Since this is always an increasing function of s, it is optimal to increase subsi-
dization as long as there is entry. But entry is deterred at:
sH = a - c - 3√F> 0 (12)
which is the optimal subsidy. The intuition for this result is the following.
Free entry pins down the equilibrium price level of the foreign firms as long as
some of them enter. This implies that the choice of the subsidy does not affect
the equilibrium price at which the domestic firm will sell its production but
increases its market share. Since there are fixed costs of production, an increase
in the market share reduces average costs and hence it increases net profits.
Consequently it is optimal to raise the market share as much as possible, which
amounts to full entry deterrence.
Summarizing:
Proposition 4. Under quantity competition and free entry, an
export subsidy is always optimal, since it helps the domestic firm to
sell more than others in the foreign market.
2.2 Optimal export subsidy with Bertrand competition
Consider our general model of price competition with an export specific subsidy,
so that the gross profit function for the leader is:
ΠH =(pH - c + s) D (pH,βH)
where we remember that D (∙) is the direct demand depending on the price of
the domestic firm pH and on the spillovers from the prices of the other firms
βH . This profit function clearly satifies Π1H3 = -p2H D1 > 0.
It is tedious to characterize the optimal export subsidy under barriers to
entry, but Appendix D solves for this in the case of the Dixit-Stiglitz specification
of the demand function. Here we will focus on the free entry case, in which the
equilibrium conditions in the second stage are:
(p - c)D1(p,β)+D(p,β)=0
(pH - c + s)D1 (pH,βH)+D(pH,βH)=0
(p - c)D(p, β) = F
where β =(n - 2)g(p)+g(pH) is the spillover received by an international firm
from the strategies of all the other firms in the market and βH =(n - 1)g(p)
is the spillover for the domestic country. This system expresses prices and the
number of firms as functions of the subsidy s, but we know from Prop. 2 that
the price of foreign firms p and their spillover β are actually unchanged with
changes in the subsidy, while pH (s) and βH (s) depend on it. Hence, assuming
12