Wholesale Prices and Cournot‐Bertrand Competition

This note considers the competing vertical structures framework with Cournot&#8208;Bertrand competition downstream. It shows that the equilibrium wholesale price paid by a Cournot (Bertrand)&#8208;type retailer is above (below) marginal costs of a corresponding manufacturer. This result contrasts with the one under pure competition downstream (i.e., Cournot or Bertrand), where the wholesale price is set below (above) marginal costs in case of a Cournot (Bertrand) game at the retail level.


I. INTRODUCTION
The papers Alipranti et al. (2014) and Rozanova (2015) demonstrate that the equilibrium wholesale prices are set above (below) marginal costs of the manufacturers in the case of Bertrand (Cournot) competition downstream. 1 Besides competition in quantities and in prices the literature considers the so called Cournot-Bertrand model (where some firms set quantities and others choose prices). The examples of these papers include Singh and Vives (1984), Tremblay and Tremblay (2011), Tremblay et al. (2011), and Tremblay et al. (2013. The aim of the current note is to study the equilibrium wholesale prices in the situation of Cournot-Bertrand competition downstream. I show that in the case of two competing vertical structures the equilibrium wholesale price paid by the retailer choosing quantity (price) is set above (below) marginal costs of the manufacturer selling the good to the retailer under consideration. In other words, at the downstream level the Bertrand-type competitor has a E13 cost advantage over the Cournot-type firm. This outcome differs from the one in case of pure (i.e., Cournot or Bertrand) competition downstream, where equilibrium wholesale price is below (above) the manufacturer's marginal costs when the retailers compete in quantities (prices).

II. MODEL
There are two manufacturers (1 and 2). Each of them sells a substitute product to its own retailer. At the downstream level retailers differ in their choice variables. Retailer 1 is a Cournot-type firm (i.e., its strategic variable is output, q 1 ). Retailer 2 is a Bertrand-type firm (i.e., it sets price, p 2 ).
The upstream firms 1 and 2 produce at constant marginal costs c 1 and c 2 , respectively. The downstream firms do not have other costs than the spending on the input from the upstream firms.
The timing is as follows: at the first stage of the game, each upstream producer bargains with its retailer over the components of the two-part tariff contract, that is over a wholesale price, w i , and a fixed fee, F i , i = 1, 2. At the second stage, the downstream firms simultaneously set their strategic variables after observing each other's contract terms.
The bargaining is modelled in the following way: it is assumed that each vertical structure (i.e., upstream firm and its downstream firm) solves the Nash bargaining problem. In vertical structure's i = 1, 2 Nash bargaining problem, upstream firm i has bargaining power β i , while downstream producer's i bargaining power is 1 − β i , where β i ∈ (0, 1].

II.1 Consumer side
Utility of the consumer depends on the goods sold by the retailers and the expenditures on the composite commodity (i.e., T ): where q i , i = 1, 2 is the quantity of commodity i consumed.
From the utility maximization problem max q 1 ,q 2 U (q 1 , q 2 ) + T s.t. p 1 q 1 + p 2 q 2 + T ≤ I (where I is the income of the consumer and p i is the unit price of good i) we get the inverse demand functions for the goods 1 and 2: Proof. Differentiation of (2) with respect to q 1 and q 2 and application of Assumptions 1 and 2 give the result of Lemma 1. Q. E. D.
Proof. Differentiating (2) with respect to p 2 (taking into account the fact that p 1 and q 2 are the functions of p 2 ) we get: From (5) and it is negative due to Lemma 1. Q. E. D.

III. EQUILIBRIUM WHOLESALE PRICES
Let (w * 1 , w * 2 ) be the Subgame-Perfect Nash equilibrium wholesale prices that the retailers 1 and 2 pay to the manufacturers 1 and 2, respectively. Proposition 1. w * 1 > c 1 , w * 2 < c 2 . Proof. At the first stage of the game each vertical structure i = 1, 2 maximizes its Nash product with respect to w i and F i . That is each vertical structure i solves the following optimization problem: where w = (w 1 , w 2 ); π U i (w) is the profit of the upstream manufacturer i without taking into account the fixed fee, while π D i (w) is the downstream firm's i payoff again without considering the fixed fee. P U i (P D i ) is a disagreement payoff of an upstream (downstream) producer i. Maximizing (11) with respect to F i we get: Plugging (12) into (11) we conclude that the equilibrium w i is found from: Vertical structure 1: For the first vertical structure 3 π U i (w) and π D i (w) are: Therefore (13) is: Assuming internal solution the first-order condition to (16) is: Assume that w * 1 = c 1 . Then (17) becomes: Due to Lemmas 2 and 3 ∂ p 1 ∂ p 2 > 0, ∂ p * 2 ∂w 1 > 0. Therefore ∂ 1 ∂w 1 | w 1 =c 1 > 0 and w * 1 = c 1 . Assume w * 1 < c 1 . Then due to Lemmas 2 and 3 ∂ 1 ∂w 1 | w 1 <c 1 > 0. Therefore w * 1 cannot be below c 1 . We conclude that w * 1 > c 1 .
The intuition for the result formulated in Proposition 1 is straightforward: When vertical structure with a Cournot(Bertrand)-type firm downstream sets the wholesale price above(below) its upstream firm's marginal costs it leads to the increase in price (reduction in quantity) of the competing downstream firm (compared to the situation where the wholesale price is equal to the marginal costs of the upstream firm). In other words, setting the equilibrium wholesale prices in the way described in Proposition 1 allows relaxing competition downstream.

IV. CONCLUDING REMARKS
The current note proves that in case of two competing vertical structures and Cournot-Bertrand competition downstream the equilibrium wholesale price paid by the Cournot(Bertrand)-type retailer is above(below) marginal costs of the corresponding manufacturer. This result suggests that if Cournot-Bertrand competition is considered in the context of vertically related markets, then the advantage that the Cournot-type firm has over the Bertrand-type producer in a one-tier framework 4 may be reduced (or even eliminated).
It is important to underline that the observability of the contracts is crucial for obtaining the result presented in the current note. One may easily verify that in case of unobservable contracts (i.e., if the downstream firm j does not observe the contract terms between the upstream firm i and the downstream firm i, where i, j = 1, 2, i = j before the second stage of the game) the equilibrium wholesale prices are equal to the marginal costs of the corresponding upstream firms (i.e., w * 1 = c 1 , w * 2 = c 2 ). The intuition for this result is the following: if the downstream firm j does not observe w i , then w i cannot affect the behavior of the downstream firm j.