Mixed duopoly: Differential game approach

• Date: 01 August 2019
• Published date: 01 August 2019
• DOI: http://doi.org/10.1111/jpet.12372

Received: 20 April 2018

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Revised: 25 January 2019

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Accepted: 9 April 2019

DOI: 10.1111/jpet.12372

ORIGINAL ARTICLE

Mixed duopoly: Differential game approach

Koichi Futagami

1

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Toshihiro Matsumura

2

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Kizuku Takao

3

1

Graduate School of Economics, Osaka

University, Toyonaka, Osaka, Japan

2

Institute of Social Science, The

University of Tokyo, Bunkyo‐ku, Hongo,

Tokyo, Japan

3

Department of Economics, Aomori

Public University, Oaza Goushizawa

Aomori City, Aomori, Japan

Correspondence

Kizuku Takao, Department of

Economics, Aomori Public University,

153‐4 Aza Yamazaki, Oaza Goushizawa

Aomori City, Aomori 030‐0196, Japan.

Email: kizukutakao@gmail.com

Funding information

Kyoto University, Grant/Award Number:

the Joint Research Program of KIER;

Japan Society for the Promotion of

Science, Grant/Award Numbers:

15H03355, 15H065240, 17K17975,

18K01500

This study formulates a dynamic mixed oligopoly model,

in which a state‐owned public firm competes against a

private firm over multiple periods. We adopt a

differential game formulated by Fershtman and Kamien

[Econometrica 55 (1987), pp. 1151–1164] and investigate

how the dynamic competition affects the optimal

privatization policy. We characterize the open‐loop

Nash equilibrium (OLNE) and Markov‐perfect Nash

equilibrium (MPNE). We show that in the MPNE, an

increase in the degree of privatization has a nonmono-

tonic effect on the price, increasing it in a wide range of

parameter spaces, which is in sharp contrast to the

result in the OLNE or static analyses. We also find that

the optimal degree of privatization is higher in the

MPNE than that in the OLNE and static equilibrium.

These results suggest that intertemporal strategic

behavior changes the optimal privatization policy.

1

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INTRODUCTION

For more than 50 years, we have observed a worldwide wave of privatization of state‐owned

public enterprises. Nevertheless, many public enterprises with significant government

ownership are still active in strategic sectors and control large portions of the world’s

resources. According to an Organisation for Economic Co‐operation and Development

(OECD) report by Kowalski, Buge, Sztajerowska, and Egeland (2013), over 10% of the largest

2,000 companies are public enterprises with equivalent to approximately 6% of the global

GDP. They are significant players in sectors, such as transportation, telecommunications,

energy, and finance in OECD countries. In planned and transitional economies, such as

China, Russia, and Vietnam, the presence of the public enterprises is further significant

(Wang & Chiou, 2016).

One classical rationale for public enterprises is to prevent private monopolies from

prevailing in natural monopoly markets with significant economies of scale. However, because

of technological improvement, many markets with public enterprises are not always

J Public Econ Theory. 2019;21:771–793. wileyonlinelibrary.com/journal/jpet © 2019 Wiley Periodicals, Inc.

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771

characterized by significant economies of scale. Indeed, a considerable number of public and

private enterprises compete in a wide range of industries (mixed oligopolies).

1

The optimal

privatization policies in these mixed oligopolies have attracted extensive attention from

researchers in fields, such as public economics, financial economics, industrial organization,

and development economics.

2

Specifically, the literature on mixed oligopolies has investigated the optimal privatization policy in

different situations. Matsumura (1998) shows that the optimal degree of privatization is never zero

unless full nationalization leads to a public monopoly. Lin and Matsumura (2012) and Matsumura

and Okamura (2015) find that the optimal degree of privatization increases with the number of

private firms and decreases with the share of foreign ownership in private firms. Matsumura and

Kanda (2005) show that in free entry markets, the optimal degree of privatization is zero when private

competitors are domestic, whereas Cato and Matsumura (2012) find that it is strictly positive when

the private competitors are foreign, increasing with the foreign ownership share in private firms. In

addition, Chen (2017) shows that the optimal degree of privatization is positive even in free entry

markets when privatization improves production efficiency. Fujiwara (2007) shows a nonmonotonic

relationship between the degree of product differentiation and optimal degree of privatization. Cato

and Matsumura (2015) discuss the relationship between the optimal trade and privatization policies,

showing that a higher tariff rate increases the optimal degree of privatization in free entry markets.

Lee, Matsumura, and Sato (2018) show that the optimal degree of privatization depends on the timing

of privatization. However, all these studies use a static model and do not capture the intertemporal

strategic interaction between public and private firms.

3

In this study, we investigate how the intertemporal strategic interaction between a public

firm (hereafter referred to as Firm 1) and private firm (hereafter referred to as Firm 2) affects

the equilibrium outcomes, given the degree of privatization in Firm 1. We also investigate the

optimal privatization policy in the presence of dynamic competition.

To investigate the dynamic competition in a mixed duopoly, we adopt a differential game

formulated by Fershtman and Kamien (1987). Differential game approaches are widely applied

to a number of studies in various fields of economics.

4

They involve two Nash equilibrium

concepts: the open‐loop Nash equilibrium (OLNE) and the Markov‐perfect Nash equilibrium

(MPNE).

5

Importantly, the MPNE is subgame perfect and considers the intertemporal strategic

behaviors of agents.

6

The seminal paper of Fershtman and Kamien (1987) characterizes the

OLNE and the MPNE of the Cournot game with a sticky price. The paper reveals that in the

MPNE, firms produce more aggressively with intertemporal strategic substitutability than that

in the static game. Subsequent studies have also shed light on the importance of intertemporal

strategic behavior for various economic problems.

1

Examples include the United States Postal Service, Deutsche Post AG, Areva, NTT, Japan Tobacco (JT), Volkswagen, Renault, Electricite de France, Japan

Postal Bank, Japan Postal Insurance, Korea Development Bank, and Korea Investment Corporation.

2

For examples of mixed oligopolies and recent developments in this field, see Amir and De Feo (2014), Pal and Saha (2014), Amerighi and De Feo (2017), Lin

and Matsumura (2018), and the works cited therein.

3

Sato and Matsumura (2019) investigate a dynamic mixed oligopoly model. However, they investigate a two‐period model, in which the first‐period

privatization policy distorts the government objective in the second period and there is no intertemporal strategic interaction between public and private firms.

4

See Long (2010) for a review of the application.

5

The MPNE is often referred to as a feedback (or closed‐loop) Nash equilibrium.

6

Cabral (2012) points out the importance of considering issues in an area of industrial organization using dynamic models. Note that the OLNE is not subgame

perfect and does not consider the intertemporal strategic behavior of agents. By definition, each player in the OLNE chooses an entire time path for his/her

action at the outset of the gamegiven the entire path strategy chosen by the other player. In contrast, each firm in the MPNE chooses its strategy at each point in

time after observing the value of its payoff‐relevant state variables. See Dockner, Jorgensen, Long, and Sorger (2000) for more detailed explanations of

differential games.

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FUTAGAMI ET AL.