Imperfect competition and Trade, Part II
Karen Helene Ulltveit-Moe
Fall 2010
Contents
1 Product di¤erentiation and Intra-industry trade
2
1.1
The e¤ects of trade in a world of imperfect competition and EoS . . .
2
1.2
Trade Equilibrium: Trade with no transportation cost . . . . . . . . .
2
1.3
Trade with transport costs: The Home market e¤ect . . . . . . . . . .
5
1.3.1
The model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.2
Demand and trade pattern . . . . . . . . . . . . . . . . . . . .
7
1.3.3
Understanding the Home market e¤ect . . . . . . . . . . . . . .
9
1.3.4
Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2 Intra-industry trade and comparative advantage
2.1
10
Characterizing the general equilibrium . . . . . . . . . . . . . . .
11
2.1.1
Inter-industry trade . . . . . . . . . . . . . . . . . . . . . . . .
12
2.1.2
Intra-industry trade . . . . . . . . . . . . . . . . . . . . . . . .
12
3 Appendix: Deriving the demand for CES preferences utility function
14
1
1
Product di¤erentiation and Intra-industry trade
1.1
The e¤ects of trade in a world of imperfect competition and EoS
In a world of economies of scale (EoS) and imperfect competition trade has two types
of e¤ects:
1. The e¤ect of trade on competition and market structure: Trade leads to
an expansion of the market, increases competition and reduces the monopolistic
distortions. This can potentially give rise to two types of gains: (a) longer
production series and lower AC; (b) reduced mark-ups for p > mc:
2. The e¤ect of trade on product selection: Expansion of the markets allows
for the introduction of product varieties that would otherwise not have been
produced, because of "insu¢ cient" demand. (see the equilibrium expression for
number of …rms)
The S-D-S model and trade:
The S-D-S model provides an explanation for intra-industry trade based on a
market failure.
The preference for variety creates intra-industry trade in di¤erentiated products
(di¤erent varieties) between countries
Economies of scale ensures that each variety is only produced by one …rm and
in one country
S-D-S can explain intra-industry trade as well as the gains from intra-industry
trade.
1.2
Trade Equilibrium: Trade with no transportation cost
Key assumptions
Two countries countries as described before open to trade with one another.
One sector of production
2
Suppose that countries are identical in technology and preferences.
The only di¤erence between the two countries is their size LH and LF for Home
and Foreign respectively.
We drop the index for varieties j for simplicity
Given constant markup, prices are equalized and therefore we have w = wH =
wF (factor price equalization).
Equilibrium scale and number of varieties in each country
Given prices, each …rm in each country produces for their domestic market but
also for the export markets.
Prices
pi =
1
wb
A …rm in country i produces, qi : Pro…ts are
i
= pi q i
wli
= pi qi w(bqi + f )
wb
qi wf
=
1
Zero pro…ts condition implies
i
=
wb
qi = q =
1
(
qi
wf = 0
1)f
b
As before, the labor market condition implies that in each country we have
Ni =
3
Li
f
Exports
We can write exports from country i to country n as
Xin = Ni pin qin
= Ni
pin
Pn
1
wLn
since
qi =
pi
Pn1
wLn
(1)
where Pn is the price index for country n:
/
1
Pn = Ni p1in +N
/ n pnn
1
1
The fraction of total income in country n spent on goods from country i is
in
=
Xin
Xn
=
Ni pi qin
Ni pin qin + Nn pnn qnn
and because wages are equalized across countries (w = wH = wF ); technologies
(thus cost functions) are the same across countries, and tastes are the same
across countries, this can be rewritten
in
=
Li
;
Li + Ln
i 6= n
Now we can write exports from country i to country n as
Xin =
Li
Li Ln
wn Ln =
wn
Li + Ln
Li + Ln
Welfare
Finally, to show that there are gains from trade we can write utility in country
i as
Ui =
f(
1)
b
LH + LF
f
1
which is increasing with trade . This is because before consumers now can
consumer more varieties than before.
4
Notice that there is no e¤ect of trade on the scale of production, nor on prices.
All gains from trade come directly through an increase in product diversity, this
is a particular feature of the S-D-S model
In the S-D-S model trade only a¤ects the number of varieties. The reason for
this is the assumption of constant elasticity of substitution. This implies that
demand elasticity also is constant and independent of number of varieties. If
we did not have constant elasticity of substitution, implying that there is not
an in…nite number of varieties, then the elasticity of demand would increase as
the number of varieties increases. As a consequence, trade would also give lower
prices, and longer production series..
1.3
Trade with transport costs: The Home market e¤ect
Key reference: Article by Krugman: Scale economies, product di¤erentiation
and the pattern of trade, in AER
Analyses the consequences of introducing transport costs into a model of imperfect competition and trade:
– Size of the Home market matters for trade patterns rates
1.3.1
The model
Two countries: Home and Foreign (all variables related to Foreign is denoted
by an star)
International trade
Trade incurs transport cost of the iceberg type, with
> 1, i.e. only a fraction
1= of the shipped good arrives.
The price paid by consumers in Foreign for alpha goods produced in Home is
thus pb = p : Same logic applies to beta goods and to the price paid by consumers
in Home for goods produced in Foreign
Note that with transport costs =) supply=demand, but demand6=consumption
if products are imported.
5
Two classes of products (alpha and beta, where a tilde distinguishes beta products from alpha products).
e
Two groups in the population in each country: L and L
– L derives its utility only from the consumption of alpha products
e derives its utility only from consumption of beta products.
–L
Assume same size of Home and Foreign in terms of labor force:
e =L +L
e =L
L+L
(2)
Di¤erent tastes across countries; for simplicity, these take the form of reversed
demand patterns:
L = gL;
L = (1
g)L
(3)
Demand and supply structures secure incomplete specialization in both countries; i.e. both countries participate in the production of both products.
Describes the features of the Home country and note that the Foreign country
is a mirror image
Utility:
U=
X
1
ci
;
U=
i
X
i
1
e
ci
;
>1
(4)
Identical cost functions for the two product groups:
li =
+ qi
(5)
lei =
+ qei
(6)
Product market equilibrium (demand equal supply)
qi = Lci
(7)
e cj
qej = Le
(8)
6
Factor market equilibrium:
n
X
n
e
X
li +
i=1
j=1
e
lj = L + L
(9)
Free entry and zero pro…ts: the sales of each industry must equal the income of
the appropriate group in the population:
npq = wL
(10)
e
n
epeqe = w
eL
(11)
Industrial structure re‡ecting consumers’preferences: Due to identical cost function and full inter–sectoral mobility of labour, we must have that w = w;
e p = pe and
q = qe, we can rewrite (10) and (11) as
1.3.2
L
n
=
e
n
e
L
(12)
Demand and trade pattern
Proposition 1 The home country will be a net exporter of the alpha industry’s products if g > 0:5; i.e. if the Home country has the larger domestic market for alpha
products and the Foreign country has the larger domestic market for beta products.
Proof
Since this is a wholly symmetrical world with countries of equal size and incomplete specialization, wage rates, output and prices of all goods will be equal:
Wages
w=w
Prices
Quantities
p = p = pe = pe =
q = q = qe = qe =
7
1
(
wb
1)f
b
A utility maximizing consumer of alpha products in Home will consume (we use
that p = p )
(b
p )
P1
p
P1
wL
wL
p
pb
=
=
p
p
=
units of a representative imported good for each unit of a representative domestic
they consume, and demand (we use that p = p )
=
(p )
P1
p
P1
wL
wL
p
p
=
1
=
1
units of a representative imported good for each unit of a representative domestic
they demand. As the same relationship applies for Foreign, we have that:
=
=
1
<1
Product market equilibrium in Home and Foreign in the market for alpha goods
is given by:
n
n
wL +
w L
(13)
n+ n
n+n
n
n
n p q =
wL +
w L
(14)
n+ n
n+n
With incomplete specialization there is production in the alpha industry in both
npq =
countries, i.e. n > 0; n > 0: This allows us to divide (13) by (14) –(left side by
left side and right side by right side). Using that p = p ; q = q ; w = w ; and
rearranging we get
n+ n
L
=
L
n+n
which in turn can be further rearranged to get
(15)
1
n
(L=L )
=
1
n
1
(L=L )
(16)
– From (16) we see that if L=L = 1; then n=n = 1; that is, if the demand
pattern of the two countries are the same, so are their production patterns.
– As the relative size of either country’s home market rises for alpha goods,
< L=L < 1= 1 (which
@( n )
is the condition for incomplete specialization), i.e. @ nL > 0
(L )
1
– If L=L <
) n = 0; the demand for alpha products in the Home
so does its domestic production, as long as
1
country is too small to sustain any alpha production
8
– If L=L > 1=
1
) n = 0; the demand for alpha products in the Foreign
country is too small to sustain any alpha production
– If a large fraction of the consumers demanding alpha products are located
in either country, then the whole output of the di¤erentiated output will
be produced in the country with the relatively larger demand.
– For
1
< L=L < 1=
1
, a disproportionate fraction of the output will
be produced in the larger country (w.r.t. demand for this particular good),
i.e.
n
n
>
L
L
– Notice that the width of the band of non-specialization depends on the
transportation cost ( ) and the elasticity of substitution ( ):
– For low transportation cost, the band shrinks, and it becomes more likely
that the production of the di¤erentiated good will agglomerate in a single
location.
– For a high elasticity of substitution (implies less importance of EoS) the
greater the band and the more likely is incomplete specialization
1.3.3
Understanding the Home market e¤ect
Country’s will tend to export products in which they face a relatively high
domestic demand.
Concentration of production in one location allows for more e¢ cient exploitation
of economies of scale, and in choosing location one prefer proximity to the larger
market in order to save on transport costs.
Note that if we had been in a HO or Ricardo world, asymmetric preferences
across countries, make countries net importer of goods where they have a relative
high domestic demand.
The possibility of incomplete specialization is greater the higher are transport
costs and the less the importance of economies of scale (measured by average to
marginal cost which in equilibrium is a function of the elasticity of substitution)
If incomplete specialization, then both countries will import and export both
product groups, but the country with the relatively larger domestic market will
be a net exporter of the respective product group.
9
1.3.4
Extensions
1. Asymmetries in demand: small and large countries Main results remain,
but wages will be lower in smaller countries, as they need to compensate for
their relative smaller market size through lower prices and thus lower wages.
2. General equilibrium: Instead of two symmetric industries, we assume one
perfect competition industry without transport costs and one industry with
imperfect competition, increasing returns to scale and transport costs. If incomplete specialization and production of in the perfectly competitive industry
in both countries, then FPE. The country with relatively larger home market
for the goods from the imperfectly competitive industry will be a net exporter
of these goods, and a net importer of the perfectly competitive goods.
3. Note that depending on market structures in product and factor markets as
well as specialization patterns, asymmetries may be re‡ected in di¤erent ways
(wages, returns to capital, capital mobility, number of …rms etc)
2
Intra-industry trade and comparative advantage
Analyses the interaction between comparative advantage and intra-industry
trade based on product di¤erentiation.
Note that we now have to pay attention to good y, as well as to the factor
market and factor prices.
Identical preferences across countries
Identical technologies across countries
A set of factors, denoted by the vector of factor endowments vh and vf which
have the price wh and wf
Utility
n
X
u = u(
'(xi ); y) =
i=1
Number of …rms: n = nh + nf
10
n
X
i=1
ci
!
=
y1
Production of good y: incurs total cost: B y (wi ); by (wi ) = AC = M C = 1 (as y
is chosen as numeraire)
Price of good y: py = 1
Production of the di¤erentiated good x incurs total cost: B(w; x) = bx (w)g(x)
The cost function B(w; x) is additive separable, and the relative use of capital
and labour is constant
Increasing returns to scale in x production: g 0 <
g
x
Market equilibrium in sector x: (described for country h, analogous for f)
ph (1 +
'00 xh
) = bx (wh )g 0 (xh )
'0
ph xh = bx (wh )g(xh )
)1+
'00 xh
g 0 (xh )xh
=
'0
g(xh )
(17)
(18)
(19)
from which follows that xh (and analogue xf ) is independent of factor prices, and
that xh = xf = x (NB, hinges on underlying assumptions of the cost function).
2.1
Characterizing the general equilibrium
by (wh ) = 1 = by wf
g(x)
g(x)
= px = b x w f
x
x
g(x)
yh byw (wh ) + nh xbxw (wh )
= vh
x
g(x)
yf byw (wf ) + nf xbxw (wf )
= vf
x
bx (wh )
yh + yf = (1
(nh + nf ) =
(20)
(21)
(22)
(23)
) wh vh + wf vf
(24)
wh vh + wf vf =px x
(25)
The general equilibrium is very similar to one where there are two sectors operating
under perfect competition and constant returns to scale.
11
2.1.1
Inter-industry trade
The theory of comparative advantage still explains net trade, i.e. net trade of
the numeraire good y for the di¤erentiated good x.
Given identical preferences and technology across countries, the theory of relative factor abundance applies, and yh ; yf ; nh ; nf are determined by relative
factor endowments.
If countries are identical, not only in terms of technology and preferences, but
also in terms of relative and absolute factor endowments, then there is zero net
trade.
2.1.2
Intra-industry trade
While CA explains net trade, it does not explain gross trade.
Even without net trade, there may still be gross trade. Let TG denote gross
trade,
= nh = (nh + nf ) be country h’s share of production of di¤erentiated
products,and
be the country h’s fraction of world income. Then with homo-
thetic preferences, the home country consumes
cyh =
(yh + yf )
of the numeraire good y, and
cxh = x
of each of the (nh + nf ) di¤erentiated goods produced.
Without loss of generality we may assume that the home country is a net exporter of di¤ erentiated goods.Suppose that it produces the …rst nh of these by
choice of labelling. The home country’s net import of the numeraire is then
cyh
yh = y f
while its exports of varieties of x are nh (1
(1
) yh ;
(26)
) px x;and import of varieties are
nf px x: Total trade is then balanced if
yf
(1
) yh = nh (1
12
) px x
nf px x:
(27)
Gross exports of di¤erentiated goods are of value
nh (1
) px x = (nh + nf ) px x (1
):
(28)
Net exports of di¤erentiated goods are
nh (1
) px x
nf p x x
= (nh + nf ) px x (1
)
= (nh + nf ) px x f (1
= (nh + nf ) px x (
(29)
(nh + nf ) px x(1
)
(1
)
(30)
) g
)
Since we have chosen labels so that the home country is a net exporter of
di¤erentiated products, then we must have that
> : For the foreign country
gross exports of di¤erentiated goods are (nh + nf ) px x(1
) :For the world as
a whole the value of gross trade, TG , in di¤erentiated products is
TG = (nh + nf ) px x f (1
) + (1
) g
(31)
while that of net trade in di¤erentiated products is, TN ; is
TN = (nh + nf ) px x (
):
(32)
The di¤erence between the two is intra-industry trade, TI ;
TI = 2 (nh + nf ) px x (1
)
(33)
We observe that
net trade is explained by conventional comparative advantage. If countries are
identical, then
=
and no net trade;
gross trade is not related to comparative advantage but to a correlation between
CA and country size. Fixing (nh + nf ) px x in (31), and varying
that TG takes on its maximum value for
= 0 and
and , we see
= 1; i.e. when a small
country has a great CA in the production of di¤erentiated goods;
intra-industry trade is most important (as a type of trade) when
is small. Since we have that
is large and
> ; then intra-industry trade must be on its
height when each of these is nearly 1=2: In other words, if the two countries are
of similar size and have no clear comparative advantage, then we will see the
predominant pattern of trade as one of intra-industry trade.
13
the total volume of trade is determinate, but its pattern is not.
FPE prevails as under perfect competition as long as identical factor proportions
across product varieties.
asymmetric preferences with a bias towards domestic varieties, reduce intraindustry trade.
3
Appendix: Deriving the demand for CES preferences
utility function
Consider the problem of the consumer with CES preferences
0
U =@
N
X
qj
j=1
subject to the budget constraint
N
X
1
1
1
A
pj q j = I
j=1
where Y is income and
measures the degree of substitutability between variety,
>1
Utility maximization implies
0
N
X
@
L=
qj
1
j=1
The FOC is
1
which implies
0
@
N
X
1
A
1
qj
j=1
1
2
N
X
4
pj q j
j=1
1
1
A
1
qj
qi
pi
=
qj
pj
Solve for qj
qj =
pi
pj
14
qi
= pj
3
I5
Now multiply for pj ; sum over j, and replace into the budget constraint to get the
demand for good j
qi =
pi
N
P
i=1
p1j
15
I