Vertical restrictions and the number of franchises: comment.

• Author: Schmidt, Torsten
• Position: Response to Owen R. Philips, Southern Economic Journal, p. 423, October 1991

In a recent issue of this Journal, Owen Phillips [9] considered a monopolist selling a manufactured good to a number of dealers. The dealers are all alike, and each unit of the final good sold to consumers requires one unit of the wholesale product. The equilibrium number of dealers N and their individual equilibrium sales levels S are endogenous and depend on the product's wholesale price P and a fee F: N = N(F, P) and S = S(F, P). With subscripts denoting derivatives, Phillips requires that [N.sub.F] [less than] 0, [N.sub.P] [less than] 0, [S.sub.F] [greater than] 0, and [S.sub.P] = 0. Product demand and dealer costs remain hidden, so that the analysis is very general. Based on these conditions alone, according to Phillips, the upstream firm will optimally set the wholesale price below its marginal cost. That appears not to be correct. This comment points out several problems with his analysis.

The assumption that equilibrium dealer output is negatively associated with the wholesale price ([S.sub.P] [less than] 0) eliminates a broad class of models from consideration. Section I presents several examples based on linear demand, quadratic costs, free entry, and various modes of competitive interaction. Section II takes up Phillips's reasoning based on the reported equations to demonstrate that, contrary to his claim, these do not require a wholesale price below marginal cost. Section III briefly reconsiders the general question of the conditions that may support such a wholesale price. An appendix corrects some errors in the reported equations, and identifies further implicit assumptions made by Phillips.

1. Examples

   The examples of this section are intended to show that the sign restrictions in Phillips's model rule out many models of interest. Consider the following setup: [A] Upstream costs are zero. [B] Dealer costs are quadratic: c(q) = [q.sup.2]. Including payments to the manufacturer, dealer costs are C(q) = [q.sup.2] + Pq + F. [C] The good is homogeneous and demand is linear: [Q.sub.d] = [Alpha] - [P.sub.d], where [Alpha] [greater than] 0. [D] The dealers are price takers. [E] Additional dealers enter until each dealer earns a zero profit.(1) This basic model will also serve as a point of reference for other examples.

   In equilibrium, the retail price must be equal to marginal cost for each dealer, and free entry assures that profit is zero. Then each dealer's output must be minimizing the average cost inclusive of payments to the upstream firm, AC(q) = q + P + F/q. The individual dealer's equilibrium output is then equal to S(F, P) = [F.sup.1/2]. The equilibrium retail price [P.sub.d] turns out to be 2[F.sup.1/2] + P, the value of average cost at its minimum. Equilibrium market output equals X(F, P) = [Alpha] - 2[F.sup.1/2] - P, and the equilibrium number of firms, ignoring the integer constraint, is N(F, P) = ([Alpha] - 2[F.sup.1/2] - P)/[F.sup.1/2].

   In this model, equilibrium dealer output is not inversely related to the wholesale price. Each adjustment of the wholesale price results in a vertically parallel shift of the AC(q) schedule, leaving the output that minimizes AC(q) unchanged, so that [S.sub.P] = 0. This model is thus excluded from Phillips's specification as he requires [S.sub.P] [less than] 0.

   For a complete solution to the manufacturer's problem of choosing (F, P), consider the reduced-form upstream profit [Pi] = FN + PX = (P + [F.sup.1/2]) ([Alpha] - 2[F.sup.1/2] - P). Because the manufacturer can extract all downstream profit with the fee, upstream profit is greatest when the product is brought to market at the lowest underlying unit cost of production. That unit cost, equal to ac(q) = q, approaches its minimum value of zero as q approaches zero. Call this dealer output the optimal output. Comparison of the equilibrium output with the optimal output implies an optimal fee that also equals zero. A strictly...