Conjectural variations equilibrium in a mixed duopoly

Abstract

We study a mixed duopoly model, in which a state-owned (public) firm maximizing domestic social surplus and a private (foreign) firm compete. Under general enough assumptions, we first justify the concept of conjectural variations equilibrium (CVE) applied to the model by demonstrating concavity of the expected profit function of each agent. Next we establish existence and uniqueness results for the conjectural variations equilibrium in the described duopoly. A particular case of a CVE with constant influence (conjecture) coefficients is also examined.

Introduction

In recent years, investigation of behavioral patterns of agents of mixed markets, in which state-owned (public, domestic, etc.) welfare maximizing firms compete against profit maximizing private (foreign) firms, has become more and more popular. For pioneering works on mixed oligopolies, see Merrill and Schneider, 1966, Harris and Wiens, 1980, Bös, 1986, Bös, 1991. Excellent surveys can be found in Vickers and Yarrow, 1988, De Fraja and Delbono, 1990, Nett, 1993.

The interest in mixed oligopolies is high because of their importance to the economies of Europe (Germany, England and others), Canada and Japan (see Matsushima et al. (2003) for an analysis of “herd behavior” by private firms in many branches of the economy in Japan). There are examples of mixed oligopolies in the United States such as the packaging and overnight-delivery industries. Mixed oligopolies are also common in the East European and former Soviet Union transitional economies, in which competition among public and private firms existed or still exists in many industries such as banking, house loan, life insurance, airline, telecommunication, natural gas, electric power, automobile, steel, education, hospital, health care, broadcasting, railways and overnight-delivery.

In the majority of the above-mentioned papers, the mixed oligopoly is studied in the framework of classical Cournot, Hotelling or Stackelberg models (cf. Matsushima et al., 2003, Matsumura and Toshihiro, 2003, Cornes and Sepahvand(2003). It is well known (cf., for instance, Figuières et al., 2004) that the Nash equilibrium (including Cournot equilibrium as a particular case) is the outcome consistent with rational agents who take the rivals’ decisions as given when they optimize. Alternately, in the Stackelberg equilibrium there are two agents who take their decisions sequentially; the first agent to move is referred to as the leader, whereas the second mover is called the follower. The Stackelberg equilibrium is an outcome consistent with the follower’s rational behavior given that she has observed the leader’s move, and the leader’s rational behavior who can infer what will be the follower’s rational reaction to her current decision.

Conjectural variations equilibria (CVE) have been introduced by Bowley, 1924, Frisch, 1933 as another possible solution concept in static games. According to this concept, agents behave as follows: each agent chooses her most favourable action taking into account that the rivals’ strategies are a conjectured function of her own strategy.

As it is mentioned in Figuières et al. (2004), the concept of conjectural variations has been the subject of numerous theoretical controversies (e.g., see Lindh, 1992). Nevertheless, economists have made extensive use of one form or the other of the CVE to predict the outcome of non-cooperative behavior in several fields of Economics. A considerable amount of empirical and econometric works also exists that evaluates what conjectures look like in particular game theoretic situations. More recently a renewal of interest for this concept has taken place using either a dynamic context, or situations of bounded (procedural) rationality, or both.

In the works by Bulavsky and Kalashnikov, 1994, Bulavsky and Kalashnikov, 1995, Isac et al., 2002, a new gamma of conjectural variations equilibria (CVE) was introduced and investigated, in which the conjectural variations (represented via the influence coefficients of each agent) affected the structure of the Nash equilibrium. In other words, we considered not only a classical Cournot competition but also a Cournot-type model with influence coefficient values different from 1 (as the influence coefficient 1 corresponds to the classical Cournot model). Various equilibrium existence and uniqueness results were obtained in the above-cited works.

In the proposed paper, we extend the model studied in Isac et al. (2002) to the case of mixed duopoly as follows. Both producers (private and public firms), instead of the classical Cournot assumptions, use the following ones:

$$G_i(\eta )=G+(\eta -q_i)w_i(G\text{,}{q}_i)\text{.}$$

Here G is the current total quantity of the product cleared at the market, $q_i$ and η are respectively the present and the expected supplies by the ith agent, whereas $G_i(\eta )$ is the total cleared market volume conjectured by the ith agent as a response to changing her own supply from $q_i$ to η. The conjecture function $w_i$ will be referred to as the ith agent’s influence quotient (coefficient). The classical Cournot model assumes $w_i\equiv 1$ for all i.

In contrast to the previous models (cf. Isac et al., 2002, Chapters 4 and 5), here we assume that the model’s participants (agents) have essentially different objective functions. If agent 1 maximizes her net profit, agent 2 tries to increase the total social surplus. Due to this difference, it is interesting to compare conditions guaranteeing the existence and uniqueness of the equilibrium with those obtained in Isac et al. (2002).

Preliminary and shorter versions of the paper were presented at the workshops ABRC’2005 and IBER’2005 and published on compact disks of the workshops proceedings (cf. Kalashnikov et al., 2005a, Kalashnikov et al., 2005b).

The paper is organized as follows. In Section 2, we specify the model and introduce the first order optimality conditions to define the equilibrium. Section 3 deals with the existence and uniqueness of the equilibrium in the model. In Section 4, the question of equilibrium uniqueness is examined for a particular case when the influence coefficients are constant: on the one hand, it makes it easier to obtain the uniqueness result; on the other hand, we get interesting upper and lower bounds on the marginal production values, which will be useful for the future investigation of the Stackelberg mixed duopoly.