See how “reciprocal dumping” can generate trade, and may or may not be in the interests
of society.
James Brander. 1981. “Intraindustry Trade in Identical Commodities” Journal of
International Economics, vol. 11: 1-14.
Consider two producers (one in Home, one in Foreign).
Each faces a fixed cost of production F and a constant marginal cost of production c
Home delivers x for sale in Home market
Home delivers u to Foreign market
Foreign delivers y to Home market
Foreign delivers v for sale in Foreign’s own market
There are “iceberg” transportation costs:
g is the fraction of goods that survive shipping (so 1-g “melts” away during transport)
Then amount of good that Foreign must produce for export in order to deliver u to the
Home market is u/g. This means that the cost of producing and delivering one more unit
to the overseas market is c/g.
Firms have “Cournot conjectures” – each takes output of competitor as given and chooses
own output to maximize own profits.
Also assume that there are no "re-imports".
Home firm maximizes total profit by solving
max P(x+y) x + P*(u+v) u - c[x+u/g] - F
x, u
where P() and P*() are inverse demand curves in the Home and Foreign output markets
respectively.
The foreign firm solves
max P(x+y) y + P*(u+v) v – c[y/g + v] – F
y, v
Since marginal costs are constant for each firm, and re-imports are not allowed, we can
divide up the profit problem for Home firm into 2 separate maximization problems:
max P(x+y) x – cx
x
and
max P*(u+v) u – c u/g
v
and can do the same for the Foreign firm:
max P(x+y) y - c y/g
y
max P*(u+v) v - cv
v
Note that we are ignoring the fixed costs, because they are independent of x,u, etc. We
include them in the model though because witout them we can't explain why there would
only be two firms producing in the first place. The firm specific fixed costs make this a
model of "internal economies of scale".
Assume that demand in each country is linear:
P(Q) = a-bQ
P*(Q*) = a-bQ*
where Q=x+y and Q*=u+v.
Since the problem is symmetric, we can restrict attention to the Home domestic market
(i.e., solve for Cournot quantities delivered to the Home market: xc and yc). Then by
symmetry we’ll have
vc = xc and uc = yc .
Solve for xc and yc
First find the Best Response Functions. Start by solving for the optimal x taking y as
given.
max [a-b[x+y]] x – cx
x
The first-order condition for an interior optimum is:
a-b[x+y] – bx – c = 0

a – 2bx – by – c = 0
a c y


x=
2b
2
This gives the Home firm’s optimal delivery of the good to Home's domestic market
given that the Foreign firm delivers y units.
Thus
a c y

x(y) =
(1)
2b
2
is described as Home firm’s “Best Response Function”.
Now do the same for the Foreign firm:
max [a-b[x+y]]y – c y/g .
y
First order condition for an interior optimum:
a-b[x+y] – by – c/g = 0

a-2by – bx – c/g = 0
a  (c / g ) x


y=
2b
2
This gives the Foreign firm’s optimal delivery to the Home market given that Home
delivers x units. Thus
a  (c / g ) x

y(x;g) =
(2)
2b
2
is the Foreign firm’s Best Response Function.
Best-response functions
y
45°
x(y) =
a-c - y
2b 2
a-c/g
2b
y(x;g) =
yc
a-c/g - x
2b
xc
2
x
a-c
2b
To solve for Cournot outputs, find xc, yc such that
x(yc) = xc
and y(xc;g) = yc
I.e. find the intersection point of the best response functions.
Do this by substituting x(yc) from equation (1) into y(x;g) for x in equation (2) and solve
for y:
y(x(y);g) =
=

y
a  c / g x ( y)

2b
2
a  c / g 1 a  c y 
 
 
2b
2  2b
2
=
a c/g a c y


2b
4b
4

4y - y

=
2a 2c / g a c

 
b
b
b b
3y =
a c
 2  g 
b gb
a  c [2  g ]
g
yc =
3b

This is the Cournot quantity of the good delivered by the Foreign firm for sale in the
Home domestic market.
Now use the Cournot value yc to solve for xc:
xc = x(yc) =
=

xc
a c 1
 yc
2b
2
a  c 1  a  c / g[2  g] 
 

b
2
3b

c
[2  g]
a c 1
g

a
=
2b
6b
6b
=
3[a  c]  a  c / g[2  g]
6b
=
3a  a  c{1 / g [2  g ]  3}
6b
=
xc
2a  c{2 / g  1  3}
6b
a  c [1  2 g ]
g
=
3b
This is Home's Cournot equilibrium output for sale in its own domestic market.
To solve for price simply plug the Cournot values into the inverse demand curve:
p = a – b[xc + yc]
 a  c / g [1  2g] a  c / g [2  g] 

= a - b

3b
3b


=a-
[2a  c / g[1  2g ]  c / g[2  g ]
3
=a=
[2a  c / g[2  g  1  2g]
3
a  c / g [1  g  2g ]
3
a  c [1  g ]
g
Pc =
3
This is the equilibrium price in the Home domestic market in equilibrium.
___________________________________________
Points to observe:
1) even though Foreign is a higher cost provider of goods to Home market, (c/g > c),
provided g is not too low, Foreign still ships goods to Home market.
2) since economies are symmetric, Home also ships some goods to Foreign market
intra-industry trade occurs even though goods are not differentiated.
3) Dumping is occurring – each country is receiving less for its goods shipped abroad
than they do for each good sold in domestic market.
Intuition for why trade occurs?





Shortage of firms means firms make positive profits.
This attracts competition from overseas.
Even though Foreign can’t supply Home market as cheaply as can Home, Foreign
still makes profits from it provided
 P(xc+yc) > c/g
i.e. Foreign can still cover its marginal cost.
Foreign willing to “flood” the Home market because opportunity cost (lost revenue)
is borne mostly by the Home firm (since it has larger share of market).
Is trade necessarily beneficial?




Home consumers gain from increased competition.
But firms (and society) loses from incurring unnecessary transportation costs.
Is unclear which dominates.
You can imagine that if g is low enough that yc  0, then a small decrease in g will
lead to some imports from abroad. This small amount of imports will have a minimal
effect on prices, but every unit that gets imported generates a large amount of waste
due to the substantial transportation costs. Is possible that each economy is worse off
from trade!
This ends the section on internal economies of scale.