that the cost of trade subsidies is given by the tax revenue necessary to finance
them, we can write the welfare of the domestic country as:
W (s)= [pH (s) - c] D [pH (s), βH (s)] - F =
= [pH(s) - c] D [pH(s),β+g(p) - g(pH)] - F
which is maximized by a subsidy satisfying the first order condition:
D (pH,βH)+(pH - c) [D1 (pH,βH) - D2 (pH,βH) g0(pH)] = 0 (13)
If we now substitute this in the equilibrium first order condition for the domestic
firm, we can derive a neater expression for the optimal export subsidy:
* = (PH - c)D^(pH1jβHλg0(pH) > о
(14)
H [-D1 (PH ,βH)]
Clearly, also this is an implicit expression, since on the right hand side PH
depends on the optimal subsidy, however, this expression makes clear our main
point: the optimal export subsidy is positive. Summarizing:
Proposition 5. Under price competition and free entry, an export
subsidy is always optimal, since it helps the domestic firm to lower
its price in the foreign market.
The result overturns common wisdom for models with strategic complemen-
tarity and barriers to entry. An accomodating behaviour is not anymore optimal
because it would just induce new firms to enter. The only chance for the gov-
ernment to increase the profits of the domestic firm is to induce an aggressive
behaviour. Then the firm will undercut the competitors gaining in market share
and will spread a low mark up over a large portion of the market, leaving the
few remaining firms with zero profits.
An explicit characterization can be obtained in the case of a Logit demand,
Di = e-ξpi / P e-ξpj with ξ>0. In this case, international firms choose the
price p = c + F + 1∕ξ and it is easy to derive that the optimal subsidy must
induce a price for the domestic firm equal to рн (s*1 ) = c + 1∕ξ, which requires
a very simple expression for the optimal export subsidy:
s*H = F>0 (15)
Another explicit result for the optimal export subsidy can be derived in
models with isoelastic demand and in the Dixit-Stiglitz model which can be
microfounded in a standard way. For instance a Dixit-Stiglitz demand:
1-θ
Di = ( P ´- 1-θ with p ≡ {XX p- a ʌ
j=1
13