Endogenous role in mixed markets: a two-production-period model.

• Author: Matsumura, Toshihiro

1. Introduction

Studies of mixed markets, in which state-owned welfare-maximizing public firms compete against profit-maximizing private firms, have become increasingly popular in recent years. (1) Mixed oligopolies are common in developed, developing, and former communist transitional economies. (2) In many countries, competition between public and private firms existed or still exists in a range of industries including the airline, rail, telecommunications, natural gas, electricity, steel, and overnight-delivery industries as well as services including banking, home loans, health care, life insurance, hospitals, broadcasting, and education. In Japanese financial markets in particular, public enterprises such as the Postal Bank (which is the largest "bank" in the world), the Development Bank of Japan, and the Public House Loan Corporation still play quite important roles. (3)

Many works on mixed oligopoly analyzed Cournot-type simultaneous-move games. Others analyzed Stackelberg-type sequential-move games. In both Cournot-type and Stackelberg-type models, the role of each firm is given exogenously. In many economic situations, however, it is often more reasonable to assume that firms choose not only what actions to take but also when to take them. (4) Whether the public firm becomes the leader or the follower in the game might be important because an alternative order of moves often gives rise to different results.

In their pioneering work, De Fraja and Delbono (1989) showed that, in quantity-setting simultaneous-move games, welfare-maximizing behavior by the public firm is not always more efficient than profit-maximizing behavior. Thus, privatization of the public firm may improve welfare even without improving the managerial efficiency of the public firm. Furthermore, Matsumura (1998a) showed that, except for monopoly by the public firm, welfare-maximizing behavior by the public firm is not optimal if we allow partial privatization and that partial privatization usually improves welfare. (5) By contrast, in the sequential-move games where the public firm is the leader, we do not find such a surprising result, and welfare-maximizing behavior is always better than profit-maximizing behavior by the public firm. (6) This result holds true when the public firm is the follower as long as the marginal cost is constant. Thus, the effect of privatization of the public firm depends on whether each firm moves simultaneously or sequentially. In Stackelberg-type sequential-move duopoly models, welfare is greater when the public firm becomes the follower than when the public firm is the leader. (7) Welfare under either of the two Stackelberg-type models (where the public firm is the follower or the leader) is larger than under the Cournot-type model. In short, an alternative order of moves in fact gives rise to different welfare implications in the context of mixed markets. (8)

In this paper, I investigate whether the public firm can play a desirable role (the role of the follower) when the role of each firm is not given exogenously. I address the issue of an endogenous order of moves in a mixed duopoly by adopting Saloner's (1987) model with two production periods. Saloner investigated a pure market duopoly model where duopolists are profit-maximizing private firms. He showed that many outcomes, including the Cournot-type and either of the two Stackelberg-type outcomes, are found to be subgame perfect equilibrium outcomes. I show that in a mixed market there is no equilibrium where the public firm becomes the leader, while the other Stackelberg-type equilibrium where the public firm becomes the follower does exist. I also show that if small inventory costs are introduced, the unique equilibrium outcome becomes the Stackelberg type where the public firm is the follower. This result indicates the importance of the Stackelberg-type model where the public enterprise is a follower. It also underlines the importance of marginal cost pricing, which is not the optimal behavior by the public firm under plausible conditions in quantity-setting simultaneous-move games. (9)

Pal (1998) also discussed endogenous timing in mixed markets using the observable delay game of Hamilton and Slutsky (1990), where each firm chooses its role before producing and cannot produce over more than one period. He showed that under plausible conditions the public firm can be the follower in an equilibrium and that in a two-period duopoly model it can also be the leader in another equilibrium. Hamilton and Slutsky (1990) presented two important endogenous timing games. One is the observable delay game referred to previously, and the other is the action commitment game, where each firm chooses to act in period 1 or to delay action until period 2. They showed that two Stackelberg outcomes are sustained as equilibrium outcomes, and this result holds both in mixed and pure duopoly models. In observable delay and action commitment games, both Stackelberg outcomes are sustainable in the mixed duopoly models. This result presents a sharp contrast to our result. In the two-production-period model, the public firm cannot play the role of the follower.

In the mixed duopoly, the public firm has an incentive to increase its rival's output. When the public firm is the leader, it strategically reduces its output so as to increase the output of the private firm. Thus, the public firm's leader output is less than its Cournot output. In both observable delay and action commitment games, a firm producing in period 1 cannot produce in period 2. Hence, the public firm can commit to this smaller output. In the two-production-period model, the public firm can add to its output in period 2, so the smaller than Cournot output in period 1 does not serve as a commitment device. This is the reason why a two-production-period model yields different results. (10)

The remainder of this paper is organized as follows. Section 2 explains the basic model with two production periods. Section 3 presents three models of fixed timing as benchmarks. Section 4 investigates the equilibrium outcomes in the basic model. Section 5 introduces inventory costs and shows that the unique equilibrium is the Stackelberg type where the public firm plays the role of the follower. Section 6 concludes the paper.

2. The Basic Model

   The firms produce perfectly substitutable commodities for which the market demand function is given by D(p) (quantity as a function of price). Let p(q) [equivalent to] [D.sup.-1](q) denote the inverse demand function (price as a function of quantity). Assume that p'

   (1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   where [x.sub.i] is firm i's output.

   The game runs as follows. In the first stage, each firm i (i = 1, 2) chooses its first-period production [x.sub.i](1) [member of] [0, [infinity]). At the end of the first stage, each firm knows [x.sub.1](1) and [x.sub.2](1). In the second stage, each firm i chooses its second-period production [x.sub.i](2) [member of] [0, [infinity]) At the end of the second stage, the market opens, and each firm i sells its total output [x.sub.i] [equivalent to] [x.sub.i](1) + [x.sub.i](2).

3. Fixed Timing Games

   Before presenting the results derived from the model with two production periods, I discuss two Stackelberg- and one Cournot-type duopoly models of fixed timing as benchmarks.

   First, consider the game where the public firm (firm 2) is the leader. Firm 2 chooses [x.sub.2] [member of] [0, [infinity], and firm 1 chooses [x.sub.1][member of] [0, [infinity] after observing [x.sub.2]. Firm 1 does not produce if and only if [x.sub.2] [greater than or equal to] D([c.sub.1]), that is, p([x.sub.2]) [less than or equal to] [c.sub.1]. Let [[PI].sub.1]([x.sub.1], [x.sub.2]) [equivalent to] p([x.sub.1] + [x.sub.2])[x.sub.1] - [c.sub.1][x.sub.1] denote the profit function of firm 1. Assume that [[differential].sup.2][PI]/[differential][x.sup.2.sub.1] = p''[x.sub.1] + 2p'

   (2) p' [x.sub.1] + p - [c.sub.1] = 0.

   Assume that [R'.sub1]

   Next, consider the game where firm 2 is the follower. Firm 1 chooses [x.sub.1] [member of] [0, [infinity]), and firm 2 chooses [x.sub.2] C [0, [infinity]) after observing [x.sub.1]. Let...