Along with prices, inventory levels — i.e., quantities — are the most fundamental decisions that must be coordinated along a supply chain.
Maintaining the single departure from the perfect competition ideal, upstream market power, we address the vertical coordination of inven-tory incentives under demand uncertainty.
Anecdotal evidence suggests strongly that incentives to maintain efficient inventory levels cannot be elicited without complex contracts.
When Corning Glass Works was constrained to simple contracts fol-lowing a case brought by the Federal Trade Commission in 1975 which ruled out resale price maintenance, one of the most detrimental effects to the firm was the cutback in inventory by downstream distribu-tors and the loss of its smaller outlets [91, p. 325]. Hourihan and Markham [87] examined a set of cases also involving the prohibition of complex contracts. These authors found as well that with the restriction of firms to simple contracts, in three of the four cases where the restric-tion was binding, the availability of the product to buyers dropped.9 This drop in availability the authors attributed to a decline in inven-tory levels and in downstream decisions to carry the product at all.
9The reader may wonder why the law sometimes restricts the set of contracts available to firms. The answer is that the law is sometimes wrong. In Section 2.4, we discussed the development of the law on resale price maintenance.
In these case studies, the contract that had supported superior inven-tory levels was resale price maintenance. The task for economic and management theory in illuminating these cases is to explain the role of resale price maintenance in generating adequate inventory incentives.
The key papers on this topic, in a competitive downstream setting, are [54, 55]. We present here simple models that capture the ideas of Deneckere et al. [55].
The starting point to any inventory incentive theory is the foun-dational newsvendor model. A firm, facing a random demand, must choose inventory before the realization of demand. All unsold inven-tory perishes. The conventional assumption is that price is exogenous.
This corresponds to an implicit assumption that all consumers value the product at some common value, v. The demand facing the firm is then perfectly inelastic up to a price equal to v, i.e., demand is rect-angular; only the size of this demand is uncertain. Taking the usual notation, we let the random demand be x, with a distribution F and continuous density f . The firm produces at a cost c < v. Thus we have a monopolist with uncertain, perfectly inelastic demand, whose product has a perfect substitute available at price v. The monopolist solves
maxy π(y) = vE min{y,x} − cy = vy − vE max{y − x,0} − cy (5.6) which yields an optimal condition
[1− F (y)]v = c (5.7)
The marginal benefit of an additional unit of inventory, [1− F (y)]v, is equated to marginal cost, c, at the optimum. This yields the classic fractile solution:
1− F (y∗) = c/v at the optimal inventory, y∗.
Ex post pricing: Suppose, however, that the monopolist must distribute its product to final buyers through the intermediation of a competi-tive retail sector. The monopolist simply sets a wholesale price w and sells to retailers, which bear no cost other than w. The retailers must choose their purchases of inventory prior to the realization of random
demand. After the realization of demand, price is determined in an ex post market. The ex post market consists of the realized demand curve (rectangular out to the realized value of x) and a supply curve that is perfectly inelastic at the amount y of aggregate inventory chosen by firms in the sector. For any realization of demand up to y, inelastic supply exceeds inelastic demand, and the market-clearing price in this ex post market is 0. For realizations of demand greater than y, the market-clearing price is v, the maximum amount that buyers will pay.
That is, given y and realized x, the market price p(x, y) =
0 : x≤ y,
v : x > y (5.8)
Industry profits are given by:
[1− F (y)]vy − wy = vy − vyF (y) − wy (5.9) Compare the classic newsvendor profit function (5.6) with the industry profits under competitive intermediation (5.9). The inventory problem under competitive intermediation is equivalent to an optimal inventory problem in which the monopolist faces the constraint that if its inventory turns out to be excessive (even by one unit), the entire revenue from the inventory must be abandoned. For the supply chain as a whole, retail competition destroys all profits if inventory is excessive even by one unit.
The downstream competitive market will order inventory, y, up to the level at which each of the price-taking firms (firms take the price, conditional upon any realization of x, as given) foresees zero expected profits. The aggregate zero-profit level is where the expected revenue from each unit equals its cost, i.e., [1− F (y)]v − w = 0.10
Incorporating this competitive market reaction as a constraint, we write the monopolist’s problem as follows
maxw,y (w− c)y (5.10)
subject to
[1− F (y)]v − w = 0
10This equation expresses the zero expected profit condition, although it is identical to the optimal inventory condition for a downstream firm facing an upstream price w.
Substituting the constraint into the objective function, the monopo-list’s problem can be written as maxy([1− F (y)]v − c)y. The first-order conditions to this problem yields the following characterization of the optimal inventory level, y∗:
[1− F (y∗)]v = c + f (y∗)vy∗ (5.11) Compare (5.11) with (5.7). Competitive intermediation adds a term to the classic fractile characterization of optimal inventory — the addi-tional term f (y)vy on the right-hand side of (5.11) as compared to (5.7). When an additional unit of inventory is added, the probability of losing the entire revenue vy is increased by the amount f (y), and this additional marginal cost is reflected in the first-order condition. This means that the optimal inventory is lower as a result of the intermedi-ation by competitive firms. Profits are reduced.
Setting aside vertical integration as a solution, we turn to the simplest contracts that resolve the incentive distortion. The natural strategy to consider, given case evidence, is (as Deneckere et al. [55]
suggest) the adoption of resale price maintenance (RPM).
Consider the imposition of a price floor at an arbitrary price p. Hold-ing p constant leads to a one-to-one relationship between the instru-ment w and the outcome y, as firms are induced to raise inventory to the point where the aggregate profits in the competitive retail sector are zero. This relationship is given by the following:
py− pE max{y − x,0} − wy = 0
Under RPM, the price does not fall to zero as soon as x is slightly below y. The monopolist’s problem becomes the following11:
maxw,y,p(w− c)y (5.12)
subject to
py− pE max{y − x,0} − wy = 0
Solving the constraint in this problem for w and substituting it into the objective function (5.12) replicates the original newsvendor maximiza-tion problem for the vertically integrated firm. The monopolist can set
11This formulation assumes that the price floor is binding; this is easily demonstrated.
p = v, and set w so that the downstream retailers choose the first-best inventory level. (Note that the w that elicits first-best inventories will be greater than marginal cost c.) Adopting RPM, therefore, allows the monopolist to achieve the first-best system profits. This is the simplest theory of the observed role of resale price maintenance in supporting inventory levels.
Ex ante pricing: In a contemporaneous paper, Deneckere et al. [54]
develop a similar model with demand uncertainty and inventory choices, but in which retailers must commit to prices before the real-ization of demand. In the simple-contract game (with no vertical restraints), the manufacturer sets a wholesale price, w, and the retail-ers then each choose price and inventory levels simultaneously. In many markets, including video markets and fashion good markets that Deneckere et al. [54] describe, the resolution of uncertainty is short, and retailers must choose prices ahead of the realization of demand.
The pricing cycle is as long or longer than the production/inventory cycle.
We can capture the basic idea of Deneckere et al. [54] with a model in which — as before — all consumers share the same value v for a unit of demand, but in which the number of consumers is random, and represented by a smooth distribution, F (x).12 But now prices are set ex ante. We assume, for simplicity, that each retailer provides one unit or none. The “inventory” decision is thus whether to carry the prod-uct or not.13Consumers are assumed to enter the market sequentially, rather than simultaneously.
The result is that the retail market is characterized by the following equilibrium, also studied by Dana [49]. With a continuum of buyers and a continuum of retailers, the result is an equilibrium with price dispersion: sellers in equilibrium set a variety of prices and buyers enter the market compare prices perfectly, each buying (upon entry) from the lowest-priced supplier still available. Retailers set prices along a continuum, trading off a higher probability of sale (at a low price) with
12Deneckere et al. [54] present a numerical example with unitary demand and common value, as we are doing here, but assume three possible states of demand.
13This assumption is inessential.
a higher realized profit conditional upon sale (at a higher price). Along the continuum, retailers each earn zero expected profits.
Consider the retail market facing a given wholesale price, w, and no other costs. For a given distribution G(p) of retailers charging various prices up to p,14 a retailer charging a particular price p will sell if x≥ G(p), an event that happens with probability 1 − F (G(p)).
The retailer’s expected profit from selling one unit is therefore [1− F (G(p))]p − w. An equilibrium price distribution is a distribution that results in zero retailer profits at each price, and thus is the solution G∗ to the following functional equation:
[1− F (G∗(p))]p− w = 0 (5.13) It is straightforward to show that G cannot have atoms, and that the support of G must be [w, v]. Note that G∗(v) is the total number of retailers.
At a given w the total inventory ordered by retailers, i.e., the input demand function facing the upstream monopolist, is given by G∗(v), the total number of retailers. The profit of the monopolist is given by (w− c)G∗(v), where G∗ is the solution to the retail equilibrium equation.
Consider, instead of the simple contract, a contract in which the upstream monopolist imposes a price floor at v. In this case, the inven-tory purchased from the monopolist is the amount that yields zero profits downstream, as earlier. That is, y is the solution to the following:
vy− vE max{y − x,0} − wy = 0 (5.14) Note that instead of a distribution of downstream prices on [w, v], all the downstream firms set price v. In terms of economics, if the G∗(v) retailers under the simple contract suddenly face a price floor at v, the quantity demanded under each realization of uncertainty is unchanged but price is higher. The higher price elicits more entry ex ante (via either additional retailers or additional inventory at each retailer). Resale price maintenance thus increases the demand for the upstream firm’s product. As before, the manufacturer can set w to elicit first-best inventory and profit levels.
14We omit the dependence of G on w to keep the notation simpler.
The Deneckere–Marvel–Peck models, the essence of which is cap-tured in our two simple settings, are in our view fundamental contri-butions to the foundations of supply chain organization. The models capture the failure of the price system to convey the optimal incentives for inventory with the introduction of the smallest departure from per-fect competition, market power upstream. And this smallest departure is enough to explain the role of resale price maintenance in correct-ing the incentive distortions. Marvel and Wang [132] add a buy-back policy to resale price maintenance and show in the model of inven-tory with pricing flexibility (the second model analyzed above) that addition of this second instrument yields first-best profits. The man-ufacturer need not even know the distribution of demand uncertainty in implementing the optimal policy, but can instead rely on retailers’
inventory incentives. Marvel and Wang [133] extend the analysis to the additional instrument of revenue sharing, and show that any two of the three instruments — a constant wholesale price, buy-backs and revenue sharing — can achieve first-best coordination.
5.3 An Aside: Perfectly Competitive versus Monopoly