Privatization of public firm and urban
unemployment in a dualistic economy
Tohru Naito
The University of Tokushima
August 27, 2012
Abstract
This paper presents a model in which the economy comprises a mixed oligopoly
in an urban area and an agricultural sector in a rural area. Using it, the eﬀect ofprivatization of public ﬁrms in the urban area on urban unemployment are analyzed. We introduce a mixed oligopoly model into Harris and Todaro model toanalyze this eﬀect. The results of our analysis show that privatization of publicﬁrms might not improve the unemployment rate in an urban area.

1Introduction

This paper presents an examination of the eﬀect of privatization of public ﬁrm on urban
unemployment rate in a dualistic economy. Recently many researchers have studied
mixed oligopoly, in which a public ﬁrm aims to maximize social welfare and private
ﬁrms seek their own proﬁt maximization. Particularly, some developing countries suchas China are proceeding with privatization of public ﬁrms. Moreover, urban areas face
severe unemployment because agricultural workers migrate from rural areas to urbanareas. Regarding urban unemployment problems, Harris and Todaro [3] elucidated amechanism that brings about unemployment caused by migration between urban areas
and rural areas in developing countries. However, they do not discuss privatization of
public ﬁrms because they deal with competitive markets in urban and rural sectors.
Regarding mixed oligopoly, many researchers have analyzed this issue: Matsumura [4],[2], and others. Nevertheless, few studies deal speciﬁcally with urban unemployment
under the mixed oligopoly model. Therefore, we extend Harris and Todaro [?] bycombining them with a mixed oligopoly model. The paper proceeds as follows. The
next section outlines the basic model. Section 3 describes the eﬀect of privatization of
public ﬁrm on the equilibrium. Finally, Section 4 presents our concluding remarks.

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2The model

2.1

Households

The economy comprises mixed oligopoly in an urban area and an agricultural sector ina rural area. All households have identical preferences. We assume that the number ofhouseholds in an economy is normalized to one. They reside in urban or rural areas,
and they are mobile between regions. Two kinds of household reside in urban area.
Although one is labor employed in the manufactured goods sector and obtains wage
income; the others are out of work. They have no income except for redistribution. The
residents in the rural area are engaged in the agricultural goods sector. They receive
wage income wy . We assume that the urban wage is ﬁxed above the market-clearing
level, and that it has downward rigidity attributable to the minimum wage system andother reason. Particularly, let wx represent the minimum wage in the urban area, which
is higher than the rural wage wy . That rural wage is determined in the labor marketof the agricultural goods sector. It is equal to the marginal product of labor in theagricultural goods sector. 1 Similar to Fujiwara [2], each household has the following
utility function Ui .
1
Ui = aq − q 2 + z, (i = x, y, u)(1)2
where q and z respectively denote manufactured goods consumption and agricultural
goods consumption, which are numeraire. Here the budget constraint of each household
iswi + I = pq + z, (i = x, y, u)(2)
where p and I respectively denote the prices of manufactured goods and redistribution
for each household. Maximizing (1) subject to (2), the demand function of manufactured goods is given as shown below.
Q=a−p

(3)

Moreover, each household’s indirect utility function is derived as follows.2
1vi = (a − p) + wi + I, (i = x, y, u)2(4)

Next we refer to urban unemployment. Let Lx and Lu respectively represent thenumber of households employed in the manufactured goods sector and unemployedhouseholds. Here we deﬁne λ as the unemployment rate in the urban area.
λ≡LuLx + Lu

1(5)Let x, y, and u respectively represent the manufactured goods sector, agricultural goods sector,and unemployment.
2
Here wu is equal to zero because they are not employed in the manufactured goods sector.

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Following Harris and Todaro [?], the expected utility in an urban area is equal tothe utility in the rural area in equilibrium. Substituting (4) into this condition, themigration equilibrium condition between regions is as follows.
(1 − λ)wx = w(6)Because we assume that the population is normalized, the population constraint isgiven as shown below.
Lx + Lu + Ly = 1(7)Combining (5) with (7), the equation (5) can be rewritten as follows.
Lx + (1 − λ)Ly = 1 − λ

2.2Production

2.2.1

Agricultural goods sector(8)

Next we examine the producer’s behavior in detail. We assume that the manufactured
good sector in the urban area is a mixed duopoly, which comprises a public ﬁrm and nprivate ﬁrms. However, the agricultural goods market in the rural area is competitive.
Although the manufactured goods sector requires only labor as input, the agricultural
goods sector requires both labor and land as input. Assuming that S̄ is the ﬁxed land
input and that it has normalized land input to one, we set up the production function
as shown below.
Y = (Ly )σ (S̄)1−σ , σ ∈ (0, 1)(9)Because the market of agricultural goods is competitive, the following wage rate inrural areas is derived because of proﬁt maximization.
w = σLσ−1y(10)Here the sum of the proﬁt in the agricultural goods sector and land rent are redistributed to each household; also, I signiﬁes the redistribution per capita.
2.2.2

Manufactured goods sector

Presuming that there is one public ﬁrm and n private ﬁrms in the manufactured goods
sector, then the market-clearing condition becomesQ = a − p = x0 +n∑

xj .

(11)

j=1

¿From (??), the inverse demand function of manufactured goods is obtained as
p = a − x0 −n∑
j=1

3xj .

(12)

Assuming that public ﬁrms and private ﬁrms have the same production technology,
that is 3Li
xi = x , (i = 0, · · · , n).
(13)mBecause the only input factor to produce manufactured goods is labor and the urban
wage rate wx is ﬁxed, the cost function of the public ﬁrm and private ﬁrms in the
manufactured goods sector is given as
Ci (xi ) = mwx xi ,(i = 0, · · · , n).

(14)Consequently, the proﬁt functions of public ﬁrm and private ﬁrms are derived as follows.
π i = a − x 0 −n∑xj  xi − mwx xi ,(i = 0, · · · , n)

(15)

j=1

Here the social welfare function comprises consumer surplus, the proﬁt of the publicﬁrm, and household income, as2nnn
∑
∑
∑
1
x0 +
xj  + π 0 +
πj + wx L0x +Ljx  + wLy + I.
2
j=1
j=1
j=1

(16)

We consider that the government owns the share of (1−θ) of public ﬁrms, as Matsumura
assumed (1998). The purpose of public ﬁrm (ﬁrm 0) is to maximize the weightedaverage of social welfare and its proﬁt, which is deﬁned as V (θ).
n∑

2n∑

n∑

1V (θ) = π0 + (1 − θ)  x0 +xj  +πj + wx L0x +Ljx  + wLy + I  (17)
2
j=1
j=1
j=1

Because the private ﬁrms maximize proﬁt and the public ﬁrm maximizes V (θ), theﬁrst order condition of private ﬁrms is derived as follows.4
a − x0 − (1 + n)x − mwx = 0(18)The public ﬁrm aims to maximize the weighted average of social welfare and its proﬁt
V (θ) with respect to production. Consequently, the ﬁrst order condition of public ﬁrm
is obtained as follows.
a − (1 + θ)x0 − nx − θmwx = 0(19)Solving (18) and (19), the Nash equilibrium outputs (x∗0 , x∗ ) are obtained as follows.
x∗0 =a + [(1 − θ)n − θ]mwx(1 + n)θ + 1(20)aθ − mwx(1 + n)θ + 1(21)x∗ =34The setting of production function follows Daitoh[1]The private firms are symmetric.

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3

Comparative statics

Because we derived the behavior of the manufactured goods sector, agricultural goods
sector, and households in the previous section, we consider the eﬀect of the privatizationof public ﬁrm on the urban unemployment rate. Let X ∗ represent the total amount of
manufactured goods production. Considering () and (), X ∗ is derived as follows.
Q∗ = x∗0 + nx∗ =(1 + nθ)a − (1 + n)mθwx(1 + n)θ + 1(22)Consequently, diﬀerentiating (22) with respect to θ, the following comparative statics
result is obtained.
a + (1 + n)mwx∂Q∗=−<0(23)∂θ[(1 + n)θ + 1]2
From (23), we know that the privatization of public ﬁrms engenders a decrease thetotal output of manufactured goods in the urban area. The necessary labor input ofpublic ﬁrm and private ﬁrm in the manufactured goods sector is given as (13), (20),and (21). Consequently, the total number of employed households in urban area L̄∗x isobtained as follows.
m[(1 + nθ)a − (1 + n)mθwx ]L̄∗x =(24)(1 + n)θ + 1
Diﬀerentiating (24) with respect to θ, we can ascertain the eﬀect of privatization ofpublic ﬁrm on urban employment.
∂ L̄∗x∂Q∗=m<0(25)∂θ∂θConsequently, privatization of public ﬁrms engenders a decrease in the required labor
input, too.
Next we refer to the eﬀect of privatization of public ﬁrm on urban unemployment.
Combining (10) with (6), the unemployed labor in the agricultural goods sector satisfying the migration equilibrium condition is the following.
[(1 − λ)wx
Ly =σ] 1σ−1(26)Substituting (24) and (26) into (8), the equilibrium unemployment rate in the urban
area is determined using the following equation.
() 1wx σ−1+λ−1=0(27)σDiﬀerentiating (27) with respect to θ, the following relation between urban unemployment and the privatization level of the public ﬁrm is obtained.
σL̄∗x + (1 − λ) σ−1∂λ= −()∂θ
σ1−σ∂ L̄∗x∂θ(1 − λ)1σ−1(wxσSumming up, we derive the following proposition.
5) 1

σ−1>0+1(28)Proposition 1 If the government proceeds in the privatization of public firms in anurban area, then its privatization decreases the total output of manufactured goods.
Moreover, the urban unemployment rate worsens because of the public firm privatization.

4Concluding remarks

As described in this paper, we construct the model by combining the Harris and Todaro
model with a mixed oligopoly model. This paper presents an examination of the eﬀectof privatization of public ﬁrms on urban unemployment. The main contribution of thispaper is the demonstration that the privatization of public ﬁrms engenders a decreasein the total output of manufactured goods and worsens the urban employment. The
results of our analysis in this paper are that it is necessary to consider the eﬀect ofprivatization of public ﬁrm on urban employment because privatization of public ﬁrms
can decrease the required labor input in an urban area. These results ﬁt well intosome reality of urban unemployment and privatization of public ﬁrms in developing
countries such as China.
We simplify the model by setting its analysis simply. Particularly, we assume alinear production function of the manufactured goods sector and a quasi-linear utility
function in this paper. Consequently, it is necessary to analyze this issue under a moregeneral production function or utility function.

References[1] Daitoh, I. (2010) “International oligopoly market and the strategic trade policy indeveloping countries: the unemployment problem in developing countries and theoptimal export policy in developed countries,” Mimeo[2] Fujiwara, K. (2006) “Trade patterns in an international mixed oligopoly,” Economics Bulletin, 6, (9), p.1–7.
[3] Harris, J.R., Todaro, M.P. (1970) “Migration unemployment and development: atwo-sector analysis,” American Economic Review, 60, p.126–142.
[4] Matsumura, T. (1998) “Partial privatization in mixed duopoly,” Journal of Public
Economics, 70, p.473–483.

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