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Factor substitutability, monopoly, and vertical integration: a heuristic analysis.

I. Introduction

A monopolist of an intermediate good can extract profit from final consumers only indirectly, since it only sells to other firms, which in turn produce final products. If those final producers substitute other inputs for the one the monopolist sells, they would not see that their substitution cuts into the monopolist's profit. Because of this "vertical externality"[10, 179], the monopolist would earn less than the maximum monopoly rent. Although the final producers would minimize their nominal costs, their input combination would be suboptimal. Vernon and Graham [11] showed that the monopolist would increase both its profit and economic welfare by vertically integrating to reverse such input substitution (for a given final output). Schmalensee[9] then investigated what portion of a perfectly competitive final product industry the input monopolist would acquire. While he found that complete integration would maximize profits, Schmalensee also discovered that the integration would affect final output and its price, but that the effect was ambiguous. Obviously, the welfare gain from lower real costs would be offset partially or entirely, if integration also reduced output. This ambiguity differed from analyses that had assumed a fixed input proportion in final production. With fixed proportions, integration never would reduce output and always would increase welfare.(1) The many refinements of Schmalensee's analysis have shown that whether integration by the monopolist would increase or decrease final production depends on the elasticities of final demand and input substitution[3; 6; 7; 12; 13]. The ambiguous effect on output that occurs when inputs are substitutable has become part of the lore about vertical integration and has been an element in the debates about antitrust policy toward vertical integration[2, 168].

This literature is very technical and does not have any single clear result comparable to that in the fixed proportions case. For that reason it is difficult to convey intuitively. This paper takes a heuristic approach that adds to this literature in several ways. First, the two components in the anatomy of integration are derived explicitly for the first time. "The first is a change in input proportions so as to reduce real costs. The second is a broadening of monopoly"[8, 27]. Each of these components can affect final output. Revealing the anatomy shows that the component due to the change in the input proportions in final production not only reduces real cost, it always contributes an increase in final output, while the component due to broadening of monopoly always tends to reduce final output. In this anatomy, the ambiguous effect of integration on output is due to ambiguity about the comparative magnitudes of the two component effects. Second, the anatomy demonstrates that an assumption that is crucial to the ambiguity about output is the pure input monopoly assumption. In the pure monopoly model, integration "broadens the monopoly" only because input substitutability affects the elasticity of the derived demand for the intermediate good relative to the elasticity of the final demand. Finally, in models of noncompetitive equilibrium in which the elasticity of final demand does not affect pricing decisions at all, the monopoly power component disappears. Then, vertical integration to reduce suboptimal input proportions has the intuitive and unambiguous result of lowering real costs and increasing final output.

II. The Model

The model here encompasses the essential traits of previous studies. Let the consumers' inverse demand for a final product Q be P = D (Q), with [D.sub.Q] < 0.(2) Assuming that Q is sold at a linear price, the marginal revenue of the final product industry is MR(Q) = (PQ)/ Q, and M[R.sub.Q] < 0 is assumed.(3)

A perfectly competitive industry produces Q from inputs X and Y. Input Y is available in perfectly elastic supply at price r. A pure monopoly produces X at constant cost k and sells it at a linear price w. Final producers are price takers with respect to both w and r. They have a linear homogeneous technology, so that input prices affect their average and marginal cost c(w, r). Explicitly, c(w, r) = [alpha](w)w + [beta](w)r, where [alpha](w) = X/Q and varies inversely with w, and [beta](w) = Y/Q and varies directly with w. Assume [c.sub.w], [c.sub.r] > 0. The real unit cost depends on the real cost of the inputs and is p(w, r) = [alpha](w)k + [beta](w)r. Units of X and Y are defined so that when w = k, then [alpha](k) = [beta](k) = I and [rho](k, r) = c (k, r) = k + r. As the monopolist sets w > k, then c(w, r) > [rho](w, r) > k + r.

Since the input monopoly is the origin of the vertical externality, the nonintegrated final product industry is kept simple by assuming it to be perfectly competitive and to have constant costs.(4) The competitive final product price is P = c(w, r). The nonintegrated input monopolist's marginal revenue mr(X) depends on the elasticity of the derived demand ([epsilon]). Consequently, the nonintegrated input monopolist has mr([X.sup.0]) = [w.sup.0](1 - 1/[epsilon]) = k. The competitive final product price then would be [P.sup.0] = c([w.sup.0], r), and final output would be [Q.sup.0] from [P.sup.0] = D([Q.sup.0]).

Vertical integration occurs by the input monopoly acquiring all of the perfectly competitive final producers.(5) The integrated pure monopoly with input cost k would produce [Q.sup.m] from MR([Q.sup.m]) = (1 - 1/[epsilon]) = k + r. Final product price would be [P.sup.m] = D([Q.sup.m]).

The ambiguity about the effect of integration on output and welfare arises from the way input substitutability affects the preintegration equilibrium ([Q.sup.0]). Since the integrated monopoly would produce [Q.sup.m], integration would raise or lower price depending on whether the preintegration equilibrium has [Q.sup.0] [is greater than or equal or less than to] [Q.sup.m]. This condition depends on the difference between the two elasticities, [epsilon] and E. The principal lesson from all the previous studies is that there is not a simple theorem here. The Allen price elasticity of derived demand is [epsilon] = (1 - [alpha])[sigma] + aE, where a = wX/cQ is the expenditure share of X in total cost, [sigma] is the elasticity of input substitution in final production, and E is the elasticity of final demand. The most inclusive condition for [Q.sup.0] > [Q.sup.m] apparently is [sigma] > E > 1 at [Q.sup.m], although Westfield concludes that "for [sigma] [is not equal to] 0, price will always rise [i.e., [Q.sup.0] > [Q.sup.m]] if the price elasticity of demand is high enough" [13, 345].(6)

III. The Anatomy of Integration

How integration by the input monopolist affects the final product price and output has two components, one due to the reduction in cost, and one due to the broadening of monopoly power. This anatomy never has been revealed analytically. It adds more intuition to the analysis. The anatomy also shows that the ambiguity about the effect of integration on output depends as much on the pure monopoly assumption as on the substitutability of inputs.

An intermediate case reveals the anatomy. Suppose the monopolist of X acquires all final producers and thereby becomes an integrated final product monopolist, but in doing so it retains the input proportion of the nonintegrated industry, X/Y = [alpha]([w.sup.0])/[beta]([w.sup.0]) < 1. Since integration switches the monopoly from the market for X to the market for Q, the monopoly now confronts final demand and MR (Q) = (1 - 1/E) rather than the derived demand and mr(X) = (1 - 1/E). This is the sense in which integration has broadened the monopoly. The effective price of X now is k rather than [w.sup.0].(7) Since the integrated monopoly keeps the initial input proportion in this intermediate case, its average (and marginal) cost of final product is [alpha]([w.sup.0])k + [beta] ([w.sup.0])r = [rho]([w.sup.0], r). Consequently, in this intermediate case the integrated monopoly would produce [Q.sup.*] such that MR ([Q.sup.*]) = [rho]([w.sup.0], r).

The equilibrium at [Q.sup.*] identifies the two components of the effect of integration on final output. First, [Q.sup.*] - [Q.sup.0] is a change in final output that occurs because integration broadens the monopoly.

Monopoly Component (MQ): For the assumptions here, and holding the input proportion constant, MQ = [Q.sup.*] - [Q.sup.0] arises strictly because integration switches the monopoly from the input industry to the final product industry.

Note that MQ exists even though the input proportion is unchanged.

The second component appears by letting the integrated final product monopoly change its input proportion to X/Y = 1, thereby reducing its unit cost to (k + r). Since MR ([Q.sup.m]) = k + r, this is the integrated equilibrium from above. Then [Q.sup.m] - [Q.sup.*] is a change in final output due to the actual substitution of inputs.

Factor Substitution Component (FQ): For the assumptions here, FQ = [Q.sup.m] - [Q.sup.*] arises strictly because integration changes input usage to the optimal proportion.

The anatomy could be extended to encompass any other components that integration might have. For example, integration might shift final demand or one of the production functions, or change the transaction costs. Whatever combination of effects integration is assumed to have in the model, intermediate cases could be defined that would isolate each effect as a component of integration, independent of MQ and FQ.

Each component MQ and FQ has the conventional sign. First, the real cost saving in the FQ component always increases final output.

Theorem 1. For the assumptions here, in vertical integration by a pure monopolist, F Q > 0.

The proof is trivial. MR([Q.sup.*]) = [rho]([w.sup.0], r) and MR([Q.sup.m]) = k + r. Since [MR.sub.q] < 0 and [rho]([w.sup.0], r) > k + r, then [Q.sup.*] < [Q.sup.m]. Since MR([Q.sup.*]) > k + r, the FQ component always adds to profit.

Second, the broadening of monopoly in the MQ component always reduces final output.

Theorem 2. For the assumptions here, in vertical integration by a pure monopolist, M Q < 0.

The proof uses the preintegration final output [Q.sup.0] as a benchmark. At [Q.sup.0], MR ([Q.sup.0]) = [P.sup.0](1 - 1/[E.sup.0]), where [E.sup.0] is the final demand elasticity at [Q.sup.0]. And the competitive equilibrium price is [P.sup.0] = [alpha]([w.sup.0])[w.sup.0] + [beta]([w.sup.0])r = c([w.sup.0], r). By adding the term ([alpha]([w.sup.0])k - [beta]([w.sup.0])k), this condition can be rewritten as [P.sup.0] - [alpha]([w.sup.0])([w.sup.0] - k) = [alpha]([w.sup.0])k + [beta] ([w.sup.0])r. Then, [P.sup.0] - [alpha]([w.sup.0])([w.sup.0] - k) reduces to [P.sup.0](1 - [[alpha].sup.0]/[[epsilon].sup.0]), where [[alpha].sup.0] and [[epsilon].sup.0] are from the Allen price elasticity at the preintegration equilibrium. When [sigma] [is not equal to] 0, then from the Allen elasticity it is clear that [P.sup.0] (1 - 1/[[epsilon].sup.0]) < [P.sup.0](1 - [[alpha].sup.0]/[[epsilon].sup.0]). Therefore, MR ([Q.sup.0]) < [alpha]([w.sup.0])k + [beta]([w.sup.0])r. Since MR ([Q.sup.*]) = [alpha] ([w.sup.0])k + [beta]([w.sup.0])r in the intermediate case, MR([Q.sup.0]) < MR([Q.sup.*]). Since [MR.sub.Q] < 0, then [Q.sup.0] > [Q.sup.*], which makes MQ = [Q.sup.*] - [Q.sup.0] < 0. Since MR([Q.sup.0]) < [rho]([w.sup.0], r), the MQ component always adds to the monopolist's profit.

The MQ and FQ components are both output effects, so that the net effect of integration on final output is MQ + FQ. On the production function in Figure 1, the intermediate case holds the monopoly on the output expansion ray [R.sup.0] that has slope [alpha]([w.sup.0])/[beta] ([w.sup.0]) < 1. Then MQ is an inward movement along [R.sup.0] from [Q.sup.0] to [Q.sup.*] as from A to B in Figure 1. Both [Q.sup.0] and [Q.sup.*] are profit-maximizing equilibria that differ because the monopoly switches from X to Q. The integrated monopoly is on the expansion ray [R.sup.1] that has slope X/Y = 1. Then FQ is an outward movement along [R.sup.1] from [Q.sup.*] to [Q.sup.m] as from C to D in Figure 1. The reduction in real cost that Vernon and Graham[11] showed is a substitution effect along an isoquant from [R.sup.0] to [R.sup.1] as from B to C on the isoquant [Q.sup.*]. In Figure 1, total costs of [Q.sup.*] fall from [C.sup.0] = [rho] ([w.sup.0], r)[Q.sup.*] to [C.sup.] = (k + r)[Q.sup.*] But this switch from [R.sup.0] to [R.sup.1] is separate from the two output effects along the rays.

IV. Input Substitutability in the Anatomy

While FQ > 0 obviously is due solely to input substitutability, so is MQ < 0. The key point is that MQ exists in the pure monopoly model only as a matter of price elasticities, [epsilon] of derived demand and E of final demand. In the Allen price elasticity, [sigma] accounts for the difference between [epsilon] and E.(8) But if the noncompetitive equilibrium was one that was not governed simply by demand elasticity, then MQ might not exist as an offset to FQ > 0.

The dependence of MQ on the demand elasticities is demonstrated by a contestable monopoly case in which MQ would not exist. Given any entry barriers into producing X, define w" < [w.sup.0] such that the incumbent monopolist would sell nothing during any period when it set w > w", making the market for X contestable at w": this w" would be the maximum sustainable price of X[1]. The competitive final product price would be P" = c(w", r). in this model, the equilibrium price of X would be determined solely by entry barriers and the pressure of potential entry and would not be influenced by the elasticity of the derived demand. Assuming that integration does not affect entry barriers, then P" = c(w", r) would be the maximum sustainable price against integrated entrants, since they still could not produce X profitably at w". In the intermediate case, the equilibrium then would be [P.sup.*] = P", which would not be influenced by the elasticity of final demand. Therefore, [Q.sup.*] = Q" and MQ = [Q.sup.*] - Q" = 0.

Since neither of the contestable equilibria that define MQ depend on the elasticity of demand, integration itself would not "broaden" a contestable monopoly when [sigma] > 0. MQ would not exist. The sole effect of integration on final output would be FQ > 0 and a lower final product price.

V. Conclusion

Input substitutability in final production is a motive for noncompetitive input producers to integrate with their customers. Models that have characterized the noncompetitive input industry as a pure monopoly have found an ambiguity about whether such integration would increase or decrease final production and its price. The ambiguity in this monopoly case has inspired a well-known literature.

The effect on output has two components, which are shown here explicitly for the first time. Each component has an unambiguous and conventional effect. The first is an increase in monopoly power that always tends to reduce output and raise final product price. The other component is the actual substitution among inputs following integration. Not only does the input substitution reduce the real costs of production, as is well-known, it always tends to increase final output. The ambiguity about how vertical integration affects output when inputs are substitutable arises from ambiguity about the comparative magnitudes of the two opposing effects.

An additional insight is that input substitutability creates a negative monopoly power component only because the pure monopoly equilibrium rests solely on demand elasticity. In models of noncompetitive equilibrium that rest on other factors, this negative component may not exist as an offset and source of ambiguity. Then, the integration would increase output and reduce final product price unambiguously.

(1.) On the fixed proportions literature and its limitations, see Hamilton and Lee[5]. (2.) A function subscripted by an argument denotes a partial derivative. (3.) This assumption places some restrictions on the parameters of final demand. (4.) In the fixed proportions models, vertical integration of two successive noncompetitive industries always would increase final output[5]. Assuming a noncompetitive final product industry would confound that effect with how input substitutability affects output. (5.) Complete integration would be optimal for the input monopoly when final production has constant costs[9]. (6.) Neither a, [sigma], nor E is assumed to be constant. Early studies of the relationship among a, [sigma] and E assumed constant elasticities[3; 6; 7; 12]. (7.) Making the effective price of the input for the integrated firm be the production cost of that input assumes that the input has no alternative use and no market price. When not all firms are integrated, however, the opportunity cost for using X is its market price[4]. (8.) When the input proportion is fixed in final production, [epsilon] = aE. With this proportion between the two elasticities, integration by the monopolist has no effect on final output[5, 116].


[1.] Baumol, William J., John C. Panzar, and Robert D. Willig. Contestable Markets and the Theory of Industry Structure. New York: Harcourt, Brace and Jovanovich, 1982. [2.] Blair, Roger D. and David L. Kaserman. Law and Economics of Vertical Integration and Control. New York: Academic Press, 1983. [3.] Chung, Kwang S., "Forward Integration by a Monopolist: Some Extensions." Southern Economic Journal, January 1984, 690-710. [4.] Hamilton, James L. and Soo Bock Lee, "Vertical Merger, Market Foreclosure, and Economic Welfare." Southern Economic Journal, April 1986, 948-61. [5.] _____ and _____, "The Paradox of Vertical Integration." Southern Economic Journal, July 1986, 110-26. [6.] Hay, George A., "An Economic Analysis of Vertical Integration." Industrial Organization Review, 1973, 188-98. [7.] Mallela, Parthasaradhi and Babu Nahata, "Theory of Vertical Control with Variable Proportions." Journal of Political Economy, Sept./Oct. 1980, 1009-25. [8.] McGee, John S. and Lowell R. Bassett, "Vertical Integration Revisited." Journal of Law and Economics, April 1976,17-38. [9.] Schmalensee, Richard L., "A Note on the Theory of Vertical Integration." Journal of Political Economy, March/April 1973, 442-49. [10.] Tirole, Jean. The Theory of industrial Organization. Cambridge, Mass.: The MIT Press, 1988. [11.] Vernon, John M. and Daniel A. Graham, "Profitability of Monopolization by Vertical Integration." Journal of Political Economy, July/august 1971, 924-25. [12.] Warren-Boulton, Frederick R., "Vertical Control with Variable Proportions." Journal of Political Economy, July/august 1974, 783-802. [13.] Westfield, Fred M., "Vertical Integration: Does Product Price Rise or Fall?" American Economic Review, June 1981,334-46.
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Author:Hamilton, James L.
Publication:Southern Economic Journal
Date:Jul 1, 1992
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