La confesión en Foe

In Chapter 4, we showed that for a monopoly selling an input to a down-stream firm, tying is economically equivalent to vertical integration if the

downstream market is competitive and the franchisee uses only this input or, equivalently, when he must use one unit of the franchisor’s input for every unit of output it produces. This type of downstream production process exhibits what are called fixed proportions. In our discussion of this result, we hinted that the situation is more complex when the franchisee uses several different inputs that he can combine in different proportions to generate his output, that is, when his production function exhibits what are called variable propor-tions. In this appendix, we show that the economic equivalence of tying and vertical integration holds under downstream competition if the franchisee’s production function is characterized by variable proportions.¹⁹⁰To achieve the vertically integrated outcome in this case, however, the franchisor must require that the franchisee purchase all inputs from him or her.

In what follows,¹⁹¹we let P (Q) represent the inverse demand function for the franchisee’s output, Q(x1, x2) stand for a linearly homogenous production function so that we have constant returns to scale downstream, and cidenotes the constant marginal cost of input xi, i = 1, 2. We limit ourselves to two inputs, for simplicity. The result is the same with more inputs, but as will be clear, the tie then must involve all of them. We assume that the production of x1is monopolized while the markets for input x2and for the final good Q are competitive. If Q(x1, x2) admits variable proportions, the full monopoly rents cannot be obtained solely from the sale of x1because the derived demand for the monopolized input will reflect both consumer and franchisee substitution in response to the supracompetitive price of x1. Franchisee substitution leads to economically inefficient use of inputs in production that the franchisor can circumvent through forward integration.

Suppose that x1is only used in the production of Q, and that the producer of x1 can successfully vertically integrate all producers of Q. This strategy would then yield the following profit function for the integrated monopolist (who would not be a franchisor),

V= P · Q(x1, x2)− c1x1− c2x2, (A1) since x1 would be priced at marginal cost internally to achieve maximum profits.

190Vernon and Graham (1971) showed that with variable proportions downstream, a monopolist that would tie only his input would not achieve the vertically integrated level of outcome. This gave rise to a new reason to vertically integrate and fueled a revival in interest for theories of vertical integration.

191This section relies on Blair and Kaserman (1978).

Alternatively, the producer of x1 could remain separate from the down-stream firms and sign a franchise agreement with each of them. The producer could then purchase x2at the competitive price c2and tie the purchase of x2

by its franchisees to the purchase of the monopolized input x1. Following this strategy, the franchisor would have the following profit function,

T = p1(x1, x2)· x1+ p2(x1, x2)· x2− c1x1− c2x2, (A2) where piis the price charged by the franchisor to all its franchisees for the i^thinput. Note that both prices need to be set optimally as a function of the amount of both types of inputs purchased to induce the franchisees to use them in the right proportions; therefore they are both functions of x1and x2. To simplify notation, however, we refer to them simply as p1and p2in what follows.

Under these conditions, vertical integration and tying are economically equivalent. That is, these strategies yield identical results with regard to both profitability and productive efficiency. This assertion can be shown by proving the following two propositions:

Proposition I. Given a monopoly in x1, vertical integration that results in the monopolization of the market for Q and tying the purchase of x2to the purchase of x1yields identical profits to the producer of x1.

Proof. We want to show that
V =
T. Canceling input costs from (A1) and (A2), this requires that

P· Q(x1, x2)= p1· x1+ p2· x2. (A3) Under the tying arrangement, the franchisees accept p1, p2, and P as given: p1

and p2are chosen by the franchisor as a function of x1and x2so as to induce the optimal input mix, and P is exogenous to the franchisee due to compe-tition. Profit maximization on the part of the franchisee requires that he set the value of the marginal product of each input equal to its price, i.e., that

P· ∂ Q

∂xi

= pi, i = 1, 2 (A4)

Substituting (A4) into (A3) and factoring P, we obtain P · Q(x1, x2)= P

which yields the desired result, namely,

P · Q(x1, x2)= P · Q(x1, x2), (A6)

as^{∂ Q}_∂x

1· x1+^{∂ Q}_∂x2 · x2= Q by Euler’s Theorem. This establishes the profitabil-ity equivalence of the alternative strategies. Now, we turn to the productive efficiency of input usage under tying:

Proposition II. Inputs x1 and x2 will be employed in efficient proportions whether the franchisor obtains control of the final-good industry through vertical integration or engages in a tying arrangement.

Proof. Efficient production requires that input proportions be adjusted such that

∂ Q/∂x1

∂ Q/∂x2 = c1

c2, (A7)

that is, the downstream firm uses the inputs in proportion to their relative marginal costs. The first-order conditions for profit maximization by the monopolist if she vertically integrates the downstream industry are



Dividing (A8) by (A9) yields expression (A7), which establishes efficient input utilization under vertical integration. To establish the equivalent result under a tying arrangement, note that (A4) describes how the franchisees will choose the amounts of inputs to use. The franchisor’s goal therefore will be to maximize
Tgiven (A4). Thus, we substitute (A4) into (A2) and differentiate with respect to x1and x2. This yields the first-order conditions:



Linear homogeneity of the production function implies that the second bracketed term on the left-hand side of (A10) and (A11) is zero. Drop-ping these terms, we factor _{∂ Q}^{∂ P} from the last two terms of the first brack-eted expression, apply Euler’s Theorem, and write conditions (A10) and (A11) as

Dividing (A12) by (A13) again results in expression (A7), thereby confirming efficient input utilization under a tying arrangement.

Propositions I and II establish an equivalence between both the private and the social effects of vertical integration and tying under input monopoly and variable proportions with downstream competition. Given such sym-metry, the firm holding monopoly power over an input for which (imper-fect) substitutes exist must select between alternative strategies on the basis of factors that lie outside our simplified model. Important considerations that will influence this choice include the number of substitutable inputs that must be tied to the monopolized input in order to ensure efficient downstream production; the frequency of changes in the costs, ci, and the cost of changing the prices to franchisees in response to those changes in costs; potential cost savings that may be available through vertical integra-tion because of transacintegra-tional efficiencies or technological inseparability of the various stages of production; the feasibility of vertical integration with the whole downstream sector; and the comparative treatment afforded the alternative strategies by the antitrust authorities. The latter is determined by the policy thrust of the antitrust enforcement agencies as well as legal precedents.

Of course, if the franchisor cannot impose requirements on all inputs, then vertical integration will dominate franchising unless there are incentive is-sues under vertical integration (e.g., paid managers have weaker incentives to provide effort than owner operators do). Moreover, while in many indus-tries manufacturers cannot reasonably vertically integrate all downstream resellers – their product is just one of a large mix of products sold at su-permarkets or in large department or discount stores – in the franchising

context, however, vertical integration is a readily available option. As we saw in Chapter 4, most franchisors operate some of their outlets directly as it is. In-creasing their reliance on vertical integration is always an option they can turn to if they find that their capacity to contract with franchisees is too constrained by the regulatory regime.

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