Discussion Papers In Economics
And Business

A Dynamic General-equilibrium Model of Mixed
Oligopoly Market with Public Firms’ Cost Reducing
R&D Investment
Suzuka Okuyama
Discussion Paper 17-16

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Dynamic General-equilibrium Model of Mixed
Oligopoly Market with Public Firms’ Cost Reducing
R&D Investment
Suzuka Okuyama
Discussion Paper 17-16
June 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Dynamic General-equilibrium Model of Mixed
Oligopoly Market with Public Firms’ Cost Reducing
R&D Investment ∗Suzuka Okuyama†
Graduate school of Economics, Osaka university

Abstract
This article presents a dynamic general equilibrium model of mixed oligopoly. In this model,
public firms engage in manufacturing diﬀerentiated goods and cost-reducing R&D investments.
This paper shows that public firms suddenly disappear from the mixed oligopoly market when
the parameter of competitiveness exceeds the threshold value. This paper also shows thatsubsidies are larger when public firms regard a representative household as more important.
Keywords: Mixed oligopoly, Public firm, R&D investment, Dynamic general equilibrium
model
JEL classification: L13, L32, O30∗ This paper was presented at a seminar of economics at the University of Toyama March 2017.

I thank seminar
participants for their comments. Also, I acknowledge many comments from Koichi Futagami, Toshihiro Matsumura,and Asuka Oura.
† Corresponding author. Suzuka Okuyama, Graduate School of Economics, Osaka University, 1-7, Machikaneyama,
Toyonaka, Osaka, 560-0043, JAPAN. E-mail: ngp008os@student.econ.osaka-u.ac.jp

1

Introduction

Privatization programs have been executed mainly in developed countries for the improvement ofpublic firms’ eﬃciency and reconstruction of public finances since the 1980s. However, mixed
oligopolies are still common, especially in developing countries. This gives the impression that thematuration of an industry makes public firms appear to be troublemakers in the market. The aim of
this paper is to discuss the eﬀects technology improvements of public firms have on the economy.
This paper presents a dynamic general equilibrium model of mixed oligopoly. Since De Fraja
and Delbono (1989), many studies investigated mixed oligopolies, where public firms maximizesocial welfare and private firms maximize their profits. Among them, Nishimori and Ogawa (2002)
and Cato (2008) considered the mixed oligopoly market where only public firms invest in researchand development (R&D) activities. They have shown that investment in public monopolies ishigher than that in mixed oligopolies. Mixed oligopolies are common in the industries of education,
finance, infrastructure, and resource development. Managements of public firms in these industries
are required to have a long-term perspective. This implies that public firms should take the future
benefits of economic agents into account. Hence, public firms in the present study consider the
future benefits of economic agents.
This paper takes an intertemporal optimization approach when considering strategies of public
firms. Public firms determine their production and investments for cost reduction consideringsocial benefits. We apply Matsumura’s (1998) approach, the first research that examined partial
privatization, to our objective functions of public firms to consider mixture ownership.
Our main findings are as follows. First, the steady state is classified into two cases according
to the degree of diﬀerentiation between public firms and private firms, and the diﬃculty of cost
reduction. If public firms manufacture similar products to those of private firms and their R&D
activities are eﬃcient, there are two kinds of steady state: one exhibits mixed oligopoly and theother exhibits pure oligopoly. However, the steady states of the mixed oligopoly may not exist if
public firms manufacture diﬀerent products from those of private firms, and their R&D activities
are ineﬃcient. This means that, in the situation, the present value of the weighted sum of utility
and aggregate profits of a mixed oligopoly is smaller than that of a pure oligopoly.,since this result
is derived from the optimization problem of public firms.
Second, subsidies to public firms increase when they regard the representative household asimportant, at least when mixed oligopoly and pure oligopoly always exist. This paper focuses on
steady state values of technology levels, a production cost over the economy, and subsidies in thissituation. The reason for the result is that the public firm undervalues costs generated by its own
activities because it is concerned about the household. This leads to lower prices of public firms
relative to that of private firms and larger R&D investments. As a result, increases in subsidies
occur. This brings about financial deficits that might be prevented by partial privatization or setting
an upper limit on the subsidy.
The first result follows De Fraja (1991) and other related literature: privatization in the suﬃciently competitive market improves social welfare. However, the present study is diﬀerent fromthat literature in that no comparative statics were utilized to obtain this result. The literature showsthat pure oligopoly is better that mixed oligopoly when the market is competitive. In the present
study, the steady state is pure oligopoly as a result of public firms solving optimization problems,
when the degree of diﬀerentiation is small. Since the small degree of diﬀerentiation implies thatthe market is competitive, this result corresponds to the previous results.
1

The second result follows Nishimori and Ogawa (2002) and Cato (2008). These studies showthat investment by a public firm in a public monopoly is larger than that in a mixed oligopoly.
Similarly, the present study shows that partial privatization has a negative eﬀect on the amountof investment. In addition, this study shows that technology improvement in public firms has anegative eﬀect on their mark-up.
The remainder of the paper is organized as follows. Section 2 sets up the model. We construct
the model by solving the optimization problem of the representative household and firms, andobtaining equilibrium. The dynamics are presented in section 2 and comparative statics is presented
in section 3. Section 4 concludes. Appendices discuss details of derivations of formulas .

2

Model

2.1

HouseholdSuppose that there is an infinitely lived representative household whose instantaneous utility atperiod t isln C(t),(1)

where C(t) is an aggregate consumption index at period t. This index is composed of m + n
diﬀerentiated goods, which can be divided into two groups according to elasticities of substitution,
which will be discussed later.
Let us denote one group as X and the other group as Y . We further denote X(Y ) group’s
consumption index as X(t) (Y (t)). Suppose the elasticity of substitution between the two group as
11−γ . Then, the aggregate consumption index is defined as
1

C(t) = [X(t)γ + Y (t)γ ] γ ,γ ∈ [0, 1].

(2)

The household can consume goods of the group X on interval [0, m]. Supposing elasticities of1, the consumption index of group X is defined
substitution between any two goods as constant 1−αasX(t) =[∫ mαxi (t) di] α1,α ∈ [0, 1],(3)0

where xi (t) is the consumption of firm i’s product in group X. Similarly, the household can consume
goods of group Y on interval [0, n] and that of group Y is defined as
Y (t) =[∫ nβy j (t) dj

] β1

,

β ∈ [0, 1],(4)0

where y j (t) is the consumption of firm j’s product in the group Y .
The budget constraint of the household is as follows. By denoting a price index of X(t) as PX (t)which is defined later, total expenditure to group X isPX (t)X(t) =∫ mp xi (t)xi (t)di,

0

2

(5)

where p xi (t) is a price of xi (t). In a similar way, by denoting a price index of group Y as PY (t), total
expenditure to the group Y isPY (t)Y (t) =∫ n

p y j (t)y j (t)dj,(6)

0

where p y j (t) is a price of y j (t). Summing up, total expenditure of the household becomes
PC (t)C(t) = PX (t)X(t) + PY (t)Y (t),(7)

where PC (t) is an aggregate price index defined later.
Since the household maximizes utility under these budget constraints, the maximization problem
consists of two stages. In the first stage, the household maximizes (2) subject to (7) to obtainallocation between goods of X group and that of Y group as follows:
1

X(t) =PX (t) γ−1γγ

PX (t) γ−1 + PY (t) γ−1

PC (t)C(t),(8)

PC C(t).

(9)

and
1

Y (t) =PY (t) γ−1γγ

PX (t) γ−1 + PY (t) γ−1

Substituting these two demand functions into (2) and solving for PC (t) , we can obtain the aggregate
price index as
γ

γ

γ−1

PC (t) = (PX (t) γ−1 + PY (t) γ−1 ) γ .

(10)With this price index, (8) and (9) can be rewritten as
1

X(t) =PX (t) γ−1γ

PC (t) γ−1

PC (t)C(t),(11)

PC (t)C(t).

(12)

and
1

Y (t) =PY (t) γ−1γ

PC (t) γ−1

In the second step, we obtain quantities of consumption per firm. Maximizing (3) subject to (5)leads to1xi (t) = ∫ m0p xi (t) α−1αp xi ′ (t) α−1 di′
3

PX (t)X(t).

(13)

Substituting this into (5) defines the price index of group X asPX (t) =[∫ m

p xi (t)

α
α−1

] α−1
α
di

.

(14)

0

Combining (11), (13), and (14), we can obtain the demand function for the product of firm i asfollows:1

p xi (t) α−1xi (t) =γγ PC (t)C(t).
αPX (t) α−1 − γ−1 PC (t) γ−1
In the same manner, we can obtain the price index of group Y as follows:

PY (t) =[∫ n

p y j (t)

β
β−1

] β−1
β
dj

.

(15)

(16)

0

and the demand function for the product of firm j as
1

p y j (t) β−1

PC (t)C(t).
(17)
β
γ
γ
PY (t) β−1 − γ−1 PC (t) γ−1
We next turn to the intertemporal maximization problem of the representative household. We
assume that the intertemporal utility function of the household is∫ ∞U(t) =e−ρ(s−t) ln C(s) ds,(18)
y j (t) =twhere ρ ∈ (0, 1) is a subjective discount rate. The household’s intertemporal budget constraint isÛ = r(t)A(t) + (1 − τ(t))[wP (t)LP (t) + w R (t)L R (t)] − PC (t)C(t),
A(t)

(19)

where r(t) ∈ (0, 1) is an interest rate and τ(t) ∈ [0, 1) is an income tax rate. The household supplies
inelastic labour L̄ and firms employ LP (t) units of labor for production at wage rate wP (t) andemploy L R (t) units of labor for R&D investment at the wage rate w R (t). The household pays thegovernment an income tax τ(t)[wP (t)LP (t) + w R (t)L R (t)] and divides after-tax income and asset
income r A(t) into saving and consumption. We first construct the current value Hamiltonian as
follows:
H ≡ ln C(t) + ν(t) [r(t)A(t) + (1 − τ(t)) {wP (t)LP (t) + w R (t)L R (t)} − PC (t)C(t)] .

(20)

From the necessary conditions of maximization, the following Euler equation obtains:ÛPÛC (t)C(t)= r(t) − ρ −.
C(t)PC (t)(21)

Normalizing PC (t)C(t) to one, we can rewrite above equation asr(t) = ρ.
This means that the interest rate is constant over time.
4(22)

2.2 Firms
This section discusses production behaviors of firms. Suppose that monopolistic competition
prevails in the diﬀerentiated goods market and there are m firms belonging to group X and n firms
belonging to group Y . Let firms in group X be public firms engaging in production and R&D
investment. On the other hand, to keep matters simple, let firms in group Y be private firms who do
not undertake R&D investment but produce goods.
Suppose that production and R&D investment require labor input. We specify their production
and R&D technologies as follows. The public firm i produces a diﬀerentiated good with thefollowing technology
xi (t) = zi (t)ε lxi (t),ε ∈ (0, 1),(23)

where lxi (t) is labor input for the production of the public firm i and zi (t) indicates the technology
level of the firm. The public firm i uses labor input lRi (t) to improve its technology level zi (t)according tozÛi (t) = lRi (t)θ − δzi (t),θ ∈ (0, 1),

δ ∈ (0, 1),(24)

where δ is an obsolescence rate of the technology level. This equation shows that technology
becomes obsolete at the rate δ.
We next consider the private firm j. The private firm j produces a diﬀerentiated good with thefollowing technologyy j (t) = ly j (t),(25)

where ly j (t) is labor input for production of the private firm j.
From (23) and (24), the profit of the public firm i becomes
π xi (t) = (p xi (t) − wP (t)zi (t)−ϵ )xi (t) − w R (t)lRi (t) + si (t),(26)

where si (t) is a subsidy to the public firm i. The government subsidies si (t) satisfy the public firm’s
budget loss because of taxation on the household. Similarly, (25) gives the profit of the private firmj as follows:
π y j (t) = (p y j (t) − wP (t))y j (t).

(27)

We now proceed to these firms’ optimization problems. The objective function of the private
firm j is∫ ∞v(t) =e−r(s−t) π y j (s) ds.
(28)t

By using (17) and (27), we can obtain the price of the private firm j as
p y j (t) =wP (t).
βWe have assumed that private firms take price indices PX (t), PY (t) and PC (t) as given.

5

(29)

Unlike private firms, the public firm i considers the household’s utility and other firms’ profits.
Hence, the objective function of the public firm i is defined as follows:
]]
[
[∫ m
∫ ∞
∫ n
−ρ(s−t)eψ ln C(s) + (1 − ψ)π xi (s)di +π y j (s)dj ds,
(30)
t

0

0

where ψ ∈ [0, 1]. The public firm i put weight ψ on the household’s utility and weight 1 − ψ on
the sum of all firms’ profits. We assume that the discount rate of the public firm i is ρ, since theowner of the public firm i is the government, which takes care of the welfare of the household. We
construct the current value Hamiltonian of the public firm i as follows:H ≡ ψ ln C(t) + (1 − ψ)[∫ i−10′π xi ′ (t)di + π xi (t) +

∫ m
i+1

′

∫ n

]

π xi ′ (t)di +π y j (t)dj
0[]+µi (t) lRθ i (t) − δzi (t) ,(31)

where µi (t) is the co-state variable of the technology level of the public firm i.
Before proceeding to the first order condition of the maximization problem, we consider derivatives of demand functions. When determining its behavior, the public firm i considers the direct
eﬀect of one unit of increase in p xi (t) on its profit and the indirect eﬀect of that on other agents’
benefits through changes in the price indices. This indirect eﬀect is generally ignored in the case of
diﬀerential monopoly. However, it is hard to ignore this eﬀect in mixed diﬀerential monopoly, since
the scale of public firms are basically large in comparison with private firms. This is the reason we
diﬀerentiate PC (t) with respect to p xi (t) when deriving the first order conditions.
Using (26) and (27), the first order conditions of maximization are derived as∂C(t)[∫ i−1) ∂ xi ′ (t) ′(∂p xi (t)∂H=0⇔ψ
+ (1 − ψ)dip xi ′ (t) − wP (t)zi ′ (t)−ε∂p xi (t)C(t)∂p xi (t)0∫ m
) ∂ xi ′ (t) ′((−ε ) ∂ xi (t)+xi (t) + p xi (t) − wP (t)zi (t)+p xi ′ (t) − wP (t)zi ′ (t)−εdi∂p xi (t)∂p xi (t)i+1∫ n(
) ∂ y (t) ]j+
dj = 0,p y j (t) − wP (t)∂pxi (t)0∂H= 0 ⇔ (1 − ψ)w R (t) = θ µi (t)lRi (t)θ−1,∂lRi (t)

(32)

(33)

∂H
= ρµi (t) − µÛi (t) ⇔ µÛi (t) = (ρ − δ)µi (t) − ε(1 − ψ)wP (t)zi (t)−ε−1 xi (t).
(34)∂zi (t)
The derivations of these derivatives in the equation (32) are explained in Appendix A.
To obtain the price of public firm i requires several steps. Taking derivatives of (13) and (17)and substituting them into (32), we obtain
{
(
}
)
1
wP (t)zi (t)−εγ
ψ xi (t) = (1 − ψ) 1 −1−
Γ(t) xi (t),
(35)
+
1−α
p xi (t)1−γ∫m∫nwhere Γ(t) ≡ 0 (p xi (t) − wP (t)zi (t)−ε )xi (t)di + 0 (p y j (t) − wP (t))y j (t)dj.
6

The left-hand side of the above equation represents the weighted marginal benefit of the household. This is the increase in the household benefit caused by one additional unit of p xi . Similarly,
the right-hand side of the above equation represents the weighted marginal profit over all firms. Thisis the change in the total profit over all firms caused by one additional unit of p xi . The price p xi (t) is
determined by the equality between the weighted marginal benefit and weighted aggregate marginal
profits. Furthermore, weighted aggregate marginal profits has two components. The component−ε1
1 − 1−α(1 − w P (t)z(t)p x (t) ) means the direct eﬀect on the profit of the public firm i by one additional unitγ
of p xi (t). The other component 1−γ
Γ(t) means the indirect eﬀect on the profit through the change
in the consumption by that amount.
Here, we check that Γ(t) represents aggregate profits of all firm except for public firms’ R&D
expenditures. Rearranging Γ(t) provides

Γ(t) =[∫ m

p xi (t)xi (t)di +

0

∫ n

]
p y j (t)y j (t)dj − wP (t)

[∫ m

−ε

zi (t) xi (t)di +

0

0

∫ n

]
y j (t)dj .

(36)

0

Substituting (5) and (6) into the above equation yields[]

Γ(t) = PX (t)X(t) + PY (t)Y (t) − wP (t)[∫ mlxi (t)di +

∫ n

0

]
ly j (t)dj .

(37)

0

By using (7)、(23) and (25), we can rewrite this equation as follows:
Γ(t) = PC (t)C(t) − wP (t)LP (t).

(38)

Defining wP (t)LP (t) as aggregate production cost of all firms, that is, CP (t) and remembering that
total expenditure is one, we can obtain
Γ(t) = 1 − CP (t).

(39)

Let us turn back to the first order condition with respect to p xi (t). Substituting (39) into (35)results in}(){
wP (t)zi (t)−εγ11−+(1 − CP (t)) xi (t).
(40)
ψ xi (t) = (1 − ψ) 1 −1−αp xi (t)1−γγFor simplicity of notation, setting (1 − α) 1−γ
as Q and setting (1 − α) 1−2ψ
1−ψ as R, the price of the
public firm i can be expressed aswP (t)zi (t)−ε.
p xi (t) =QCP (t) + 1 − Q − R
1
The mark-up QCP (t)+1−Q−R
is an increasing function of CP (t).

7

(41)

2.3 Equilibrium
In this section, we examine equilibrium conditions of the goods market, the labor market and acapital market. The aggregate production cost over firms will play an important role in theseequilibrium conditions.
2.3.1

Goods Market Equilibrium

Let us derive the aggregate production cost over m + n firms to obtain the goods market equilibrium
condition. Because of symmetry across public firms and across private firms, combining (10), (14),(15), (16), (17), (29), and (41) yields
γ

PX (t) γ−1
1 QCP (t) + 1 − Q − Rx(t) =γγ ,mwP (t)z(t)−εPX (t) γ−1 + PY (t) γ−1

(42)

and
γ

1 β
PY (t) γ−1y(t) =γγ .
n wP (t) PX (t) γ−1 + PY (t) γ−1(43)

It is obvious that these output levels are functions of wP (t), z(t), CP (t), PX (t), and PY (t).
From (23) and (25), production of these diﬀerentiated goods requires the following labor inputs:γ

1 QCP (t) + 1 − Q − R
PX (t) γ−1lx (t) =γ ,γmwP (t)PX (t) γ−1 + PY (t) γ−1

(44)

and
γ

1 β
PY (t) γ−1ly (t) =γ .
γn wP (t) PX (t) γ−1 + PY (t) γ−1(45)Because LP (t) = mlx (t)+nly (t) and CP (t) = wP (t)LP (t), (44) and (45) give the aggregate production
cost asγ

CP (t) =γ(QCP (t) + 1 − Q − R)PX (t) γ−1 + βPY (t) γ−1γγ

PX (t) γ−1 + PY (t) γ−1

.

(46)

Rearranging this equation, we obtain
(
) γ−1
γ
PX (t)CP (t) − β=.
PY (t)(1 − Q)(1 − CP (t)) − R(47)

By using this ratio, we can rewrite (42) and (43) as follows:
x(t) =(QCP (t) + 1 − Q − R) (CP (t) − β)11,−εm wP (t)z(t)QCP (t) + 1 − Q − R − β

and
8

(48)

y(t) =

2.3.2((1 − Q)(1 − CP (t)) − R
1 1.
n wP (t) (QCP (t) + 1 − Q − R − β) β−1(49)

Labor market equilibrium

Suppose that labor can freely move between production and R&D activities, the labor market
equilibrium condition becomeswP (t) = w R (t) ≡ w(t),(50)

LP (t) + L R (t) = L̄.

(51)

and

Utilizing CP (t) = w(t)LP (t) and L R (t) = mlR (t), (51) can be rewritten as
CP (t)+ mlR (t) = L̄.
w(t)

(52)

From (33) and (52), we can obtain the equilibrium wage rate as
[

(

1 ]
) 1−θ
θ µ(t)CP (t) = w(t) L̄ − m.
(1 − ψ)w(t)(53)

This implicitly defines w(t) as a function of the aggregate production function cost and the co-statevariable. We denote the equilibrium wage rate as w(CP (t), µ(t)). In addition, (52) and (53) give theequilibrium labor input for R&D activities as1] 1−θθ µ(t)lR (t) =.
(1 − ψ)w(CP (t), µ(t))

[

2.3.3

(54)

Capital market equilibrium

Private firms pay their profit to the household as dividends. From (27), (28), (29), and (49), we canobtain the non-arbitrage condition asvÛ (t) = rv(t) −1 − β (1 − Q)(1 − CP (t)) − R,
n QCP (t) + 1 − Q − R − βwhere v(t) is a stock price of the private firm.

9

(55)

2.3.4

Constraints for the aggregate production cost over all firms

Let us discuss conditions guaranteeing the existence of firms. To prove concavity of objectivefunctions of public firms, we must check that the second order condition of (31) is negative. This
imposes the following constraint on the aggregate production cost over firms:
CP (t) > β,(56)

when m > 1−γ
γ . Its proof is given in Appendix B.
The equilibrium profits of private firms give the free entry condition of private firms as
1−Q−R
.
(57)
1−Q
Accordingly, (56) and (57) conclude that the mixed oligopoly exists if the aggregate production costover firms satisfies the following inequality.
CP (t) ≤β < CP (t) ≤

1−Q−R
.
1−Q

(58)

Moreover, it is easy to check that the market is a pure oligopoly when CP (t) is β. 1 Now, we havecompleted the discussion of equilibrium conditions.

3

Dynamics

3.0.1

The structure of dynamics

We now examine the dynamics of the system. Substituting (52) into (24) yields
[

(
)] θ
1
CP (t)
zÛ(t) =− δz(t),L̄ −mw(CP (t), µ(t))(59)

Substituting (48) into (34) yields the following:
ρ=δ+

Û
µ(t)
ε(1 − ψ) 1 (QCP (t) + 1 − Q − R)(CP (t) − β) 1
+
.
µ(t)
m
z(t)
QCP (t) + 1 − Q − R − β
µ(t)

(60)

The left-hand side of this equation is the cost incurred by giving up a unit of consumption forinvestment and the right-hand side represents the return on R&D investment. The right-hand side
is composed for three terms: compensation for technological obsolescence, the rate of the marginalvalue change of the technology, and cost-reducing eﬀects of R&D investment.
We next explore the relationship between z(t) and CP (t). Equilibrium price indices can bederived by (14), (16), (29) and (41). Substituting these indices into (47) and solving this for z(t)gives
z(t) = ϕ1ε[CP (t) − β(1 − Q)(1 − CP (t)) − R

1See Appendix.B

10

] 1−γ
εγ

(QCP (t) + 1 − Q − R)− ε ,
1

(61)

(
where ϕ ≡

α−1

m α

β−1

)
β . This equation indicates that CP (t) is a function of z(t). Let us denote then βfunction as CP (z(t)).

Equation (61) reveals that the diﬀerential equations (59) and (60) have two
endogenous variables z(t) and µ(t).
Consequently, we can summarize the structure of dynamics as follows. The dynamics iscomprised of three diﬀerential equations (55), (59) and (60) and two equations (53) and (61). Theseequations involve three endogenous variables z(t), µ(t), and v(t).
3.0.2

Steady states

Before proceeding to the phase diagram analysis, we solve for the steady states value of the dynamics.
Û = 0 must be satisfied. Thus, the following holds at the steady state.
At the steady state, zÛ(t) = µ(t)( )θ []θεθδ(QCP∗ + 1 − Q − R)(CP∗ − β)
1 L̄
z =
.
δ m
(ρ − δ)(QCP∗ + 1 − Q − R − β) + εθδ(QCP∗ + 1 − Q − R)(CP∗ − β)∗(62)Using (61), we can obtain the following equation for the steady state value of CP∗ ,1−γ(CP∗ − β) εγ −θ
ϕ
{
} 1−γ
1
(1 − Q)(1 − CP∗ ) − R εγ (QCP∗ + 1 − Q − R) ε +θ] −θ( )δ [
1 L̄
ρ−δ
∗
∗
∗
∗
.
=
(QCP + 1 − Q − R − β)CP + (QCP + 1 − Q − R)(CP − β)
δ m
εθδ
1
ε

(63)

The existence of the steady state can be proved by graphical investigation of (63) in β ≤ 1−Q−R
1−Q .
Diﬀerentiating the right-hand side of (63) with respect to CP∗ shows that it is a downward sloping
curve and takes non-negative values in the domain of CP . In the same way, we can find that the
1
1
and U-shaped curve if γ ≥ 1+εθ.
left-hand side of (63) is an increasing function of CP if γ < 1+εθ
1∗These results provide figure 1(a) which proves the existence of CP in the case of γ < 1+εθ and figure
1.
1(b) which shows the case of γ ≥ 1+εθ
1Figure 1(a) depicts the case of γ < 1+εθ
. In this case, there is one intersection in the interval,
1−Q−R∗
β < CP < 1−Q . However, this graph implies that there are two steady states. It is clear that theleft-hand side of (61) and the right-hand side of (62) are zero when CP is β. It follows that β is also
the steady state value of CP . Hence the number of steady points is the number of intersections on
the figures plus one.
1
. Figure 1(b) shows that the number ofSimilar arguments can be applied to the case of γ ≥ 1+εθ
steady points for this case can be zero, one or two, depending on parameters. When L̄ is suﬃciently
large, there exits two steady states.
Using implicit function theorem, we can find that the zÛi (t) = 0 locus goes through the origin
1, theand has a positive slope. Similarly, we can find characters of the µÛ i (t) = 0 locus. If γ < 1+εµÛ i (t) = 0 locus has a negative slope and asymptotically approaches the horizontal and vertical axes,1, the µÛ i (t) = 0 locus is an inverted U-shape and goes through the origin andrespectively. If γ ≥ 1+ε
asymptotically approaches the horizontal axis. The characters are explained in Appendix E.
11

1
Figure 1: (a) The number of steady state in γ < 1+εθ1(b) The number of steady state in γ ≥ 1+εθ121Figure 2(a) shows the phrase diagram of γ < 1+ε. There are two steady points which are E0 ,the origin and E1 , an intersection of two loci. The value of z ∗ is zero at E0 which means that public
firms do not exist in this steady point. Then the market is a pure oligopoly market, where only
private firms operate. Even if this steady point has no transition path, once a suﬃcient upward
jump of µ occurred, public firms would begin to operate and the economy would reach steady point
E1 . At the steady point E1 , z ∗ is no longer zero, that is, CP∗ is greater than β. This implies that themarket is a mixed oligopoly in E1 .
1
1
The phase diagram of 1+ε≤ γ < 1+εθ
is represented as figure 2(b). This figure is also the phase
diagram of one intersection case on figure 1(b). If there are two intersection points on figure 1(b),the two loci have three intersections on the phase diagram. Figure 2(c) shows this situation. A
similar argument as before can be applied to these cases. The steady points E0 of these two figures
exhibit the pure oligopoly and the steady points E1 of these figures exhibit the mixed oligopoly,
respectively. 2
However, the steady state E1 disappears if there is no intersection on the figure 1(b). When twoloci have no intersection on figure 1(b), the phase diagram is shown as figure 2(d). It is obviousthat there is only one steady point E0 corresponding to the pure oligopoly. This implies that mixed
1
oligopoly does not always exist when γ ≥ 1+εθ.
Hence, we conclude that the steady state diﬀers from γ as follows.

Proposition 1 In this economy, the market can be either a mixed oligopoly or a pure oligopoly,depending on the initial variables z(0) and µ(0). However, there is a set of parameters which make1.
the market never be the mixed oligopoly market when γ ≤ 1+εθAs mentioned before, γ is the parameter referring to the elasticity of substitution between X(t) andY (t). It is obvious that the larger γ indicates the more competitive market. Moreover, larger ε andθ imply that one unit of labor input has a bigger eﬀect on cost reduction. Therefore, the market
is more competitive and R&D activities are more productive when parameters γ, ε, and θ satisfy1
γ ≥ 1+εθ.

4

WelfareIn the previous section, we have clarified that there always exists the steady point of the mixed
1
oligopoly when γ < 1+εθ
. We now proceed to the steady state eﬀects of increasing ψ under the1
condition γ < 1+εθ .
By (63), we define1−γΦ(CP∗E1, ψ)

≡ ϕ

1
ε

(CP∗E1 − β) εγ −θ(QCP∗E1 + 1 − Q − R)− ε −θ
1

1−γ
εγ

{(1 − Q)(1 − CP∗E1 ) − R}( )δ [
ρ−δ
1 L̄
(QCP∗E1 + 1 − Q − R − β)CP∗E1
−
δ m
εθδ
] −θ
∗E1
∗E1
+ (QCP + 1 − Q − R)(CP − β) ,2We suppose that midpoint of three intersections is unstable point.

13

(64)

1
Figure 2: (a) The phase diagram in γ < 1+ε1
11≤ γ < 1+εθor in γ ≥ 1+εθwith one intersection case on figure 1(b)(b) The phase diagram in 1+ε1(c) The phase diagram in γ ≥ 1+εθ with two intersections case on figure 1(b)1(d) The phase diagram in γ ≥ 1+εθwith no intersection case on figure 1(b)

14

∗E

∂Φ(CP 1 ,ψ)Figure 3: The sign of∂R

where CP∗E1 refers to the sate value of aggregate production cost at steady state E1 . Using the implicit
function theorem to the above equation yields
∂Φ(C

∗E1
,ψ)

P
dCP∗E1
∂ψ
=−
.
∗E
dψ
∂Φ(CP 1 ,ψ)

(65)

∗E

∂CP 1
∗E

∗E

∂Φ(CP 1 ,ψ) ∂R∂Φ(CP 1 ,ψ)can be rewritten as∂ψ∂R∂ψ by the chain rule and∗E∂Φ(CP 1 ,ψ)∂R
is negative by the definition of R. The task is now to find the sign of.
that the sign of ∂ψ
∂R∗E1
Diﬀerentiating (64) with respect to R, we can find that the sign is negative if CP satisfies theWe can see at once that the molecule

following inequality.
βγ
>
[

ρ−δ
{(1 − Q)(1 − CP∗E1 ) − R} CP∗E1[δ(1 − γ)(QCP∗E1 + 1 − Q − R)+ρ−δ+ 1)CP∗E1 − β}(QCP∗E1
{(
δεθ

+

γ{(1 − Q)(1 − CP∗E1 ) − R}]]
ρ − δ ∗E1
1 − Q − R) −
βCP .
δεθ

(66)

To check that CP∗E1 satisfies the inequality, we draw the left-hand side and the right-hand side of(66). The left-hand side has a negative coeﬃcient of CP∗E1 2 and is zero when CP∗E1 are 0 or 1−Q−R
1−Q .

The right-hand side of (66) has a positive coeﬃcient of CP∗E1 3 and three icon points: two of them
are smaller than β and one of them is larger than 1−Q−R
1−Q . Depicting the graphs of these equations,
we can find that these two curvatures have three intersections, A, B, and C as shown in figure 3.
∗E1
Since A is smaller than 0 and C is larger than 1−Q−R
1−Q , (66) is not satisfied when CP is between 0
and B and is not satisfied when CP∗E1 is between B and 1−Q−R
1−Q . Substituting β into (66) shows that15

(66) is obtained only if (1 − Q)(1 − β) − R + (1 − γ − γεθ)β > 0. Hence, B is smaller than β when1
(1 − Q)(1 − β) − R + (1 − γ − γεθ)β > 0. This condition is satisfied at least when γ < 1+εθ, since1
we set R less than (1 − Q)(1 − β). In conclusion, (66) is satisfied only if γ < 1+εθ .
∗E

Now, we turn to the denominator of (65). Figure 1(b) indicates the sign of∂Φ(CP 1 ,ψ)
∗E

∂CP 1

is positive.

Because of (63) and (64), we can find that the diﬀerence of gradients of each curve at the steady∗Epoint E1 on figure 1(b) correspond to the sign of the denominator. Therefore,∂Φ(CP 1 ,ψ)
∗E

∂CP 1

has positive1
sign when γ < 1+εθ.
1
. Moreover,
These results reveal that increases in ψ have positive eﬀects on CP∗E1 when γ < 1+εθ
Appendix C shows that increase in CP (t) has a positive eﬀects on z(t). Hence, we establish thefollowing proposition.
1
Proposition 2 Increase in ψ has positive eﬀect on z ∗1 and CP∗E1 under the condition γ < 1+εθ.

This result mainly comes from the fact that public firms undervalue their costs of production and
R&D investment compared to the household’s utility as ψ is increasing. This underestimation of
cost leads to larger R&D investments and lower prices relative to private firms’ prices. Since thenegative eﬀect on CP∗E1 by technology improvement is smaller than the positive eﬀect on CP∗E1 due
to cheaper prices of the public firms, an increase of ψ has a positive eﬀect on aggregate production
cost.
Last, we discuss the subsidy to public firms. Substituting (41), (48) and (54) into (26) gives
[
]
∗E
∗E
1 (−QCP 1 + Q + R)(CP 1 − β)m(δz∗ )−θ∗E1∗
CP + s∗ .
(67)
πx =
−
∗E
∗
−θ
1
m
L̄ − m(δz )QCP + 1 − Q − R − β
The government sets the subsidy to compensate the public firms’ budget losses as follows:(sum o f subsidies) =(−QCP∗E1 + Q + R)(CP∗E1 − β)m(δz∗ )−θ∗E1
C−.
L̄ − m(δz∗ )−θ P
QCP∗E1 + 1 − Q − R − β(68)which is derived by (67) It is obvious that this equation is an increasing function in CP∗E1 . Considering
proposition 2 and this result gives following proposition.
1
Proposition 3 An increase in ψ leads to larger subsidies if γ > 1+εθ

That is, an increase in ψ leads to an increase in the technology level and the aggregate production
cost, and then, the amount of subsidy increases. The public firm that assigns more importance thehousehold’s utility is more aggressive in investing and sets lower prices compared to private firms.
Consequently, the existence of such public firms leads to financial pressure.

5

Conclusion

This paper proposes a dynamic general equilibrium model of the mixed oligopoly with public firms’
R&D activities. We formulate public firms’ intertemporal optimization problem.
16

We find that the steady state of the mixed oligopoly may not exist if public firms manufacture
similar products to those of private firms and R&D activities are eﬃcient. This result follows De
Fraja (1991) and other related literatures. However, we obtain this result without comaparative
statistics.
We show that investments increase as public firms regard the household as important. This
follows Nishimori and Ogawa (2002) and Cato (2008). However, we also show that this leads tolower prices of public firms relative to that of private firms and larger subsidies to public firms.
In this paper, we ignore R&D activities of private firms. The model with both kinds of firms’
R&D activities would yield very diﬀerent results. It is important to examine how the results change.

6

AppendixA

Derivatives of demand functions
1

Let us obtain derivatives in the first order condition (32). Diﬀerentiating p xi (t) 1−α in the numeratorof (15) and PC (t) in its denominator provides the derivative of the public firm i’s demand functionasγ∂p xi (t) α−1
p xi (t) α−1 ∂PC (t)− γ−11∂ xi (t)
=
+
,
γ
γ
γ
α
α
∂p xi (t)PX (t) α−1 − γ−1 PC (t) γ−1 ∂p xi (t)PX (t) α−1 − γ−1 ∂p xi (t))(
1γ ∂PC (t)−1−1= −p x (t) +PC (t) xi (t).
1−α i1 − γ ∂p xi (t)
1

1

(69)

Diﬀerentiating PC (t) in (15) obtains the derivatives of other public firms’ demand functions asγ

p xi ′ (t) α−1 ∂PC (t)− γ−1∂ xi ′ (t)=,γα∂p xi (t)PX (t) α−1 − γ−1 ∂p xi (t)γ ∂PC (t)=PC (t)−1 xi ′ (t).
1 − γ ∂p xi (t)1(70)

Diﬀerentiating private firm j’s demand function (17), we obtain1γ

p y j (t) β−1 ∂PC (t)− γ−1∂ y j (t)=,γβ∂p xi (t)PY (t) β−1 − γ−1 ∂p xi (t)γ ∂PC (t)=PC (t)−1 y j (t).
1 − γ ∂p xi (t)(71)

The partial derivative of PC (t) with respect to p xi (t) becomes∂PC (t)= xi (t)PC (t),∂p xi (t)17(72)

from (10) and (14).
The derivatives of the household’s utility function with respect to p xi (t) in (32) can be derived
as follows. Totally diﬀerentiating PC (t)C(t) = 1 by p xi (t), we obtain∂PC (t)∂C(t)C(t) + PC (t)= 0.
∂p xi (t)∂p xi (t)

(73)

∂C(t)
C(t)−1 = −xi (t).
∂p xi (t)(74)

Combining this and (72) yields

Substituting these derivatives into (32) yields
[
{
(
)
}]
wP (t)zi (t)−εγ∂H1= −ψ + (1 − ψ) 1 −1−+
Γ(t) xi (t).
∂p xi (t)1−αp xi (t)
1−γ

B

(75)

Second order conditions

In this appendix, we derive second order conditions to check concavity. Using (1), (2), and (3), thehousehold’s first order derivative with respect to xi (t) is
1 ∂C(t) ∂ X(t)∂ ln C(t)== C(t)−γ X(t)γ−α xi (t)α−1,∂ xi (t)C(t) ∂ X(t) ∂ xi (t)(76)

and then the second order derivative becomes][∂ 2 ln C(t)∂ X(t)∂ ln C(t) ∂C(t)
−1
−1
−1
.
γ
C(t) + (α − γ)X(t) + (1 − α)xi (t)=−∂ xi (t)∂ xi (t)∂ xi (t)∂ xi (t)2(77)

Hence, the following condition is suﬃcient for concavity.
α > γ.

(78)

Similar arguments can be applied to the case of y j (t). We can obtain the condition
β > γ,(79)

to ensure that the second order condition of y j (t) is a concave function.
We next turn to public firm i. Its first order condition has been derived in Appendix A. With
the envelope theorem, the second order condition of p xi (t) becomes[]1 − ψ wP (t)zi (t)−ε
∂Γ(t)
∂2 H
=−
−Q
) xi (t).
1−α∂p xi (t)∂p xi (t)2
p xi (t)218(80)

Using Γ(t), (15), (17), (23) and (25), the partial of Γ(t) with respect to p xi (t) in the above equation
can be derived. Substituting this into (80) can be rewritten as
[
(
)
]
∂2 H
1 − ψ wP (t)zi (t)−εQwP (t)zi (t)−ε=−−− QCP (t) xi (t) xi (t).
(81)
1−α
1−α
p xi (t)∂p xi (t)2
p xi (t)2γ
Substituting (41) and (48) and Q ≡ (1 − α) 1−γ
into (81), we can obtain

[(
)
1−ψ
1
∂2 H
γ
= −
−
xi (t)
1 − α p xi (t) 1 − γ∂p xi (t)2]
1 Q2
CP (t)(CP (t) − β)1+
(QCP (t) + 1 − Q − R)xi (t). (82)m 1 − α wP (t)z(t)−ε QCP (t) + 1 − Q − R − β
The task is now to find conditions which make (82) negative. Assuming xi (t) is a positive value,
we must show the signs of terms in the parentheses. It follows that
1 − Q > 0,(83)

because of (78). This inequality and the fact that the maximum value of R is 1 − α brings
1 − Q − R > 0.

(84)

These conditions show that the mark-up of public firms is non-negative as follows:
QCP (t) + 1 − Q − R ≥ 0.

(85)

It is natural to assume the mark-up of the private firm i is larger than that of the public firm. Thus,（29) and (41) give
QCP (t) + 1 − Q − R − β ≥ 0.

(86)

Consequently, the parentheses in (82) would be negative only if
CP (t) > β.

(87)

under the conditions (84), (85), (86), and m > 1−γ
γ 3.

CThe relation of z(t) and CP (t)Diﬀerentiating (61) with respect to CP (t) provides()
γ
Q(CP (t) − β)
(1 − Q)(1 − β) − R≥.
(1 − Q)(1 − CP (t)) − R
1 − γ QCP (t) + 1 − Q − R
3This condition can be derived by (41), (42), (47), and (57).

19

(88)

∂z(t)>0
Figure 4: ∂CP (t)Figure 5: z(t) = z(CP (t))

We will prove that this inequality is established under β ≤ CP (t) < 1−Q−R
1−Q by depicting a graph ofthese equations. Both sides of (88) have positive slopes in the constraint. Introducing β into CP (t),we can obtain that the left hand-side is 1 and the right hand side is 0. Hence, the left hand-sideis larger than the right hand-side when CP (t) = β. Introducing 1−Q−R
1−Q into CP (t), we can obtainthat the left hand-side is infinity but the right hand-side is finite in the neighborhood of 1−Q−R
1−Q .
These results enable us to depict both sides of (88) as figure 4 and then the graph of (61) can be
represented as figure 5. 4 Since the graphs intersect the horizontal line at β and increase to infinity
in the domain, it follows that an increase in z(t) leads to an increase in CP (t).

D CP (t) of a pure oligopoly
We prove that aggregate production cost over all firms of the pure oligopoly where only private
firms exist is β. Since there are no public firms, (2) and (4) can be rewritten as
C(t) = Y (t) =

[∫ n

β

y j (t) dj

] β1

.

(89)

0

Lack of R&D activities provides the intertemporal budget constraint as
Û = r A(t) + wP (t) L̄ − PY (t)Y (t),
A(t)

(90)

by (7) and (19). Therefore, a demand function for private firm j is1

y j (t) = ∫p y j (t) β−1

β∞β−1 dj.
p
(t)
y
j
0

.

4For simplicity, we ignore that two graphs never cross under β ≤ CP (t) < 1−Q−R
1−Q on figure 4.

20

(91)

From (27) and (91), we can obtainwP (t).
(92)
βAssuming a symmetric equilibrium , we obtain the following goods market equilibrium condition.
p y j (t) =wP (t),
βfrom (89), (91), and (92). The labor market equilibrium condition becomesY (t) =CP (t) = wP (t)ny(t) = β,

(93)

(94)

from（25 and (93). Hence, we complete the derivation.

E Preparation for phase diagrams
In this appendix, we discuss loci of (59) and (60). Setting zÛ(t) = 0, (59) can be rewritten as
[

(
)] θ
1
CP (z(t))
L̄ −− δz(t) = 0.
mw(CP (z(t)), µ(t))(95)

Here, we define
[

(
)] θ
1
CP (t)
Θ(z(t), µ(t)) ≡L̄ −− δz(t),mw(CP (t), µ(t))
and apply the implicit function theorem to this equation as follows:(96)∂Θ(z(t),µ(t))

dµ(t)
∂z(t)
= − ∂Θ(z(t),µ(t)) .
dz(t)

(97)

∂ µ(t)

Diﬀerentiating (96) with respect to z(t) yields[ ()] θ−1∂Θ(z(t), µ(t))
1
CP (z(t))
L̄ −
= −θ
∂z(t)
m
w(CP (t), µ(t))()
dCP (z(t))
w(CP (z(t)), µ(t)) − CP (z(t)) ∂w(C∂CPP(z(t)),µ(t))(z(t))
dz(t)
− δ,
(98)
w(CP (z(t)), µ(t))2()
CP (t)
1
where m L̄ − w(t) equals lR (t). The derivative of (53) with respect to w(t) and (54) become∂CP (z(t))
CP (z(t))−,∂w(CP (z(t)), µ(t))
w(CP (z(t), µ(t))
)
(
(
)
θ
mlR (t) − L̄ − mlR (t) ,= L̄ +1−θ1
mlR (t) ≥ 0.
=
1−θ
21

(99)

Rearranging this inequality leads to w(CP (z(t)), µ(t)) − CP (z(t)) ∂w(C∂CPP(z(t)),µ(t))> 0. Hence, (98) and(z(t))(99) show∂Θ(z(t), µ(t))
≤ 0.
∂z(t)
Turing to the denominator of (97), diﬀerentiating (96) with respect to µ(t) leads to[ ()] θ−1∂Θ(z(t), µ(t))θ 1
CP (z(t))=
L̄ −∂ µ(t)m mw(CP (z(t)), µ(t))
CP (z(t))∂w(CP (z(t)), µ(t)).
2∂ µ(t)
w(CP (z(t)), µ(t))

(100)

(101)

(z(t))
It is obvious that L̄ − w(CCPP(z(t)),µ(t))
is positive from (52). We must examine the partial ofw(Cp (z(t)), µ(t)) with respect to µ(t). Solving (53) for µ(t) gives[] 1−θ
(1 − ψ)CP (z(t))
w(CP (z(t)), µ(t)) L̄ −.
µ(t) =
θmw(CP (z(t)), µ(t))(102)

The derivative of (102) with respect to w(CP (z(t)), µ(t)) is[] −θ
(1 − ψ)∂ µ(t)CP (z(t))=
L̄ −∂w(CP (z(t)), µ(t))
mθw(CP (z(t)), µ(t))[]
CP (z(t))
L̄ − θ
.
w(CP (z(t)), µ(t))(103)

Using (54), it is easy to see that (103) is non-negative. Therefore, the following inequality isestablished.
∂Θ(z(t), µ(t))
≥ 0.
(104)
∂ µ(t)
From (100) and (104), the locus of z(t) has positive slope. Incidentally, (59) and (61) show thatthe locus goes through the point (z(t), µ(t)) = (0, 0).
We now turn to the locus of µ(t) which is defined asε(1 − ψ) 1 (QCP (z(t)) + 1 − Q − R)(CP (z(t)) − β)
= 0.
m
z(t)
QCP (z(t)) + 1 − Q − R − βfrom (60). Denoting the left-hand side of (105) as(ρ − δ)µ(t) −Ω(z(t), µ(t))≡(ρ − δ)µ(t)ε(1−ψ)m

1 (QCP (z(t)) + 1 − Q − R)(CP (z(t)) − β),z(t)QCP (z(t)) + 1 − Q − R − βand applying the similar method as in the case of z(t), we can obtain
22

(105)

(106)

∂Ω(z(t),µ(t))∂ µ(t)∂z(t)= − ∂Ω(z(t),µ(t)) .
∂z(t)

(107)

∂ µ(t)The derivative of (106) with respect to z(t) shows that CP (t) satisfies[]
d ln z(t)d(QCP (z(t)) + 1 − Q − R)(CP (z(t)) − β)>ln,dCP (z(t)) dCP (z(t))
QCP (z(t)) + 1 − Q − R − β(108)

if the condition, ∂Ω(z(t),µ(t))> 0 holds. Let us calculate both sides of the above inequality, respec∂z(t)tively. Here, (61) provides the left-hand side of this inequality as
1
ε

[(

)
]
1−γ
(1 − Q)(1 − β) − RQ−,γ{(1 − Q)(1 − CP (z(t))) − R}(CP (z(t)) − β) QCP (z(t)) + 1 − Q − R(109)

and that of the right-hand side asQ(1 − Q)(1 − β) − R+.
QCP (z(t)) + 1 − Q − R (QCP (z(t)) + 1 − Q − R − β)(CP (z(t)) − β)Therefore, by rearranging these, we obtain

(110)

[
{(1 − Q)(1 − β) − R} (QCP (z(t)) + 1 − Q − R) (1 − γ)(QCP (z(t))]+1 − Q − R − β) − γε {(1 − Q)(1 − CP (z(t)) − R} > γ(1 + ε)Q(CP (z(t)) − β)(QCP (z(t)) + 1 − Q − R − β){(1 − Q)(1 − CP (z(t))) − R} .

(111)

Let us define the left-hand side of (111) as the function f (CP ) and the right-hand side as thefunction g(CP ) . The function f can be depicted as a graph on figure 6, since its coeﬃcient of CP (t)2
is positive and − 1−Q−R
and −(1−γ−εγ)(1−Q−R)+(1−γ)βsatisfies f (CP ) = 0. Similarly, the function gQ(1−γ−εγ)Q+εγ
can be depicted as a graph on figure 7. The coeﬃcient of CP (t)3 is negative and g equals zero when
, β, or 1−Q−R
CP (t) is − 1−Q−R−βQ
1−Q .
Let us compare five values of CP that make f = 0 or g = 0. It is obvious that−

1−Q−R
1−Q−R− β
<−
.
Q
Q

(112)

Since (58) provides (1 − Q)(1 − β) − R > 0, we can get
1−Q−R− β
1−Q−R
<β<
,
Q
1−Q

(113)

1 − Q − R − β −(1 − γ − γε)(1 − Q − R) + (1 − γ)β 1 − Q − R<<.
Q(1 − γ − γε)Q + γε
1−Q

(114)

−
and
−

23

Figure 7: Function g

Figure 6: Function f
Moreover, we can get−(1 − γ − γε)(1 − Q − R) + (1 − γ)β(1 − γ − γε)Q + γε−(1 − γ − γε)(1 − Q − R) + (1 − γ)ββ≤(1 − γ − γε)Q + γε

β>

if
if

1
,
1+ε
1
γ≥
.
1+ε

γ<

(115)
(116)

1
, that is,With this result, figure 6 and figure 7 show that f > g under the condition γ > 1+ε∂Ω(z(t), µ(t))
> 0,
∂z(t)

(117)

1
1
if γ ≥ 1+ε. Now, we turn to the case of γ ≥ 1+ε. Assuming CˆP as the variable satisfies f = g
1−Q−R
in β ≤ Cp (t) ≤ 1−Q , figure 6 and figure 7 shows that f < g if β ≤ CP (t) ≤ CˆP and f > g if
CˆP < CP (t) < 1−Q−R . 5 This implies that1−Q∂Ω(z(t), µ(t))
∂z(t)

{

<0
>0

i f β < CP (t) < CˆP,
i f CˆP ≤ CP (t) < 1−Q−R
1−Q .

(118)

By the way, it is obvious that∂Ω(µ(t), z(t))= ρ − δ > 0.
∂ µ(t)
1
Therefore, (107), (117) and (119) give the following results. If γ < 1+ε,

5We eliminate the case that f unction f and f unction g have more than three intersections.

24

(119)

dµ(t)
< 0.
dz(t)

(120)

1
If γ ≥ 1+ε
,

dµ(t)
dz(t)

{

>0
≤0

if
if

β < CP (t) < CˆP,CˆP ≤ CP (t) ≤ 1−Q−R
1−Q .

(121)

Having discussed the slope of the locus of µ(t), we now turn to its intercept. Introducing (61)
into Ω(µ(t), z(t)) = 0 and setting µ(t) as 0 yield{
CP (z(t)) =

1−Q−R
1−Q
β, 1−Q−R
1−Q ,1
i f γ < 1+ε,1
i f γ ≥ 1+ε
.

(122)

In conclusion, (120), (121) and (122) show that 61Û• If γ < 1+ε, the µ(t)= 0 locus has a negative slope and asymptotically approaches thehorizontal and vertical axes.
1
Û = 0 locus is an inverted-U shape going through the origin and asymptot• If γ ≥ 1+ε, the µ(t)ically approaching the horizontal axis.

References[1] Anderson,S,P. de Palma,A. Thisse,J,F.(1997)Privatization and eﬃciency in a diﬀerentiatedindustry. European Economic Review. 41,1635-1654.
[2] Cato, S. (2008). Mixed oligopoly, productive eﬃciency, and spillover. Economics Bulletin,
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[3] De Fraja,G. Delbono,F. (1989) Alternative Strategies of a Public Enterprise in Oligopoly.
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Press. Cambridge, Massachusetts, London, England.
6We adopt 00 for convenience of explanation.

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[7] Ishibashi,I. Matsumura,T. (2006) R&D competition between public and private sectors. European Economic Review. 50,1347-1366.
[8] Matsumura,T. (1998) Partial privatization in mixed duopoly. Journal of[9] Nishimori,A. Ogawa,H.(2002)Public Monopoly, Mixed Oligopoly and Productive Eﬃciency.Australian Economic Papers.41(2)185-190.
[10] Peretto,F,P. (1999) Cost reduction, entry and the interdependence of market structure andeconomic growth. Journal of Monetary Economics.43,173-195.

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