and we know that both prices must be decreasing in both the subsidy and the number
of firms (since strategic complementarity holds). The optimal trade policy maximizes:
PH (s) 1-θ [pH (s) - c] „
1 F
n - 1)p(s)- ɪ + PH(s)-ɪ
A closed form solution for this problem does not exist, but one can derive an approx-
imate solution in the case firms choose their prices ignoring their impact on the price
index, which is reasonable for n big enough. In this case we have the equilibrium
prices p = c∕θ and pH = (c — s)/θ. Substituting in the welfare function we have:
W(s)=
W(s)=
( c-s ) 1-θ [c - s - cθ]
1
θ [(n - 1)( c )- 1-θ + ( c-s )- 1¾]θ
-F
whose maximization delivers the optimal negative export subsidy as:
sH (n) `
c - sH - θ
θ ■
1+(n - 1) (c-cH´ 1-θ
<0
(26)
Let us now solve for the exact optimal export subsidy under price competition and
free entry. The price of the foreign firms P(s) and of the domestic firm PH (s), and
the number of firms n(s) solve the system of equilibrium conditions (24), (25) and the
free entry condition:
1
θ - θ
θ ɪ, Γ θ 1 ^∣
F = p- i-θ - cp-i-θ (27)
(n - 1)p-τ-θ + pH1-θ
From (25) and (27) one can derive the price of the other international firms implicitly
defined by the smallest root of:
c Fθ (p - c)1 θ p1-θ
P = θ + θ
which is independent of s. The optimal subsidy maximizes:
W(s) = ph(s)-θ [pH(s) - c]__F = ph(s)-θ [pH(s) - c] f - F
h[n(s) - 1]p-τ-θ + pH(s)-τ-θi [p 1-θ - cp 1-θ]
where I used (27) in the second line. It is immediate to verify that the optimal
subsidy must satisfy the first order condition ph (s) = c∕θ. Substituting for ph in
the equilibrium condition (24) one obtains the optimal subsidy:
*
H=
1+θ
____________(O 1-θ - c___________
θ θ S— / θ 1 ∖ θ „
1 + (c) 1-θ (p 1-θ - cp 1"θ) F-θ
>0
(28)
which can be rewritten as (16).
27