Exclusive trading contracts in successive differentiated duopoly.

• Author: Dobson, Paul W.

1. Introduction

   In the case of most consumer purchases, from a bottle of mineral water to a household appliance, there are two usually distinct considerations--what product to buy (e.g., Perrier or Evian), and where to purchase it (e.g., Wal-Mart or Safeway)--with competition taking place at two vertically related levels between, on the one hand, manufacturers' brands and, on the other, retailers' services associated with its characteristics or location. However, for some products, such as new automobiles, even though manufacturing and retailing operations remain formally separated, the distinction is blurred because consumers are able to purchase only one manufacturer's products from a single retailer, thus effectively merging the two forms of competition.(1) This may be the result of a deliberate policy by firms to use contractual relations to limit both interbrand and intrabrand rivalry. Nevertheless, whilst it appears to restrict consumer choice and dampen competition, the practice may not necessarily be detrimental to social interests, especially since the tie may allow the contracting parties to enter into a more efficient trading relationship.

   This paper considers the mutual incentive for firms to enter into exclusive trading relationships which restrict a manufacturer to distributing its product through a single retailer and the corresponding retailer to purchasing goods only from this manufacturer. The incentive for adopting exclusive trading against unrestricted trading rests on two conflicting sets of factors. Advantages to be gained from having an exclusive trading contract may include using side payments between the two parties to internalise externalities which adversely affect joint profits due to independent successive price setting, as well as limiting direct interbrand and intrabrand competition when retailers specialise in selling different products, thus avoiding head-to-head competition and cannibalizing sales. Against these arguments, by signing such an agreement, the retailer necessarily limits its product range, and the manufacturer similarly limits where its product can be distributed. Consequently, potential sales may be lost by either party.

   Most of the existing literature examining the strategic effects of vertical restraints on interbrand and/or intrabrand competition, including Bonanno and Vickers [3], Rey and Stiglitz [13;14], Lin [7], O'Brien and Shaffer [11] and Besanko and Perry [2], have considered this issue in the context of manufacturers effectively imposing restraints on retailers, and having the power to extract the entire generated surplus through franchise fees. In contrast, this paper, following Chang [4], allows for the possibility of market power to exist at both the manufacturing and retailing levels, and consequently examines restraints in terms of their mutual advantage to both parties.(2)

   Our paper considers the conditions which offer a vertical pairing higher joint profits from entering into a contractually binding exclusive trading relationship, where contracting parties can use side payments (e.g., two-part tariffs) to distribute profits.(3) As with Chang's analysis, the issue is examined in a successive competition setting where two upstream firms (manufacturers) face two downstream firms (retailers). However, our analysis is distinguished by modelling behaviour in terms of price-setting competition(4) and, more importantly, by allowing both products and retailer services to be differentiated. It is thus considerably more general. This two-dimensional differentiation, which is captured in terms of a simple parameterisation of representative consumer demand, enables us to examine the private and social desirability of exclusive trading contracts across a range of interbrand and intrabrand rivalry conditions.

   The analysis shows that firms are jointly better off with exclusive trading only when there is relatively little differentiation between products and between retailers' services. However, the individual incentives facing each pair ensure that with moderate degrees of differentiation, the firms may be caught in a prisoners' dilemma, where the equilibrium outcome involves both pairs adopting exclusive trading, but the firms would be better off from unrestricted trading. With high levels of differentiation, the penalty in foregoing sales from restricting product range is sufficient to overcome this problem and the equilibrium outcome has firms using non-exclusive trading contracts.

   The welfare analysis shows that, when there is little differentiation at both stages, social welfare levels are higher with unrestricted competition. This is in direct conflict with private incentives for exclusivity. The conflict, that is private desirability but social undesirability, is mirrored in Rey and Stiglitz [14] and implicitly in Besanko and Perry [2] (given our assumptions on costs) and Bonanno and Vickers [3]. Yet it is not present in our model for all degrees of differentiation, as we show later. Thus we would argue that a blanket ban on exclusivity would be inappropriate.

   Our analysis is developed through a specific linear model, which nevertheless is generalised somewhat in order to examine its robustness. Section II develops this model, in particular the pricing stages of the game, after which equilibria are considered in section III and welfare comparisons are made in section IV. After a brief analysis of an alternative case involving manufacturer collusion, the paper closes with some concluding comments in section VI.

2. The Model

   We develop a three-stage subgame-perfect equilibrium model in which each stage is a Nash strategy choice. There are two manufacturers and two retailers. In the first stage, contract offers detailing the intermediate price and trading arrangements are exchanged between the manufacturers and retailers, resulting in each manufacturer-retailer pair simultaneously making a contract decision between (i) a formal exclusive distribution contract, strategy D, (ii) a contract covering both exclusive distribution and exclusive purchasing, strategy E, or (iii) trading with a non-exclusive distribution contract, strategy N. Each firm is constrained to enter into at most one exclusive distribution relationship (thus ensuring that all four firms are active in the market), and then given that the strategy set facing each trading pair is {D, E, N}, there are four basic contractual configurations to consider: (N, N) (which is also the default), (E, E) (which is an equivalent outcome to (D,E), (E,D) and (D,D)), (D,N) (or its counterpart (N,D)) and (E,N) (or equivalently (N,E)).

   The second and third stages characterise behavior at both levels of the vertical structure by successive duopolistic price competition. In the second stage, manufacturers set transfer prices to retailers, and in the final stage retailers set market prices. These stages are solved in the standard recursive manner, by initially considering the retailers' decisions in terms of the given transfer prices, and then using Nash equilibrium outcomes from this stage to determine the actions of the manufacturers in the previous stage. We proceed by discussing the ingredients of these two stages in this section, then moving on to consider the first.(5)

   The two upstream manufacturers, [M.sub.1] and [M.sub.2], indexed by i,j = 1, 2, i [not equal to] j, each produce their own branded product, and there are two downstream retailers, [R.sub.1] and [R.sub.2], indexed by h,k = 1,2, h [not equal to] k, capable of selling these products.(6) The manufacturers supply the products to the retailers at a constant unit price (though, under an exclusivity agreement a side-payment may also be involved), where the transfer (intermediate) price between manufacturer i and retailer h is [r.sub.ih] for quantity [q.sub.ih], which is then sold on to consumers at the unit price [p.sub.ih]. The manufacturers produce under constant returns to scale, incurring a common unit cost [c.sub.M], which to aid exposition, and without further loss of generality, is set to zero. As well as paying the transfers, the retailers incur common retailing costs at constant per unit level [c.sub.R], which is again for convenience set to zero.

   The retailers are assumed to face a large number of consumers. The representative consumer maximizes U([q.sub.11], [q.sub.12], [q.sub.21], [q.sub.22], y) subject to the budget constraint I = y + [q.sup.T] p. Here [q.sup.T] = [[q.sub.11] [q.sub.12] [q.sub.21] [q.sub.22]] and [p.sup.T] = [[p.sub.11] [p.sub.12] [p.sub.21] [p.sub.22]] .For much of the paper we assume U to be quadratic and strictly concave with the following form:

   (1) U = y + [i.sup.T] q - ([q.sup.T] Zq)/2

   where y is the quantity of the numeraire commodity consumed, i is a unit vector and

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

   This is equivalent to having consumers maximize

   [i.sup.T]q - ([q.sup.T] Zq)/2 - [q.sup.T]p

   since the Lagrange multiplier will equal one. Hence the demand structure is

   p = i - Zq

   For example, if both products are sold by both retailers then inverse demand for product i from retailer h is a simple linear function of the four traded quantities:

   (2) [p.sub.ih] = 1 - [q.sub.ih] - [Beta] [q.sub.ik] - [Gamma] [q.sub.jh] - [Delta] [q.sub.jk] 0 [equal to or less than] [Beta], [Gamma], [Delta] [equal to or less than] 1

   In cases where demand is "rationed," in the sense that both products are not sold by both retailers, the relevant elements of the q vector are set to zero. Later, we engage in some selective generalisation to the inverse demand function

   (3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

   in order to check results for sensitivity.

   The parameter [Beta] measures the degree of intrabrand rivalry, that is how similar the retailers' services are perceived by consumers to be when selling the same product, which, for example, may have to do with differences in the retailers' locations...