2008-49
Should civil servants be restricted inwage bargaining?
A mixed-duopoly approach
石田 潤一郎松島 法明

Should civil servants be restricted in wage bargaining? A
mixed-duopoly approach∗

Junichiro Ishida
Osaka School of International Public Policy, Osaka University
andNoriaki Matsushima†
Graduate School of Business Administration, Kobe University
October 7, 2008

Abstract
Should civil servants (employees in the public sector) be allowed to bargain collectively?
To answer this question, we construct a model of unionized mixed duopoly and examine
the regulatory framework of public institutions, especially focusing on a wage regulation
imposed on the public firm. The wage regulation turns out to yield critical welfare implications as it gives rise to two opposing strategic effects: the wage regulation intensifies
downstream-market competition while it loosens upstream-market competition. The overall welfare effect is ambiguous, depending crucially on the degree of product differentiation
between the firms. We also show that, in contrast to the popular belief, granting the rightto bargain collectively to civil servants would not necessarily help them because they tend
to demand excessively high wages when they are allowed to bargain collectively. Finally,
we briefly discuss a new perspective on the role of profit motives in public institutions
when the wages are determined endogenously.

JEL classification numbers: H42, J31, J38, L13.
Keywords: Mixed oligopoly, Wage bargaining, Wage regulation, Labor unions, Strategic
complements.
∗

We thank two anonymous referees, the coeditor of the journal and seminar participants at University ofTokyo for many comments and suggestions which vastly improved the paper. All remaining errors are ourown. Financial support from the Japanese Ministry of Education, Culture, Sports, Science and Technology(Grant-in-Aid for Young Scientists (B)) is gratefully acknowledged.
†
Corresponding author: Noriaki Matsushima, Graduate School of Business Administration, Kobe University,
2-1 Rokkodai, Nada, Kobe, Hyogo, 657-8501, Japan. Phone: +81-78-803-6981. E-mail: nmatsush@kobe-u.ac.jp

1

Introduction

Mixed markets, where state-owned public firms compete against private firms, are fairly common in developed, developing, and former communist transitional economies. In many countries, public firms exist and compete with private firms in a range of industries such as theairline, rail, telecommunications, natural gas, electricity, steel, and home-delivery industries,
as well as services such as banking, home loans, health care, life insurance, hospitals, broadcasting, and education.1 In fact, even those typically considered as national monopolies in anarrowly defined sense can be regarded as competing in mixed markets with product differentiation, because there are often other firms or industries which produce substitutable goods
and services.2 Accordingly, the study of mixed markets has become increasingly popular inrecent years.3
In standard mixed-market models, public firms are typically assumed to maximize social
welfare while private firms are to maximize profit.4 Under this setup, it is often supposed thatthe role of public firms is to maximize social welfare by correcting inefficiencies arising frommarket failures. There is no convincing reason to believe, however, that employees in the public
sector (to whom we refer somewhat loosely and inclusively as civil servants throughout the
paper, mostly for expositional purposes) share the same goals; it rather seems more natural toassume that they aim at maximizing their own utility just like employees in the private sector.5
1

Among developed countries, mixed markets are more prevalent in Europe, Canada, and Japan. Although

they are less significant in the United States, there are examples of mixed oligopolies such as the packaging andovernight-delivery industries.
2

For instance, De Fraja (1993) points out that although a national rail system is usually considered as anational monopoly, it does have some oligopolistic interaction with private coach companies.
3

Merrill and Schneider (1966) and Harris and Wiens (1980) are the pioneering works in this field. See alsoBös (1986, 1991), De Fraja and Delbono (1990), and Nett (1993) for excellent surveys.
4

See, for instance, De Fraja and Delbono (1989), Cremer et al. (1991), Fjell and Pal (1996), Anderson etal. (1997), Pal (1998), Pal and White (1998), and Matsushima and Matsumura (2003).
5

Of course, one can argue that mission-oriented public institutions tend to attract more ‘motivated agents.’

For instance, Besley and Ghatak (2005) emphasize the importance of motivation and preference matching ina mission-oriented, rather than profit-oriented, sector. Still, there is no convincing reason to believe that eventhe goals of those ‘motivated agents’ coincide precisely with the goals of pubic institutions.

1

If the goals of civil servants are not to maximize social welfare, their behavior may need tobe regulated in some ways. What becomes crucial along this line is how wages in the public
sector should be regulated because the wage level virtually determines the firm’s productive
efficiency, ultimately leading to serious welfare implications.6 If wages in the public sector are
set in an inappropriate way, any subsequent effort to increase efficiency might totally be wiped
out. It is this aspect of public firms that we focus on in this paper.
In reality, the way civil servants are regulated is surprisingly diverse across countries. In
many countries today, civil service salaries are determined largely through collective bargaining
between the union and the government, although there are often some job- and title-specificrestrictions. In contrast, there are still some countries such as Germany (for the case of
Beamte), Japan and the US where civil servants are regulated and restricted in some senses.
When civil servants are restricted in their right to bargain collectively, their salaries are moreor less tied to the average wages in the respective industries (the ‘equal pay for equal work’
principle) or those in the private sector as a whole. In Japan, for instance, civil servants are
typically not allowed to bargain collectively, and their wages are instead determined based on
the advise of the National Personnel Authority, called Jinji-In-Kankoku, with its particularemphasis on the equalization between the private and public sectors.7
Because of its potential impact on national welfare, the regulatory framework of civil
servants has naturally been a topic of heated debate. Although different countries have theirown regulations and it is certainly difficult to draw a clear line,8 a recent trend is clearly shifting
6

The productive efficiency of public and/or private firms in mixed markets is examined by many researchersin other contexts. See Nett (1994), Nishimori and Ogawa (2002), Corneo and Rob (2003), Ma (2004), andMatsumura and Matsushima (2004) for theoretical discussions on this issue. Using the data on Japanese parcel
companies, Mizutani and Uranishi (2003) empirically examine the efficiency of public firms. Meggison andNetter (2001) provide an excellent survey.
7

Although the advice formally covers only national government employees, it typically sets the baseline andhence has strong impacts on salaries for local government employees.
8

For instance, in Britain, civil service salaries are based on the recommendations of independent review

bodies rather than on collective bargaining. The review bodies, however, receive detailed evidence from tradeunions and employees, and their recommendations are usually adopted by the government. There is also a
country like France where the legal status of collective bargaining is somewhat uncertain. In France, although
unions are recognized as capable of conducting negotiations, these negotiations are not compulsory, and the2towards granting civil servants the right to organize and bargain collectively, mostly from theviewpoint of protecting workers’ rights. For instance, a report by the Committee on Freedom of
Association, one of the six Governing Body committees of International Labour Organization(ILO), points out that the current Japanese system is not consistent with international labor
standards, and urges the Japanese government to engage in full consultations with the tradeunions with the view to amend the current legislation that denies civil servants the right tobargain collectively (Cases 2177 and 2183, November 2002). The Committee has also issued
a recommendation against the US state of North Carolina, which as well as in some other US
states, strictly prohibits any collective bargaining in the public sector, that the government
should promote a framework allowing collective bargaining in the public sector (Case 2460,
March 2007). The stance taken by ILO reflects an emerging global consensus that the rightto bargain collectively (and even strike) constitutes an important part of workers’ rights thatshould be respected even for civil servants.
Despite this recent trend, however, welfare consequences of allowing civil servants to bargain collectively are not necessarily straightforward, especially in a mixed market. This paper
thus examines the regulatory framework of public institutions with a particular focus on thewage regulation imposed on their employees. To this end, we consider two distinct regulatory
regimes in a mixed duopoly. In the first, civil servants are allowed to form a union to represent themselves and, more importantly, to bargain collectively just as workers in the private
sector.9 In the second, civil servants are prohibited to bargain collectively, and their wages aretightly regulated by some wage-setting rule (unilateral regulation), which more or less abides
by the ‘equal pay for equal work’ principle. We compare these two regimes and explore welfare
implications of the wage regulation imposed on civil servants.
government is free to disregard any agreement that may have been reached. See Bordogna (2003) for moredetail.
9

Canada is one of those countries which provide several examples of mixed unionized oligopoly: unionism inthe public sector is very active, as represented most notably by Canadian Union of Public Employees (CUPE),
while at the same time, public firms play a very large role in the Canadian economy. One prime example of thiscase is Canada Post which competes with private enterprises such as FedEx. Employees at Canada Post are
represented by the Canadian Union of Postal Workers (CUPW) – one of the most active trade unions known
for a series of high profile strikes.

3

The analysis reveals that the way civil service salaries are determined indeed leads tosubstantial welfare consequences. As is normally the case in this type of setup, the welfare
effect of the wage regulation hinges critically on its impact on the equilibrium wages (or the
prices of intermediate goods): roughly speaking, the wage regulation is welfare-enhancing if itlowers the wages. First, the wage regulation, which totally deprives civil servants of the right tobargain collectively, can have a direct effect as it places an exogenous cap on the public firm’s
wage. In a situation where the firms and their respective unions interact strategically, however,there are two additional indirect effects, one positive and one negative, which work throughstrategic market interactions. The first effect, a positive one, is that the wage regulation by
design gives a cost advantage to the public firm, which forces the private firm’s union to lower
its wage in order to compete in the downstream market. The second effect, a negative one,is that it also releases the private firm’s union from stiff wage competition as it can now set
its wage unilaterally, knowing that the public firm’s wage is automatically set according to
the rule. We can thus summarize these observations as follows: the wage regulation intensifies
downstream-market competition while it loosens upstream-market competition. The overall
welfare impact is indeed ambiguous due to this tradeoff and can be either positive or negative,
depending on the degree of product differentiation between the two firms.
While the main purpose of the paper is to provide some insight on the regulatory frameworkof public institutions, it also raises two additional implications of some interest. First, themodel shows somewhat paradoxically that although the private firm’s union always loses withthe wage regulation, the public firm’s union may sometimes benefit from it. In other words,
while the union can take advantage of the public firm’s inherent tendency to expand output,
this incentive can be excessively strong even from the union’s viewpoint. This result implies
that there are situations where the union would be made better off by strategically giving up
the right to bargain collectively in wage bargaining.
Second, the model also provides a new perspective on the role of budget constraints on
public institutions. The public firm must reduce output in pursuit of more profit when a tighter
budget constraint is imposed, and a reduction in output normally entails a welfare loss. In the4present setup, however, this can actually be welfare-enhancing because the tighter constraint
is instrumental in inhibiting the union from becoming overly aggressive in wage bargaining
and thus controlling its behavior. We argue that this aspect provides a new perspective on
the role of profit motives in public institutions when the wages are set endogenously.
As stated, there are increasingly many works on mixed oligopoly where a welfare-maximizing
public firm competes with profit-maximizing private firms. At the same time, the literatureon unionized oligopoly have also flourished almost independently.10 There are, however, surprisingly few works that integrate these two strands and examine unionized mixed oligopoly.11
Moreover, we also believe that the regulatory framework of civil servants is an important
agenda with immense policy implications. The main task of this paper is hence to shed lighton this aspect of mixed markets which, in our opinion, has not been examined enough in arigorous context. To this end, the paper proceeds as follows. The basic setup is briefly outline
in the next section. The market equilibrium under each regime is characterized in sections 3
and 4, and the welfare analysis follows in section 5. Some extensions of the model are discussed
in section 6. Finally, some concluding remarks are made in section 7.

2

The Model

2.1

Environment

We formulate a mixed-duopoly model, in which a welfare-maximizing public firm competes
against a profit-maximizing domestic private firm. For expositional purposes, we refer to the
public (private) firm as firm 0 (firm 1) and its union as union 0 (union 1).12 The basic structure
10

See, for instance, Horn and Wolinsky (1988), Naylor (2002), Lommerud et al. (2003), and López andNaylor (2004).
11

An exception, to the best of our knowledge, is De Fraja (1993) who explicitly incorporates wage bargaining

in mixed oligopoly in an attempt to examine the effect of privatization on the wages. Besides several differencesin the setup (which we will mention later), the present paper crucially differs from De Fraja (1993) as our focus
is primarily on welfare consequences of the wage regulation.
12

In this paper, we do not allow the government to nationalize more than one firm. As pointed out by Merrill

and Schneider (1966), the most efficient outcome is achieved by the nationalization of all firms, if nationalization
does not change the costs of firms. The need to analyze a mixed oligopoly arises, however, because it is often
impossible or undesirable, for political or economic reasons (e.g., X-inefficiency), to nationalize an entire sector.

5

of the model follows a standard product differentiation model, e.g., Singh and Vives (1984).
Each firm produces a differentiated good. A representative consumer’s utility is given by
U = x0 + x1 −x0 2 + 2γx0 x1 + x1 2
,
2

(1)

where γ ∈ (0, 1) represents the degree of product differentiation.13 This specification implies
the following inverse demand functions: for positive demands and i = 0, 1,
pi = 1 − xi − γxj , j 6= i.

(2)

The firms are homogeneous with respect to the productivity. Each firm adopts a constantreturns-to-scale technology where one unit of labor is turned into one unit of the final good.
The price of labor, i.e., the wage, that firm i has to pay is denoted by wi .

2.2

Wage setting

The private firm is unionized, and its wage is determined as a consequence of (Nash) bargaining
between the firm and its union.14 Let w̄ denote the competitive wage. Taking this as thereservation wage, the union sets the wage w1 to maximize the following utility function:
u1 = (w1 − w̄)θ x1 , θ > 0,(3)where θ is the weight the union attaches to the wage level. This setup implies that the union
possesses full bargaining power.15 For most part, mainly to simplify the analysis, we restrict
our attention to the case where θ = 1. We also set w̄ = 0, as its level is inconsequential in aqualitative sense.
13

This is a departure from De Fraja (1993) where the public and private firms produce a homogeneous good(perfect substitutes). Needless to say, the case of perfect substitutes is a special case of our specification (γ = 1).
14

In this paper, we do not consider the case of a monopoly union. As can be expected, although workers

are made better off, the presence of a monopoly union leads to higher wages and is generally welfare-reducing,which might present a compelling case against it.
15

This specification allows us to focus on a case where the union possesses sufficiently strong bargaining

power. As the union’s bargaining power becomes smaller, the model naturally converges to the standard
mixed-oligopoly case where strategic interactions in the input market cannot be discussed. For the sake of theanalysis, therefore, we consider the other end of the spectrum to emphasize the impact of wage regulations.

6While the private firm’s wage is determined by collective bargaining, we consider two
distinct wage-setting regimes for the public firm. In one regime, no regulation is imposed on
the public firm so that the public firm’s union (union 0) is allowed to bargain collectively. Since
the public firm is unionized just as the private firm, the union sets the wage w0 to maximize
u0 = (w0 − w̄)θ x0 ,(4)

where w̄ = 0 as above.
In the other regime, on the other hand, the public firm’s union is prohibited to bargain
collectively. In the absence of collective bargaining, the wage w0 is determined according
to some wage-setting rule along the line of the ‘equal pay for equal work’ principle. Morespecifically, the public firm’s wage is given by
w0 = kw1 + (1 − k)wpr ,(5)

where wpr is the average wage for the entire private sector. An important point in thisspecification is the fact that wpr is exogenous. Since its level is inconsequential in a qualitative
sense, we again set wpr = 0 for analytical simplicity in what follows.16
In this setup, k ∈ (0, 1] is an exogenously given variable which determines how closely w0
should follow w1 .17 The ‘equal pay for equal work’ principle is tightly implemented when k is
close to one. In this paper, we take this variable k as exogenous, mostly reflecting conditionsin the outside labor market as well as various institutional factors. When the two firms aretechnologically identical and workers are fully mobile, for instance, k would have to be fairly
close to one in order to attract any worker. We in general consider a realistic case where k issufficiently large and close to unity.
16

Setting wpr = 0 directly implies wpr < w1 in equilibrium. Although the level of wpr is not consequential,

this fact (that wpr < w1 ) is qualitatively important. We focus on this case since the average wage for unionizedworkers is normally higher than that for the entire private sector which necessarily includes non-unionizedworkers.
17

Of course, k can in principle take any positive number. We restrict attention to the case where k ≤ 1

because it is unrealistic and perhaps politically infeasible to institute a rule that sets civil service salaries above
the market average.

7

2.3

TimingWe consider a two-stage game, and the timing of the game is as follows:
1. [the upstream market] If the public firm’s union is allowed to bargain collectively, each
union chooses its wage wi simultaneously. If not, union 1 unilaterally sets the wage w1 while
w0 is determined by the wage-setting rule (5).
2. [the downstream market] Each firm chooses its quantity xi simultaneously to maximize its
respective object.

3

The output market

We first consider the problem faced by the public firm. The public firm aims at maximizing
the social welfare,18 which is defined as
W = (U − p0 x0 − p1 x1 ) + (π0 + π1 ) + (u0 + u1 ).

(6)

Taking w0 as given, the public firm’s problem is defined asmax
x0

W, s.t. (p0 − w0 )x0 ≥ 0.

The constraint implies that there is some lowerbound for the public firm’s profit, i.e., thepublic firm faces a budget constraint.19 This simply means that even the public firm is notallowed to incur an unlimited amount of loss. For now, we assume that the lowerbound is zeroso that the public firm must yield nonnegative profits.
18

In De Fraja (1993), the public firm is assumed to maximize the weighted sum of consumer surplus, profit

and the union’s utility where the weight attached to the union’s utility is exogenously given. The assumption
is necessary to obtain an equilibrium as, without it, the union would be able to unlimitedly raise its wage. Seefootnote 19 for more on this point.
19

Without this restriction, the public firm’s union can unlimitedly raise its wage because the optimal output

level of the public firm is independent of the wage, which only affects the distribution of wealth between thefirm and the union. Since no firms, including public entities, possess an unlimited amount of resources, it isnatural to assume that the public firm faces a restriction of this kind. In Japan, for instance, the Postal Law(Article 3) stipulates that postal fees must be set at an appropriate level that is sufficient to cover all necessary
expenses.

8

Similarly, consider the problem faced by the profit-maximizing private firm. Taking w1 asgiven, the private firm’s problem is defined asmax (p1 − w1 )x1 = (1 − x1 − γx0 − w1 )x1 .
x1(7)

Solving these two problems yields the following result.
Lemma 1 In the output market equilibrium, the budget constraint for the public firm is always
binding. The output levels are given byx0 =2 − γ − 2w0 + γw1
1 − γ − w1 + γw0, x1 =.
2 − γ22 − γ2

Proof: See Appendix A.
The lemma states that the budget constraint for the public firm is always binding. The
public firm thus sets the quantity at a level that yields zero profit in equilibrium. As we will
see later, this indicates an important role played by the budget constraint imposed on the
public firm.

4

The market equilibrium

4.1

The unregulated public firm

We first consider a case where no regulation is imposed on the public firm, and its wage isdetermined as a result of collective bargaining between the firm and the union. Because ofthis, we need to consider two independent maximization problems simultaneously. First, theproblem for union 0 is defined asmax
w0

w0 θ x0 =w0 θ (2 − γ − 2w0 + γw1 ).
2 − γ2

Similarly, the problem for union 1 is defined asmax
w1

w1 θ x1 =w1 θ (1 − γ − w1 + γw0 ).
2 − γ29

Lemma 2 Suppose that no regulation is imposed on the public firm, and its union is allowed
to bargain collectively. Then, the equilibrium wage and output levels, denoted as wiN and xNi ,are given by2θ(1 + θ) − θγ − θ2 γ 22θ(1 + θ) − 2θγ − θ2 γ 2
N
,
w
=
,
1
2(1 + θ)2 − θ2 γ 22(1 + θ)2 − θ2 γ 22(2(1 + θ) − γ − θγ 2 )2(1 + θ) − 2γ − θγ 2 )
N
xN
=
,
x
=
.
0
1
(2 − γ 2 )(2(1 + θ)2 − θ2 γ 2 )(2 − γ 2 )(2(1 + θ)2 − θ2 γ 2 )w0N =Proof: The first-order conditions are given by
w0 =
w1 =

2 − θγ + θγw1,2(1 + θ)θ(1 − γ + γw0 )
.
1+θ

(8)
(9)

Solving and substituting them into the output levels derived in lemma 1 then yields the results.
Q.E.D.

4.2

The regulated public firm

We now shift our attention to the case where the public firm is tightly regulated in order to see
how the wage regulation affects the equilibrium outcome. In this case, the public firm’s wage
is determined automatically by the wage-setting rule (5). Taking the equilibrium outcome inthe output market as given, union 1 maximizes
u1 = w1 x1 =w1 θ (1 − γ − w1 + γw0 ).
2 − γ2(10)

Knowing that the public firm simply follows the wage-setting rule, union 1 solves
max
w1

w1 θ (1 − γ − w1 + γkw1 ).
2 − γ2

Lemma 3 Suppose that the wage regulation is imposed on the public firm, and its union isprohibited to bargain collectively. Then, the equilibrium wage and output levels, denoted as wiRand xRi , are given by
w0R =
xR
0 =

kθ(1 − γ)
θ(1 − γ), w1R =,
(1 + θ)(1 − γk)(1 + θ)(1 − γk)2(1 + θ) − γ − θγ 2 − (2θ + 2γ − (1 + θ)γ 2 )k
1−γ
, xR
.
1 =
2
(1 + θ)(1 − γk)(2 − γ )(1 + θ)(2 − γ 2 )10Proof: Solving the problem for union 1, we obtain the equilibrium wages. Substituting theminto the output levels derived in lemma 1 then yields the equilibrium output levels.

5

Q.E.D.

The regulatory framework of public institutions

The underlying theme of this paper is whether and how civil servants, especially their wages,
should be regulated. With the equilibrium wage and output levels, we are now ready to assess
the impact of different wage-setting regimes on social welfare. In what follows, we restrict ourformal analysis to the case where θ = 1, because the impact of a change in θ is fairly intuitive
although the analysis becomes highly complicated computationally. We later briefly discuss
what would happen when θ diverges away from unity.
The economy consists of three components: the unions, the firms and consumers. To see
the welfare impact of the wage regulation, however, we first need to pay close attention to how
the wage regulation affects the equilibrium wages as the welfare impact on each component
hinges critically on them. We then examine each component in turn in order to identify who
gains and who loses with the wage regulation.
The equilibrium wages: The equilibrium wage levels have critical welfare implications as
they exert direct influences on the output levels. We first establish that the wage regulation
can either increase or decrease the wages, especially depending on γ.
Proposition 1 There exists some threshold w̄0 (γ) such that the wage regulation raises union
0’s wage, i.e., w0R > w0N , for k > w̄0 (γ). Similarly, there exists some threshold w̄1 (γ) suchthat the wage regulation raises union 1’s wage, i.e., w1R > w1N , for k > w̄1 (γ).
Proof: It directly follows from lemmas 2 and 3. With some algebra, the thresholds can beobtained asw̄0 (γ) ≡4 − γ − γ22(4 − γ − γ 2 )
and
w̄
(γ)
≡
.
1
8 − 3γ 2 − γ 32(4 − 2γ − γ 2 )Q.E.D.
[Figure 1 about here]
11

Figure 1 depicts w̄0 (γ) and w̄1 (γ), showing when the wage regulation raises the equilibrium
wages. There are two parameters of interest, namely k and γ. The impact of the wage
regulation with respect to k is relatively straightforward as can be seen from the figure. Noticethat when k is small, the public firm has a cost advantage, and that puts some pressure on
the private firm’s union to lower its wage. Conversely, a higher k implies better coordination
between the unions, which substantially weakens wage competition and allows the private
firm’s union to demand a higher wage. The impact is therefore monotonic: the larger k is, thehigher the equilibrium wages are under the wage regulation.
On the other hand, the impact of the wage regulation with respect to γ is less clear, since
there are three different effects, one direct and two indirect, at force. First, the wage regulation
places an exogenous cap on the public firm’s wage. We refer to this as the “direct effect” of
the wage regulation because its effect is present without any market interactions and hence
independent of γ. When γ is relatively small and market interactions are less of a factor,
the direct effect dominates, and the wage regulation tends to lower the public firm’s wage.
Evidently, the direct effect does not affect the private firm’s wage.
Beside this direct effect, the wage regulation also gives rise to two additional indirect effects
which work through strategic market interactions. The first indirect effect, which we refer toas the “downstream-market effect,” arises from the fact that the public firm by design ends up
with a cost advantage under the wage regulation (when k < 1). Since the public firm is able toprocure its labor input at a lower price, this virtually forces the private firm’s union to lower its
wage in order to compete in the downstream market. This aspect of the regulation is welfareenhancing because a lower wage generally leads to more output, which illustrates a virtue ofthe wage regulation: by placing an exogenous cap on the public firm’s wage, the regulation
intensifies downstream-market competition and realizes more output and lower market prices.
This does not necessarily mean, however, that the wage regulation invariably lowers theequilibrium wages. Although the wage regulation certainly has a bright side, there is alsoa cost associated with it because the private firm’s union can now set its wage unilaterally
without any strategic concerns. Knowing that the public firm’s wage is set according to12

the wage-setting rule, this releases the union from stiff wage competition and hence loosens
upstream-market competition. This aspect of the regulation is welfare-reducing as it tends toraise the equilibrium wages, leading to less output. We refer to this as the “upstream-market
effect” of the wage regulation.
The overall effect is determined largely as a result of this tradeoff. Both of the indirect
effects become weaker, but the upstream-market effect eventually dominates as γ decreases.
Due to this, the wage regulation tends to raise the private firm’s wage when γ is relatively
small. As for the public firm’s wage, on top of it, we also need to take into account the direct
effect. Since the direct effect is independent of γ, it becomes relatively more important as γ
decreases and the indirect effects become weaker. As a consequence, the impact of the wage
regulation on the public firm’s wage is generally non-monotonic with respect to γ: the wage
regulation tends to raise the public firm’s wage when γ is neither too large nor too small. Since
the public firm generally occupies a larger market share, this force is especially important when
we discuss welfare implications of the wage regulation.
Union utility: Notice that a higher wage does not always make the union better off, nordoes a lower wage always make it worse off. We now examine under what conditions each
union is made better off under the wage regulation.
Proposition 2 There exists some threshold ū0 (γ) such that the wage regulation makes union
0 better off for k > ū0 (γ), while it never makes union 1 better off.
Proof: See Appendix A.
The proposition shows two results that are somewhat counterintuitive. First, the wage
regulation never makes the private firm’s union better off although it tends to raise its wage
for a wide range of γ. This is mostly due to the fact that the private firm would lose the costadvantage under the regulation and is hence forced to cut down its output or, equivalently,
employment. Although the wage regulation allows the private firm’s union to set its wage
unilaterally, and this will often be reflected in a higher wage, it never benefits the union as it
loses a competitive edge in the market.
13

Second, the wage regulation often makes the public firm’s union better off although it tends
to decrease its wage: in this sense, there is a situation where the public firm’s union would
be made better by strategically giving up the right to bargain collectively. This is certainly
paradoxical, given the fact that the right to bargain collectively is granted to civil servants inmany countries in order to protect their human rights. Figure 2 depicts ū0 (γ), showing when
the wage regulation benefits the public firm’s union.
[Figure 2 about here]
Firm profit: We now focus on how the wage regulation imposed on the public firm affects
the profit made by each firm. This turns out to be a simple exercise since the public firmalways makes zero profit in equilibrium (see Lemma 1), which allows us to restrict attention
to the private firm.
Proposition 3 The wage regulation always lowers firm 1’s profit.
Proof: See Appendix A.
The proposition states that the wage regulation imposed on the public firm always lowers
the private firm’s profit. The result is fairly intuitive. Even without the wage regulation,
the public firm tends to operate on a larger scale because of the difference in the objectives,
which generally crowds out the private firm. This factor is somewhat alleviated when the wage
regulation is not imposed, because the cost advantage then belongs to the private firm withthe public firm’s union becoming excessively aggressive. When the wage regulation is imposed,on the other hand, the situation is totally turned around where the cost advantage now shifts
to the public firm. On top of the difference in the objectives, this forces the private firm tolower its output. Along with proposition 2, this implies that the private firm and its union areunambiguously on the losing side of the wage regulation.
Consumer surplus: Another important component of social welfare is consumer surplus.
In general, the more the firms produce, the more consumers benefit since more production
14

results in lower market prices. A critical factor here is again the public firm’s wage which
determines the cost of production.
Proposition 4 There exists some threshold CS(γ) such that the wage regulation enhances
consumer surplus if k < CS(γ).
Proof: See Appendix A.
Figure 2 illustrates CS(γ). As we have already seen, since the upstream-market effectdominates, the wage regulation tends to raise the public firm’s wage when γ is in some intermediate range. Conversely, this implies that the wage regulation is more likely to benefit
consumers when γ is closer to either end of the spectrum.
Social welfare: The public firm aims at maximizing social welfare. Likewise, the government’s ultimate concern is supposedly to maximize social welfare. We say that the wage
regulation is efficient if it enhances social welfare.
Proposition 5 There exists some threshold W (γ) such that the wage regulation is efficient(enhances social welfare) if k < W (γ).
Proof: See Appendix A.
As stated, unambiguous losers of the wage regulation are the private firm and its union
because the regulation by design gives the public firm a cost advantage, which would otherwise
belong to the private firm. If the wage regulation is to be efficient, any positive effect arisingfrom it must overcome the cost incurred by the private firm. Again, the primary factor inthis is the public firm’s wage. As can be seen from Figure 2, the wage regulation tends to bewelfare-reducing when γ is in some intermediate range.
Figure 2 also depicts how social welfare, consumer surplus and the utility of the public
firm’s union are related to each other. The figure indicates several logical relationships among
them. In particular, it shows that any inefficient wage regulation, the one that reduces social

15

welfare, always benefits the public firm’s union. This result points out an inherent tension between welfare-maximizing public firms and utility-maximizing civil servants. This observation
ultimately amounts to the following statement.
Proposition 6 The wage regulation can never be Pareto-improving by itself under any circumstances.
Proof: To show this, we need to show that the wage regulation is neither Pareto-improving
nor Pareto-worsening (if the regulation is Pareto-worsening, then everyone can be made betteroff by removing the regulation). First, since the private firm and its union inevitably loseon the wage regulation, it can never be Pareto-improving. The wage regulation can never be
Pareto-worsening either because an inefficient regulation always benefits the public firm’s union(see figure 5). This proves that there exists no regulatory framework that is Pareto-improving
under any circumstances.

6

Q.E.D.

Extensions

We have thus far restricted our attention to cases that are relatively tractable in order toillustrate fundamental forces behind our setup. In this section, we relax some of the restrictions
to show that our main logic is robust to some alterations to the basic model in a qualitative
sense.20

6.1

The budget constraint

One important role of public firms is to correct inefficiencies that arise in oligopolistic markets
because firms have some control over the market prices, which leads them to reduce output toan inefficient level. In this sense, intuition certainly suggests that it is not desirable to force
the public firm to pursue more profit by imposing a tighter budget constraint.21 This intuition20

As can easily be expected, the analysis becomes quite complicated computationally with additional pa-

rameters, and most of the discussion in this section is hence based on numerical examples. Results are available
upon request.
21

Here, we consider a situation where the government places some budget constraint on the public firm.

An alternative way to incorporate this aspect is to generalize the public firm’s objective function and place16

certainly holds true when the wage regulation is imposed on the public firm. The reason why
a tighter budget is welfare-reducing when the public firm’s union is not allowed to bargain is
fairly straightforward. A primary effect is on the public firm’s output. Under a tighter budget,the public firm needs to reduce its output in order to raise the market price. Moreover, thischange in the output level gives rise to a secondary effect. While a tighter budget reduces the
public firm’s output, it provides a room for the private firm to increase its output. This allows
the private firm’s union to demand a higher wage, which consequently raises the public firm’s
wage as well. An increase in the wages reduces the total output, which inevitably reduces
consumer surplus and social welfare.
The same conclusion does not necessarily hold, however, when the public firm’s unionis allowed to bargain collectively. In this setting, there arises a virtue of a tighter budget
because it effectively inhibits the union from becoming overly aggressive in wage bargaining.
Moreover, this yields an additional effect as strategic complementarity between the unions
lowers the private firm’s wage as well. A decrease in the wages all works to improve welfare byincreasing the total output, lowering the market price and hence enhancing consumer surplus.
The analysis thus points out a channel through which imposing a tighter budget constraint
would actually enhance welfare when the public firm’s union is allowed to bargain collectively,
thereby shedding light on a new perspective on the role of profit motives in public institutions.

6.2

The unions’ objectives

Up to this point, we have restricted our attention to the case where θ = 1, but his may be atight restriction because there is no reason to believe a priori that the objective of the unions
is to maximize the wage bill wi xi . It is often argued, for instance, that unions are generally
more eager to protect employment rather than to raise wages. This situation corresponds
closely to the case where θ < 1 as the unions attach greater importance to employment thana different weight on the public firm’s profit. We chose not to follow this approach for two reasons. First,
pure-strategy equilibria may fail to exist in the bargaining stage for some parameter values in this generalized
setting. Second, whether the government actually abides by the budge constraint is easier to verify, while what
objective function it has in mind is not. For this reason, we think that it is more practical to impose a budget
constraint as an explicit restraint placed on the government.

17

to wages. As already stated, however, the impact of a decrease in θ, away from unity, is fairly
straightforward, and all of the results obtained thus far hold in a qualitative sense. Here, wewill briefly discuss the role of the unions’ objectives in this setting.
The equilibrium wages: Since a decrease in θ implies that the union places less emphasis
on wages and instead focuses more on employment, it generally decreases the equilibrium
wages, for both of the firms. This means that the wage regulation is less likely to lower thewages as θ decreases from unity. Since the welfare impact of the wage regulation is closely
related to its impact on the equilibrium wages, the following result virtually sums up the
impact of a change in θ.
Proposition 7 There exists some threshold w̄0 (γ, θ) such that the wage regulation raises union
0’s wage, i.e., w0R > w0N , for k > w̄0 (γ, θ). Similarly, there exists some threshold w̄1 (γ, θ) suchthat the wage regulation raises union 1’s wage, i.e., w1R > w1N , for k > w̄1 (γ, θ). Moreover,both w̄0 and w̄1 are increasing in θ.
Proof: See Appendix A.

Union utility: The situation becomes increasingly messy here, but the results are again
fairly straightforward: a decrease in θ enlarges the region where the wage regulation benefits the
public firm’s union. To see why, recall that when the public firm’s union is allowed to bargain
collectively, it tends to demand excessively high wages, which result in less employment. A
decrease in wages is evidently more detrimental when the union attaches more importance towages compared to employment. Because of this, the wage regulation tends to work more forthe public firm’s union as θ becomes smaller.
Firm profit: One can show that proposition 3 holds independently of the union’s mission,
meaning that the wage regulation always lowers the private firm’s profit. The reason why thewage regulation is always detrimental to firm 1 is the same as in the case where θ = 1: thewage regulation gives the public firm a cost advantage, which would otherwise belong to the18private firm. As a result, the fact that the wage regulation lowers the private firm’s profit isfairly robust to changes in θ.
Consumer surplus: A decrease in θ gradually shrinks the region where the wage regulation
works for consumers. In other words, the wage regulation tends to make consumers worse off
as the unions place more emphasis on employment. The result follows almost directly fromproposition 7, as a decrease in the wages normally benefit consumers. When the unions careless about the wages, a vice of wage bargaining becomes less significant because it raises thewages to a lesser extent.
Social welfare: The wage regulation is less likely to lower the equilibrium wages as θ decreases from unity. This almost directly implies that the wage regulation tends to be more
efficient when θ is closer to unity. The result is again intuitive because it simply states that
placing a cap on wages works better when the unions have a stronger incentive to raise them.
The logic discussed in the previous section remains relevant, and a change in θ only yields
quantitative changes.

7

Conclusion

The paper presents a model of unionized mixed duopoly and examines the regulatory framework of public institutions in order to gain some policy implications. In particular, perhaps
not surprisingly, we show that the way civil service salaries are determined has critical welfare
implications. The analysis reveals, however, that welfare effects of the wage regulation are
subtle and certainly not straightforward as the regulation gives rise to two opposing indirect
effects that operate through strategic market interactions. The overall welfare impact is indeed
ambiguous, depending crucially on the degree of product differentiation between the firms. In
general, the wage regulation tends to be welfare-reducing when the degree is neither too large
nor too small. As a practical implication, this means that the wage regulation is more desirable
in industries where products are more homogenous: the finance industry might be one example of this case. In contrast, the wage regulation may not function well in more differentiated19industries such as public transport and home delivery (where spatial differentiation is both
possible and critical).
Besides this, the paper also raises several interesting issues. First, in contract to the popularbelief, the wage regulation imposed on the public firm may actually benefit its workers because
they tend to demand excessively higher wages when they are allowed to bargain collectively.
In this sense, the wage regulation functions virtually as a commitment device to keep themaway from stiff wage competition, which ultimately leads to more employment. Conversely
speaking, we can argue that granting the right to bargain collectively to civil servants would
not necessarily help them, despite its original intention. It is also pointed out that the wage
regulation is unambiguously detrimental to the private firm and its workers because it gives
the public firm a competitive edge in the market.
Second, the paper also raises a new perspective on the role of budget constraints on the
public firm. A tighter budget constraint implies that the public firm must shift attention fromwelfare maximization to profit maximization. In ordinary circumstances, this is not desirable
as the public firm must decrease its output, resulting in higher market prices. When civil
servants are allowed to bargain collectively, however, imposing a tighter budget constraint canactually be welfare-enhancing because it is instrumental in regulating their behavior. The logic
also implies another virtue of partial privatization, much in the spirit of Matsumura (1998).
As a final note, since the model is highly stylized to deliver the main messages in a relatively
clear way, there are admittedly several potential avenues to extend the current analysis. One
limitation of our analysis is that we consider only a limited class of regulations, focusing onwhether to impose a particular wage regulation. Moreover, we take the wage-setting rule asexogenous, because it is supposedly subject to conditions in the outside labor market. Since
the wage-setting rule yields quite a critical welfare impact, however, an attempt may be madeto endogenize it. This extension adds an extra dimension to the model because one needs
to explicitly consider interactions in the labor market to thoroughly characterize the optimal
wage-setting rule.22 This implies that there are now three markets – the upstream, downstream
22

A possible way to model this is to assume that the marginal cost of the public firm is a decreasing functionof k since a smaller k implies a larger wage disparity between the firms, and the public firm consequently faces

20and labor markets – to be considered. We sidestep this process because our main focus is on
the tradeoff between the upstream-market and downstream-market effects, which would not
be affected in a qualitative sense by the extension. It is nonetheless of some interest to pursue
this issue in future by examining a situation where the public and private firms compete forworkers in the labor market in order to endogenize the wage-setting rule.
[2008.10.7, 894]

Appendix A: Proofs
Proof of Lemma 1: The Lagrangian for the public firm’s problem is formulated as
L = x0 + x1 −x20 + 2γx0 x1 + x21+ λ(1 − x0 − γx1 − w0 )x0 .
2

Taking w0 as given, the first-order condition is given by∂L= 0 ⇔ 1 − x0 − γx1 + λ(1 − 2x0 − γx1 − w0 ) = 0.
∂x0
If the constraint is slack, and there exists an interior solution, the optimal quantity for firm 0 is
x0 = 1 − γx1 .

(A.1)

If the constraint is binding, on the other hand, the public firm sets the quantity so that the resulting
profit is zero. The constraint and the optimal quantity in this case are
1 − x0 − γx1 − w0 = 0, x0 =(1 + λ)(1 − γx1 ) − λw0.
1 + 2λ(A.2)

The problem for the private firm is, on the other hand, much simpler. The optimal quantity forthe private firm is then given byx1 =1 − γx0 − w1
.
2

(A.3)

Given these results, we now obtain the output level for each firm. Solving the first-order conditions(A.2) and (A.3), we obtain
x0

=

x1

=

λ

=

2 − γ − 2w0 + γw1,
2 − γ2
1 − γ − w1 + γw0,
2 − γ2(2 − γ 2 )w0.
2γ − 2w0 + γw1

a difficulty attracting high-quality workers.

21

(A.4)
(A.5)
(A.6)

Notice that the constraint is always binding if w0 > 0, which is always the case as long as k > 0. Q.E.D.
Proof of Proposition 2: Using the results obtained thus far (see Lemma 3), we can compute theutility of each union under the regulation:
R R
uR
0 = w0 x0 =k(1 − γ)(4 − γ − γ 2 − 2(1 + γ − γ 2 )k) R(1 − γ)2, u1 = w1R xR
.
1 =
2
2
4(1 − kγ) (2 − γ )4(1 − kγ)(2 − γ 2 )(A.7)

Similarly, we can also compute the utility under no regulation (see Lemma 2):
N N
uN
0 = w0 x0 =2(4 − γ − γ 2 )2
(4 − 2γ − γ 2 )2
N
N N
,
u
=
w
x
=
.
1
1
1
(8 − γ 2 )2 (2 − γ 2 )(8 − γ 2 )2 (2 − γ 2 )

(A.8)

N
Comparing uRi and ui for i = 0, 1, we can show that the wage regulation makes union 0 better off ifp(4 − γ − γ 2 )(64 − 32γ 2 + γ 4 − γ 5 − γ(1 − γ)(8 − γ 2 ) 16 + γ 2 )k>≡ ū0 (γ).
4(64 − 80γ 2 + 32γ 3 + 5γ 4 − 8γ 5 + 2γ 6 + γ 7 )Q.E.D.
Proof of Proposition 3: Using the results obtained thus far (see eq(7), Lemmas 1, 2, and 3), we
can compute firm 1’s profit under each regime:(1 − γ)2,4(2 − γ 2 )2(A.9)

(4 − 2γ − γ 2 )2.
(8 − γ 2 )2 (2 − γ 2 )2

(A.10)

π1R =
π1N =

With some algebra, we can show that π1N > π1R for any k and γ.
Proof of Proposition 4:

Q.E.D.

Let CS denote the consumer surplus. The following relationship thenholds:CS = U − p0 x0 − p1 x1 = U − π1 − u0 − u1 .

(A.11)

Given this, we can compute the consumer surplus under each regime as follows:
17 − 2γ − 16γ 2 + 2γ 3 + 3γ 48(1 − γk)2 (2 − γ 2 )2
(1 + γ)(2(8 + γ − 11γ 2 + 3γ 3 + γ 4 )k − (4 + 4γ − 3γ 2 − 7γ 3 + 4γ 4 )k 2 )−, (A.12)8(1 − γk)2 (2 − γ 2 )280 + 16γ − 80γ 2 − 12γ 3 + 17γ 4 + 4γ 5.
(A.13)=
2(8 − γ 2 )2 (2 − γ 2 )2CS R =CS N

Comparing these two and solving for k, we can show that the wage regulation enhances consumer
surplus ifk<

512 + 256γ − 832γ 2 − 336γ 3 + 472γ 4 + 133γ 5 − 90γ 6 − 24γ 7 + 4γ 8 + γ 9
256 + 512γ − 320γ 2 − 832γ 3 + 116γ 4 + 472γ 5 − 19γ 6 − 90γ 7 − 3γ 8 + 4γ 9
(1 − γ)(8 − γ 2 )(2 − γ 2 )+2256 + 512γ − 320γ − 832γ 3 + 116γ 4 + 472γ 5 − 19γ 6 − 90γ 7 − 3γ 8 + 4γ 9p× 256 + 192γ − 304γ 2 − 208γ 3 + 131γ 4 + 50γ 5 − 16γ 6 − 2γ 7 + γ 8 .

22(A.14)

Q.E.D.
Proof of Proposition 5:

Noting that W = U when θ = 1, we can compute the social welfare

under each regime:
WR

WN

=

=

23 − 4k 2 − 2(7 + 23k − 4k 2 )γ − (12 − 28k − 23k 2 )γ 28(1 − γk)2 (2 − γ 2 )22(3 + 12k − 11k 2 )γ 3 + (1 − 12k − 9k 2 )γ 4 − 2k(1 − 4k)γ 5+,8(1 − γk)2 (2 − γ 2 )2304 − 144γ − 256γ 2 + 92γ 3 + 67γ 4 − 12γ 5 − 6γ 6.
2(8 − γ 2 )2 (2 − γ 2 )2

(A.15)(A.16)

Comparing these two and solving for k, we can show that the wage regulation is efficient if√
γK − (1 − γ)(8 − γ 2 )(2 − γ 2 ) Kk<,256 − 512γ − 320γ 2 + 960γ 3 − 76γ 4 − 504γ 5 − 101γ 6 + 102γ 7 − 15γ 8 − 8γ 9whereK ≡ (4 − γ − γ 2 )(64 − 64γ − 28γ 2 + 37γ 3 + 5γ 4 − 5γ 5 − γ 6 ).
Q.E.D.
Proof of Proposition 7:

Given the equilibrium wages under each regime, it is conceptuallystraightforward, although highly tedious, to derive the threshold values:
2 − γ + (4 − γ − γ 2 )θ + (2 − γ 2 )θ2,
2 − γ 2 + (4 − γ 2 − γ 3 )θ + (2 − γ 2 )θ2(A.17)(2 − γ)θ + (2 − γ 2 )θ2.
2(1 − γ) + (4 − 2γ − γ 2 )θ + (2 − γ 2 )θ2(A.18)w̄0 (γ, θ) ≡w̄1 (γ, θ) ≡To show that both are increasing in θ, it suffices to show that∂ w̄0 (γ, θ)(1 − γ)γ(2(1 − γ) + 2(2 − γ 2 )θ + (1 + γ)(2 − γ 2 )θ2 )
=
> 0.
∂θ
(2 − γ 2 + (4 − γ 2 − γ 3 )θ + (2 − γ 2 )θ2 )2(A.19)∂ w̄1 (γ, θ)(1 − γ)(2(2 − γ) + 4(2 − γ 2 )θ + (2 + γ)(2 − γ 2 )θ2 )
=
> 0.
∂θ
(1 + θ)2 (2 − γ + 2θ − γ 2 θ2 )2(A.20)
Q.E.D.

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25

k
0.2
0.9

0.4

0.6

0.8

1

Γ


w0 HΓL

0.8
0.7
0.6

w1 HΓL
0.5
Figure 1: Wages w̄i (γ)
Note: the wage regulation raises wi if k > w̄i (γ).

k
0.2

0.4

0.6

0.8

1

Γ

0.95
0.9

CSHΓL

0.85

WHΓL

0.8


u0 HΓL

0.75
Figure 2: The public firm’s union, consumer surplus, and social welfare.
Note: the wage regulation enhances (u0 , CS, W ) if (k > ū0 (γ), k < CS(γ), k < W (γ)).

26