which has an interior solution for the optimal subsidy (without entry deterrence)
if goods are poor substitutes or if marginal costs are increasing enough. If such
an interior solution exists, it must satisfy the first order condition:
p (z(s), βH) + z(s) [p1 (z(s),βH) - p2 (z(s), βH) h0(z)] = c0(z)
which is a complicated implicit expression. However, if we substitute this in the
equilibrium first order condition for the domestic firm, we can derive a neater
expression for the optimal export subsidy:
Sh = [-P2 (z, вн ) h0 (z)] z > 0 (10)
Under perfect substitutability, this becomes:
sH = pε > 0 (11)
which is increasing in the elasticity of demand (notice that p is independent
from the subsidy). Moreover, it implies that domestic firms produce until their
marginal cost equates the equilibrium price (p = c0(z)) and enjoy positive profits
because returns to scale are decreasing for their level of production. Notice that
the optimal subsidy would be the same in presence of other domestic firms:
there is not a terms of trade effect because the equilibrium price is independent
from the subsidy, while domestic firms crowd out foreign ones. It is simple to
derive the optimal ad valorem subsidy in this case: for instance, with perfect
substitutability (11) implies that the optimal rate of export subsidization would
be simply equal to the elasticity of demand.
The role of trade policy is the same as with barriers to entry, but here it is
always optimal to induce an aggressive behaviour of the national firm, which is
done through subsidization. If there is low substitutability between goods and
the marginal costs are constant or decreasing (or even not too much increasing),
it is even better to set a subsidy which deters entry of international firms. Such
a subsidy has to satisfy the free entry condition for n slightly smaller than 2,
which implies that just one firm (the domestic firm) can profitably remain in
the market:
xp [x, z(sH)] — c(x) = F
For instance, consider the linear example. Here, imagining that there is
entry in equilibrium and imposing the free entry condition for a given subsidy
s, we obtain the equilibrium production for each international firm x = FF
and the number of firms n = (a — c — s) /√F — 1, which imply total production
Q = a — c — FF. Consistently with Prop. 2, the subsidy does not affect the level
of production of the other firms but decreases their number. The equilibrium
production of the subsidized firm is instead z = FF + s, which generates profits
∏H = (VF + s)2. The government maximizes profits net of the tax revenue
necessary to finance the subsidies:
W(s) = √F √F+ + s) — F
11