1. Introduction

Fierce competition generally brings down price and raises social welfare. Recognizing this, antitrust authorities implement competition policies to intensify market competition. An index that can measure the competition intensity is the type of competition, that is, whether Cournot or Bertrand competition prevails. Previous studies focus on other indicators such as the number of firms and the degree of product differentiation, but regardless of the indicator is employed, tougher competition is desirable for society. Thus, several papers focus on a situation where equilibrium social welfare decreases with the competition intensity (Lahiri and Ono, 1988; Mukherjee and Zhao, 2009; Fanti, 2013). Thus, the literature on industrial organization compares outcomes under Cournot competition with those under Bertrand competition and finds that in many cases, Bertrand competition yields outcomes that are more desirable for society.

For example, a classic study that compares Cournot competition with Bertrand competition is by Singh and Vives (1984). They show that the equilibrium price in Bertrand competition is lower than that in Cournot competition. Studies that followed indicate that this relationship is reversed (e.g., Delbono and Lambertin, 2016a; Häckner, 2000; Zanchettin, 2006).

Some recent studies consider an intermediate competition type between Cournot and Bertrand competitions. One of them is supply function competition (Grossman, 1981; Klemperer and Meyer, 1989). Some recent studies analyze properties of supply function competitions (e.g., Delgado and Moreno, 2004; Ciarreta and Gutierrez-Hita, 2006; Menezes and Quiggin, 2012; Delbono and Lambertini, 2015, 2016b). Under the supply function competition, it is common wisdom that the supply function equilibrium is intermediate between that under Cournot and Bertrand (Delbono and Lambertini, 2015). However, Delbono and Lambertini (2016b) challenge this well-known result. They consider a market with quadratic cost and show that the supply function competition creates the largest social welfare among the competitions.

Following Delbono and Lambertini (2016b), we reconsider a welfare ranking between Cournot, Bertrand, and the supply function competitions. There are some differences between our model and theirs. In particular, we consider a vertically related market where an upstream firm sequentially contracts with two downstream firms. Our formulation of a sequential contract is the same as that of Kim and Sim (2015). In other words, we introduce the supply function competition into the model with the sequential contract presented by Kim and Sim (2015).

Because the supply function competition leads to Cournot and Bertrand equilibria as special cases, considering a supply function competition model suffices to examine whether Bertrand is the best outcome and Cournot, the worst. When comparing the two competition structures with other cases in the supply function competition, we show that Cournot and Bertrand competitions do not lead to the worst outcomes. That is, at parameter values where equilibrium outcomes are not the same as those under Cournot and Bertrand competitions, we obtain the minimum social welfare. This result is quite different from that in Delbono and Lambertini (2016b).

The paper proceeds as follows. The next section presents the model. Section 3 calculates the equilibrium and provides the main results. Section 4 concludes the paper.

2. The Model

We consider a market with upstream firm and two asymmetric downstream firms To produce one unit of product, each downstream firm must purchase one unit of input from the upstream firm. We assume that the upstream firm can choose different prices for the downstream firms. We denote the wholesale price for downstream firm by Let denote the output of downstream firm We assume that the marginal cost of upstream firm is zero, that of downstream firm is also zero, and that of downstream firm is Following the literature on supply function competition, we assume that downstream firm uses a linear supply function , where is the price, intercept is an endogenous variable chosen by each firm, and is an exogenous parameter.

Menezes and Quiggin (2012) and Delbono and Lambertini (2015) show that as converges to zero, equilibrium outcomes under the supply function competition converge to those under Cournot competition; however, as diverges to infinity, equilibrium outcomes converge to those under Bertrand competition. Therefore, to compare equilibria under Cournot, Bertrand, and the supply function competitions, it suffices to consider the supply function competition with Moreover, as an increase in moves equilibrium outcomes closer to those in Bertrand competition, we can interpret as the index of competition intensity.

As downstream firm is relatively inefficient, for large the equilibrium output of would take a value of zero. To guarantee a positive outcome, we assume

We assume that the inverse demand function is As we assume a linear supply function, substituting into the inverse demand function and solving it for we have

Substituting this equation into we have the following supply functions:

Using the above equations, we define the profits of downstream and upstream firms as follows:

Consumer surplus and social welfare

Following Kim and Sim (2015), we consider the sequential contracts between the upstream and downstream firms. In the first stage, the upstream firm decides wholesale price and in the second stage, downstream firm chooses in its supply function. After these decisions, in the third stage, the upstream firm offers wholesale price and in the fourth stage, downstream firm determines in its supply function. We could justify this timing of the game as follows. When the upstream firm already creates a long-term relationship with downstream firm the contract may express a commitment effect. Then, given the contract with downstream firm the upstream firm and downstream firm must negotiate their contract. Hence, a Stackelberg timing structure is a natural assumption here.

We assume complete information. The model is solved by backward induction. Only pure strategies are considered throughout this paper.

3. Calculating Equilibrium and Condition Yielding Worst Welfare

The profit maximization problem of downstream firm is

The first-order condition yields

where superscript represents that the outcomes are in stage 4. Substituting this outcome into and we have and

Substituting the above results into the profit of the upstream firm, the maximization problem is

The first-order condition leads to

where superscript represents that the outcomes are in stage 3. Substituting this outcome into and we have and

Substituting the above results into the profit of downstream firm the maximization problem is

From the first-order condition we have

where superscript represents that the outcomes are in stage 2. Substituting this outcome into and we have and

Substituting the above outcomes into the profit of upstream firm, the maximization problem is

The first-order condition leads to

where superscript represents that the outcomes are in equilibrium. Summarizing the above results, we obtain the following proposition:

Proposition 1. The equilibrium social welfare is

Where

Differentiating with respect to we have

where

and

After a tedious calculation, we obtain the following proposition:

Proposition 2. When we can freely choose a value of the equilibrium social welfare is minimized at the following

such that satisfies and satisfies where

Proof.

First, using the discriminant of the numerator for we show that the value of that minimizes does not exist in As the sign of denominator is positive, the sign of first derivative is the same as that of numerator. The numerator is the quadratic function of and the coefficient of is positive Then, if the sign of discriminant is non-positive, we will have Numerically solving for we have Hence, the value of that minimizes does not exist in

Next, we consider the case with We will show that the equilibrium social welfare increases with if or As the discriminant takes a positive value, we have two solutions: and which satisfy Solving for we have

Then, if we have Here, from numerical calculation, we show that and This result implies that the maximum value of such that at some is derived from and the minimum value of such that at some is obtained from . That is, and Hence, if or then the social welfare is minimized at

Finally, we consider the case with From numerical calculation, we have Hence, is a strictly convex function of Then, the first-order condition characterizes the value that minimizes That is, by solving the first-order condition for we have and From the discriminant we have Then, given the value of that minimizes equilibrium social welfare is implicitly determined as follows:

where satisfies and satisfies Therefore, we obtain the proposition.

We can depict the result of Proposition 2 in Figure 1. In this figure, there are three regions. In the right area, as we have we omit this parameter range; in the shadow area, we have in the bottom area, we have Hence, given the boundary between the shadow and bottom areas determines that leads to the minimum social welfare.

Figure 1. Effect of

This proposition means that if is in the range , the equilibrium social welfare is minimized neither at nor for In other words, under the supply function competition except where and the equilibrium social welfare is smaller than those under Cournot and Bertrand competitions.

Now, we explain an intuition behind Proposition 2. Since we consider the case of a sequential contract, follower faces a residual demand after leader decides its output. Thus, the follower behaves less aggressively than in the case with a simultaneous contract. Then, to encourage the follower’s production, the upstream firm reduces the wholesale price for follower As leader knows this action, it reduces its own output. If a decrease in output dominates an increase in output, the total output reduces. Since in this case, an increase in reduces social welfare and in the other case, the rise of decreases the social welfare, there exists a positive such that social welfare is minimized.

4. Conclusions

We consider the supply function competition structure in a vertical market with a sequential contract. We show that if the technological difference between downstream firms is moderate, an intermediate competition intensity yields the minimum social welfare. This result indicates that a competition policy that enhances competition, measured by the type of competition, is not desirable for society.

ACKNOWLEDGEMENTS

I would like to express my cordial gratitude to Tomomichi Mizuno for his sincere encouragement, constructive and helpful comments, and in-depth discussion during the various stages of preparation. Needless to say, all remaining errors are mine.