Restricted Export Flexibility and Risk Management with Options and Futures

variables in this case. Then, the first-order conditions for an optimum become

E [U^,(Π:) (Pd + Pf max(S - Pd/Pf, 0) - √(Q;))] = 0,             (12)

[. . ~ . . . ~ 1

U^,(Π:)(F - S)J = 0,

where an asterisk (^:) again indicates an optimal level. The second-order conditions for
the unique maximum, (Q*, H*), are satisfied given risk aversion and the convexity of the
cost function. It is still assumed that Q* > Qf + Q_d so that the firm has some degree of
export flexibility at the optimum.

The firm’s optimal production decision is analyzed first. Rewriting condition (12)

[. . ~ . ~ . . 1
^u '^{(n ,}        ' ^- P f, 0)

c (^Qo) = P^{d +} P^f--------EU(⅛)i--------

Inspection of condition (14) reveals that, in general, the firm’s optimal output, Q*, de-
pends on the firm’s attitude toward risk and on the nature of the underlying exchange
rate uncertainty. This implies the following result.

Proposition 3 If currency futures are the only hedging instrument available to the re-
stricted export flexible firm, its optimal output, Q*, is neither separable from the firm’s
attitude toward risk nor from the distribution of the exchange rate.

Since the available currency derivatives do not allow for complete elimination of ex-
change rate risk from marginal revenue, the firm’s willingness to assume risk and the
characteristics of the exchange rate distribution have an adverse impact on the firm’s
optimal production decision.

Now, the firm’s optimal futures position is characterized. If the currency futures
market is unbiased, F = E[S], condition (13) can be written as

Cov [u ^,(Π X),S?] =0.                                (15)

Based on condition (15), the following proposition can be established where a proof is
given in Appendix A.

16

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