Privatization in a mixed oligopoly: Productivity,
market concentration, and the optimal degree ofprivatization
Shinjiro MIYAZAWA∗
Graduate School of Economics, Kobe University
2-1, Rokkodai, Nada-ku, Kobe 657-8501, Japan
October 1, 2012

Abstract
This paper investigates the optimal degree of privatization for a public firm in a homogeneous mixed oligopoly. I show that full privatization is optimal when a public firm
has a severe productivity disadvantage or competes with many private firms. The optimal degree of partial privatization is increasing in the degree of productivity disadvantage
and the number of private firms. I further show that partial privatization can be optimal
for a public firm even when full privatization would completely remove any productivity
disadvantages.
Keywords: quantity-setting competition, partial privatization, mixed oligopoly
JEL classification: L13, L32, L33.

∗

Tel.: +81 78 803 6801; Fax.: +81.78.803.6801. E-mail: miyazawa@econ.kobe-u.ac.jp1

1

Introduction

The privatization of public firms is a significant issue in most countries. The standard framework for understanding this issue is a mixed oligopoly model since public firms often competewith private firms in these countries.1 In an early study on mixed oligopoly, De Fraja andDelbono (1989) show that when public firms compete with many private firms, they should
maximize profits rather than welfare in order to improve overall social welfare. This result implies that full privatization is socially preferable when the market is sufficiently competitive.
However, they do not consider the possibility of partial privatization, in which public firms
respect both social welfare and their own profits. Matsumura (1998) explicitly considers thepossibility of partial privatization and shows that partial privatization is optimal in a mixed
duopoly when a public firm is as productive as its private competitor. The literature now contains many analyses on the optimal degree of privatization, following Matsumura (1998).
The most relevant studies to this article, in addition to Matsumura (1998), are Matsumura
and Kanda (2005) and Fujiwara (2007).2 Matsumura and Kanda (2005) generalize Matsumura(1998) to a mixed oligopoly setting with more than two private firms and show that at least
partial privatization is socially preferable in the short run. Fujiwara (2007) investigates a differentiated mixed oligopoly and shows that partial privatization is socially preferable in the
short run.3 However, the literature lacks general analyses on the relationships between firms’
productivity, market concentration, and the optimal degree of privatization. I investigate these
relationships in a homogeneous mixed oligopoly and show that full privatization is optimal
when public firms have severe productivity disadvantages or compete with many private firms
and that the optimal degree of partial privatization is increasing in the degree of productivity
disadvantage and the number of private firms.
The results in this study, similar to Matsumura (1998), imply that a degree of privatization
is socially preferable even when privatization does not improve the productivity of public firms
but that full privatization is not socially preferable if public firms are as productive as private
firms. In other words, full privatization is socially preferable only when a public firm has someproductivity disadvantages. The relationship between productivity and privatization is very
important since most people suspect that X-inefficiency arises in public firms and expect thatprivatizing such firms removes the X-inefficiency and consequently improves social welfare.4
I explicitly consider the productivity improvements derived from privatization, and I show thateven if full privatization makes public firms as productive as private firms, full privatization
is not always socially preferable. Partial privatization is more preferable when the marginal
productivity improvements from additional privatization are negligible when the firm is close
1

The earliest papers on mixed markets include Merrill and Schneider (1966), Harris and Wiens (1980), andBeato and Mas-Colell (1984). See De Fraja and Delbono (1990) and Nett (1993) for a survey of this literature.
2
Other studies concerning the optimal degree of privatization include Ishibashi and Kaneko (2008), Ohnishi(2010), and Wang and Chen (2011).
3
The main goal in Matsumura and Kanda (2005) and Fujiwara (2007) is to investigate the optimal degree ofprivatization in the long run, when private firms can enter and exit the mixed market, which is not the aim of thisarticle.
4Some existing studies, such as De Fraja (1993), Schmidt (1996), Hart, Shleifer, and Vishny (1997), Corneo
and Rob (2003), Matsumura and Matsushima (2004), Ishibashi and Matsumura (2006), and Miyazawa (2008), investigate productivity or cost differences endogenously, but these studies do not consider the relationship between
productivity improvements and the degree of privatization.

2

to full privatization, when the number of private firms is sufficiently small, and when the
productivity disadvantage itself is minor.
Finally, I present a uniform and intuitive explanation for these results. Privatization has
three effects on the equilibrium social welfare in mixed oligopoly. Because privatization reduces the concern for social welfare and therefore the output of public firms, it can increase
the deadweight loss in the oligopolistic market. At the same time, however, privatization can
mitigate inefficient substitutions from private production to public production, and because it
can mitigate the X-inefficiency of public firms, privatization can also improve their productivity. A higher degree of privatization is socially preferable when the latter two effects are
significant and the former is not, which is the case when there are many private competitors.
The remainder of this article is organized as follows. In Section 2, I formulate the model.
Next, in Section 3, I investigate the optimal degree of privatization. Finally, in Section 4, I
present concluding remarks.
2 ModelConsider the market for a homogeneous commodity. The inverse demand function is given byp(q), where p ≥ 0 is the price of the commodity and q ≥ 0 is its total quantity. n + 1 firms
compete in this market. Firm 0 is owned by either the state, private investors, or a combinationof both, whereas firms 1 to n are owned only by private investors. Let α ∈ [0, 1] be the state’s
share in firm 0. α also represents the degree of privatization. I refer to firm 0 as a public firm
when α > 0 and as a fully privatized firm when α = 0 and to firm j, j ∈ {1, . . . , n}, as a
private firm.
Given the ownership structures of these firms, each firm chooses its output quantity xi ≥ 0
simultaneously. Firm i’s cost function is given by ci (xi ), and its∑profit is written as Πi (x) =p(X) − ci (xi ), where x is an output quantity vector and X = ni=0 xi , i ∈ {0, . . . , n}. For
now, suppose that the ownership structure of a firm does not affect its cost.5
The payoffs of firm 0 and firm j, U0 and Uj , are given by
U0 = αW (x) + (1 − α)Π0 (x),Uj = Πj (x),where W (x) is the social welfare. That is,∫ X
p(q)dq − pX +W (x) =
0n∑

∫ X
p(q)dq −Πi (x) =
0

i=0

n
∑

ci (xi ).

i=0

Firm 0 maximizes the weighted sum of social welfare and its own profit in proportion to thestate’s share of ownership.6
Assumption 1 p(q) is twice differentiable and satisfies p′ (q) < 0 for any (p, q) such that p > 0
and q ≥ 0.
5

At the end of the next section, I investigate the case in which the state’s share of the firm. α, affects the
public firm’s cost.
6
The assumption of proportionality is just for simplicity. See Matsumura (1998) and subsequent papers formore details.

3Assumption 2 ci is twice differentiable and is strictly increasing in xi for any xi ≥ 0.
The first-order conditions for the firms’ maximization problems are(1 − α)p′ x0 + p − c0 ′ = 0
p′ xj + p − cj ′ = 0.

(1)
(2)

Assumption 3 The relevant second-order conditions for (1) and (2) are satisfied.
Let R0 (X−0 ; α) and Rj (X−j ) be the reaction functions for firm 0 and firm j, respectively,
so that,R0 (X−0 ; α) = arg max U0 (xi , X−i ; α)
x0 ≥0

and
Rj (X−j ) = arg max Uj (xj , X−j ),xj ≥0where X−i = X − xi .
Let xEi be the equilibrium quantity of output for firm i. Then, the equilibrium social welfare
is written as
∫ XE
n
∑
E
W ≡
p(q)dq −ci (xE
(3)
0 ),
E

∑n

0

i=0

E
i=0 xi .

where X =Additionally, assume the following.
′
E
E
′′ E
Assumption 4 For any α, (i) c0 ′′ (xE0 ) ≥ (1 − α)p (X ) when x0 > 0, and (ii) cj (xj ) >′E
Ep′ (X E ) and p′′ (X E )xE
j + p (X ) < 0 for any j such that xj > 0.
EThis assumption guarantees that −1 ≤ R0 ′ (X−0; α) and −1 < Rj ′ (XjE ) < 0, where R0 ′ (X−0 ; α) =∂R0 (X−0 ;α). Assumptions 3 and 4 are satisfied for the standard set of assumptions that p′′ (q) ≤ 0∂X−0
for any (p, q) such that p > 0 and q ≥ 0 and that ci ′′ (xi ) ≥ 0 for any xi .

3

Results

I investigate the optimal level of α and thus the optimal degree of privatization in this section.
Let A∗ = arg maxα∈[0,1] W E be the set of optima. Further, let α∗ be the optimum if A∗ is asingleton. First, consider the optimality of privatization. Suppose that xE
i > 0 for any i. Then,the effect of privatization on social welfare is captured by differentiating (3) as follows.
)
n (
∑
dX E
dxE
dW E
i
′ E
E
(4)
=
p(X ) −
ci (xi )
dα
dα
dα
i=0
}
n {
∑
)
(
dxE
i
′ E
=
p − ci (xi )
(5)
dα
i=0
[
{
}]
n
E
E
∑
dx
dx
j E
+
= −p′ (X E ) (1 − α) 0 xE
xj
,
(6)
dα 0
dα
j=1
where the derivation from (5) to (6) follows from the first-order conditions, (1) and (2). The
following results about the effect on the equilibrium quantities hold.
4

dxE

dxE

E

j
dX
0
Lemma 1 Suppose that xE
i > 0 for any i. Then, dα > 0, dα > 0 for any j, and dα > 0.

The explanation of these results is similar to those in Matsumura (1998) and Matsumura andKanda (2005). An increase in α increases the incentive for firm 0 to increase output to improvesocial welfare. Because of the private firms’ strategic substitutability, that is, Rj ′ < 0, itEfollows that the increase in α increases xE
0 and decreases xj . In addition, because of the′private firms’ lower substitutability, that is, Rj > −1, the above effects result in an increasein X E .
E
When α = 1, following from (6), dW
is rewritten as
dα
}
{
n
E
∑
dx
dW E
j E
(7)
= −p′ (X E )
xj .
dα
dα
j=1
E
Let xEi (0) and xi (1) be the equilibrium quantities when α = 0 and when α = 1, respectively.
Then, the following result holds.
E
Proposition 1 (privatization of state-owned firms) Suppose xE
0 (0) > 0 and xj (1) > 0 for∗nsome j. Then, 1 ∈/ A holds irrelevant to {ci }i=0 and n.

This proposition states that public firms should be at least partially privatized if they competewith some private firms in a homogeneous commodity market. This result is a generalizationof the results in Matsumura (1998) and Matsumura and Kanda (2005). Matsumura (1998),
which is a seminal study in this literature, shows that public firms should be at least partiallyprivatized when the market is a duopoly, that is, when there is one private firm. Matsumura
and Kanda (2005) find a similar result in the oligopolistic setting where a public firm competes
with more than two private firms with an identical cost function. In this proposition, I showthat a similar result holds even when private firms are heterogeneous.
The intuition is as follows.7 When α = 1, the public firm is a welfare maximizer and consequently sets its quantity so that marginal cost equals price, following from (1). This behaviorof the public firm is clearly socially preferable if there is no strategic interaction between public and private firms, but this is no longer the case if strategic interaction matters. The private
firms, which maximize their profits, set their quantities so that their marginal costs are equal
to their marginal revenues, which are less than the price, following from (2). It follows that aslight decrease in x0 and a subsequent increase in xj results in an efficient product substitution and consequently improves the social welfare. When the public and private firms decide
their quantities simultaneously, the benevolent public firm overproduces since it fails to takethe strategic effect of its choice on the private firms’ decisions into account. The result showsthat at least partially privatizing the public firm is socially preferable because the public firm
behaves less aggressively, as follows from Lemma 1.8
7

See also Matsumura (1998) and Matsumura and Kanda (2005) for an intuitive explanation of this result.
The fact that the public firm decides to overproduce in a simultaneous quantity-setting competition in a mixed
market is well known in this literature. For example, Beato and Mas-Colell (1984) show that the equilibrium
quantity in a sequential quantity-setting competition where the public firm is the Stackelberg leader is lower
than that in a simultaneous quantity-setting competition and that the equilibrium social welfare of the former
competition is higher than that of the latter one in a mixed duopoly.
8

5

E

Next, consider the optimal degree of privatization. When α = 0, following from (6), dW
dα
is rewritten as
{
}
n
∑
dW E
dxE
i
= −p′ (X E )
xE
.
(8)
i
dα
dα
i=0
Then, the following result holds.
Proposition 2 (partial privatization of efficient public firms) Suppose xE
0 (1) > 0. Then,
∗
n
E
E
0∈
/ A holds irrelevant to {ci }i=0 and n if x0 (0) ≥ maxj∈{1,...,n} xj (0).
This proposition states that full privatization is not socially preferable if the public firm canhave the largest market share when it is fully privatized. Since the market share is determinedby relative productivity, as is implied in (1) and (2), the proposition implies that full privatization is not socially preferable if the public firm is at least as productive as the most productive
private firm. This result is a generalization of the result in Matsumura (1998), who shows thatfull privatization is not socially preferable when the market is a duopoly and the public firm is
as productive as the private firm, or, equivalently, when a fully privatized public firm produces
as much output as the private firm does in equilibrium. In this proposition, I show a similar
result when the market is an oligopoly and the public firm produces as much output as thelargest private firm does in equilibrium. This proposition clearly implies that a similar result′′
holds in the case of identical private firms, that is, 0 ∈/ A∗ if xE
0 (1) > 0 and c0 (x) = cj (x) for
any x ≥ 0 for any j.
The intuition for this result is as follows. When Assumption 4 holds, the private firms’
reactions to the other firms’ quantity setting are less substitutive, that is, Rj ′ > −1, and consequently the equilibrium aggregate quantity is increasing in the equilibrium quantity of the
public firm.9 Moreover, the equivalence of the equilibrium quantities of the fully privatizedfirm and a private firm implies that these firms produce the commodity at an equivalent margin,p − ci ′ (xi ). Therefore, a slight increase in x0 from xE
0 (0) increases both the consumer surplus
and the producer surplus.
Now, I investigate the relationships among the cost functions, the number of private firms,
and the optimal degree of privatization. In order to present meaningful results, I employ thefollowing specification.10
Specification 1 The inverse demand function is p(q) = a − bq, and the cost functions arec0 (x) = 21 (k + ∆)x2 and cj (x) = 12 kx for all j, where a > 0, b > 0, k > 0, and ∆ ≥ 0.
This specification that the inverse demand function is linear and the cost functions are quadratic
is fairly standard in this literature. In addition, for later discussion, I employ a specification
similar to that of De Fraja and Delbono (1989).11 ∆ ≥ 0 represents the public firm’s degree of
9

These results are explicitly shown in the proof of Lemma 1.
The properties of ci and n change the equilibrium quantities depending on the relationships between p′ , p′′(if differentiable), ci ′ , and ci ′′ (if differentiable) in general. Investigation in a general setting is very complicatedand does not produce any fruitful results.
11
As noted by De Fraja and Delbono (1989), specifying a quadratic cost function is useful since it precludes
the possibility that no private firm produces any output. I can present a similar result below in the text underanother specification with constant marginal costs.
10

6

productivity disadvantage. As is shown in Proposition 2, if the public firm had a productivity
advantage, that is, ∆ < 0, then 0 ∈/ A∗ would hold regardless of the parameters.
Then, the following results hold.
Proposition 3 (productivity, number of private firms, and privatization) Given Specifica¯ > 0 (a finite n̄ ≥ 1) such thattion 1, (i) A∗ is a singleton, (ii) there exists a finite ∆∗∗¯¯ (n ≥ n̄), (iii) ᾱ is decreasing
α = ᾱ ∈ (0, 1) if ∆ < ∆ (n < n̄) and α = 0 if ∆ ≥ ∆¯ (n̄) is decreasing in n (∆).
in ∆ and n, and (iv) ∆This proposition states that a higher degree of privatization is socially preferable for public
firms with severe productivity disadvantages and for those with many private competitors. This
proposition fills a gap in the current literature. De Fraja and Delbono (1989) show that a public
firm should maximize profits, in other words, should be fully privatized when the number ofprivate firms is sufficiently large, which is similar to this proposition. However, they do notconsider the possibility of partial privatization and productivity differences between public andprivate firms. This proposition fills the gap between full nationalization and full privatizationin the analyses of privatization in a mixed oligopoly. In fact, when productivity is symmetricamong public and private firms, as in De Fraja and Delbono (1989), full privatization is notsocially optimal if partial privatization is possible.
Matsumura (1998) shows that full privatization is socially preferable when the productivity
disadvantage is sufficiently high.12 However, he does not state the tendency of the optimal
degree of privatization, nor does he provide any implications about the relationship between
the number of private firms and optimal degree of privatization. Fujiwara (2007) shows that the
optimal degree of privatization is increasing in the number of private firms in a differentiatedmixed oligopoly. However, since he does not consider productivity differences, he does notprovide any implications about the relationship between the productivity disadvantage and the
optimal degree of privatization.
In addition to filling the theoretical gaps in the literature explained above, I provide a
uniform and intuitive explanation for these results is as follows. Privatization, or a decreasein α, has two opposite effects on social welfare due to the drop in the public firm’s output.
First, it mitigates inefficient production substitutions from private firms to the public firm,
but second, it aggravates the deadweight loss in the oligopolistic market. On the one hand,when the productivity disadvantage is severe, the first effect is significant and a high degreeof privatization is socially preferable. On the other hand, when the number of private firms islarge, the second effect is not significant and again, a high degree of privatization is socially
preferable.
The results in Proposition 3 imply the following results.
Corollary 1 When ∆ = 0, α∗ ∈ (0, 1) and limn→∞ α∗ = 0 for any level of the parameters.
Matsumura (1998) shows the optimality of partial privatization when the public firm is as
productive as the private firms. This result is the simplest generalization of his result, and it
also implies that the finding of De Fraja and Delbono (1989) which implies possible optimalityof full privatization of public firms is due to the disregard of partial privatization.
12

Matsumura (1998) uses cost functions with constant marginal costs in his explanation.

7

I have not considered the possibility that the privatization could improve the public firm’s
productivity thus far. However, most people suspect that X-inefficiency arises in public firms
and that privatizing the firms removes the X-inefficiency in the public firm and consequently
improves social welfare. To take the possibility that privatization improves productivity into
account, I employ the following specification instead of Specification 1.
Specification 2 The inverse demand function is p(q) = a − bq and the cost functions arec0 (x) = 21 {k + µα + (∆ − µ)α2 }x2 and cj (x) = 12 kx for all j, where a > 0, b > 0, k > 0,
∆ > 0 and µ ∈ [0, ∆].
The terms in the brace in c0 (x) are the degree of productivity of the public firm, which simplifies to k + ∆ when α = 1 and to k when α = 0. This specification represents the assumptionthat full privatization completely removes the X-inefficiency of the public firm denoted by ∆.
It also represents another assumption that the level of productivity improvement effect, whichis represented by −{µ + 2(∆ − µ)α}, is decreasing (if µ < ∆) or constant (if µ = ∆) in the
degree of privatization, or in other words, is increasing in α. µ is the level of the productivity
improvement effect when α = 0. Since I am interested in the optimality of full privatization,
the level of µ is significant, as is shown below.
Then, I present the following result.
Proposition 4 (productivity improvement and privatization)
Given Specification 2, full pri√vatization is not socially optimal when b >(n+4)µ+kµ+(n+4)2 µ2.
4This proposition states that full privatization is not socially preferable even when privatization
improves the productivity of public firms if the number of private competitors n is sufficiently
small and the productivity improvement effect almost disappears as the firm approaches fullprivatization, or in other words, if µ is sufficiently small. Furthermore, full privatization isnever socially preferable when the productivity improvement effect completely disappears,
that is, µ = 0. Finally, it implies that full privatization is not socially preferable when the levelof X-inefficiency ∆ is small if the level of the productivity improvement effect is constant, thatis, µ = ∆.
The intuition is as follows. Under Specification 2, privatization has a third effect on social
welfare in addition to the first two in the explanation of Proposition 3, because it improves
productivity. Full privatization is not socially preferable when µ is sufficiently small and consequently this third effect almost disappears, as is already stated, when n is sufficiently small
and consequently the second effect of aggravating deadweight loss is large, and when µ = ∆and ∆ are sufficiently small and consequently the first effect of mitigating inefficient productsubstitution is not significant.
4

Concluding Remarks

I investigate the optimal degree of privatization and its relationship with productivity and market concentration in a homogeneous mixed oligopoly to fill gaps in the literature. Then, I
show that full privatization is optimal when a public firm has a severe productivity disadvantage or competes with many private firms and that the optimal degree of partial privatization isincreasing in the degree of productivity disadvantage and the number of private firms.
8

The relationship between productivity and privatization is very important because mostpeople suspect that X-inefficiency arises in public firms and expect that the privatization of
the firms removes the X-inefficiency and consequently improves social welfare. I explicitly
consider the productivity improvements derived from privatization and show that even if fullprivatization completely removes the X-inefficiency in public firms and consequently makes
fully privatized firms as productive as private firms, full privatization is not socially preferable
when the productivity improvement effect almost disappears as the firm approaches full privatization, when the number of private firms is sufficiently small, and when the productivity
disadvantage itself is minor.
I present a uniform and intuitive explanation for these results. Privatization has three effects on the equilibrium social welfare in a mixed oligopoly. The output of public firms falls
because they are less concerned with social welfare, which aggravates the deadweight loss
in the oligopolistic market. At the same time, however, this drop in output mitigates inefficient product substitutions from private firms to public firms. Furthermore, privatization can
mitigate X-inefficiency and consequently improve the productivity of public firms. A higher
degree of privatization is socially preferable when the latter two effects are significant and theformer effect is not significant, which is the case when there are many private competitors.
Needless to say, the results in this article are restrictive because I only investigate theequilibrium outcome in a homogeneous market in the short run. I also neglect the existenceof foreign firms in the same market. Investigations in more general settings are objectives for
future research.
Appendix A
A.1

Proof of Lemma 1

Following from (1),
∂R0
p′ x0
=
> 0,
∂α
(1 − α)p′′ x0 + (2 − α)p′ − c0 ′′(A1)where x0 = R0 , for R0 > 0. The sign of this inequality follows from Assumptions 1 and 3.
Then, the two below equations follow from (1) and (2).
( E
)
dxE
∂R0
dX
dxE
0
0
′
=
+ R0
−
dα
∂α
dα
dα
(
)
E
E
E
dxj
dxj
dX
= Rj ′
−
dα
dα
dα

9

Solving these equations gives
/(

E

n
∑

′

)

dX
1 ∂R0
Ri
=
1−
>0
′
dα
1 + R0 ∂α
1 + Ri ′
i=0
(
)/ (
)
n
n
′
′
∑
∑
1 ∂R0
Rj
Ri
dx0
=
1−
1−
>0
′
′
dα
1 + R0 ∂α
1 + Rj
1 + Ri ′
j=1
i=0
/(
)
n
∑
dxj
1 ∂R0 Rj ′
Ri ′
=
1−
< 0,
dα
1 + R0 ′ ∂α 1 + Rj ′1 + Ri ′i=0
in which the signs of inequalities follow from Assumption 4 and (A1).
Proof of Proposition 1
E
When xE
0 (0) > 0, x0 (1) > 0. Without loss of generality, I can neglect private firms suchExj (1) = 0, that is, I can focus on the competition among the public firm and private firms
dW Esuch that xE
j (1) > 0. Then, from Lemma 1 and (7), dα < 0.

Proof of Proposition 2
Suppose maxj∈{1,...,n} xE
j (0) > 0. Focusing on the competition among the public firm andprivate firms such that xE
j (0) > 0 gives equation (8). Then, following from Lemma 1 and
E
xE
≥
max
x
(0),
j∈{1,...,n} j
0
}
n {
E
∑
dxE
dW E
i
′
E
E
′
E dX
> −p (X )x0 (0) = −p (X )xE(0) > 0.
dα
dα
dα 0
i=0
E
Suppose maxj∈{1,...,n} xE
j (0) = 0. Then, xj (α) = 0 for all j and for any α. When this is thecase, α = 1 is the unique optimum since I suppose xE
0 (1) > 0.

Proof of Proposition 3
Given Specification 1,xE
0 =a(b + k)
,
G

xE
1 =

a{(1 − α)b + k + ∆}
,
G

where
G = {(1 − α)b + k + ∆}(b + k) + b(b + k) + nb{(1 − α)b + k + ∆}.
E

Let ᾱ be a solution of dW= 0. Then, there is a unique solutiondαᾱ =(b + k)2 − nb∆.
(b + k)2 + nbk
10

(A2)

Note that α is in [0, 1] and that ᾱ < 1 holds. Further note that α = 1 is not the optimum, as
is shown in Proposition 1. Then, the optimal level of α is max{ᾱ, 0}, which is unique (claim(i)).
(b + k)2¯ᾱ ⋛ 0 ⇔ ∆ ⋚
≡∆
nb
and
(b + k)2ᾱ ⋛ 0 ⇔ n ⋚
≡ n̄
b∆
¯
dᾱ
∆
dn̄
(claim (ii)). Furthermore, d∆< 0 and ddnᾱ < 0 (claim (iii)), and ddn< 0 and d∆< 0 (claim(iv)).
Corollary 1
2(b+k)
When ∆ = 0, ᾱ = (b+k)
2 +nbk ∈ (0, 1), following from (A2). Then, limn→∞ ᾱ = 1.

Proof of Proposition 4
Given Specification 2,(n + 4)µ +dW E (0)
>0⇔b>
dα

√

kµ + (n + 4)2 µ2.
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