2.1 Optimal export subsidy with Cournot competition
Consider the general model of quantity competition which allows for imperfect
substitutability between goods. The gross profit of the domestic firm in presence
of an export specific subsidy is:
ΠH = z [p (z, βH)+s] - c(z)
where we remember that z is now production of the domestic firm, p (∙) is the
inverse demand which depends on the spillovers from the production of other
firms βH , c (∙) is the cost function and s is its subsidy. This profit function is
clearly characterized by Π1H3 =1 > 0. The equilibrium first order conditions in
the second stage where nash competition takes place in the foreign market are:
p(x, β)+xp1 (x, β) = c0(x)
s +p(z,βH)+zp1(z,βH)=c0(z)
where β =(n - 2)h(x) + h(z) is the spillover received by an international firm
from the strategies of all the other firms in the market and βH ≡ (n - 1)h(x) is
the spillover received by the domestic firm. If the number of firms is given, it
is standard to derive the optimal trade policy. For instance, in case of perfectly
substitute goods we have
* Pε
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sH(n) 1 I ι / г f 1 M
H 1+1/[n(1 -χ)]
where ε ≡ -zp0/p is the elasticity of demand (with respect to domestic pro-
duction) and χ ≡ -xp00/p0 is the elasticity of the slope of the inverse demand
which represents the degree of convexity of demand. As well known, in the
linear case with demand p = a - xj and marginal cost c we have χ =0and
sHH (n) = (a — c)(n — 1)∕2n > 0 but, if demand is convex enough, an export tax
may become optimal.
Let us now consider free entry. In the second stage we have also the zero
profit condition:
xp(x, β)=c(x)+F
The equilibrium system expresses production levels and the number of firms as
functions of the subsidy s, but we know from Prop. 2 that the production of
foreign firms x and their spillovers β are actually unaffected by changes in the
subsidy, while z(s) and βH (s) depend on the subsidy. Hence, we can write the
welfare of the domestic country as the profits of the domestic firm net of the
tax revenue necessary to finance the subsidy:
W(s) = z(s)p (z (s), βH (s)) — c(z) — F =
= z(s)p [z(s), β + h(x) — h(z)] — c(z) — F
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