Urban unemployment, privatization policy, and a differentiated mixed oligopoly1
The University of Tokushima

Tohru Naito

1. Introduction
This paper combines a traditional dualistic economy model with a differentiated mixed oligopoly model. In
development economics, Harris and Todaro (1970), which is one of pioneer studies in dualistic economy,
describes a dualistic economy by assuming the downward rigidity of wage in urban area and explains theoccurrence of unemployment endogenously. Though there is no room for doubt that Harris and Todaro model
is one of most important studies about dualistic economy analysis, it is hard to say that the setting
necessarily accords with reality. Harris and Todaro model has been extended from the various viewpoints.
Corden and Findlay (1975) extend Harris and Todaro model by taking account of mobile capital between
regions.
As for a mixed oligopoly, De Fraja and Delbono (1990) construct a mixed oligopoly market, where private
firms compete with a public firm in the homogeneous goods market. De Fraja and Delbono (1990) compares
the mixed oligopoly, which a public firm is full nationalized, with the pure oligopoly, which a public firm isfull privatization. Matsumura (1998) considers the model where the public firm maximizes the weighted
average of social welfare and profit as the objective function and show that the partial privatization of public
firm is optimal. Matsumura and Kanda (2005) take account of allow free entry of private firms in the mixed
oligopoly.
Naito (2012) introduces a mixed duopoly into Fukuyama and Naito (2007) and analyzes the effect ofprivatization of public firm on urban employment. However, Naito (2012) deals with not differentiated goods
but homogeneous goods. In the actual world, it is not natural to assume the homogeneous goods produced
in urban area.
Fujiwara (2007) consider the quasi-linear utility function constructed by Ottaviano, Tabuchi, and Thisse(2002) and introduces product differentiation into a mixed oligopoly model. Though Fujiwara (2007) adopts
the quasi-linear utility function of Ottaviano, Tabuchi, and Thisse (2002) in his paper, he does not takeaccount of monopolistic competition. Thus, we combine a traditional dualistic economy model with adifferentiated mixed oligopoly model to construct the model which enable to analyze the urban
unemployment and the privatization of public firm of privatization under a differentiated mixed oligopoly.
Thus, we combine a traditional dualistic economy model with a differentiated mixed oligopoly model toconstruct the model, which enables to analyze the urban unemployment and the privatization of public firmof privatization under a differentiated mixed oligopoly. We adopt Ottaviano, Tabuchi, and Thisse (2002) andFujiwara (2007) to describe a product differentiation and a mixed oligopoly market. On the other hand, we
use Harris and Todaro (1970) to describe a traditional dualistic economy. We consider the effect ofprivatization of public firm on urban unemployment or social welfare in this model.
The remainder of the paper is organized as follows. Section 2 presents the model and discusses the1 This study is supported by Grant-in-Aid for Scientific Research A (25245042) from Japan Society of thePromotion of Science.

properties of market equilibrium. Based on section 2, section 3 clarifies the effects of public firm privatization
on migration between urban area and rural area, urban unemployment, and social welfare. The last section
presents concluding remarks.

2. The Model
We consider the economy consists of two regions in our model. One of them is urban area, the other isrural area. Following Harris and Todaro (1970), every household in the economy can migrate between
regions without any cost. We consider the three types of household in this model. Indexes c, r, and u denotethe household employed by the manufactured goods sector, those employed by agricultural goods sector, andthose who are not employed in both sectors and reside in urban area, respectively. Now we consider two
kinds of products: agricultural goods and manufactured goods. The manufactured goods and agricultural
goods are produced in urban area and rural area, respectively. Though manufactured goods sector require
labor and capital as input factors to produce their own products, the agricultural goods sector requires only
labor as input factor. Though each manufactured goods are differentiated and are produced by one public
firms and n-private firms in urban area, the agricultural goods are homogeneous.

2.1 Household
We assume that the urban wage is fixed above the market-clearing level of labor market, and that it has
ö (= wc) represent
downward rigidity attributable to the minimum wage system and so on. Particularly, let w

the minimum wage in the urban area, which is higher than the rural wage wr . That rural wage isdetermined in the labor market of the agricultural goods sector and is equal to the marginal product of laborin the agricultural goods sector. Here we assume that households who reside in urban area and are not
employed in manufactured goods sector do not have wage income, that is, wu is equal to zero. Let Lc; Lr
and L u represent the number of each household type. Moreover, we normalize total population in theeconomy as one.
Since we assume that all households have common preference, they have the following same quasi-linear
utility function as well as Ottaviano, Tabuchi, and Thisse (2002).

ð
ñ
ð
ñ h
i2
RN
RN
RN
ìàí
í
Ul = ë ql0 + 0 ql(i)di à 2 [ql0]2 + 0 [ql(i)]2di à 2 ql0 + 0 ql(i)di + z;

(l = c; r; u) (1), where ql0; ql(i), and N denote the consumption of goods produced by public firm, that produced by i-th
private firm, and agricultural goods, the number of private firm in manufactured goods sector, respectively.2
Since all households has one unit of labor and ök units of capital, they allocate their wage income and rkö
to consumption of manufactured goods and agricultural goods. Thus, the budget constraint of eachhouseholds are given byRö = p0 ql0 + N p(i)ql(i)di + z
wl + rk0(2)

Maximizing (1) subject to (2), demand functions of manufactured goods is as follows.

qãl0 = a à [b + c(N + 1)]p0 + cP
2 We assume thatë > 0 andì > í > 0. As for detailed explanation of this utility function, seeOttaviano, Tabuchi, and Thisse (2002).

(3)

(4)

qãl (i) = a à [b + c(N + 1)]p(i) + cP;

where a, b, and c are ë=[ì + íN] , 1=[ì + íN], and í=(ì à í)[ì + íN], respectively. Moreover, priceindex P is defined as follows; P ñ [p0 +function is derived as follows.
a2(N+1)vl = 2b à a

à c2

RN
0

p(j)dj] Substituting (3) and (4) for (1), the indirect utility

ð
ñ
ð
ñ
RN
RN
b+c(N+1)
2
2
p0 + 0 p(i)di + 2[p0] + 0 [p(i)] di

h
i2
RN
ö
p0 + 0 p(i)di + wl + rk

(5)

Though we assume that all households are mobile between regions, the wage in manufactured goods
sector has downward rigidity attributable to the minimum wage system and so on and higher than thewage in agricultural goods sector. Thus, each household compares the expected utility in urban area withthe utility in rural area. Now we define õ as the unemployment rate in urban area, that is,
L

(6)

u
õ ñ Lc+Lu

All households have no incentive to migrate between urban area and rural area when the expected utility
in urban area is equal to the utility in rural area. Thus, migration equilibrium condition is given by thefollowing equation.

(1 àõ)vc +õvu = vr , (1 àõ)w
ö = wr(7)

Since we assume that total population in the economy is normalized, the population constraint is as follows.

Lc + Lu + Lr = 1(8)

Combining (6) with (8), the population constraint in the economy is revised as follows.

L c + (1 à õ)L r = 1 à õ(9)

2.2 Production
2.2.1 Agricultural goods sector
We consider that two kinds of goods are produced in this economy. One of them is the agricultural goods
and produced in rural area. On the other hand, the other is the manufactured goods and done in urban area.
We assume that the market of agricultural goods is competitive. The agricultural goods sector has
decreasing returns of scale with respect to labor. Particularly we specify the production function ofagricultural goods sector as follows.

Z = (Lr)û;û 2 (0; 1)(10)

The agricultural goods sector is competitive and the agricultural goods are homogeneous. Moreover, we
assume that the wage has no downward rigidity in rural area. So the wage is equal to the marginal productin rural area.

wr = û(Lr)ûà1(11)

2.2.2 Manufactured goods sector
We assume that the manufactured goods sector is the differentiated mixed oligopoly. In the mixed oligopoly
market, the public firm competes with private firms in the common market. Here we consider that the
manufactured goods market is the monopolistic competition as well as Ottaviano, Tabuchi, and Thisse(2002). Supposed that the number of private firm is so large, the behavior of firms in urban area does notaffect the price index in the manufactured goods market regardless of public firm or private firms. First ofall, we derive the total demand of each manufactured goods, which is denoted by Q0; Q(i) . Since thedemand for each manufactured goods is independent of household's income due to quasi-linear function,
each total demand function of manufactured goods sector is given by

Q0 = a à [b + c(N + 1)]p0 + cP
Q(i) = a à[b +c(N +1)]p(i) +cP;

(12)
(13)

i 2 [0; N]

Here we assume that Qj units of production require Ljc=m units labor and one unit of capital following
Naito (2012). Though the private firms pursue their profit in mixed oligopoly, the public firm determines
his production to maximize the social welfare. Thus, we define the social welfare function to analyze thebehavior of public firm. Let W represent the social welfare in this model as follows.

W=a2(N+1)
àa
2b

ð
ñ
ð
ñ
RN
RN
b+c(N+1)
2
2
p0 + 0 p(i)di +[p]+[p(i)]
di
0
2
0

h
i2
RN
RN
RN
à c2 p0 + 0 p(i)di + ù0 + 0 ù(i)di + w
ö(L0c + 0 Licdi) + ùr + wrLr + r(N + 1) (14)where ùr is the profit of rural sector. Since we assume that the number of private firm is large as well as
Ottaviano, Tabuchi, and Thisse (2002), both public firm and private firms do not affect the price index of
manufactured goods market. Here we consider that the government owns the share of (1 à ò) of public
firm as well as Matsumura (1998). Thus, the purpose of public firm (firm 0) is to maximize the weighted
average of social welfare and its profit, which is defined byV(ò).
(15)

V(ò) ñ ù0 +(1 àò)W

Since each private firms determine the price to maximize their own profit function, the first order condition
for profit maximization is given by
@ù(i)
ö =0
@pi = a à 2[b + c(N + 1)]pi + cP + [b + c(N + 1)]mw(16)

On the other hand, since the public firm maximizes the weighted average of social welfare and its profit,
the first order condition is as follows.
@V(ò)ö [b + c(N + 1)] = 0@p 0 = a à (1 + ò)[b + c(N + 1)]p0 + cP + òmw

Solving (16) and (17), we derive the following equilibrium prices of each goods.3òapã0 = (1+ò)[b+c(N+1)]ó(ò+1 )2 cN+2 [(1+ò2 )c+(1+ò)b ]2 (1+ò)b+(2+(1+ò)N)c(1+ò)að+2ò[(1+ò)b(1+ò)(N+1)c]
2 (1+ò)2 b+(2+(1+ò)N)c

1+ò b+(N+1)c mw
ö]pã = 2(1+ò)b+[2+(1+ò)N]c + 2((1+ò))[b+[2+(1+ò)N
]c

(17)

ñ
mw
ö

(18)
(19)

Moreover, substituting (18) and (19) for price index P, we derive the equilibrium price index as follows.

ð

ã

P =

(2ò+(1+ò)N)2(1+ò)b+(2+(1+ò)N)c

ñ

ð

a+

[b+c(N+1)]á[(2+N)ò+N]
2(1+ò)b+(2+(1+ò)N)c

ñ

mw
ö

(20)

Finally, we refer to the capital market. Since every firm requires one unite of capital as a fixed input for3 Here we assume that each private firm are symmetric;p = p(i)i 2 (0; N)production. Total number of manufactured goods sector is equal to N+1. On the other hand, every

k units of capital as initial endowment and total number of households ishousehold in the economy has ö
one. Thus, the following equation must hold due to market clear condition in equilibrium.4ö
N+1 = k3. Migration between urban area and rural area

(20)

In the previous section, we analyze the behavior of households and firms in the model. Using equilibrium
derived in previous section, we consider the effect of privatization of public firm on unemployment and social
welfare. Taking account of (12), (13), (18), (19), the definition of price index manufactured goods, andö is as follows.
production function, the labor demand of manufactured goods sector L

h
ð
ñi
ð
ñ
(2ò+(1+ò)N)[b+c(N+1)]á[(2+N)ò+N]ö
Lc = N + 1 à b 2(1+ò)b+(2+(1+ò)N)c am à b 2(1+ò)b+(2+(1+ò)N)c m2w
ö(21)

Combining (7) with (11), we can derive the following equation.
1

Lr = (1 à õ) ûà1

1ð ñûà1

(22)

w
ö
û

Substituting (21) and (22) for (9), the equilibrium unemployment in urban area is determined by thefollowing equation.
1
h
ð
ñi
ð
ñ
ð ñûà1û
2ò+(1+ò)N)(b+c(N+1))â((2+N)ò+N)w
2
ûà1
N + 1 à b 2(1+ò( )b+(2+(1+ò)N
am
à
b
m
w
ö
+
(1
à
õ)
= 1àõ
2(1+ò)b+(2+(1+ò)N)c
û
)c

(23)

Applying the implicit function theorem to (23) to analyze the effect of privatization of public firm on urban
unemployment, dõ=dò is derived as follows.
dõdò = à

n

1
û
1àû

1
ö ûà1
( 1àõ) w
+1
û

( )(

)

â

à[4bm(b+c(N+1))]â[(b+c(N+1))mw
ö +a]
2
2
1+ò
b+
2+(1+ò)N
c
[ (
) (
) ]

o
>0

(24)

Since both denominator and numerator in (24) are positive, the sign of (24) is determined uniquely. So we
derive the following proposition.
【Proposition 1】

The progress of public firm's privatization in urban area leads to deteriorate urban unemployment.
Next we consider the effect of public firm's privatization on social welfare in equilibrium. Let õã represent

õ to hold (23). Substituting õã for (22), the equilibrium rural employment are given by1ð ñûà11öãã ûà1 w.
Lr = (1 à õ )û

(25)

Since we define (14) as social welfare function, we substitute (18), (19), equilibrium price index, and (25) for(14) to derive the equilibrium social welfare function. Let W* represent the equilibrium social welfare
function. Thus, the equilibrium social welfare function is as follows.
òh i2ó
a2 (N+1)
b+c(N+1)ããã 2
W = 2b à
p0 + N[p ] + c2 [Pã]2 + [Lãr ]û
2(26)

4 Thus, the number of private firms in manufactured goods sector is determined by the household’s initialendowment of capital.

Differentiating (26) with respect toò , the effect of public firm's privatization on social welfare in
equilibrium is as follows.
@Wã@ò = à [b + c(N + 1)]

ò

@p ã
@p ã
ã 0
p0 @ò + Npã @ò

ó

á
ã
@ à
ã û
+ cPã@P
+
(L
)
r
@ò
@ò

(27)

Here we define the first term, the second term, and third term as price effect, price index effect, and rural
employment effect, respectively. Generally, the sign of (27) is not determined uniquely. Though we knowthat the effect of privatization of public firm on price index effect and rural employment effect are positive
with easy calculation, the effect of privatization of public firm on price index is ambiguous. If price index
effect and the price effect are superior to rural employment effect, the progress in privatization of public
firm leads to improve social welfare. Thus, we derive the following proposition.
【Proposition 2】

When the price effect and the rural employment effect, are superior (inferior) to price index effect, theprogress in privatization of public firm leads to improve (make worse) social welfare.

4. Concluding remarks
We combine the traditional Harris and Todaro model with mixed oligopoly and introduce monopolisticcompetition into this model. As the results of our analysis, we show that the progress in privatization of
public firm make worse urban unemployment. Moreover, the privatization can improve social welfare if
the price effect and rural employment effect is large relatively. For simplification, we do not refer to capital
market. Thus, this point is our subject in future research,

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