Core-Periphery model: does perfect competition
matter?
Sergey Kichko
National Research University Higher School of Economics, pr. Rimskogo-Korsakova, 47, St.
Petersburg 190068, Russia, +7 (812) 994 83 11, [email protected], corresponding
author
Evgeny Zhelobodko (1973-2013)
National Research University Higher School of Economics, pr. Rimskogo-Korsakova, 47, St.
Petersburg 190068; Novosibirsk State University, ul. Pirogova 2, Novosibirsk 630090, Russia
Abstract
We consider a two-sector agglomeration model a-la Krugman but replace the second
perfect competition sector by a monopolistic competitive one. Under this assumption
bifurcation diagram is the same as in classical Core-Periphery model.
Moreover,
assuming Cournot or Bertrand oligopoly in the second sector does not change the
classical result.
There are two key-factors for our result: separation of the markets
under Cobb-Douglas-over-CES utility, and independence of second sector trade ows
from its characteristics. We conclude that the dispersion force in the Core-Periphery
model is the immobile workers' demand.
Keywords: Core-Periphery structure, agglomeration, dispersion forces, long-run equilib-
rium.
JEL codes: F12, F15, R12, R13
1
Core-Periphery model: does perfect competition matter?
1
Introduction
The the role of the perfectly competitive sector in the classical Core-Periphery model is
not discussed in the literature.
We pose a question, whether the type of competition in
the second sector matters for bifurcation, the most interesting feature in the model.
Is
perfect competition assumption crucial for obtaining the same tomahawk pattern of longrun equilibria as in Krugman (1991)?
Tomahawk diagram means that on the path of globalization (trade cost reduction) rst
both countres keeps the manufacturing, then at some stage switch to specialization, i.e.,
the mobile factor (say, skilled workers) ew away from one of the countries, which country
remains ambiguous.
tomahawk diagram.
According to Baldwin et al.
(2003), three eects generate the
Among the agglomeration forces, the market-access eect means
that rms tend to locate near the bigger market, whereas the cost-living eect is the
consequence of rms' location choice, being expressed as the price index changing with
rms' relocation.
The region that attracts more rms has lower price index because of
tougher competition in the local market. By contrast, the market-crowding eect stands
for dispersion forces: it is desirable for the rms to locate in the market with less competition.
But what are the key-factors of dispersion in the Core-Periphery model?
Krugman
(Krugman, 1998) suggests an answer that a dispersion force stems from the immobility of
unskilled workers. Meanwhile, the Starett theorem (Starett, 1977) provides a hint that perfect competition could matter for prevalence of dispersion forces over the agglomeration one
under high trade cost. Under perfect competition, autarky in each region is the only stable
long-run equilibrium.
This result suggests that perfect competition could be a dispersion
force that exceeds agglomeration forces under high trade cost.
Moreover, Combes et.
al
(2005) show that in a two-sector model with homogeneous goods only dispersion equilibrium
is stable. But under increasing level of product dierentiation it is more likely that CorePeriphery structure arises. The reason for this is that a decrease in toughness of competition
between rms leads to a reduction in market-crowding eect.
To shed more light on these questions, we modify the model proposed by Krugman (1991),
introducing monopolistic competition instead of perfect competition in the second sector.
Trade in goods produced by the second sector is still costless and workers are immobile.
Our main nding is that long-run equilibria have the same structure regardless of the
type of competition (whether it is perfect, monopolistic or oligopolistic). This unexpected
2
result stems from the Cobb-Douglas-over-CES preference specication and the free entry assumption. Thus, in a sense, we outline the weakest so far set of assumptions which
generates the Krugman's core-periphery result.
2
The Model
We assume that the world economy includes two countries named Home and Foreign and two
monopolistically competitive sectors in each country. The only production factor is labor.
Labor is heterogeneous across the sectors.
The consumer side includes
required in sector
1,
and
L2
L1
consumers supplying inelastically one unit of labor
consumers doing the same for sector
call the former (latter) agents of type 1 (2). Let
λ
and
(1 − λ)
2.
In what follows, we
stand for the shares of the
rst type agents in Home and Foreign. Each country is assumed to accommodate the same
number of second type agents.
NlH
Let
and
NlF
stand for the number of rms in Home and Foreign in sector
l = 1, 2;
xij
lm (k) is individual consumption of variety k produced in sector l in country i and consumed
ij
ij
in country j by agent type m; plm (k) is the price of xlm (k).
All consumers are identical in preferences. More precisely, each consumer of type m = 1, 2
in Home seeks to maximize their utility given by
ˆ
H
U (Xm
)
N1H
≡
ˆ
σ1 −1
σ1
di +
xHH
1m (i)
ˆ
N1H
ˆ
pHH
(i)xHH
1
1m (i)di +
0
where
H
Xm
!µ· σ1
σ1 −1
σ1 −1
σ1
FH
di
x1m (i)
ˆ
N2H
σ2 −1
σ2
xHH
2m (i)
ˆ
di +
0
0
0
s.t.
N1F
N1F
ˆ
H
FH
pF
1 (i)x1m (i)di +
0
N2H
N2F
H
xF
2m (i)
σ2 −1
σ2
!(1−µ)·
di
0
ˆ
pHH
(i)xHH
2
2m (i)di +
0
N2F
H
FH
H
pF
2 (i)x2m (i)di ≤ wm ,
0
is the whole vector of Home consumption;
H
wm
is the wage of type
m
agent in
Home.
Since CES preferences individual consumption of dierent type agents in the same country
xij
l (k) is the aggregate
demand for variety k produced in sector l in country i and consumed by all agents in
are aggregated. So we introduce aggregate demands in each country:
country
j.
λL1 xHH
l1 (k)
For example, aggregate Home consumption of domestic varieties equals xHH
(k) ≡
l
+ 0.5L2 xHH
l2 (k).
Type 1 consumers' rst order conditions yield the following inverse relative demands:
xHH
1
H
xF
1
!−
1
σ1
pHH
= 1F H ;
p1
F
xF
1
xHF
1
!−
1
σ1
=
F
pF
1
pHF
1
Mirror-image formulas hold for the second type of consumers.
On the production side, both manufacturing sectors present homogeneous rms. Each
rm faces a xed cost of
F
units and a marginal cost of
3
c units of labor.
The total production
σ2
σ2 −1
cost is
sector
C(y) = F wli + cywli , where y is output and wli is the wage in country i ∈ {H, F } and
l ∈ {1, 2}. Second-sector-produced good requires zero trade cost. By contrast, τ > 1
is the iceberg-type trade cost for the rst-sector-produced good. Labor of the second sector
in Foreign is chosen to be the numeraire, and the corresponding wage rate is normalized to
1.
Each rm produces one unique variety (each is produced by single rm). Furthermore,
since both sectors admit a continuum of rms, the number of rms in each sector is big
enough to ignore impact of each rm on the market.
Each Home rm in sector
l
maximizes its prots given by
πlH = pHH
xHH
+ pHF
xHF
− wlH cxHH
− τ wlH cxHF
− F wlH .
l
l
l
l
l
l
Firms' problems in Foreign are symmetric, and thus skipped.
The monopoly pricing rule together with iso-elasticity of demands faced by rms in each
sector imply that prot-maximizing prices for producers in Home are given by
pHH
=
1
w1H · c · σ1
,
σ1 − 1
pHH
=
2
pHF
=
1
w2 · c · σ2
,
σ2 − 1
w1H · τ · c · σ1
,
σ1 − 1
pHF
=
2
w2 · c · σ2
,
σ2 − 1
where w2 = w2H /w2F .
ij
ij
Equilibrium. We dene trade equilibrium as a bundle ({xl }, {pl }, NlH , NlF , wli i, j={H, F }, l={1, 2} )
satisfying consumer's maximization problem, producer's maximization problem, balances on
labor markets and trade ows, consumer's budget constraints and zero-prot condition. In
what follows, we focus on equilibria where both countries produce goods in both sectors.
Mobile workers choose (not) to migrate if and only if they gain (loose) in terms of individual welfare or, equivalently, the real wage, which is the ratio of nominal wage
w1i
to the
average price index across sectors. Formally, it is captured by the ad-hoc dynamic equation
λ̇ = λ(1 − λ)
where
Pji
is price index of
j = 1, 2
w1F
w1H
µ
1−µ −
µ
1−µ
H
H
F
P1
P2
P1
P2F
sector good in country
!
,
(1)
i = H, F .
The long-run equilibria are fully determined by the properties of the equation (1), which
implies our main result.
Proposition. The bifurcation diagram, and therefore the pattern of long-run equilibria, are
independent of whether the second sector is governed by perfect or monopolistic competition.
Proof.
It suces to show that (i)
H
model, and (ii) P2
=
w1H , w1F , P1H
and
P2F , and both are independent of
4
λ.
P1F
are the same as in Krugman
Standardly, we denote the freeness of trade as
ϕ ≡ τ 1−σ1
and relative wage in sector 1 as
w ≡ w1H /w1F .
Lemma. The equilibrium relative wage in sector 1 solves the following equation:
F (w) = 0,
where
(1 + µ)
(1 + µ)
F (w) ≡ (1 − λ) 1 −
− ϕw1−σ1 .
1 − ϕ2 − ϕwσ1 − λ w 1 −
1 − ϕ2
2
2
See Appendix A for the proof.
The function
F (w),
and therefore the relative wage in sector 1, are exactly the same as
in the classical Core-Periphery model (see Sidorov, Zhelobodko, 2013).
Next, we note that the markets are separated in the sense that (i) two separate sectoral
sub-problems of consumer's choice can be considered and (ii) wages, and, therefore, total
expenditure are independent of
σ2 .
Hence, it is readily veried that
P1H
and
P1F
are the
same as in Krugman model.
Last, we consider the price indices of sector 2.
second-sector wage
w2
We show in Appendix B that relative
equals to one. So far, price indices do not depend on wages in both
sectors as well as trade cost. It is readily veried that sector
and Foreign
P2F
are equal, which we denote
P2 =
=
P2F
price indices in Home
P2H
P2 :
σ2
σ −1
P2H
2
σ 2
·
=c· 2
σ2 − 1
F
L2
1
σ2 −1
.
In the classical model, price index of the second sector is equal to one. When we instead
consider monopolistic competition with costless trade in the second sector, price index in
H
this sector (P2 and
P2F ) is a constant (though not necessarily equal to one) and depends on
exogenous parameters only: the elasticity of substitution in this sector and the sectoral size.
Both indices are independent of trade cost
τ,
as well as of
λ.
Thus, price indices of the
second sector are the same for all possible long-run equilibria.
To conclude, the dynamic equation (1) is the same as in classical Krugman model up
to a positive multiplier
P2µ−1 ,
which is independent of
λ.
Hence, the patterns of long-run
equilibria is always the same. Q.E.D.
Discussion.
What makes the above result to be true?
Basically, the only dierence
from classical Core-Periphery lies in the values of trade ows and stems from consumers'
love for variety of the second-sector-produced goods in our model. Love for variety implies
bilateral trade ows in these goods. In a nutshell, only the vector of these ows can aect
the bifurcation diagram.
5
By a trade ow
T ij
we mean gross value of imports from country
i
to country
j.
We
show in Appendix B that trade ows in sector 2 are given by
TFH =
T HF =
L2
2
(1 − λ)Lw1F +
L2
2
1−µ
·
2
1−µ
·
2
λLw1H +
,
.
This implies that the trade ows are independent of any characteristics of the second
sector, in particular, of the elasticity of substitution
σ2
between varieties produced in this
sector. The classical model is obtained from ours as a limiting case when
σ2 → ∞.
This result still holds under asymmetric quantities of second sector workers, since total
volume of trade does not depend on distribution of the second sector workers across countries.
Moreover, under free entry the same long-run equilibrium trade-o arises under any type
of competition in the second sector. This result nds its origin in the nested Cobb-Douglasover-CES preferences, under which the markets are separated.
To explain why free entry
assumption is crucial, note that the equilibrium price and elasticity of substitution under
Cournot and Bertrand oligopoly (see Baldwin et. al, 2003) are given by
p=
cε
;
ε−1
1
εCournot
=
1
1
·
+
σ2
N
1
1−
;
σ2
εBertrand = σ2 −
σ2 − 1
,
N
N stands for the total number of rms in the second sector. Free entry provides the
same N in the economy independently on spatial distribution of employment in both sectors.
where
Consequently, prices for the second-sector-produced goods in both countries are equal to each
other and constant.
The same holds for second sector goods' price indices.
So, oligopoly
in the second sector results in the same dierential equation for real wages as in Krugman
(1991), and hence the same pattern of long-run equilibria.
3
Conclusion
We have studied the question whether perfect competition in the second sector is crucial
for obtaining a tomahawk diagram in the Krugman's Core-Periphery model.
Surpris-
ingly, we have found out that the set of long-run equilibria has the same structure under
perfect or monopolistic competition. Moreover, under free entry this result still holds in an
oligopolistic environment.
Apart from free entry, another key-factor of Krugman's Core-
Periphery result is the Cobb-Douglas-over-CES specication of preferences, which leads
to markets' separation. However, this result is very unlikely to hold under alternative specications and/or oligopoly settings without free entry. Thus, seems that we have outlined
the weakest set of assumptions when the Core-Periphery structure of long-run equilibria is
Krugman-type.
6
Acknowledgements
The authors thank Philipp Ushchev for his kind assistance in preparation of the manuscript.
We specially thank Jacques-François Thisse for a series of valuable suggestions and comments. This study was carried out within The National Research University Higher School
of Economics Academic Fund Program in 2013-2014, research grant No. 12-01-0176. We
also gratefully acknowledge the nancial support from the Russian Federation Government
under Grant No.
11.G34.31.0059 and from the Russian Foundation of Basic Researches,
Grant No. 13-06-00914-a.
References
[1] Baldwin, R., Forslid, R., Martin, P., Ottaviano G.I.P., Robert-Nicoud F. 2003. Economic
Geography and Public Policy. Princeton University Press, Princeton, New Jersey.
[2] Combes, P.-P., Mayer T., Thisse J.-F. 2008. Economic Geography. Princeton University
Press, Princeton, New Jersey.
[3] Krugman, P. 1991. Increasing Returns and Economic Geography, Journal of Political
Economy, 48399.
[4] Krugman, P. 1998. What's New About the New Economic Geography?, Oxford Review
of Economic PolicyJournal of Political Economy, Vol. 14, N 2, 7-17.
[5] Sidorov, A., Zhelobodko, E. 2013. Agglomeration and Spreading in an Asymmetric World,
Review of Development Economics, Vol. 17, issue 2, 201-219.
[6] Starrett, D.A. 1977. Measuring returns to scale in the aggregate and the scale eect of
public goods, Econometrica, Vol.45, N 6, 14391455.
4
Appendices
4.1
Appendix A.
Proof of Proposition.
Aggregate consumption of the rst sector good are given by
H
xF
=
1
xHF
=
1
τ σ1 −1 wσ1 − 1
(σ1 − 1)F
,
·
cτ
τ 2(σ1 −1) − 1
τ σ1 −1 − wσ1
(σ1 − 1)F
,
· σ 2(σ −1)
cτ
w τ 1
−1
7
=
xHH
1
(σ1 − 1)F τ σ τ σ−1 wσ1 − 1
,
· σ 2(σ −1)
cτ
w τ 1
−1
F
xF
=
1
(σ1 − 1)F τ σ1 τ σ1 −1 − wσ1
,
·
cτ
τ 2(σ1 −1) − 1
which is veried by elementary but tedious algebra.
Using labor market balance we get equilibrium number of rms:
N1H =
λL1
,
F σ1
N2H =
L2
,
2F σ2
N1F =
(1 − λ)L1
,
F σ1
N2F =
here
w = w1H /w1F
L2
,
2F σ2
is the relative wage in the rst sector.
Trade balance amounts to
HF
HF
H FH
H FH
N1H pHF
+ N2H pHF
= N1F pF
+ N2F pF
1 x1
2 x2
1 x1
2 x2
or
HF
H FH
F
N1H pHF
= N1F pF
+ N2 p2 (xH
1 x1
1 x1
2 − x2 )
HF
H FH
N1H pHF
= N1F pF
+
1 x1
1 x1
(1 − µ)(λLw1H − (1 − λ)Lw1F )
2
Plugging the equilibrium values into the last formula after simplication yields
h
i
h
i
λ 2wτ σ1 −1 − wσ1 +1 (2 + (1 − µ)(τ 2(σ1 −1) − 1)) = (1 − λ) 2w2σ1 τ σ1 −1 − wσ1 (2 + (1 − µ)(τ 2(σ1 −1) − 1))
and denoting
ϕ = 1/τ σ1 −1
(1 − λ) 1 −
we get
(1 + µ)
(1 + µ)
1 − ϕ2 − ϕwσ1 − λ w 1 −
1 − ϕ2
− ϕw1−σ1 = 0,
2
2
Q.E.D.
4.2
Appendix B
First order condition for producers' and consumers' problems and zero-prot condition are
H
xHH
= w2−σ2 xF
2
2
F
xF
= w2σ2 xHF
2
2
8
=
+ xHF
xHH
2
2
(σ2 − 1)F
c
F
H
xF
+ xF
=
2
2
(σ2 − 1)F
c
We solve this system and after simplication we get
w2 = 1.
So, in equilibrium wages in second sector are the same and
H
xHH
= xF
2
2 ;
F
xF
= xHF
2
2 .
Total revenue of the second-sector rms equals total expenditure on the second-sectorproduced goods:
L2
H FH
,
N2H pHH
xHH
+ N2F pF
= (1 − µ) λLw1H +
2
2
2 x2
2
L2
F FF
HF
N2F pF
+ N2H pHF
= (1 − µ) (1 − λ)Lw1F +
.
2 x2
2 x2
2
H = xH , xF F = xHF = xF , N F = N H = N and pHH = pHF = pF H = pF F = p we nd
Using xHH
= xF
2
2
2
2
2
2
2
2
2
2
2
2
2
2
trade ows
T ij
that given by
TFH =
T HF =
L2
2
(1 − λ)Lw1F +
L2
2
1−µ
·
2
1−µ
·
2
Q.E.D.
9
λLw1H +
,
,