A Variety Expansion Model of Growth with Rent
Seeking Activities
Zhiwei CENyand Junko DOIz
December 2006 (preliminary)

Abstract
The purpose of this paper is to examine how rent seeking activities a¤ect on
economic growth. We construct a model which incorporates two types of household based on Caselli and Ventura (2000) into a variety expansion model. The
economic growth rate depends on the market share of a main …rm in intermediate goods sector. Therefore, …rms engage in rent seeking activities in order tokeep their market share or monopolistic pro…t forever. Then, the bureaucracy
can increase their utility from gift provided by the …rms. Whether rent seeking
activities may increase the rate of economic growth or not depends on source ofrent seeking.

JEL Classi…cation code: O4, L12
Keywords: rent seeking, monopolistic competition, R&D-based growth model

We thank Dinopolus, Junichi Itaya, Noritaka Kudo, Toru Maruyama, Kazuo Mino and Kazuhiro
Yuki, and seminar participants at Hokkido University and Keio University.
yKyoto Sangyo University, Kamigamo Motoyama, Kita-ku, Kyoto, 603-8555, Japan, Email:cen25@cc.kyoto-su.ac.jpzKansai University, 3-3-35 Yamatecho, Suita, Osaka, 564-8680, Japan, E-mail:jkdoi@ipcku.kansaiu.ac.jp1

1

Introduction

The purpose of this paper is to examine relationship between rent seeking1 and economic growth. Rent seeking activities have a long history, and it has been repeated ateach era, each place and each situation. Even recently, we remember Microsoft as an
example. The article said that Microsoft, which is one of the biggest companies in theworld, o¤ers a big political contribution to Washington in 1997. When the administration of justice tries to strictly apply the Antimonopoly law to the company, in order to…ght this, President Bill Gates o¤ers about $200 millions for lobby activities in 1997.
It is said that the sum of such activities is 67% up compared with 1996. This showsthat the lobby activity is still useful for keeping their pro…t.
Therefore, many researchers have shed light on this problem since 1960’ (Tullock(1967), Krueger (1973), Baumol (1990), and so on). However, it seems that theliteratures disagree with the e¤ect on economic growth of rent seeking. One view ofthese discussion is that rent seeking activities and corruption can promote economic
growth, because such activities give bureaucracies incentives to remove cumbersome
regulations or work speedy. Bardhan (1997) introduces historical episodes which rent
seeking activities can enhance economic growth and Lui (1985) and so on argue such
positive e¤ect from theoretical aspect. The other point of view is that such activities
lower growth rate, because they distort the allocation of resource. Mauro(1995) …nds
that such activities have a negative impact on the growth rate from evidence. Murphy
et al.(1993) and Angeletos and Kollintzas(2000) show that the growth rate declines
by rent seeking, because a part of workers is placed to nonproductive section, whichis rent seeking activities, lobby activities and so on, ine¢ ciency in this economy increases. Therefore, Hall and Jones(1997) say that corruption can be treated as one ofextortionately tax.
While we …nd a great number of papers have been written on the subject, there are a
few literature which argue rent-seeking and corruption in the framework of endogenous
growth models. We …nd some paper based on the second view, that is, the distortion of
resource allocation through rent seeking and corruption leads to the decline of economic
1

We know similar words as rent seeking. That is, corruption, rent seeking and lobby activities.
Wikipedia, the free encyclopedia by internet, says the di…nition of three words as follws. Politicalcorruption is the misuse of public o¢ ce for private gain. Corruption arises in both political andbureaucratic o¢ ces and can be petty or grand, organized or unorganized.
The phenomenon of rent-seeking was …rst identi…ed in connection with monopolies by Gordon
Tullock, in a paper in 1967. It takes place when an entity seeks to extract uncompensated value fromothers by manipulation of the economic environment –often including regulations or other government
decisions.
Lobbying is the practice of private advocacy with the goal of in‡uencing a governing body, in orderto ensure that an individual’s or organization’s point of view is represented in the government.

2

growth. What is resource for the growth in literatures of this topic? Sarte(2001) andJalali-Naimi and Karimi (2003) consider it government spending, then they …nd thereason why the smaller provision of government spending. That is, agency problem
between government and private sector result in reducing it, so that economic growthgoes down compared with the case without such problem. Using the model based onhuman capital2 , Angeletos and Kollintzas(2000) argue that the existence of rent seeking
and corruption distorts equilibrium allocation, as a result, the growth rate also goes
down. In our paper, we consider the engine of growth as technological progress in orderto analyze how rent seeking activities a¤ect economic growth. We construct our model
based on a variety expanding model, Romer (1990) and Grssman and Helpman(1991).
One of the most important characterization of discussion about rent seeking andcorruption is that much existing literatures assume that just as if the resource disappear
from the economy by such activities. That is, productive resource in the economy
change in unproductive. For example, let us see Angeletos and Kollintzas. In theirpaper, intermediate good …rms employ productive and unproductive workers, which
engage in rent seeking activities and get demand from …nal goods …rm. Since rent
seeking activities reduce productive activities, the level of intermediate goods decreases.
So that output of …nal goods also decreases, and the rate of growth rate goes down.
Certainly, rent seeking activities and corruption are the extra cost for the …rm whichburdens it. However, since people who can receive it may increase their utility by somepresents from rent seeking activities, it is possible to say that such activities give some
contribution to the economy. From this point of view, in our model, we introduce
heterogenous agents based on Caselli and Ventura(2000). That is, there are two types
of households which can or cannot get gift from rent seekers. The former people workat the government sector and the latter at the private. The former people have a power
to keep market share of a …mr or determine which …rm has in…nite monopolistic right.
Therefore, …rms in private sector o¤er some bribe to the o¢ cials to keep their pro…t.
On the other hand, the bureaucracy request some gift or presents to …rms in order
to raise his utility. Like this, the most di¤erent point in existing literatures is that
we consider rent seeking activities and corruption as not unproductive activities but
productive activities for a part of agents in this economy.
In this setting, the second purpose of this paper is to analyze dynamics of heterogeneous agent model based on Caselli and Ventura(2000). We show the economy has
a unique balanced growth path, and investigate the characterization by comparative
analysis.
This paper is organized as follows. In section 2, we construct the base model.
2
They also argue the case where technological progress is the engine of growth, however, that isthe extention of base model, and not comolete.

3

In section 3, we investigate the characterization of balanced growth path (BGP). In
section 4, we expand the base model to incorporate three cases which are di¤erent
source of rent seeking activities. In section 5, we examine the characterization of BGP
which reduced from section 4. In section 6, concluding remarks will be shown.

2

The Base Model

We expand a variety expanding model to include heterogeneous agents and rent seeking
activities. We assume that there are many in…nitely lived consumers, indexed by j inthis economy. We allow two types of agents, which are distinguished by working sector, workers in private and government sectors. We assume that the number of workers
in the government sector are …xed. People in government sector, the bureaucracy orpoliticians, can get some contributions from intermediate goods …rms. Since o¢ cials
have a power to determine which …rm should have the in…nite patents to monopolistically produce ith good, the …rm desiring to have the right are willing to o¤er a part of
pro…t as bribe. Such gifts or presents increase the o¢ cials’utility.

2.1

HouseholdWe assume that the number of population in this economy, L, is large, constant andthat each consumer is small in the sense that his or her choices have negligible e¤ects onmarket equilibrium. Following Casse and Ventura (2000)3 , the consumer j maximizes
his or her utility as follows,
Uj =

Z 1
0

(cj +

jg

1

)1

e

t

dt

(1)

where cj is consumption of consumer j, and g is the special service that some consumers
receive from rent seeking …rms. There are two kinds of consumer heterogeneity in thiseconomy. Firstly, only people working at government sector can get some gift.
isparameter which stands for how degree the receiver takes it into his or her utility.
Larger means that o¢ cials feel happier than smaller when they get some bribe.
Moreover, the second source of consumer heterogeneity is also represented by the degreeof social power, as measured by j . The higher j ; the stronger social power he has.
That is, if he is a high class bureaucracy, he can get more large amount of special service
provided by rent-seeker. On the contrary, if he is an young or lower class bureaucracy
who does not have so much power, then j is smaller. We assume that the sum of j
3

Although g is public goods in Casse and Ventura(2000), we de…ne g as gift which only bureaucracies can get in this paper.

4

RL
is equal to 1, that is 0 j dj = 1: Since this service is not provided through market,
consumers face to the following budget constraint,s_ j = rsj + wlj(2)

P cj ;

where sj denotes the asset holding of the household j, w is wage rate and P denotes
the price of consumption good. We assume that in each moment household supplies
l units of labor inelastically to private sector. The number of workers in government
sector is …xed, L; and they do not get any payment. Therefore, they have only asset.
Setting and solving Hamiltonian as usual, the optimal conditions for this problem givethe Euler equation,
c_j =

r

(cj +

j

g )

j

g

1

g_

(3)

_

where = PP : This Euler equation shows that the growth rate of consumption depends
on the degree of the social power. If he works at the private sector, in the case of
j = 0; then the Euler equation comes back usual form.

2.2

Producers

2.2.1

Final goods

Final goods, Y is produced by intermediate goods. The production function is givenby"
Z A " 1
" 1
"
Y =
xi di
;" > 1(4)0

where xi is intermediate good i 2 [0; A] and " denotes the elasticity of substitution
among intermediate goods. The range of intermediate goods, A, is expanding throughR&D activities.
Considering cost minimization problem of …nal goods sector, we get the following
inverse demand function,P"xi = " Y:(5)piwhere P is price index, P =

2.2.2

1
RA 1 "
1 "
p
di
:
i
0

Intermediate goods sector

Firms producing intermediate goods produce their goods based on the design createdby the R&D …rm. We assume that one unit of intermediate good is produced with one

5unit of labor. Each …rm in intermediate goods sector engages in rent seeking activities
in order to keep patent or permits necessary for the conduct of business or their marekt
share. Even if the …rm is a monopolistic producer for good i until t period, since the…rm always faces to potential competitors which aim to get rid of the monopolistic
pro…ts. If a …rm have almost market share of the goods, he also is always in danger to
be took his market share from the competitors. Therefore, the monopolistic producer
has incentive to engage in rent seeking activities, that is, they o¤er some type of gift
to bureaucracies. If o¢ cials receive gifts, they accord facility for leaders of ith goods
within that period. Therefore, the leader or a monopolistic …rm are willing to burden
this cost each period in order to keep his monopolistic pro…t forever.
Existing Firm Firstly, we consider the …rm which is leader. The …rm always faces
to potential competitors which aim to get rid of the monopolistic pro…ts. Therefore,
the monopolistic producer has incentive to o¤er some type of gift to bureaucracies tokeep their monopolistic pro…ts. Following Jacquemin(1985), we consider the pro…ts ofintermediate goods sector as follows,i = (piw) (xini )biwhere ni is the output level of potential competitors and bi is the rent seeking costto keep their monopolistic pro…ts, in other words, bribe. If the other …rm enters themarket, the existing …rm looses his monopolistic power, and his pro…t is also reducedby ni .
Potential Firm To determine the bribe level, we see the potential competitors. The
potential competitors consider the pro…ts after they can succeed in the entry of market,
so we assume that the most simplest example of the entry rate isn_ i = pi ni

wli

bi ;

where n_ i is the entry rate. The increase in pro…ts raises the incentive of competitors.
For entering this market, the competitors have to pay the extra bribe (bi times ;> 1) than existing …rm. Since the existing …rms set a bribe level in order to keep
their market share unchanged each period, the competitor have to pay more bribes to
get rid of the monopolistic position.
The leader set a bribe level to keep their market share. The all demand for ith
goods is xi : Therefore, the leader sets a bribe level which the entry rate is equal to thegrowth rate of demand. That is,

6

n_ i
x_ i
bi
=
= pi w:nixi
ni
Then, we can know the rent seeking cost for existing …rm as follows,
bi =

ni

pi

w

x_ i
xi

(6)

:

Using this bribe level, the pro…t of the existing …rm is
i = (pi

w)(xi

ni )

ni

x_ i
xi

pi

w

w)

ni 0
pi = 0:Then, pro…t maximization gives@ i= (p0i )(xi@xini ) + (piSince cost minimization of …nal goods sector gives the inverse demand function, pi =1
1xi " P Y " ; the price of ith is given by
"
pi =
w:
" 1 + 1 + 1 nxii
where nxii means the market share of competitors on ith goods. If their market share
increases, then the price of leader will decrease.
Assuming symmetric equilibrium, we can eliminate subscript, i: Therefore, pro…tof the leader is
9
80
1
1 n
<
1
1+
x
n
n
n x_ i =
A
@
h
i
1 w(1
)+
:
(7)
=x
:
x
x
x xi ;"
1
1+ 1 nx

2.3

R&D

The research and development sector creates the new design for the intermediate goods
and is assumed to be competitive. The knowledge creation function is
A_ = LR A;(8)

where LR stands for labor engaged in R&D activities and is the productivity of R&D.
Pro…t ( R = v A_ wLR ) maximization in this sector gives
w = Av;(9)where v denotes the patent price. Moreover, the arbitrage condition is
r=

v_
+ :
v v

7

(10)

2.4

Government

All things government in this economy do is to determine whether to give o¢ cialapproval or not. We assume that the payment for the bureaucracy is the same rate
as private sector and this wage is determined by law. In Japan, when the National
Personnel Authority determines the wage of o¢ cials, it refers to that of private sectors.
Here, we assume that a gift is something to increase the o¢ cials’utility. We do not
specify gift as goods or services, because there is no gift market. It is important thatsuch present can increase utility of a part of people. Therefore, we assume that whenbureaucracies put some source into a black box, they can increase their utility. However,since we con…rm the existence of steady state in this economy, we use following typeof black box. That is, this is a gift function.
Z A

g=

1

1

bi

di

;

>1

(11)

0

where g stands for the total gifts in this economy and is parameter. If the bureaucracyor politicians get some gifts at a period, they permit the right which the briber canmonopolistically produce the good within the period.o

2.5Market Equilibrium

Now, note that market equilibrium in this economy. We assume that the number of
population, L, is constant in each moment. Therefore, full employment condition is
LI + LG + LR = L;

(12)

where LI is labor for producing intermediate goods and is equal to Ax in symmetric equilibrium. LG is the amount of the bureaucracy. In order to make our model
analytically simple, we …x the amount of workers in government sector. That is,
LG = L;
where (0 < < 1) is the constant ratio of workers in government sector in terms oflabor supply.
The market clearing condition of …nal goods is
Z L
Y =cj dj;
(13)
0

where

RL
0

cj dj = c, c is aggregate consumption in this economy.

8

2.6

Dynamic System

Let us show dynamic system in this economy. We set the wage rate, w, as numerarire.
When we examine the movement of this economy, we must consider not individual
variables but the entire economy. As a result, we can analyze this model in the framework of representative consumer model. Equation (10) can be rewritten using (25),(7)and (12), (9),

r=

a+

1
V

a

L

80
1
<
1
1 + 1 nx@hi A (1
1 n
:
"
1
1+
x

_
_
where V = Av, a = A=A= LR and LLII =Therefore, the interest rate can be expressed bynx

LR L_ R:L LR LR

n
n
)+
x
x

!9
_LI =
LI ;

a

(14)
(14) is the interest rate.

r = R(V; LR ):We de…ne V = Av as the total asset in this economy. Assets are distributed to eachhousehold, Av=L = sj , hence using the budget constraint, the movement of total asset
is
V_ = R (V; LR )V + V L q;
(15)
where letting q = P cj L. V is only state variable in this economy.
Denoting m = gc ; the movement of gift is derived by taking the time derivative of(4), (11) and (13), and combining them
g_
=
g

1

LR +

1
(1 + m)r(1 + m) +

1
("

1)

LR (1 + m)

1

m+

"
"

1
(16)

LR

Di¤erentiating m with respect to time and using (3) gives
m
_
=
m

g_
g

c_
c
(
1) (1 + m) r=
(1 + m)(1 + m)(1 + m)+(1 + m)
(" 1)

m
"
+
1 " 1

1
("

1)

1

(1 + m)LR(17)

Following the same step for m, di¤erentiating q with respect to time and using (16)
gives
1
c_
q_
=
a+
q
1 "
c
q
=
(m; LR )9(18)

Moreover, the movement of labor employed in R&D sector is determined by the full
employment condition and the de…nition which is that one unit of intermediate goods
is produced by one unit of labor, that is, xi = LAI : We obtain the movement as follows;
L_ R
=
LR

(1

)L LR
LR
r
1
(1 + m)(1 + m) +1("(1 + m) LR

1)

m
"
+
1 " 1

LR

+ LR :(19)

These four variables, V; m; q and LR ; show dynamics in this economy.

3

Characterization on Balanced Growth Path (BGP)

In this section, we consider characterization of this economy along balanced growth
path.

3.1

The Long-Run Growth Rate

Let see the long run growth rate on each case. This economy shows that the various
quantities ‡ow at constant rates, for this, (15), (17) (18) and (19) must be zero at thesteady state. At the steady state, the existing …rm pays the cost to keep his or her
market share constant. That is, the entry rate is equal to a: From (6), the cost isdetermined asnibi =(pi w + a)
Proposition 1 There are two steady states in this economy. One of steady states isstable, the other is unstable.
Proof. Substituting (14) into m_ = 0 condition,n 2
f (a) =ax

1 + (1

n
)L +
x

B2
" B

+

where
B=1

1+

1
"

1

1

n
x

(

1) a+ (1

)L

B2
=0
" B
(20)

The coe¢ cients of a is positive and the constant term is also positive. Therefore,10we can …nd two growth rates4 to be satis…ed with (20). The growth rates are follows:
h
i
n
1
B2
1 + (1)L x + " B w + " 1 (
1)
rh
i2
2
2
1 + (1)L nx + "B B w + " 1 1 ()L "B B1)
4 nx (1
a=
:
2 nx
(21)

Corollary 2 On the stable equilibrium, market share of the leader is larger, then thegrowth rate may increase.
We show the numerical example in Table 1. If the leader has a small market share,
then he has to set a lower price. Therefore, he get relatively small pro…t, and thislowers incentives to promote technological progress in R&D sector5 .

4

Source of rent seeking

4.1

Intermediate goods sector

In above section, we show that if the leader have more lager part of the market shareof the goods, then the growth rate increases. So, from following section, we consider
the leader engage in rent seeking activities from the starting point. Therefore, they
already have 100% share on the goods. Then, we consider how di¤erent of source onrent seeking activities a¤ects economic growth.
Each …rm in intermediate goods sector must o¤er some type of bribes to bureaucracies in order to keep patent or permits necessary for the conduct of business6 . Even if
the …rm is a monopolistic producer for good i until t period, since the …rm always faces
to potential competitors which aim to get rid of the monopolistic pro…ts. Therefore,
the monopolistic producer has incentive to o¤er some type of gift to bureaucracies. Ifo¢ cials receive gifts, they do not give others permits to produce ith goods, in otherwords, they prevent others from entering the market of ith goods within that period.
i2
h
2
1)
4 nx [ (1We assume that 1 + (1)L nx + "B B w + " 1 1 (5
If the existing …rm has the market share 100%, then the growth rate is
4

a =
6

1
"+(

1)

[ (1

)L

("

1) ] :

This is also along the line which corruption imposes an extortional tax.

11

)L

] > 0:

Therefore, the …rm which already has a patent must pay this cost each period in orderto keep his monopolistic pro…t forever.
The existing …rms set this bribe level by various ways. We investigate three scenarios, according to how the …rm producing intermediate goods o¤ers bribes to thebureaucracies. In other word, the di¤erence among three scenarios depends on thesource of gifts.
Case 1: extra labor
Case 2: a part of intermediate goods
Case 3: a part of sales
In Case 1, we may think the extra labor make more goods to present to the bureaucracies or they work for increasing in o¢ cials’utility, for example, they go gol…ng witho¢ cials. Case 2 is that a constant quantity of intermediate goods o¤ered to o¢ cials. Then, …rms increase prices to compensate pro…ts because of reducing their goods
which they can freely sell in the market. So that the price for …nal goods sector andproducer price become di¤erent. This type of bribes is like consumption tax. Case 3
is that bureaucracies request a part of sales to …rms
Case 1 Following scenario 1, the existing …rms determine his bribe level, then, bi =1 li: Then, the pro…t is written by
1i = p1i xi

w(1 +1 )li ;

(22)

1 wli is the extra cost for the …rm having the ith patent. If the …rm can protect his

pro…t without paying extra cost, it would not be a rent-seeker. Therefore, the condition
for willing to burden 7 is
Z 1
Z t
Z 1
rt
C
rt
rt
d +
d <
d ; = 1; 2
(23)i ( )ei ( )ei ( )e00twhere i is the monopolistic pro…t without any bribe and Ci denotes the pro…ts which
the …rm get in the case where he does not o¤er any gift. From pro…t maximization ofeach …rm, we can show the optimal pricing formula such that
p1i =

"
"

1

(1 +

1 )w:

(24)

This shows that the price of good i is determined by marginal cost of labor input, w;
times (1+ 1 ) , multiplied by the mark-up ratio, " " 1 : That is, the cost of rent seeking
7

In both cases, we asssume that the rate is constant over time for analytical simplicity.

12

is shifted on the intermediate good price, and the marginal revenue is equal to themarginal cost. An increase in price leads to a decreasing demand for intermediate
goods from (5). As a result, the production of …nal goods also decreases. Therefore,
rent-seeking activities, in addition to distortion derived from this monopolistic market,
lead to more distortion to resource allocation in this economy. The fact that corruption
reduces the output level of intermediate goods is the same mechanism as Angeletos andKollintzas.
Case 2 Let us consider Case 2. Similarly to Case 1, the monopolist determines abribe in order to prevent any …rms from getting rid of his market. Then, he decides abribe level as follows,
s2i = 2 xi
Then, the pro…t of the …rm producing intermediate good i is given by
2i = (p2i2 )x2iwi li :

(25)

The …rm burdens rent seeking cost on each goods. In this case 2 represents the same
bribe rate imposed on all goods. That is, this type of bribe resembles consumption
tax. Then …rms can receive pro…ts (pi
2 )xi . From pro…t maximization of each …rm,
we can show the optimal pricing formula such thatp2i ="w + 2:" 1(26)This shows that the price of good i is determined by marginal cost of labor input,
w; plus the rent-seeking costs, 2 : That is, the cost of rent seeking is shifted on theintermediate good price, and the marginal revenue is equal to the marginal cost.
Case 3 Let us consider Case 3. As said above, in this case, the bribe level is s3i =3 p3i x3i :Therefore, the pro…ts of this type are
3i = p3i (13 ) x3i

wx3i

(27)

This means that bribes are a some ratio of sale. In this case 3 represents the bribe
rate imposed on goods. The bribe rate is the same for all goods. Then …rms can get
pro…t p3i (13 ) xi . From pro…t maximization of each …rm, we can show the optimal
pricing formula such that
p3i =

"
"

1
1

1

13

w:
3

(28)

This shows that the price of good i is determined by marginal cost of labor input, w;
multiplied by the mark-up ratio, " " 1 : Since the …rm decreases intermediates which hecan sale, the price increases by 1 1 : That is, the cost of rent seeking is shifted on the3intermediate good price, and the marginal revenue is equal to the marginal cost.
In all cases, the price is independent of i, so we eliminate subscription i from now.

4.2

The other sectors

Final goods, government and R&D sectors behave as the same in the base model,
respectively. Therefore, conditions for market equilibrium are also unchanged. Note
that, ( = 1; 2; 3) represents Case 1, Case 2 and Case 3. respectively.

4.3

Dynamic System

Let us show dynamic system in this economy. The di¤erence between base model andthree cases in the following model is only intermediate goods sector. Therefore, it isshown in the interest rate in dynamic system. Equation (10) can be rewritten using(22),(25),(27) and (12), (9),
r1 =

1+

r2 =

1+

r3 =

1+

1
"

1

a+

1
"

1

(1

) L;

(29)

1

( V + 2)
1
( V + 2)
a+
(1
) L;
(30)
" 1
V
" 1
V
" 3+1
1
" 3+1
1
a+
(1
) L (31)(" 1) (1V(" 1) (1V
3)
3)

These four variables, V; m; q and LR ; show dynamics in this economy.

5

Characterization on Balanced Growth Path (BGP)

In this section, we consider characterization of this economy along balanced growth
path.

5.1

The Long-Run Growth Rate

Let see the long run growth rate on each case.

14

Case 1 This economy shows that the various quantities ‡ow at constant rates, for
this, (15), (17) (18) and (19) must be zero at the steady state. Substituting (29) intom
_ = 0 condition gives us the rate of technological progress at the balanced growth
path,
(1
) L
(" 1)
a1 =
:
(32)
"+(
1)
In order to de…ne a feasible steady state, the parameter values should satisfy thefollowing conditions,(1) L(" 1) > 0;and"+(1) > 0:Under these conditions, we ensure the technological progress expressed by (32) is
positive. From (32), we …nd that if the number of bureaucracy that are unproductive
workers in this model increases, then the growth rate decreases. This may imply thatthe small government will be better than big one for the economic growth rate. We…nd that the long run growth rate in Case 1 is independent 1 :Case 2 Similarly to Case 1, substituting (30) into m_ = 0 condition gives us the rateof technological progress at the balanced growth path,
a2 =

(1 +

2 ) (1) L("
f" + 2 + (
1)g

1)

:

(33)

When the feasible condition is satis…ed in Case 1, at the same time, (33) is positive.
To analyze the characterization of the steady state, we assume > 18 and use thefollowing lemma.
Lemma For the existence of feasible steady sate, = " 11 is required.
Proof. From q_ = 0 condition, the following relationship is allowed at the steady state,
c_
1
=
a:
c
" 1

(34)

As technological progress proceeds, the economic growth rate increases from …nal goods_
market clearing condition, YY = cc_ . The less competitive the market of intermediate
goods is, the faster the growth rate increases. At the BGP, from (17) m_ = 0 condition
says that the growth rate of gift is times as that of consumption goods, gg_ = " 1 1 a.
8This condition is required on welfare analysis.

15

The growth rate of gift is derived from di¤erentiating (11) with respect to times. Thus,
we obtain
g_
=
g
=

1
1
1

a+

x_ i
xi

a:

Substituting (34) and (35) into m_ = 0 condition, we …nd thatexistence of the BGP.

(35)
= " 11 is required for

Let us consider = 1 is the benchmark case. Smaller (< 1) means that the degreeof contribution in their utility is small. For example, suppose that an o¢ cial is givenbeer by the …rm. If the person prefers wine over beer, then his utility will not likely
increase. We may say that small may re‡ect this situation. Similarly to this, when
the bureaucracy receives chocolates, and like them very much their utility will increase
more than the …rms expected. This case may be captured by bigger (> 1):

Case 3 Similarly to Case 1, substituting (31) into m_ = 0 condition gives us the rateof technological progress at the balanced growth path,a3 =( " 3 + 1) (1) L (" 1)
:
f" (1
1)g
3) + ((36)

When the feasible condition is satis…ed in Case 1, at the same time, (36) is positive.

5.2

The E¤ect of rent-seeking activities

Next, let us consider how the higher the rate of bribe a¤ects on the economic growthrate. We can obtain the following proposition between rent seeking level and economic
growth.
Proposition 3 When the …rm o¤ers more bribes to the bureaucracy, whether it canpromote economic growth or not depends on the type of bribe.
(1)Rent-Seeking activities have no impact on economic growth, when intermediate goods…rms give bureaucracy more workers as a bribe (Case 1).
(2)Rent-Seeking activities have a positive impact on economic growth, when intermediate goods …rms give bureaucracy a part of intermediate goods as a bribe(Case 2).
(3)Rent-Seeking activities have a negative impact on economic growth, when intermediate goods …rms give bureaucracy a part of their sales as a bribe (Case 3).

16

Proof. Firstly, we …nd that the long run growth rate in Case 1 is independent 1 in(32): Therefore, the increase of rent seeking cost cannot a¤ect economic growth in Case
1.
Next, di¤erentiating a2 with respect to

2 ; we obtain

@a2
(1
)L(
=
@ 2
f" + 2 + (

2)
1)g2

(37)

To analyze (37), let us focus on the numerator. Even if (1
)L(
2)
< 0,
(1 )L( 2)then " =is given by f (" ) = 0 is always smaller than 1, therefore[ (1 )L+ ](37) is always positive, because we assume that " > 1: Then, we can draw Figure 1.
When the …rm o¤ers more bribes to the bureaucracy, it can promote economic growth.
However, the impact is getting smaller as 2 is larger.
Finally, di¤erentiating a3 with respect to

3 ; we obtain@a3( ") f (1) L (" +
=
@ 3
f" (1
3) + (

2) + ("
1)g2

1) g

< 0:

So, in this case, the more bribe has a negative impact on economic growth.
What occurs when 2 increases? Then, the economy is a¤ected by two channels:
the positive and negative impact. Price of intermediate goods increases, dp2i = " " 1 d 2 ,and this leads to decrease in demand, dxi = "P " pi " 1 Y dp2i from …nal goods sector.
Decrease in demand leads to decrease in labor demand. That is, labor employed inintermediate goods sector decreases. As a consequence, labor will shift from the production activities to R&D sector, which can promote economic growth. This is the
positive impact on the long run growth rate. On the other hand, these change a¤ect
the pro…t of intermediate goods sector. The pro…t decreases by
d 2i =

xi d 2 :Decrease in pro…t of intermediate goods sector makes the value of this …rm decrease.
This leads to a decrease in the wage rate of labor in the R&D sector. Then, labor will
shift R&D sector to production sector. This is negative impact on economic growth.
The e¤ect on rent seeking activities depends on which e¤ect dominates in this economy.
In this monopolistic competitive economy, the …rst impact always dominates the seconde¤ect. However, the size of the impact depends on market structure. Using (12) and(33); we obtain labor in intermediate goods sector at the steady state as follows:
LI =

[" +

2] (1
)L
("
f" + 2 + (
1)g
17

1)

:

(38)

where LI > 0: If rent seeking cost increases, the labor in this sector decreases. The
impact in less competitive economy is larger than more competitive: Therefore, theformer e¤ect may easily dominate in less competitive economy rather than more one.
In the standard variety expansion model, increasing in price of intermediate goods
through a rise in wage decreases pro…t of this sector, and it reduces demand from…nal goods sector. However, a rise in wage in R&D sector increases in incentive to
move to this sector, so that the growth rate will go up. If we assume that wage isnumerarie in the standard model, then we do not analyze a rise in productive costin intermediate goods sector. In the rent seeking economy, if increase in price occurs
through the change in o¢ cials’attitude, then growth rate increases in more competitive
rather than less one. Since the bribe in this economy is required in order to keep hismonopolistic position, it is the barrier to entry.
Note that Case 1 and Case 3. From above discussion, we …nd that the bribe canreallocate labor from intermediate goods to R&D sector9 . However, when extra labor
is required for rent seeking activities, the e¤ect of reallocation is cancelled out. Since
rent seeking cost is treated as unproductive factors in existing literatures, one moreunit of cost will make the growth rate decrease. On the other hand, in this paper,
such cost is productive cost for a part of people, it may increase in the growth rate.
Therefore, the positive and negative e¤ect is cancelled out, so that bribe does not haveany e¤ect on economic growth.
In Case 3, since the existing …rm loose a part of sales, their monopolistic pro…ts
decreases. This leads to decreasing their value and wage for labors. Therefore, labors
move to more attractive …rms, which is intermediate goods sector. Therefore, growthrate will decrease.
Corollary 4 The less competitive economy, the larger the impact on economic growthis.
Proof. Moreover, let us focus on " in Case 2.
@
@"

@a2
@ 2

=

2 f" +

1)g f (1)L(f" + 2 + (
1)g2

2+(

2)

g

(39)

We can obtain Figure 2, that is, the bigger " is, the smaller the value of @@a2 is. This2
shows that the economy with the less competitive market can enjoy more rapid growthrate when they o¤er one more unit bribe to bureaucracies.
9

Futagami and Doi (2004) have shown that an increase in commodity tax rates reduces the demandfor consumption goods and reallocates laobr from production of the goods to R&D activities, as aresult, the growth rate in long run increases.

18

5.3

Dynamics

Let us examine the stability property of our dynamic system.
Proposition 5 The stability of the long run equilibrium is saddle point stability.
Proof. This economy has one state variable V , and three jump variables (m; q; LR ).
The variable V is always constant from de…nition, V = 1: Moreover, the dynamic
behavior of m; q;and LR are independent of variable q. Since we know the steady statevalue of q using V_ = 0 and the growth rate at the steady state derived from conditionsm_ = 0 and L_ R = 0. We may focus on the dynamics of m and LR : Applying Taylor
expansions to equations (17) and (19) around the steady state (m; LR ) and rearranging
the terms, we obtain the following :dz_
dd= 11 12_d21 d22
dLR

m
LR

m
;
LR

where

d11 =

d12

LR

r

1
(1 + m)+(1 + z)1+m=((1 + z)

d21 =
d22 =
r10 =
r20 =
r30 =

LR +

1

1
"

LR +
1)

r0

1
(1 + z)

r
+

LR

1

+

1
("

1
("
r

1)
+

1+ z
("
0
r
1
+
(1 + m) (1) < 0;
(" 1)
"
;
" 1
1
1+
(1 + ) :
" 1
" 3+1
1
1+
(" 1) (1
V
3)

LR

1)

1

LR ;
"

+

1

1
1)

LR

1

"
1

1

;

LR ;

It is easy to …nd that the eigenvalues of this system are 0 and negative, evaluated thesteady state. In this economy, we have one state variable and three jump ones. Now,
since we …nd one negative eigenvalues, this system exhibit saddle point stability in longrun equilibrium.

19

6Conclusion

We have constructed a model based on a variety expansion model incorporating two
types of agents, simple workers and the bureaucracy. We assume that some people whohave a special social power, like the bureaucracy or politicians, can get some gifts fromrent seekers, but others cannot get any gift. Utility of the bureaucracy is created bysuch gift and consumption subject to their budget constraint, on the other hand, since
the other people who do not have any special power (simple workers) cannot get any
gifts, their utility is from only consumption goods.
Based on this setting, we …rstly have considered that how rent seeking activitiesa¤ect on economic growth, when the leader tries to keep their market share constant. If
the market share of leader is small, the pro…t is also small, and the rate of technologicalprogress is smaller. Therefore, the existing …rm try to keep their market share using
rent seeking activities.
Second, we have investigated that how rent seeking activities a¤ect on economic
growth according to three scenarios. That is, …rms in intermediate goods sector pays
rent seeking cost as workers (Case 1), intermediate goods or something like a speci…ctax (Case 2) and a part of sales (Case 3). This economy has a unique balanced growth
path in three cases. Although such activities do not have any impact on the longrun economic growth in Case 1, Case 3 shows that they have negative impact on theeconomic growth. However, we …nd that such activities have a positive impact on thegrowth rate along the path in Case 2. The less competitive market the economy is,
the bigger impact they have. That is, rent seeking cost plays a role of reallocationof labors. This may be the adverse result from existing literatures which consider
rent seeking as unproductive activities for the economy. In other words, they consider
that just as if some resource may disappear from the economy. In this paper, we
think rent seeking as not unproductive resource but productive for a part of people
in the economy. In this setting, while resource does not disappear from the economy,
it is also the barrier for the potential competitor who try to get rid of monopolistic
pro…ts, and such activities play a role to reallocate labors. On the stability of longrun equilibrium, Caselli and Ventura (2000) con…rm the saddle point stability underRamsey-Cass-Koopmans technology. We also show that the stability is kept under avariety expansion model.
In this model, we treat the o¢ cials request as exogenous. How do propositions
change introducing bureaucracies’optimization? This is left for the further work.

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21

[15] Sarte, P.D. (2001),“Rent-seeking bureaucracies and oversight in a simple growthmodel, ”Journal of Economic Dynamics and Control 25, 1345-1365.
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22

Numerical Example

Table 1:a2

a180.00160.00140.00120.00100.00
80.00
60.00
40.00
20.00
0.00
0.00

μ
0.20

η
1.50

n/x
0.20

ε
13.00

2

0.40

0.60

0.80

θ
1.5

ρ
0.05

w

L

1.00

10.00

φ

Figure 1a2

εFigure 21