The first-order conditions for program (4) are given by
e[u'(Π*)(Pd + Pf max(S - Pd/Pf, 0) - c'(Q*))] =0, (5)
e[u'(Π*)(F - S?)] =0, (6)
e[u'(Π*)(C - max(S - Pd/Pf, 0))] = 0, (7)
where an asterisk (*) indicates an optimal level. The second-order conditions for the
unique maximum, (Q*, H*, Z*), are satisfied given risk aversion and the convexity of the
cost function. For ease of exposition, Qd , Qf , the distribution of S? and the cost function
are assumed to be such that the firm possesses some degree of export flexibility at the
optimum, Q* > Qd + Qf .7
3 Optimal production and risk management
with futures and options
This section characterizes the firm’s optimal production and risk management decisions
on the premise that the firm can trade both currency futures and currency call options.
First, examine the firm’s optimal production decision. Rewriting condition (7) as
E[U,(∏*)]C = E[U,(∏*) max(S - Pd/Pf, 0)] and substituting this into condition (5) yields
e[u'(Π*)][Pd + PfC - c,(Q*)] = 0. (8)
Since U,(∏) > 0 for all Π, equation (8) implies cz(Q*) = Pd + PfC. Thus, the following
separation result is established.
7The case where Q* < Qd + Qf is a completely different problem from an economic point of view. In
order to exclude this case, the condition that Q* ≥ Qd + Qf had to be considered in the optimization
problem. Hence, assuming the existence of some flexibility in the optimum only avoids lengthy discussions
of Kuhn-Tucker conditions and the corner solution (Q* = Qd + Qf ) which is of no interest since there is
no flexibility at all.