ECON 4415 International trade
Material for the seminar on 10 May 2004 (Arne Melchior)
Questions:
1. Discuss whether regional trade integration may undermine the WTO.
Hint: Read Chapter 10 in Hoekman and Kostecki, and you may find
other supplementary material.
2a. (Optional, not mandatory.) Depart from the “home market effect”
model as described in lecture note 5 or Chapter 10, Section 4 in the
Helpman-Krugman book, or in the seminar note on the model. Now
develop a similar model but for three countries of identical size (Li=L is
the same for all the three). Trading costs for X goods are t12 between
countries 1 and 2, and t between country 3 and the other two countries,
1< t12t. If t12=t, the situation is symmetrical with no regional integration
between countries 1 and 2, and all the three countries must have equal
shares of X production. Discuss the changes in production for the three
countries if t12 is reduced, i.e. that regional trade integration occurs
between countries 1 and 2. Hint: See lecture note 5, or seminar note on
the home market effect.
2b. (Optional, not mandatory.) Discuss the welfare implications in
problem 2a above. Hint: See lecture notes 5.
Illustration on question 2.
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The analysis can be undertaken in three steps:
Define the model, as in the lecture notes, and derive the pricing
condition and firm size x. As there, assume that all countries are
diversified (i.e. they all produce B as well as X), and then you have pB as
well as the wage in all countries equal to 1. Then the total income of
each country is simply L.
Then set up the three equations for how individual firms in each
country allocate sales across markets. These are:
x11 + x12 + x13 = x
x21 + x22 + x23 = x
x31 + x32 + x33 = x
where x, firm scale, is given by your former solutions. Now you may use
the demand equations
xji = Yi p- tji1-Pxi-1
to express all x’s in the first column as functions of x11, all in the second
column as functions of x22, and all in the third column as functions of
x33. You simply divide demand for x21 by demand for x11 and so on,
using the appropriate levels of trading costs (=1 for home market sales).
After doing this, you have three unknowns. But you also can use that
countries 1 and 2 must be identical (same size, same trading costs), so
x11=x22. Then the whole thing reduces to two equations with two
unknowns, you may solve for x11 and x33 (also substituting the solution
for firm size x), and then derive all the other x’s.
Finally, you set up the equations for market clearing for X goods
in each country. The consumption share for X is  and the income in all
countries is L, so you have
n1px11 + n2 p x21 + n3 p x31 = L
n1px12 + n2 p x22 + n3 p x32 = L
n1px13 + n2 p x23 + n3 p x33 = L
Here p is known from the pricing condition, the x’s you have derived
above, and you know that n1=n2 since countries 1 and 2 are identical
(country 3 has the same size, but faces different trading costs). Using
this, the first two equations become identical, and you may solve for n1
and n3. Having obtained the solutions for the number of firms, you may
analyse how these expressions respond to changes in t12. If t12=t, the
three countries are obviously in the same situation. All the way you have
to assume that t12 as well as t are larger than 1, but you may have t12<t.
As an option (not required), as question 2, you may also check how
regional integration affects welfare for the three countries. Then you
have to set up the price indexes, which must take the form
Px1 =
n p
1
1
 n 2  p t12
1
 n3  pt 
1

1
1
Since n1=n2, the expression for country 2 will be similar. For country 3
we must have:
2
Px3 =
n  p t 
1
1
 n2  p t 
1
1
 n3 p

1
1
Using the solutions for the number of firms, you may then analyse the
bracketed expressions, as indicated earlier. The larger they are, the
higher is welfare. Since p is the same everywhere, you may put it outside
the brackets and forget it.
Solution to the trade bloc exercise (Question 2)
Model set-up:
- Three countries, labelled 1, 2 and 3.
- One factor of production, labour.
- The three countries are of equal size, Li=L for all i.
- Two sectors:
- Sector B: “Agriculture”, constant returns to scale, one labour unit
required to produce one unit of B.
- Sector X: “Manufactures”, produced with increasing returns to scale,
differentiated goods, each firm producing one variety of the good.
- No trade costs between countries for good B.
- “Iceberg” trade costs tij>1 for trade between countries i and j.
- CES sub-utility function for X goods, as in lecture 5.
- Cobb-Douglas utility function Ui=XiCBi1- for B (CBi=consumption
of B in country i) and X aggregate, giving consumption shares 1- for B
and  for X.
- Simplifying assumption: All countries produce X as well as B. With
free trade in B, prices for B as well as wages will then be equalised
between countries, and we choose units so that wi (the wage) = PBi (the
price for B goods) =1 in all countries. Total income in each country is
then L, and total consumption of X goods in all the three countries is L.
Consumption of B is (1-)L. Since the price for B is 1, this is also the
demand function for B goods.
Now following lecture notes 5, we know that the demand
function for an individual variety from country i in country j is
x*ij = L p*ij- Pj-1
where Pj is the price index for X goods in country j, and we have used
asterisks * to indicate that here, it is the quantity demanded and the “real
price” paid by consumers that enter. Given iceberg trade costs tij, some
goods “melt away” in shipment and we have x*ij=xij/tij, where xij is the
quantity shipped by the firm. Similarly, if firms receive the price pij for
the goods, the real price paid by consumers is higher since they receive
less than xij, hence we have pij=pij*tij. Using this to express demand in
terms of producer-side quantities and prices, we obtain the “corrected”
demand functions
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xij = L pij- tij1- Pj-1
(which are important to remember).
X producers have fixed costs f and constant marginal costs c in
all countries, hence their cost functions are
Ci = f + c j xij
where you may note that because of iceberg trade costs, trade costs do
not enter (since they are reflected in the quantities xij). Profits are then
i = j pij xij - f - c j xij
Maximising profits with respect to xij gives the first-order condition
pij + pij/xij xij – c = 0
or
pij + pij/xij xij = c
Putting pij outside brackets on the left hand side gives
pij ( 1 + pij/xij xij/pij ) = c
and we see that the second term in brackets is the inverse of the demand
elasticity faced by the firm, i.e.
pij ( 1 + 1/eij ) = c
Now we introduce the assumption about competition; there is
monopolistic competition so that firms neglect the impact of their actions
on the whole market. The true elasticity of demand with the given
demand structure is --(1-)sij, where sij is the market share of a firm
from country i in market j for X goods (shown in earlier lecture,
remember also that >1). With monopolistic competition, we assume
that there are many firms so that sij is small, and the firm neglects the
second term of the expression. Hence with monopolistic competition, the
perceived elasticity eij is equal to -. Using this, the first-order condition
above becomes
pij ( 1 - 1/ ) = c
or equivalently
pij = c/(-1)
i.e. the price is a mark-up on marginal costs, determined by the elasticity
of substitution. Given that  and c are the same for all pij, we then know
that all producers charge the same producer prices, which we denote
p=pij for all i,j. Given that we have
4
p = c/(-1)
in all cases, we may in the following drop subscripts on prices.
With monopolistic competition, we also assume that there is free
entry and exit so at equilibrium, firms earn no pure profits. Setting
profits equal to zero and using pij=p we obtain
i = j p xij - f - c j xij = 0
which determines the total output quantity of the firm, which is
j xij = f/(p-c)
Given that f, p and c are the same for all firms, the output quantity must
be the same for all firms, and we denote it by x. Using the solution for p,
we have
x = f (-1)/ c
The above provides the first steps when solving any model of this kind.
Next, we turn to the trade bloc model.
Now we assume that between countries 1 and 2, trade costs are
t12. Between country 3 and the two former, trade costs are t. We assume
that countries 1 and 2 may form a trade bloc by lowering t12, hence we
assume t12t. Observe that this is the only thing that can create any
difference between the three countries; they are identical in all other
respects. Observe also that countries 1 and 2 are identical in all respects;
hence we may simplify in the following by assuming that all solutions
are similar for firms in countries 1 and 2.
The next step when solving the model is to determine how firms
allocate their sales across the three markets. This is given by the three
equations:
x11 + x12 + x13 = x
x21 + x22 + x23 = x
x31 + x32 + x33 = x
where x, firm scale, is given by the solution above (but we use x for
simplicity in order to save notation). Now we use the demand functions
xij = Yi p- tij1-Pj-1
to express all x’s in the first column as functions of x11, all in the second
column as functions of x22, and all in the third column as functions of
x33. We simply divide x12 and x32 by x22 and so on, and obtain:
x11
x11t121-
x11t1-
+ x22 t12 1-
+ x22
+ x22 t 1-
+x33 t 1-
+x33 t 1-
+x33
=x
=x
=x
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Given that countries 1 and 2 are identical, we must have x11=x22, and we
may use this to simplify further. We then easily find the solutions
1
1 t
x11 = x22 = x

1 t 2 t
1
2  2
12
1
1
1  t  2t
x33 = x

1 t 2 t
12
1
2  2
12
The solutions for other xij’s are then easily found by using the
relationships x12=x22t121- and so on, as above. Hence we have
determined how much each firm sells in all markets.
By examining the solutions above, we easily find that if t12<t we
must have x33>x11=x22. Hence firms in the trade bloc export more, and
sell less in their home market. Formally, it is easily established that the
derivative of x11 with respect to t12 is positive, while the derivative of x33
with respect to t12 is negative. These results will also be relevant for the
welfare analysis, see the last page of the note.
The final step in solving the model is to determine the number of
firms in each country. For this purpose, we set up the three marketclearing equations, denoting the number of X firms in each country by
n1, n2 and n3. Also this time, we know that n1=n2, and this can be used to
simplify the calculations. Sales by all firms in each market must add up
to total consumption in each market:
n1 x11 p + n2 x21 p + n3 x31 p = L
n1 x12 p + n2 x22 p + n3 x32 p = L
n1 x13 p + n2 x23 p + n3 x33 p = L
Using that n1=n2 and x32=x31 due to the symmetry between countries 1
and 2, we may simplify into two equations:
n1 (x12 + x22) + n3 x32 = L/p
n1 (x13 + x23) + n3 x33 = L/p
This gives
n1 = L/p
x x
x (x  x )  x (x
33

x
23
)
x x x x
x (x  x )  x (x  x
23
)
33
n3 = L/p
32
12
12
33
12
22
22
22
32
13
32
13
23
13
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The denominator in these expressions, which we denote by Z, can (using
x12=x22t121- and so on, as well as x11=x22) be simplified to
Z = x11 x33 ( t121- + 1 – 2 t2-2)
Also simplifying the numerators and completing the (somewhat tedious)
calculations, we obtain the solutions
1


L 1

t
n1 = n2 =


1
1
1 
p x 1  t
1  t12  2 t 

and
1

1 

t12
L 1
t
2


n3 =

1
1
1 
p x 1 t 2 t
1  t 
12

If t=t12 we easily find that n1=n2=n3. The total number of firms in the
world is 2n1+n3. Using the solutions above, we find this to be 3L/(px)=
3L/(f). f is firm size (in value), L is the size of each market for X,
and the world market is three times as large.
In order to examine what happens if t12<t, we may take the
derivative

n1/t12 =
1
L

px

(1  t
1
12
2
t
1
)
2


 1
 t (1   ) t12


which is negative since >0. Hence the lower is t12, the larger is the
number of firms in countries 1 and 2. Hence we have shown that
regional integration will increase the number of X firms in the
integrating countries. Given that world demand for X goods is constant,
it is then evident that regional integration among countries 1 and 2 must
lead to a lower number of X firms in country 3. Hence regional
integration leads to “production-shifting”; X production is transferred
from country 3 to the integrating countries.
In order to show the welfare impact, note from lecture note 5 that
welfare is inversely related to the price indexes Pi. The lower price level
for X goods in a country, the higher is welfare. Now, as a trick to
simplify the welfare analysis, write down the demand for x ii; i.e. a firm’s
sales in the domestic market:
xii = L p- Pi-1
Now, if the welfare in country i is inversely related to Pi, it must also be
inversely related to xii, since p may here be treated as a constant
(common to all firms in all countries). Hence we may derive the welfare
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impact of integration directly by studying how changes in t12 affect xii.
We have already established that regional integration will decrease home
market sales for firms in the integrating countries, and increase home
market sales for firms in country 3. Hence regional integration will
increase welfare in countries 1 and 2, and reduce welfare in the nonparticipating country 3.
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