified assumption rules out the stochastic cost effect on the firm's dumping behavior under uncertainty, a subject examined in some details by Eckel and Smith [3] in a different context.

4. The possibility of having multiple solution was ruled out in BC [1] by assumption.

5. Notice that the foreign demand function in BC consists of a non-stochastic constant term: [q.sub.f] = a + q([p.sub.f])v. Assume that a = 0. Fact 1 together with footnote 10 in their paper implies that [p.sub.d] = [p.sub.f]. There is no dumping. Here, we show that firms will still dump even if demand consists of no non-stochastic constant term.

6. It is clear from (7) and (8) that there exist four possible solutions. Let us write (7) as [A][B] = 0 and (8) as [C][D] = 0. This yields four different combinations: (a) A = 0 and C = 0; (b) A = 0 and D = 0; (c) B = 0 and C = 0; (d) B = 0 and D = 0. If [R.sub.a] = 0, B [not equal to] 0 and D [not equal to] 0. Thus, case (a) must result. In this case, uncertainty has no effect on the pricing behavior (trivial case). If [R.sub.a] [greater than] 0 and [[Sigma].sub.j] [not equal to] 0, cases (b)-(d) may result. Cases (b) and (c) are similar to conditions (11) and (12) which result in dumping above marginal cost. Our analysis below focuses on case (d). This is the condition that has not been dealt with in the literature and is the most general and interesting one among four cases above.

7. It is clear from (1) and (2) that for a given price, the demand variance is equal to [q.sup.2](p)[[Sigma].sup.2]. The non-stochastic part of the demand function q(p) has the square effect. Thus, in the case where the elasticity is high, a change in price will have a substantial impact on the variance. The firm will balance this impact by charging more (to decrease the q[(p).sup.2] part) in the market with higher variance.

8. According to the Capital Asset Pricing Model (CAPM), the expected return on asset like gold with negative correlation with the market portfolio (i.e., asset with negative CAPM beta) is less than the return on the risk-free asset. However, risk averse individuals may still hold such assets as gold in their portfolios in order to achieve risk diversification.

References

1. Blair, Roger D. and Leonard Cheng, "On Dumping." Southern Economic Journal, January 1984, 857-65.

2. Davies, Stephen S. and Anthony J. McGuinness, "Dumping at Less Than Marginal Cost." Journal of International Economics, February 1982, 169-82.

3. Eckel, Catherine C. and William T. Smith, "Price Discrimination with Correlated Demand." Southern Economic Journal, July 1993, 58-65.

4. Ethier, Wilfred J., "Dumping." Journal of Political Economy, June 1982, 487-506.

5. Hillman, Arye L. and Eliakim Katz, "Domestic Uncertainty and Foreign Dumping." The Canadian Journal of Economics, August 1986, 403-16.