Governments and Producers in Multinational Markets: A Mixed Oligopoly Analysis

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Subsidization and Privatization: In a Multinational

Mixed Oligopoly

Ali DadpayCollege of Business, Clayton State University, 2000 Clayton State Blvd., Morrow, GA 30260, USA.

E-mail: [email protected]

This paper studies a single multinational market where private and public firms fromdifferent nationalities compete. The model allows the domestic government to subsidizeits firms. In contrast to previous studies in this model, the subsidy varies with the marketstructure. Privatizing either the domestic public firm or the foreign public firm or both ofthem unambiguously increases the optimal subsidy rate and domestic social welfare.Nonetheless, both countries would benefit from simultaneous privatization.Eastern Economic Journal advance online publication, 3 September 2012;doi:10.1057/eej.2012.23

Keywords: subsidization; privatization; mixed oligopoly; multinational markets

JEL: L13; L32

INTRODUCTION

An increasing number of authors have contributed to the studies on mixed oligopolysince the model’s introduction in the 1960s [Merril and Schneider, 1966]. Thedefining characteristic of mixed oligopoly markets has been the presence of awelfare-maximizing public firm competing with profit-maximizing private firms.The early work focuses on analysis of a domestic market, where a public firm inter-acts with private ones [Harris and Wiens 1980; Cremer et al. 1989, 1991; DeFrajaand Delbono 1989, 1990; Fershtman 1990; Sertel 1988; Nilssen and Sorgard 2002].More recently, authors have considered mixed oligopoly in a domestic market opento foreign competition [Fjell and Pal 1996; Pal and White 1998, 2003; Fjell andHeywood 2004]. Within both strands of work, there has been interest in theinfluence of government subsidization. In mixed oligopoly models, where foreigncompetition is present, subsidization is considered an element of strategic tradepolicy. The purpose of this paper is to extend the examination of subsidizationbeyond the models of domestic markets with foreign competition to a model of asingle market that crosses national boundaries and includes competing public firms(see Dadpay and Heywood 2006).

The present study is motivated by the commercial airline industry in the PersianGulf region. This region includes countries such as Iran, United Arab Emirates(UAE), Kuwait, Saudi Arabia, Qatar, and Bahrain. Their mutual business interestsand cultural ties have created a growing commercial airline industry. Several publicflag carriers and private airlines connect the region’s capitals and its major cities. Onboard these flights, citizens of different nationalities share the consumers’ surpluswhile governments maximize their respective social welfares and entrepreneursmaximize their profits. The governments of the region differ in their policiesregarding the airline industry. Some subsidize the industry through discounted fueland public funds for its operations, as in the case of Iran Air; and some encourage

Eastern Economic Journal, 2012, (1–21)r 2012 EEA 0094-5056/12

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tourism and hotel industries but do not subsidize the airline industry directly, asis the case of the UAE and the Emirates. In particular, this paper is motivated bythe airline markets for the routes between Iran and the UAE.1

This paper considers a mixed oligopoly market with a foreign public firm anda domestic public firm, as well as foreign and domestic private firms with con-sumers from different nationalities, to extend the existing analysis. This paperis also motivated by the fact that market subsidies have been much debated byauthors studying government involvement in such markets. Many [Myles 2002; Fjelland Heywood 2004] have suggested that subsidization recovers first-best pricingin a mixed oligopoly. The present study introduces the idea of a subsidy rate ina single multinational market and investigates whether such is the case in this kindof market. The findings depart from the existing literature in suggesting that theunilateral privatization in multinational mixed oligopoly markets does notnecessarily involve a prisoners’ dilemma, as suggested previously.

A series of previous papers consider a multi-stage game in which the governmentsets a subsidy in the first stage in an attempt to eliminate the inefficiency associatedwith market power and to maximize the social welfare function. Later stages involvequantity setting in a variety of mixed oligopoly structures. Poyago-Theotoky [2001]shows that in a mixed oligopoly closed to foreign competition, the optimal subsidyrate is identical regardless of whether the public firm acts as a Stackelberg leader,moves simultaneously with other firms, or maximizes its profit like privatefirms. Moreover, the optimal subsidy achieves first best with marginal cost equalto price. As the equilibrium is identical with either a profit-maximizing private firmor a welfare-maximizing public firm, privatization has no influence on welfare inthe face of subsidization. Myles [2002] shows that these results hold for a widevariety of cost structures as long as some modest conditions on functional formare met. Sepahvand [2002] argues that privatization of mixed oligopoly does notimprove domestic welfare even in the presence of foreign firms. He introducesadjusted marginal cost pricing based on the relationship between foreign firms’output and domestic firms’ output, showing that regardless of the objective functionof the public firm, there is an identical optimal subsidy rate, which adjusts marginalcost pricing.2

The present study finds that the optimal subsidy rate is a function of the numberof firms in the market, their cost structure, and the domestic country’s share ofconsumers’ surplus. It shows that the optimal subsidy does not result in marginalcost pricing. As a consequence, privatization is found to yield ambiguous economicresults. The direction of changes in equilibrium values under privatization differswith respect to the share of consumers’ surplus, relative number of private firms inthe market, and cost structure. The study also shows that unilateral privatizationenhances welfare in the country where public industry is subsidized. This departsfrom previous literature. Pal and White [1998] show that in a single domesticmarket, privatization reduces welfare. Fjell and Heywood [2004] confirm this forthe case of a domestic market with a Stackelberg leader, and Pal and White [2003]argue it for the case of intra-industry trade markets. Dadpay and Heywood[2006] show that governments are trapped in a prisoners’ dilemma in a multinationalmixed oligopoly market, and that neither benefits from unilateral privatization.This study argues that governments can evade this prisoners’ dilemma throughsubsidization and can benefit from unilateral privatization.

The “Model” section presents the model, and the “Characteristics of theEquilibrium” section describes the equilibrium and simulation results. The

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“Privatization” section examines privatization effects in this model. The“Conclusion” section summarizes the conclusions.

MODEL

Assume that a multinational market is shared by two countries. As the multinationalmixed oligopoly structure is developed by extending the domestic mixed oligopolymarket with foreign competition to include a foreign government as well, it iscommon to refer to one country as the domestic country and to the other one as theforeign country. There is one domestic public firm, one foreign public firm, mdomestic private firms and n foreign private firms in this market. Assume thatall firms produce a homogenous commodity for which there is a demand byconsumers from both countries. All firms use the same technology in producingthis commodity, so the cost structure is the same for all of the firms. Several authors[Pal and White 1998; Fjell and Heywood 2004; Dadpay and Heywood 2006] haveused quadratic cost functions to study mixed oligopoly markets. It also mustbe noted that quadratic cost functions are often assumed in studies of the aviationindustry [Borenstein and Rose 1994]. It also allows a linear marginal cost withslope k>0:

CðqÞ ¼ 0:5�k�q2 þ fð1Þ

Private firms maximize profits, and public firms maximize the social welfarefunctions of their respective societies. The governments pay subsidies to both privateand public firms. These subsidies are a function of the firms’ outputs. In the existingmixed oligopoly literature, authors usually assume that subsidies are proportionalto the firm’s outputs [Poyago-Theotoky 2001; Myles 2002]. Thus, the total subsidythat a firm receives is a linear function of its output:

Sd ¼ sd�qð2Þ

where sd is the domestic subsidy rate.There is no report of governments paying non-linear subsidies in the aviation

industry. To avoid further asymmetry in the model, I assume that the domesticgovernment subsidizes public and private firms alike. This is not an unrealisticassumption since in many industries, including aviation, public firms use similartechnology and their expenses are close to that of the private sector.

Demand in market is linear and can be written as follows:

P ¼ a� qdo þ qfo þXmi¼1

qdi þXnj¼1

qfj

!ð3Þ

In equation (3), P is price, qod is domestic public firm output, qo

f is foreign public firmoutput, qi

d is domestic private firm i’s output (i¼ 1,y,m), and qjf is foreign private

firm j’s output (j¼ 1,y, n). On the basis of the demand function, consumers’surplus in this multinational market can be written as follows:

CS ¼ 0:5 qdo þ qfo þXmi¼1

qdi þXnj¼1

qfj

!2

ð4Þ


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Using equation (4), domestic consumers’ surplus and foreign consumers’ surplus aredefined as follows:

CSd ¼ 0:5y qdo þ qfo þXmi¼1

qdi þXnj¼

qfj

!2

ð5Þ

CSf ¼ 0:5ð1� yÞ qdo þ qfo þXmi¼1

qdi þXnj¼1

qfj

!2

ð6Þ

where y is the domestic customers share of consumers’ surplus, which is also referredto as market share in the rest of this paper.

As domestic government subsidizes domestic firms, both private and public,their profit functions include the subsidy rate as an exogenous parameter. Thepublic domestic firm’s profit is as follows:

pdo ¼ Pqdo � f� 0:5�kðqdoÞ2 þ sqdoð7Þ

The private domestic firm’s profit function is as follows:

pdi ¼ Pqdi � f� 0:5�kðqdi Þ2 þ sqdi ði ¼ 1; . . . ;mÞð8Þ

The introduction of the subsidy also changes the domestic social welfare function.Its social welfare function is written as the sum of consumers’ surplus and totalprofits, and since a subsidy is paid, it must be deducted from the sum:

Wdo ¼ 0:5�y qfo þ qfo þ

Xmi¼1

qdi þXnj¼1

qfj

!2

þpdo þXmi¼1

pdi � sd qdo þXmi¼1

qdi

!ð9Þ

Assume initially that the foreign government does not subsidize foreign firms for thetime being. The foreign public firm’s profit function is as follows:

pfo ¼ Pqfo � 0:5kðqfoÞ2 � fð10Þ

The foreign private firm’s profit function is as follows:

pfj ¼ Pqfj � 0:5kðqfjÞ2 � fð11Þ

And the foreign country’s social welfare function is

Wfo ¼ 0:5ð1� yÞ qdo þ qfo þ

Xmi¼1

qdi þXni¼1

qfj

!2

þpfo þXnj¼1

pfið12Þ

First, the domestic government chooses a subsidy rate to maximize domestic socialwelfare function. Then domestic and foreign public firms choose their output tomaximize their respective social welfare functions and the private firms, domesticand foreign, choose their outputs to maximize their own profits. The firms, publicand private, move simultaneously, after domestic government sets a subsidy rate.The results are equilibrium values for the outputs of both the private and publicfirms.


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Equilibrium

Assuming there is only one domestic private firm and one foreign private firm inthe market, m¼ n¼ 1, then the domestic private firm’s output is as follows:

qd�

i ¼sd½ðaþ 1Þk2 þ ðaþ 4Þkþ 2�ðkþ 1Þðk2 þ 5kþ 2Þð13Þ

And the private foreign firm’s output is as follows:

qf�

j ¼ak2 þ ðak� 1Þksdðkþ 1Þðk2 þ 5kþ 2Þð14Þ

Note that the private firm quantities are identical across countries in the absenceof subsidization. The domestic private firm’s output always exceeds the foreignprivate firm’s output and their difference is as follows:

qd�

i � qf�

j ¼sd

kþ 1ð15Þ

The expressions for the outputs of the public firms are more complicated:

qd�

o ¼kðkþ 2Þðak� sd þ 2ayÞ þ ayðk2 þ 2Þ

kðkþ 1Þ½k2 þ 6kþ yðaþ sdÞ þ 1� þ kð2� sd þ yaÞð16Þ

qf�

o ¼sdkð1� k� yÞ þ akðkþ 1Þðk2 þ 4yk� 5yþ 2a� 2ayþ 4Þ

kðkþ 1Þ½k2 þ 6kþ yðaþ sdÞ þ 1� þ kð2� sd þ yaÞð17Þ

Total output and equilibrium price in the market will be

Q� ¼ 2að2kþ 1Þ þ sdk

k2 þ 5kþ 2ð18Þ

P� ¼ aðk� 1Þ � sdk2 þ 5kþ 2

ð19Þ

The equilibrium full expressions are presented in Appendix A.In the first stage of the game, the domestic government maximizes the social

welfare function by choosing the domestic subsidy rate with perfect knowledgeof the private and public firms’ outputs. The domestic optimal subsidy rate is afunction of all parameters in the market

s� ¼ fða; k;m; n; yÞ ¼ aA5k

5 þ A4k4 þ A3k

3 þ A2k2 þ A1kþ A0

k6 þ B5k5 þ B4k4 þ B3k3 þ B2k2 þ B1kþ B0ð20Þ

for a¼ 1, k¼ 1 & m¼ n¼ 1 the domestic optimal subsidy rate is simplified to

s�d ¼�12y2 þ 76y� 11

aðy2 � 5yþ 5Þ þ 246ð21Þ

where A0,y,A5 and B0,y,B5 are the functions of model parameters. Their fullexpressions are presented in Appendix B. The subsidy rate serves as a device for


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the government to increase social welfare. Using sd*, I derived optimal outputsfor the domestic public and foreign firms, and for foreign private and publicfirms. The characteristics of the optimal subsidy rate and the resulting equilibriumvalues are examined in the next section.

CHARACTERISTICS OF THE EQUILIBRIUM

This section characterizes the equilibrium built around two subsections. I presentthree simple propositions describing the equilibrium and contrasting it with resultsprovided by existing literature. The second subsection is a direct comparison ofthis model of a single international market with firms from two competingcountries with the model of Pal and White [1998] who examine foreign firms ina single domestic market.

Characterizing the equilibrium

Proposition 1: The optimal subsidy does not achieve the first best as price does notequal marginal cost.

Proof: By substituting sd¼ sd* in equations (13) and (14) for private firms andequations (16) and (17) for public firms and equation (19) for the price, I can writeequilibrium price and marginal cost as functions of the number of foreign anddomestic firms and the market share. For simplicity, I show the values when a¼ 1and k¼ 1:

P� ¼ 2ð2y2 � 15yþ 33Þð9� 2yÞ2 þ 226

ð22Þ

qd�

o ¼2ð2y2 þ 34yþ 13Þð9� 2yÞ2 þ 226

ð23Þ

For k¼ 1, MC¼ qd�

o , as it is apparent, despite the identical denominator,equilibrium price does not equal marginal cost.3 The optimal subsidy does notachieve the first best in this market. This remains true for all values of k and a with ademonstration available from the author. Simulations of the difference betweenprice and marginal cost for combinations of m and n show that there is a marketshare, y* for each pair of m and n, for which qd

�o ¼P* and for y>y*, P*>MC. For

many combinations of m and n, y*o0.5, which shows that to be able to enforce aprice more than marginal cost, domestic economy does not need to have dominantmarket share in the presence of a subsidy.

Although this proof is based on numerical simulations, it departs from previousmodels examining mixed oligopoly, whose findings are based on cases where thereare only one private firm and one public firm in the market. Using these simulations,this paper investigates the dynamics within mixed oligopoly markets further. Itdemonstrates that unlike in previous studies where subsidization recovers first best,in a multinational single market, this does not happen. The reason is that thedomestic public firm’s attempt to maximize domestic welfare is met with an overseas


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response by the foreign public firm. This makes the relationship between the optimaloutputs and the optimal subsidy non-linear. As a result, the market does not havemarginal cost pricing.

Proposition 2: Entry of domestic private firms reduces the optimal subsidy rate, andentry of foreign private firms increases the optimal subsidy rate.

Proof: These follow unambiguously from the derivatives of s* with respect to thenumber of domestic firms, m, and with respect to the number of foreign firms, n.These derivatives for the simplified case of a¼ 1, k¼ 1 are as follows:

qs�dqm¼ ðy� 1Þ½8ðnþ 4Þy3 þ 12y2ðnþ 6Þ2� þ 6nðnþ 12Þ2ðy2 � 1Þ � 24nðnþ 6Þ2y

ðmþ 4Þn2 þ ð24� 4yÞmnþ 96mþ 48nþ 4y2m� 32yþ 144o0ð24Þ

qs�dqn¼ 8my4 þ ð2nþmþ 4Þ2y2ð1� yÞ þ ½2mðnþ 24Þ2 þ 12mn�ð1� y2Þðmþ 4Þn2 þ ð24� 4yÞmnþ 96mþ 48nþ 4y2m� 32yþ 144

40ð25Þ

In equation (24) all positive expressions are multiplied by (y�1)o0. This identifiesthe derivative of the optimal subsidy rate with respect to m as a definitely negativeterm. Meanwhile, all expressions, which are positive, in equation (25) are multipliedby (1�y)>0. This identifies the sign of the derivative of the optimal subsidy ratewith respect to n as positive. Simplifying equations (24) and (25) for m¼ n¼ 1, onecan rewrite them as follows:

qs�dqm¼ 40y4 þ 548y3 þ 426y2 � 1176y� 1014

4y2 � 36yþ 317

qs�dqn¼ 8y4 � 49y3 � 1225y2 þ 1274

4y2 � 36yþ 317

It could be shown that these two expressions are, respectively, negative and positivefor all legitimate values of y.

As the subsidy is designed to increase output in the market, and the entry of adomestic firm increases the output, the government reacts by reducing the subsidyrate. When a foreign private firm enters the market the aggregate output increases,changing the balance of profit losses and consumers’ surplus gains. In this case, thegovernment increases the subsidy for domestic firms to compensate for the profitlost to the other country. Also, the entry of foreign private firms introduces newcompetitors and decreases the domestic private firms’ profits. Government herereacts to the reduction in private firms’ profits by increasing the subsidy. Thisproposition also explains why in markets that have a large number of privatecompetitors, the entry of one firm will not greatly alter subsidization.

Proposition 3: An increase in demand increases (decreases) the subsidy rate whenthe domestic market share is larger than half (smaller than half) and the number ofdomestic private firms is smaller (larger) than the number of foreign private firms.


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Proof: At issue is the influence of a change in a on sd*. From equation (20):

s�d ¼aA5k

5 þ A4k4 þ A3k


k6 þ B5k5 þ B4k4 þ B3k3 þ B2k2 þ B1kþ B0

) qs�dqa¼ A5k

5 þ A4k4 þ A3k



The sign of the derivative varies as follows:

If mon; y40:5 thenA5k

5 þ A4k4 þ A3k



If m4n; yo0:5 thenA5k

5 þ A4k4 þ A3k


k6 þ B5k5 þ B4k4 þ B3k3 þ B2k2 þ B1kþ B0o0

An increase in demand forces the government to increase the subsidy rate if andonly if, the domestic market share is significant and the number of domestic privatefirms is smaller than that of foreign ones. Any shift in demand, causing theequilibrium price to increase, will be followed by government increasing the subsidyrate to prevent any inconvenience because of an increased price, and to enable firmsto produce the demanded quantity at a low price.

Proposition 4: If the domestic market share y is less than half and the initial valueof the slope of marginal cost is large enough (greater than 4), then an increase in theslope of marginal cost, k, decreases optimal subsidy rate.

Proof: The derivative of s*d with respect to k will be

qs�dqk¼ a

P10i¼0

Ciki

ðk6 þ B5k5 þ B4k4 þ B3k3 þ B2k2 þ B1kþ B0Þð26Þ

The denominator is positive, so the sign of derivative depends on the Ci’s sincek>0. Ci’s are complicated non-linear functions of m, n, and y (the expressions forCi’s are reported in Appendix C), but numerical simulations available from theauthor provide the following categories:

Perfect symmetry: m ¼ n; y ¼ 0:5) ðqs�d=qkÞo0For example, for m¼ n¼ 3, a¼ 1, y¼ 0.5, equation (26) can be written as follows:

qs�dqk¼ � 400ðk10 þ 7k9 þ 5k8 þ 2k7 þ k6 þ k5 þ 2:4k4 þ 10k3 þ 1127k2 þ 13kþ 2

ð4k6 þ 74k5 þ 443k4 þ 986k3 þ 1127k2 þ 470kþ 64Þ2o0

This is negative for all of the positive values of k. Numerical results provide thefollowing conditions for the sign of derivative:

Asymmetric:

y4o

0:5; ko4

4) qs�dqk

4o

0

If k increases, the total cost of production increases for all levels of output. Anincrease in k can be treated as a rotation of MC to the left. This moves the profitmaximization point to the left, reducing the optimal output. When the domestic


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country’s share of consumers’ surplus is large, this loss of output negatively affectsthe domestic social welfare, and the government reacts by increasing the subsidy.When k is large and the domestic share of consumers’ surplus is small, profit isrelatively more important, so the government reduces the subsidy rate since it is lessconcerned with lowering the price to increase consumers’ surplus.

Proposition 5: Entry of a new domestic (foreign) private firm increases (decreases)domestic welfare and decreases (increases) foreign social welfare.

Proof: The equilibrium levels for domestic and foreign social welfare functions canbe obtained by substituting the optimal subsidy rate and equilibrium outputs inequations (9) and (12). It can be shown that

qW�dqm

40;qW�dqn

o0

qW�fqm

o0;qW�fqn

40

This follows intuition, as the entry of a domestic firm increases the share domesticprofit and increases domestic consumers’ surplus. A foreign entrant creates off-setting effects on profit and consumers’ surplus.

Model comparisons

The equilibrium characteristics for this model diverge from those found in previousstudies. Dadpay and Heywood [2006] show that in a mixed oligopoly withoutsubsidy, results are symmetric and total output, equilibrium price, and private firms’outputs are independent from market share. In the absence of subsidization, allprivate firms produce the same output. Introduction of a subsidy creates a gapbetween private firms’ outputs and causes an increase in domestic private firms’outputs. Private firms’ outputs are no longer independent from market share, sincethe optimal subsidy rate depends on it. This results in sensitivity of equilibrium priceto changes in the market share. A multinational mixed oligopoly market with adomestic subsidy is a far less robust market than a domestic mixed oligopoly with asubsidy as outlined by Pal and White [1998] and a multinational mixed oligopolywithout a subsidy as discussed by Dadpay and Heywood [2006]. This fact highlightsthe new importance of equilibrium total output and price in this model.

Though the results are different, entry of new firms is treated as before.Introduction of domestic subsidization does not change the positive contribution ofan increase in the number of domestic firms to domestic welfare and its negativeeffect on foreign welfare. Results are symmetric and entry of foreign private firmsaffects domestic and foreign welfare in the opposite direction.

PRIVATIZATION

Several authors have studied privatization in mixed oligopoly markets and itsconsequences. Pal and White [1998] analyze the effects of privatization in a domesticmixed oligopoly market open to foreign competition. They argue that the optimallevel of subsidy decreases with privatization, and that social welfare increasesbecause of it. These results are based on the assumption that a privatized public firm


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operates more efficiently as the value of k, the slope of the marginal cost ofproduction, falls with privatization. Poyago-Theotoky [2001] shows that optimalsubsidy is identical for all market structures in a closed domestic market withoutdifferences in efficiency between public and private firms. Sepahvand [2002] provesthat even in a domestic market open to foreign competition, privatizing thedomestic public firm will not change the optimal subsidy rate or welfare. He arguesthat with subsidization, governments can achieve any level of welfare in a mixedoligopoly market, even that of a liberalized market. In their study of a singlemultinational mixed oligopoly market, Dadpay and Heywood [2006] concludedthat while simultaneous privatization by both countries increases welfare, neithercountry has the incentive to privatize its public firm individually as suchprivatization reduces their own social welfare and increases that of their rival.This section examines the effect of privatization on the optimal subsidy rate andforeign and domestic welfare.

It must be noted than when a public firm is privatized, the number of private firmsincreases, as in the case of the domestic public firm from m to mþ 1 and in the caseof the foreign public firm from n to nþ 1. Privatizing both foreign and domesticpublic firms increases the number of private firms to mþ nþ 2 from mþ n. There isno change in the cost structure, and the newly privatized firm maximizes its profitlike any other private firm. Now in the first stage, the domestic government choosesthe optimal subsidy rate. In the second stage, private firms from both countriesmaximize their profits, and the remaining public firm maximizes its respective socialwelfare simultaneously.

Analyzing the equilibrium values, we notice that privatization does not change thegap between the outputs of the respective country’s private firms. It remains as inequation (15):

qd�

i priv � qf�

j priv ¼s�d priv

kþ 1

It is difficult to offer a closed form proof for any proposition because of thenumber of parameters and polynomials involved. In order to study and to analyzethe market dynamics, this paper utilizes numerical simulations, a practice commonin mixed oligopoly analysis. These simulations allow researchers to study complexmarket structures, which makes it possible to enrich the existing literature, whereauthors have simplified the markets significantly. For example, Pal and White [1998]consider a mixed oligopoly market with only one domestic private firm and oneforeign private firm. This study demonstrates how their findings are valid only forthat very special case. Inspired by the international commercial aviation industry thenumerical simulations cover the potential market structures, which are created bythe combinations of one to five domestic private firms and from one to five foreignprivate firms as suggested by Dadpay and Heywood [2006]. As for the slope ofmarginal cost, k, we assume values from 1 to 10, which includes the effects ofdoubling and tripling k. The findings are not reported as propositions but assimulation results, emphasizing their numerical nature.

Simulation Result 1: Privatizing any or both public firm generally increases theoptimal subsidy rate.


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Analysis: While it is difficult to present an unambiguous closed form solution,numerical simulations show that for all reasonable combinations of m and n, asmentioned before, and the difference between the following:

s�d priv � s�d40

s�f priv � s�d40

s�sim priv � s�d40

Tables 1a, 2a, and 3a summarize the effects of privatizing the domestic publicfirm, the foreign public firm, and both public firms on the optimal subsidy rate fora¼ 1, k¼ 1, y¼ 0, 0.25, 0.5, 0.75, and 1. Note that all the values are positive andthey remain positive for every choice of k from 1 to 10.

When the domestic public firm is privatized as m increases in a column, thedifference between sd_priv* and sd* decreases for the given y. If m is held constant andn increases, the difference between the optimal subsidy rates decreases. If m and nstay constant, the difference between the two subsidy rates increases as y increases.This means that when the domestic share of the market is large; privatizing thedomestic public firm causes the government to increase the subsidy rate substantiallyto keep the domestic social welfare maximized. It is interesting to note that thechanges in the optimal subsidy rate become less sensitive to market share, y, as thenumber of firms increases in two countries. This can be explained by rememberingthat the subsidy is paid to increase the output of firms and when the number of firmsin the market is large enough, the domestic government does not need to increase thesubsidy rate substantially to ensure greater production.

Table 2a shows the change in the subsidy rate associated with privatizing theforeign public firm. When the number of domestic private firms is less than thenumber of foreign private firms, the optimal subsidy rate increases with y. The valueof y has relatively little influence on the change in the subsidy when m¼ n. Ingeneral, the market share plays a smaller role in determining changes in the subsidyfor foreign privatization than it did for domestic privatization. This finding isjustified by keeping in mind that the domestic welfare function includes bothconsumers’ surplus and producer surplus. Privatizing the foreign public firmincreases output and consumers’ surplus. If m o n, then the weight of domesticfirms’ profit in the domestic social welfare function is small, and increases in themarket share enables domestic consumers to benefit from foreign privatization.Thus, domestic welfare increases due to the gained consumers’ surplus. When n¼m,the relative importance of profit in the two countries is the same, and variation inmarket share has less influence on the changes to the subsidy elicited by foreignprivatization.

Table 3a presents the effects of simultaneous market privatization on the optimalsubsidy rate for combinations of m and n varying from 1 to 5 and y¼ 0, 0.25, 0.5,0.75, and 1. For all combinations of m and n the change in optimal subsidy rate isincreasing with y. It also is increasing with respect to m when n is constant. Thechange in the optimal subsidy rate is decreasing with respect to n if y>0.5 when m isheld constant. When y is near either 0 or 1, the increase in the optimal subsidy withrespect to m is small. For all combinations of m and n, privatization requires ahigher subsidy for domestic firms. When the domestic market share increases, theamount of change in the optimal subsidy rate grows because of the increasedimportance of consumers’ surplus in the social welfare function. As the market


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Table 1 Domestic privatization effects

N 1 2 3 4 5

M

(a) Domestic privatization effects on the optimal subsidy rate

1 0.05, 0.14, 0.25, 0.35, 0.45 0.06, 0.12, 0.2, 0.25, 0.32 0.07, 0.1, 0.15, 0.2, 0.24 0.08, 0.1, 0.13, 0.16, 0.2 0.09, 0.1, 0.11, 0.13, 0.15

2 0.05, 0.13, 0.23, 0.33, 0.44 0.06, 0.11, 0.17, 0.24, 0.32 0.07, 0.11, 0.15, 0.2, 0.24 0.07, 0.1, 0.13, 0.16, 0.2 0.08, 0.1, 0.11, 0.13, 0.16

3 0.05, 0.13, 0.21, 0.3, 0.43 0.06, 0.1, 0.17, 0.23, 0.3 0.06, 0.1, 0.14, 0.2, 0.24 0.07, 0.1, 0.12, 0.16, 0.2 0.07, 0.1, 0.11, 0.13, 0.16

4 0.05, 0.12, 0.2, 0.3, 0.43 0.05, 0.1, 0.1, 0.23, 0.3 0.06, 0.1, 0.13, 0.18, 0.24 0.06, 0.1, 0.12, 0.15, 0.2 0.06, 0.1, 0.1, 0.13, 0.16

5 0.05, 0.1, 0.2, 0.3, 0.42 0.05, 0.1, 0.15, 0.22, 0.3 0.05, 0.1, 0.13, 0.18, 0.24 0.05, 0.08, 0.11, 0.15, 0.2 0.06, 0.08, 0.1, 0.13, 0.16

(b) Domestic privatization effects on the domestic welfare

1 0.00, 0.1, 0.2, 0.2, 0.3 0.01, 0.1, 0.16, 0.24, 0.3 0.02, 0.1, 0.16, 0.23, 0.32 0.02, 0.1, 0.16, 0.24, 0.32 0.02, 0.1, 0.16, 0.24, 0.32

2 0.00, 0.1, 0.2, 0.3, 0.4 0.01, 0.1, 0.2, 0.27, 0.4 0.02, 0.1, 0.17, 0.26, 0.37 0.02, 0.1, 0.17, 0.26, 0.36 0.02, 0.1, 0.17, 0.26, 0.36

3 0.01, 0.1, 0.19, 0.31, 0.45 0.02, 0.09, 0.19, 0.3, 0.42 0.02, 0.1, 0.19, 0.28, 0.4 0.03, 0.1, 0.19, 0.28, 0.39 0.03, 0.1, 0.19, 0.28, 0.39

4 0.01, 0.1, 0.2, 0.32, 0.48 0.02, 0.1, 0.19, 0.3, 0.44 0.02, 0.1, 0.19, 0.3, 0.42 0.03, 0.11, 0.19, 0.3, 0.4 0.03, 0.1, 0.19, 0.29, 0.41

5 0.01, 0.1, 0.2, 0.33, 0.5 0.02, 0.1, 0.2, 0.32, 0.46 0.03, 0.1, 0.2, 0.31, 0.44 0.03, 0.11, 0.2, 0.31, 0.43 0.03, 0.11, 0.2, 0.3, 0.42

(c) Domestic privatization effects on the foreign welfare

1 0.4, 0.3, 0.25, 0.1, 0.01 0.5, 0.41, 0.32, 0.2, 0.1 0.6.49, 0.39, 0.27, 0.15 0.7, 0.56, 0.45, 0.34, 0.2 0.74, 0.63, 0.5, 0.4, 0.3

2 0.4, 0.3, 0.26, 0.1,�0.02 0.5, 0.42, 0.31, 0.18, 0.02 0.6, 0.48, 0.36, 0.23, 0.07 0.66, 0.55, 0.42, 0.28, 0.1 0.73, 0.6, 0.47, 0.33, 0.2

3 0.41, 0.3, 0.3, 0.1,�0.04 0.5, 0.42, 0.3, 0.16,�0.02 0.59, 0.48, 0.35, 0.2, 0.02 0.66, 0.54, 0.4, 0.24, 0.1 0.72, 0.6, 0.44, 0.27, 0.1

4 0.41, 0.36, 0.3, 0.1,�0.1 0.5, 0.42, 0.3, 0.15,�0.05 0.59, 0.47, 0.3, 0.2,�0.02 0.66, 0.53, 0.38, 0.2, 0.02 0.72, 0.58, 0.42, 0.25, 0.1

5 0.42.36, 0.3, 0.1,�0.1 0.5, 0.42, 0.3, 0.14,�0.07 0.59, 0.47, 0.3, 0.2,�0.05 0.66, 0.5, 0.37, 0.2,�0.02 0.71, 0.57, 0.4, 0.22, 0.02

a=1, k=1, y=0, 0.25, 0.5, 0.75, 1.

Note: Negative numbers are bold to show that privatization does not have the same effect.

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Table 2 Foreign privatization effects

N 1 2 3 4 5

M

(a) Foreign privatization effects and optimal subsidy rate

1 0.32, 0.28, 0.26, 0.27, 0.30 0.35, 0.30, 0.27, 0.25, 0.24 0.37, 0.31, 0.27, 0.23, 0.20 0.38, 0.32, 0.27, 0.22, 0.18 0.39, 0.33, 0.27, 0.21, 0.17

2 0.33, 0.35, 0.36, 0.39, 0.46 0.39, 0.37, 0.35, 0.36, 0.37 0.4, 0.37, 0.35, 0.33, 0.33 0.42, 0.38, 0.34, 0.32, 0.3 0.42, 0.38, 0.34, 0.30, 0.28

3 0.38, 0.39, 0.42, 0.48, 0.57 0.41, 0.40, 0.41, 0.43, 0.47 0.42, 0.41, 0.40, 0.40, 0.41 0.44, 0.41, 0.39, 0.38, 0.38 0.44, 0.41, 0.39, 0.37, 0.38

4 0.4, 0.42, 0.47, 0.54, 0.65 0.42, 0.43, 0.45, 0.48, 0.54 0.44, 0.43, 0.44, 0.45, 0.47 0.45, 0.44, 0.43, 0.43, 0.43 0.45, 0.44, 0.42, 0.41, 0.40

5 0.41, 0.44, 0.50, 0.59, 0.72 0.43, 0.45, 0.48, 0.53, 0.59 0.45, 0.45, 0.47, 0.49, 0.52 0.46, 0.45, 0.46, 0.46, 0.48 0.46, 0.45, 0.45, 0.44, 0.45

(b) Foreign privatization effects on the domestic welfare

1 0.2, 0.21, 0.25, 0.3, 0.56 0.18, 0.22, 0.27, 0.33, 0.57 0.17, 0.25, 0.30, 0.37, 0.6 0.17, 0.28, 0.33, 0.4, 0.6 0.16, 0.31, 0.37, 0.44, 0.6

2 0.33, 0.42, 0.48, 0.57, 0.8 0.32, 0.43, 0.5, 0.59, 0.79 0.31, 0.44, 0.52, 0.62, 0.8 0.3, 0.45, 0.54, 0.64, 0.8 0.29, 0.46, 0.55, 0.65, 0.8

3 0.45, 0.6, 0.7, 0.83, 1.01 0.43, 0.61, 0.71, 0.84, 0.98 0.42, 0.6, 0.71, 0.84, 0.97 0.41, 0.62, 0.73, 0.85, 0.1 0.41, 0.63, 0.74, 0.86, 1.0

4 0.54, 0.75, 0.9, 1.07, 1.2 0.53, 0.76, 0.9, 1.06, 1.2 0.52, 0.76, 0.9, 1.05, 1.1 0.51, 0.77, 0.9, 1.05, 1.1 0.51, 0.77, 0.9, 1.05, 1.1

5 0.6, 0.89, 1.07, 1.3, 1.4 0.6, 0.9, 1.06, 1.26, 1.3 0.6, 0.9, 1.06, 1.24, 1.3 0.6, 0.9, 1.05, 1.23, 1.3 0.6, 0.9, 1.05, 1.22, 1.3

(c) Foreign privatization effects on the foreign welfare

1 0.5, 0.4, 0.35, 0.3, 0.02 0.59, 0.53, 0.46, 0.4, 0.07 0.67, 0.64, 0.56, 0.47, 0.1 0.7, 0.7, 0.65, 0.56, 0.2 0.8, 0.84, 0.75, 0.65, 0.2

2 0.5, 0.4, 0.33, 0.22,�0.1 0.6, 0.5, 0.39, 0.28,�0.05 0.6, 0.56, 0.4, 0.33,�0.01 0.7, 0.62, 0.5, 0.4, 0.02 0.74, 0.7, 0.5, 0.42, 0.05

3 0.5, 0.43, 0.3, 0.2,�0.2 0.6, 0.5, 0.34, 0.2,�0.2 0.6, 0.5, 0.4, 0.2,�0.13 0.7, 0.55, 0.4, 0.24,�0.1 0.7, 0.6, 0.4, 0.3,�0.14 0.54, 0.45, 0.3, 0.1,�0.3 0.6, 0.5, 0.3, 0.1,�0.2 0.6, 0.5, 0.3, 0.1,�0.2 0.6, 0.5, 0.3, 0.1,�0.2 0.7, 0.5, 0.3, 0.1,�0.25 0.6, 0.5, 0.3, 0.1,�0.3 0.6, 0.5, 0.3, 0.02,�0.3 0.63, 0.4, 0.2, 0.01,�0.3 0.6, 0.4, 0.2, 0.01,�0.3 0.6, 0.4, 0.2,�0.04,�0.3

a=1, k=1, y=0, 0.25, 0.5, 0.75, 1.


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Table 3 Simultaneous privatization effects

N 1 2 3 4 5

M

(a) Simultaneous privatization effects on the optimal subsidy rate

1 0.07, 0.12, 0.2, 0.36, 0.57 0.08, 0.12, 0.2, 0.28, 0.42 0.08, 0.11, 0.16, 0.23, 0.33 0.08, 0.1, 0.14, 0.19, 0.26 0.08, 0.09, 0.12, 0.15, 0.2

2 0.05, 0.1, 0.2, 0.35, 0.57 0.06, 0.1, 0.18, 0.28, 0.42 0.07, 0.1, 0.16, 0.23, 0.33 0.07, 0.1, 0.14, 0.19, 0.26 0.07, 0.09, 0.12, 0.16, 0.22

3 0.03, 0.1, 0.2, 0.33, 0.56 0.05, 0.1, 0.17, 0.27, 0.41 0.06, 0.1, 0.15, 0.22, 0.32 0.06, 0.1, 0.13, 0.19, 0.26 0.07, 0.09, 0.12, 0.16, 0.22

4 0.02, 0.1, 0.18, 0.33, 0.56 0.04, 0.09, 0.16, 0.26, 0.41 0.05, 0.09, 0.14, 0.22, 0.32 0.06, 0.09, 0.13, 0.19, 0.26 0.06, 0.09, 0.12, 0.16, 0.22

5 0.01, 0.1, 0.17, 0.32, 0.56 0.04, 0.08, 0.15, 0.26, 0.4 0.05, 0.09, 0.14, 0.21, 0.31 0.05, 0.08, 0.13, 0.18, 0.26 0.06, 0.08, 0.12, 0.16, 0.22

(b) Simultaneous privatization effects on domestic welfare

1 0.05, 0.08, 0.15, 0.25, 0.4 0.03, 0.08, 0.16, 0.26, 0.4 0.02, 0.08, 0.17, 0.27, 0.4 0.02, 0.09, 0.18, 0.29, 0.42 0.01, 0.1, 0.18, 0.3, 0.43

2 0.05, 0.1, 0.17, 0.3, 0.47 0.04, 0.1, 0.18, 0.29, 0.45 0.03, 0.09, 0.18, 0.3, 0.45 0.02, 0.09, 0.19, 0.3, 0.45 0.02, 0.1, 0.19, 0.3, 0.45

3 0.06, 0.1, 0.18, 0.31, 0.5 0.04, 0.10, 0.19, 0.31, 0.48 0.03, 0.1, 0.19, 0.3, 0.47 0.03, 0.1, 0.19, 0.32, 0.47 0.02, 0.1, 0.2, 0.32, 0.47

4 0.06, 0.1, 0.19, 0.32, 0.54 0.04, 0.1, 0.19, 0.32, 0.5 0.03, 0.1, 0.2, 0.32, 0.49 0.03, 0.1, 0.2, 0.33, 0.48 0.02, 0.1, 0.2, 0.33, 0.48

5 0.06, 0.1, 0.2, 0.34, 0.57 0.05, 0.11, 0.2, 0.33, 0.53 0.04, 0.1, 0.2, 0.33, 0.51 0.03, 0.1, 0.21, 0.33, 0.5 0.02, 0.1, 0.21, 0.33, 0.49

(c) Simultaneous privatization effects on the foreign welfare

1 0.46, 0.4, 0.33, 0.2,�0.1 0.59, 0.5, 0.39, 0.24,�0.04 0.68, 0.58, 0.45, 0.29, 0.1 0.76, 0.64, 0.5, 0.34, 0.15 0.82, 0.69, 0.54, 0.38, 0.2

2 0.47, 0.4, 0.32, 0.16,�0.1 0.59, 0.5, 0.37, 0.2,�0.1 0.68, 0.56, 0.4, 0.23,�0.01 0.74, 0.61, 0.46, 0.27, 0.04 0.8, 0.66, 0.49, 0.3, 0.1

3 0.47, 0.4, 0.3, 0.14,�0.2 0.59, 0.5, 0.35, 0.16,�0.13 0.67, 0.55, 0.39, 0.2,�0.1 0.74, 0.6, 0.43, 0.22,�0.05 0.8, 0.6, 0.46, 0.25,�0.014 0.47, 0.4, 0.3, 0.13,�0.2 0.59, 0.49, 0.34, 0.14,�0.2 0.67, 0.5, 0.38, 0.16,�0.14 0.73, 0.59, 0.4, 0.2,�0.11 0.8, 0.62, 0.43, 0.2,�0.15 0.5, 0.42, 0.3, 0.1,�0.25 0.59, 0.49, 0.34, 0.1,�0.22 0.67, 0.5, 0.36, 0.13,�0.19 0.73, 0.58, 0.39, 0.15,�0.2 0.8, 0.6, 0.4, 0.2,�0.1

a=1, k=1, y=0, 0.25, 0.5, 0.75, 1.


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becomes more competitive when n or m increase, the larger amount of consumers’surplus requires a smaller adjustment in the optimal subsidy rate for a given marketshare.

Simulation Result 2: Privatizing the domestic public firm generally increases bothdomestic welfare and foreign welfare

Analysis: While it is difficult to present an unambiguous closed form solution due tocomplexity of the resulting functions, numerical simulations show that for allreasonable combinations of m and n, Wd�

0priv �Wd0 >0 and Wf�

0priv �Wf0 >0.

Tables 1b and 1c show numerical simulation results for the effect of privatizationon domestic and foreign social welfare when a¼ 1, k¼ 1 and y¼ 0, 0.25, 0.5, 0.75,and 1. All values in the table are positive. Note that all the values remainpositive for any choice of k that was tried. Privatizing the domestic publicfirm increases domestic social welfare and foreign welfare for all combinationsof m and n over y. Privatizing the domestic public firm increases domesticsocial welfare more than foreign social welfare when mþ 1 is greater thann or when the domestic market share, y, is 40.5. However, when n is greaterthan mþ 1, the gain in foreign welfare is greater than that for domesticwelfare. The gain in domestic welfare from privatization is greater with larger mbut decreases with larger n. The foreign welfare gain from privatizing thedomestic public firm increases with respect to m, but only if the domestic shareof market is small. It always increases with respect to the size of the foreignprivate sector, n.

Simulation Result 3: Privatization of the foreign public firm generally increasesdomestic welfare but decreases foreign welfare when mXn�1 and y is close to 1.

Analysis: Numerical simulations show that for all reasonable combinations of mand n, the difference between Wd�

0 f�priv �Wd040. However, for Wf�

0 f�priv �Wf0

numerical simulations identify values of y and combinations of m and n, forwhich privatization reduces foreign welfare. Privatizing foreign public firmsdecreases foreign social welfare for several combinations of m and n when y isclose to one.

Tables 2b and 2c present the effect of foreign privatization on domestic welfareand on foreign welfare, respectively. The gain in domestic welfare associated withprivatization increases with y for all combinations of m and n. The domesticwelfare gain also increases with m when n is held constant. Thus, depending on themarket share, the domestic social welfare will increase because of the foreigngovernment’s decision to privatize its public firm. However, the foreign socialwelfare may either improve or decline. It can be concluded that when the domesticmarket share or the domestic private sector is large, the foreign government maynot have any incentive to privatize its public firm. However, if y is small and n>m,then the foreign government may consider privatizing its public firm to improvethe social welfare.

Simulation Result 4: Simultaneous privatization of both public firms increasesdomestic welfare and it decreases foreign welfare if y is close to one and mXn�1.


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Analysis: Simulation for several combinations of m and n shows that Wd�

0 sim�priv �Wd

040 and Wf�0sim�priv �Wf

0o0 if y close to one and mXn�1.Tables 3b and 3c present the changes in the domestic and foreign social welfare

levels caused by simultaneous privatization. For all combinations of m and n thedomestic welfare change associated with simultaneous privatization increases andforeign welfare change decreases with y. Also, the domestic welfare gain increaseswith respect to m, while the foreign welfare change decreases. When m is heldconstant, the domestic welfare gain decreases, and the foreign country’s welfare gainincreases with respect to n. For y close to one and mXn� 1, the foreign socialwelfare decreases as a result of simultaneous privatization. In this case, bysubsidizing its industry, the domestic government denies foreign country anypotential gain from a simultaneous privatization.

When the domestic market share is dominant, the domestic welfare level increaseswhile the foreign welfare level declines. When the foreign market share is dominant,foreign welfare gain is larger than that of the domestic economy from simultaneousprivatization. The gap between the foreign welfare gain and the domestic welfaregain increases as n increases. When domestic country’s share of market is notdominant for different combinations of m and n, both countries benefit fromsimultaneous privatization. This represents a limit on the domestic government’sability to use subsidization to gain the upper hand and to deny foreign governmentany gain in a privatization process.

Despite the generally beneficial results of simultaneous privatization, onecountry’s best response to another’s privatization may not be privatization.Numerical simulations show that the domestic welfare gain associated with theforeign privatization is greater for all y than its gain because of simultaneousprivatization. On the other hand, domestic privatization may actually increaseforeign welfare less than simultaneous privatization, when y is close to 0.

The ability of the domestic government to engage in subsidization does notreplace its need for a public firm to regulate the market. If a foreign countryprivatizes its public firms, the domestic country will do better with a combinationof a public firm and subsidization than it would with just subsidization. This is adramatic departure from the existing literature. Those results emphasize thatsubsidization makes the public firm irrelevant because the first best can beachieved with or without privatization. In a model with competing public firms,subsidization does not allow the government to privatize without potentialnegative welfare effects.

This summarizes the remaining free rider problem within the market. Thedomestic government would prefer foreign privatization to its own unilateralprivatization, or even simultaneous privatization. On the other hand, the foreigncountry may suffer reduced welfare from any type of privatization if the value of y islarge enough and m>n. Under these circumstances, the likely outcome will be theprivatization of the domestic public firm as the domestic government knows aforeign privatization will be less likely. In other cases, the foreign country faces asituation similar to that of domestic country. It would prefer domestic privatizationto either its own unilateral privatization or simultaneous privatization. In this case,each country may wait for the other to move first, resulting in no privatization or innegotiation to privatize simultaneously.

This paper’s findings are fundamentally different from those of Pal and White[1998]. They concluded that privatization improves welfare when domestic firms aresubsidized and that the optimal subsidy decreases with privatization of the domestic


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public firm. This is not true in a multinational market, where the number of firms isgreater than one. The current model could be simplified to that of Pal and White byassuming n¼ 0, y¼ 1 and privatizing the foreign public firm. In this case, the modelconfirms earlier findings. Yet theirs is a special case as numerical simulations exhibitthat simply increasing the number of foreign private firms above one cause theoptimal subsidy rate to increase because of privatization. The current analysis alsodeparts from those of Dadpay and Heywood [2006]. They demonstrate thatprivatizing the domestic public firm decreases domestic social welfare. They alsoargue that when both countries privatize their public firms simultaneously, thensocial welfare improves in both countries. The present paper shows that thedomestic government evades this prisoners’ dilemma by subsidizing its industry. Thepresence of subsidization improves the ability of the domestic government toprivatize its public firm without fearing any welfare loss. It must be noticed thatsubsidization does not eliminate the possibility that a domestic country does betterwhen a foreign government privatizes its public firm.

CONCLUSION

The presence of a domestic subsidy brings about a change in the output of bothdomestic and foreign private firms. The private firms become conscious of theirrespective countries’ share of consumers’ surplus once subsidization is implemented.Without subsidization, a private firm’s output is independent of market share andreflects only competition and demand. As a consequence of this new dependence,total output and the equilibrium price also depend on the market share. In turn, thiscreates a model that is more sensitive to changes in exogenous parameters.

Domestic subsidization enables the domestic economy to benefit from unilateralprivatization of its public firms while still benefiting more from the foreigngovernment’s decision to liberalize its industry. Without the subsidy, the domesticsocial welfare level declines because of the privatization of the domestic public firm.The presence of a subsidy does not, however, generate first best outcomes as suggestedby the previous studies. In those models, the issue of privatization was irrelevant aseither with or without the public firm; the subsidy could ensure maximum welfare.Thus, the role of subsidization to recover the first best is less robust than the mixedoligopoly models to date have suggested. The domestic government’s decision tosubsidize its industry alters the foreign country’s welfare gain in a simultaneousprivatization. The foreign government may not have any incentive to agree to asimultaneous privatization anymore. This is the result of the asymmetry that thedomestic government introduces to the market. To prevent any welfare loss, theforeign government relies on its public firm and does not have any incentive toprivatize unilaterally. Nonetheless, for some cases, foreign social welfare increases withsimultaneous privatization even as it would not increase for unilateral privatization.

Future work might include allowing one or both of the public firms to beStackelberg leaders in their respective countries. Future cases also could includemultinational markets where governments act as Stackelberg leaders and set publicfirms’ outputs and subsidy rates simultaneously in the first stage, to be followed byprivate firms in the next stage. Finally, one might imagine a delegation game inwhich governments provide optimal incentive contracts to the public firms thatchange their objective functions, or they can exercise discrimination in dealing withprivate firms.


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Acknowledgements

The author would like to thank John Heywood, Gilbert Skillman and two anon-ymous referees for their insightful comments and suggestions. Any shortcoming ishis and his alone.

Appendix A

Firms’ outputs equilibrium values when domestic government subsidizes domesticfirms unilaterally.

Domestic private firm output:

qd�i ¼sd½ðkþ 1Þðkþ 2Þ þ nk� þ akðkþ 1Þðkþ 1Þðk2 þ nkþmkþ 3kþ 2Þ

Foreign private firm output:

qfj� ¼k½aðkþ 1Þ �msd�

ðkþ 1Þðk2 þ nkþmkþ 3kþ 2Þ

Domestic Public firm output:

qd�o ¼kðkþ 2Þðak�msd þ 2ayÞ þ ayðk2nþ 2Þ

kðkþ 1Þ½ðmþ nþ 5Þðkþ 1Þ þ kðk� 1Þ þ ymðaþ sdÞ� þ kðany�mnsd þ 2Þ þ ðakþ 2Þð1�mÞ

Foreign public firm output:

qf�o ¼kmsdðn� k� yÞ þ akðkþ 1Þ½ð2aþmÞð1� yÞ þ 3ð1þ ykÞ þ kðkþ yÞ � 2yðnþ 1Þ�

kðkþ 1Þ½ðmþ nþ 5Þðkþ 1Þ þ kðk� 1Þ þ ymðaþ sdÞ� þ kðany�mnsd þ 2Þ þ ðakþ 2Þð1�mÞ

Total output:

Q� ¼ a½ðmþ nþ 2Þkþ 2� þ sdkm

k2 þ ðmþ nþ 3Þkþ 2

Equilibrium price:

P� ¼ aðk� 1Þ �msdk2 þ ðmþ nþ 3Þkþ 2


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Appendix B

s� ¼ aA5k

5 þ A4k4 þ A3k



A0 ¼ �4yA1 ¼ 2y2 � 2yðmþ 2nÞ þ 2m� 19y

A2 ¼ y2ðmþ nþ 6Þ � yðn2 þ 11nþ 35Þ þmnþ 7m� n

A3 ¼ y2ð2mþ 2nþ 6Þ � yðn2 þ 11mþ 11nþmnþ 31Þ þmnþ 9m� 2n

A4 ¼ y2ðmþ nþ 2Þ � yð5mþ 5nþ 13Þ þ 5m� n

A5 ¼ �yðmþ nþ 2ÞB0 ¼ 4mþ 4

B1 ¼ 4mnþ 4nþ 18m� 3ymþ 20

B2 ¼ y2m� yð10mþ 2mnÞ þ n2ðmþ 1Þ þ 10mnþ 14nþ 41

B3 ¼ 2y2m� yð12mþ 2mnÞ þ 2n2 þ 8mnþ 28mþ 18nþ 44

B4 ¼ y2m� 6ymþ n2 þ 2mnþ 10nþ 12mþ 26

B5 ¼ 2m� ymþ 8

Appendix C

qs�

qk¼ a

P10i¼0

Ciki

ðk6 þ B5k5 þ B4k4 þ B3k3 þ B2k2 þ B1kþ B0Þ2

C0 ¼ 8ym2 þ 16ymn� 8m� 4y

C1 ¼ y2ð24m� 8n� 8m2 þ 8nmÞ þ yð213m2 þ 88mþ 16mnÞ� 56m2 þ 104mþ 20mn

C2 ¼ y3ð�45m� 5nmÞ þ y2ð2n2 � 3m� 8n� 44m2 þ 19nm� 110Þþ yð213m2 þ 268mþ 16mn� 235Þ � 170m2

C3 ¼ y2ð2n2 þ 134mÞ þ yð�8n3 þ 12n3mÞ � 14m2nþ 48mn

C4 ¼ 12my4 þ 6m2y3 þ y2ð2n2 � 111m2 � 28mnðmþ nÞ þ 19nmþ 54Þþ yðn3ð12m� 27Þ þ 13m2n2Þ � 310m2 � 86mn2 � 2m2n2


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C5 ¼ 2m2ny3 þ y2ð2n4 � 192n� 8mnð3mþ 4n�mn� 10Þ þ 212Þþ yðn3ð3m� 8Þ þ 13m2n2Þ � 208m2 � 86mn2 � 6m2n2 þ 30n

C6 ¼ y2ð2n4 � 192n� 12mnðmþ 2n� 9mn� 10Þ þ 212Þþ yðn3ð12m� 14Þ �m4 þ 10n2Þ þmn3 � 34m2n� 8mn2 � 69mþ 48mn

C7 ¼ y2ð8n2 þ 60nþ 74mþ 6mn� 2nm2 � 2n2mÞþ yð14m2n� 184n� 460Þ � 12m2nþ 84mþ 20mnÞ

C8 ¼ y2ð18nþ 19mþ 3mnþ 34Þ þ yðm2 � 57m� 53nþ 3mn2 þ 2mnÞþ 2nð4n2 � nþm�mnÞ

C9 ¼ 2y2ðmþ nþ 2Þ � yð10mþ 12nþ 26Þ � 2n

C10 ¼ �yðmþ nþ 2Þ þm

Notes

1. For details on the international commercial airline industry, see Doganis [2001].

2. Fjell and Heywood [2004] point out that these results depend critically on the assumption of the timing

of moves after privatization. They show that privatization affects the optimal subsidy if the public firm

remains a leader after privatization.

3. Neither are price and marginal cost related linearly as Sepahvand’s [2002] adjusted marginal cost

pricing indicates.

References

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Organization, 9(1): 43–53.

Dadpay, A., and J.S. Heywood. 2006. Mixed Oligopoly in a Single International Market. Australian

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Pal, D., and M.D. White. 1998. Mixed Oligopoly, Privatization, and Strategic Trade Policy. Southern

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Governments and Producers in Multinational Markets: A Mixed Oligopoly Analysis

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