Journal of Inrrrnufiond Money and Finance ( 199 I), 10, 108-l 17

Exporting firm and forward markets: the multiperiod case

ITZHAK ZILCHA AND RAFAEL ELDOR*

Department of Economics, Tel-Aviv University, Tel Aviv, Israel

We consider a model with a competitive risk-averse exporting firm who faces uncertain exchange rates in a multiperiod analysis. The capital stock (or fixed input) has to be determined at the outset while the variable input (labor) is chosen optimally at the beginning of each period, but before the realization of the exchange rate. We show that introducing unbiased currency forward markets decreases the capital/labor ratio in all periods. We also show that such a lirm tends to ‘overhedge’ compared to the one-period cases. In some cases the introduction of unbiased forward markets results in higher investments and production in all dates.

This paper studies the behavior of a competitive exporting firm which faces uncertain exchange rate in a multiperiod framework. This firm chooses at the outset its optimal level of capital stock, which remains fixed throughout all the production periods. However, at the beginning of each date it chooses the optimal level of the variable input before the realization of the uncertain exchange rate.

This paper combines two strands of literature. The first is the behavior of a competitive firm under uncertainty with fixed and variable factors of production (see, for example Turnovsky, 1973; Stewart, 1978; Wright, 1984). The second is the literature concerning the impact of forward markets upon the firm’s optimal production and hedging decisions (e.g., Feder et al. (1980) Holthausen (1979) Ethier (1973), Benninga et al. (1985), and Kawai and Zilcha (1986)).

Our model differs from those contained in the first type of literature in several respects. The variable input in our case must be chosen before the realization of the exchange rate (in each period). Moreover, we introduce risk sharing arrangements in the form of forward currency markets. We differ from both strands of literature by treating a model with muftiperiod production and hedging decisions.

As a result of our more general framework, in particular, the multiperiod analysis, we obtain results which are different from certain well-known properties when we consider the long-run. Under constant returns to scale the firm’s optimal capital/labor ratio declines (in all periods) in the presence of forward markets and these ratios are independent of the utility function and its subjective beliefs. However, the usual ‘separation property,’ regarding the absolute level of

* Financial support from the Foerder Institute for Economic Research is gratefully acknowledged. We

would like to thank Gideon Fishelson and Simon Benninga for their helpful comments.

OX-5606/91/01,/0108-10 6 1991 Butterworth-Heinemann Ltd

ITZHAK ZILCHA AND RAFAEL ELDOR 109

capital and labor, does not hold since the employment of the variable input in the future periods, and hence its output, remains uncertain.

In the case of unbiased forward markets, one usually finds that a risk-averse firm fully hedges its output (see, for example, Holthausen, 1979; Feder et al., 1980). In our model this risk-averse firm sells currency forward more than the value of its output (except in the last period), i.e., overhedges. This result follows from our assumption that there is a positive (or zero) correlation between the firm’s future profits and the exchange rates in the intermediate periods.

Under certain assumptions about the utility function and the production technology, we show that the introduction of unbiased currency forward markets increases both capital investment and production (exports).

In the next section we introduce the model and derive the optimality conditions. Section II considers the optimal hedging behavior. Section III deals with a special case. We conclude the paper with a discussion of our results.

I. Optimal production in the presence of currency future markets

Consider a competitive risk-averse firm that exports all its output. This firm is a price-taker in international markets and faces uncertain exchange rates. We assume that the firm produces and exports in several periods and thus faces uncertain revenues in a few dates. To simplify our analysis let us assume that the number of periods is two.

The firm employs capital K and labor L using a neoclassical production function F(K, L). We assume that F(K, L) is homogeneous of degree one, and satisfies FK>O, FL>O, FKK<O, F,,tO, F,,>O.

The capital in this model represents all the fixed inputs to be determined at the beginning of the production periods, while labor is a variable input; however, the latter is determined at the beginning of each period before the resolution of uncertainty for that period. Denote by r the capital price in period 1 and by wi, w2 the labor wages in periods 1 and 2. The firm is a price-taker in the factors markets in the domestic country.

When a currency forward market exists, we denote by e{ the forward exchange rate from date t =0 to date t= 1, and by e{ the forward exchange rate from date t= 1 to t=2. This firm maximizes the expected utility of the sum of its profits (in both periods) denominated in local currency. Let U represent its von-Neumann Morgenstern utility function; hence it satisfies, U’>O and U”<O. The firm’s decision about labor to be employed in period 2 will depend upon the realization of the exchange rate P, for period 1, i.e., Lz = L,(e,). Similarly, since e{ and L2 depend on the realization of P,, the forward sale of foreign currency in period 1, X,, will be dependent upon the value of e,. Therefore the firm chooses inputs and forward commitments such that (E is the expectation operator):

(1)

(2)

max EU(ii) K.L1,Lz(e1) XI.xz(el)

s.t.

ii = E,P,F(K, L,)+(e{-~l)X,-rK-w,L,

110 E.rportiny firm und forward markets

where it is the total profit of the firm (in these two periods), Zr is a random variable representing the spot exchange rate at the end of date t, P, the product price in date t denominated in foreign currency, X, is the forward sale of foreign currency by the firm in date t - I to be delivered in date t, t = 1, 2, and d is the discount factor for date 1. profits. We assume that P,, Pz, wr, wz, and t- are such that the firm produces a positive output in both periods.

An important feature in this model is that the conditional distribution of the random variable ZZ may depend upon the realization of Zr. We assume that the mean of P2 given e,, denoted Z,(e,), is nonciecreasing in e,. This certainly includes the case of independence. However, we have chosen to analyze the case of either zero or positive correlation since we believe that this is the more likely case when each period is not ‘too long,’ but this is not based upon empirical evidence. A similar analysis can be conducted under the assumption of negative correlation between Zr and P,. As a result of our assumption, the currency forward price from date 1 to date 2, e{, will also depend upon the realization of e’,, i.e., e{=ei(e,). Denote by ?,=Et,. We say that the forward markets are unbiased if e{=Z, and e{(e,)=Z,(e,) for all e,. Finally, we denote by E, the expectation with respect to P,. E, stands for the conditional expectation on ZZ for a given value of e,, i.e., the expectation with respect to t,(e,).

Necessary and sufficient conditions for the optimization (l), denoting by * the optimum levels, are

(3) E[EtPrFk(K*, L:)+GZ,P,F,(K*, LT(e,))-r]U’(rr*) = 0,

(4) EIZIP,FL(K*, I,:)-\v,]U’(n*) = 0,

(5) E2C~2(e,)P2FL(K*, Lf(e,))-wJU’(n*) = 0, for all e, ,

(6) E(e{-Z,)U’(z*) = 0,

(7) E,(e~(e,)-E,(e,))U’(n*) = 0, for all e,.

Combining equations (4) and (6) we derive

(8) e{P,F,(K*, Lf) = wr.

Similarly from equations (5) and (7) we obtain

(9) e{(er)PJJK*, G(e,)) = *v2, for all e, .

Conditions (8) and (9), together with the homogeneity (of degree 0) of FL, imply that the firm determines the optimal labor/capital ratio regardless of its attitude towards risk and regardless of its beliefs about the distributions of Z1 and P2. However, in determining the absolute levels of capital and labor, the firm takes into account its attitude towards risk and the distribution functions of Zr and P2. This can be seen from condition (3) where U’(7t*) is an integral part of this condition.

The widely recognized separation result does not hold in this model even though the production function is deterministic. The reason is that the second period output depends upon the employment of labor in the second period, which in turn depends upon the realization of Z1. Therefore, when the firm determines its optimal capital stock, the output in the second period is uncertain and hence its attitude toward risk and the distribution of the random variables must be incorporated into its decision process.

ITZHAK ZILCHA AND RAFAEL ELDOR 111

Without loss of generality, we shall take 6= 1 from now on. Let us denote the optimal levels of inputs when forward markets do not e.yist

(i.e., X, ~0, X2 G 0 in problem (1)) by R, e,, L2(el). Now let us show that production becomes more labor intensive as a result of introducing forward markets.

Theorem 1: The introduction of unbiased currency forward markets decreases the capital/labor ratio in all periods for industries where capital is determined once and for all and labor is a variable factor. Formally,

(a)

@I K* rz

G(eJ < $0’ for all values of e, .

Proo$ The necessary and sufficient conditions, when no forward markets are available, in the long-run case are

<LO> E[Z,P,FK(R, L,)+Z,P,FK(R, L,(e,))-r]U’(Tz) = 0,

<11> E[Z,P,F‘(R, L,)--w,]U’@) = 0,

(12) EZC22(el)P2FL(K L(e,))-w,lU'(ir) = 0, for all e, .

But &(er) is an increasing function of e,, hence Cov(Z,, E2U’(7i)) ~0 since U’ is a decreasing function. Thus from equation (11) and e{=.FI we obtain:

(13) Z&F&, t,)- WI >o.

By the exact same argument, since et/(er)=e,(e,), we have

(14) P2(e,)P,F,(R,2,(e,))-wz>0, for all e,.

Comparing equations (13) and (14) to equations (8) and (9) and using the homogeneity of degree zero of FL, we easily derive (a) and (b).

The proof of Theorem 1 shows that when forward markets are not available, as a result of the uncertain exchange rates the expected value of the marginal product of labor exceeds the wage rate. However, when the firm uses forward contracts it equates the value of the marginal product of labor (using the forward prices) to the wage rate.

II. Optimal hedging

The purpose of this section is to study the optimal engagements in forward contracts of a competitive firm with some fixed and some variable inputs. We derive here that the optimal hedging rules are different significantly from the case where all inputs are variable. One main reason for this result is that the returns attributed to the fixed input are correlated with the first period exchange rate. To emphasize this difference let us consider tirst the unbiased forward markets case.

112 E.uportiny firm and forrcard markets

Theorem 2: ilssume that the currency forward markets are unbiased, and that the mean of Z2, i.e., Z2(e,), is increasing in e,, then

(15) xy > PIF(K*, LT),

(16) XT(e,) = P2F(K*, Lf(e,)), for all e,.

We observe that although currency forward markets are unbiased, the first period forward sale is not a full hedge if P, and Pz are correlated. However, in the second period (in general, on the last date) the firm fully hedges its output.

Proojl Let us show first that equation (16) holds. From equation (5) we obtain, given e:(el)=E(P,Je,)=~,(e,), that the firm chooses XT(e,) such that

(17) CovG2(e,), U’(n*(e,))le,) = 0, for all e,,

where

(18) n*(ei) = P,CPiF(K*, LT)-X:J+~z(e,)CP,F(K*, LT(e,))-XT(e,)]

+Z,XT+Z,(e,)XT(e,)-w,L:-w,L:(e,)-rK*.

Thus rr*(ei) is either monotone increasing in P, (when P2F(K*, L~(e,))>X~(ei)) or non-increasing (if P,F(K*, Lz(e,))<XT(e,)). But from equation (17) it is clear that n*(ei) cannot be either increasing in E,(e,) or decreasing. Therefore P,F(K*, L~(e,))=X~(e,) for all values of e,.

To show that (15) holds we obtained from equations (4) and (8) that when e{=e, the firm chooses XT such that

(19) Cov(P,, u’(n*)) = 0.

But, using equation (16) of this theorem, rc*(ei) reduces to

rr*(el) = e,[P,F(K*, L:)-Xr]+Z,(e,)P2F(K*, LT(e,))--w,LT(e,)

+e,P,XT-w,LT--rK*.

The derivative of A(e,)=P,Z2(e,)F(K*, LT(e,))-wzLT(el) with respect to e, is,

p Wed dG(ed 2 ~+CJ’22(el)F,W*7 L3el))-w21 de.

1 1

However, from the F.O.C. we find that A’(e,)=P,[dC,(e,)/de,]. Since @,(e,) is increasing in e, whenever P,F(K*, LT)-Xy20 it follows that n*(ei) is a monotone increasing function of e,. Hence the covariance of P, and U’(n*) should be negative which is a contradiction to (19). Therefore PIF(K*, LT) < XT.

To understand the overhedging in period 1, let us analyze the firm’s position after one period. In the case where the spot exchange rate is lower than e{=?, the firm profits from its position in the forward market and its level of output

by (XT -P,F(K*, Ly))(Cl-e,)>O. However, low realization of e, implies lower expected profits in the following period. Thus, the overhedge, X: -PIF(K*, LT), can be considered as a hedge against uncertain returns in the second period. Note that if all factors of production are variable, in particular K can be chosen for the second period after the realization oft,, it can be shown that XT must equal the value of output P,F(KT, LT), using the unbiased forward exchange rate market. Hence the first period overhedging is attributed to the uncertain return to the fixed input (i.e., capital) in the second period which is correlated with the

ITZHAK ZILCHA AND RAFAEL ELDOR 113

realization of P,. This rationale holds also in commodity forward markets. For example, if an agricultural product’s prices p, and p2 are correlated, a farmer may use this period forward commodity market not only to hedge its profits in the current period, but also against the impact of the realization of P, upon the profits in the following periods. The present value of all future profits equals to the market value of his farm (= his assets). Thus, in using the current forward market, he is hedging against change in the value of his assets.

In the case of biased forward markets, if e{>2, then part (a) of our theorem clearly holds since there is an additional incentive to sell forward more than the output. In the normal backwardation case (i.e., e{<Z,) the difference between X: and P,F(K*, L:) depends on the tradeoff between the premium that the tirm pays in the forward market and the value of hedging future profits.

III. A special case

In general, one cannot strengthen Theorem 1 in the sense of indicating what precisely happens to the level of capital invested as a result of introducing unbiased forward markets. We can derive such a result under the following assumption:

Assumption (A./). The utility function U(X) exhibits decreasing absolute risk aversion and XV’(X) is concave. Assumption (A.l) holds for example when U(X) = BY, 0 <z < 1, or when V(X) is a logarithmic or quadratic function. Now we prove, assuming (A.l), that introducing unbiased forward markets results in higher capital investment and higher labor, hence the output increases in both periods. We shall also assume that F(K, L) is a Cobb-Douglas production function for the sake of simplifying the proof of the next theorem. This result can be proved, assuming (A.l), for most CES production functions. Although our assumptions are sufficient conditions we believe that this theorem does not hold without some additional restrictions on the utility and the production functions.

Theorem 3: Assume that (A. I) holds and that F(K, L) is a Cobb-Douglas production function. Introducing unbiasedforward markets results in higher capital investment and higher production; i.e., higher exports in both periods. In particular, K* > I?, L:>L, and Ll(e,)>:@z(e,) for all values of e,.

Since the proof of Theorem 3 is relatively long we bring it in the Appendix.

IV. Discussion

The floating of major world currencies in recent years has caused an increase in the activity of forward (and futures) currency markets. A number of countries (including Finland, F rance, Germany, Israel, Italy, Spain, and Sweden) have instituted exchange rate guarantee programs. 1 These arrangements are designed to aid firms which produce for export markets to face exchange rate risks. This paper is another contribution to analyzing the impact of such forward markets upon the competitive firm’s production and optimal hedging decisions in a multiperiod model.

Let us show that the firm’s overhedging behavior in the first period holds even if one introduces unbiased long-term forward markets. Suppose that in our model

111 Exporriny firm and forward markers

we add an unbiased forward market from date 0 to date 2. This will not affect the firm’s optimal production decisions and its first period overhedging behavior for the following reasons. First, at time 0 the second period output is uncertain and secondly the firm can hedge against the impact of low realization of the exchange rate in the First period on profits in all periods only through one-period forward markets. Thus an optimally chosen two-period hedge makes the covariance between the marginal utility of profits and the second-period exchange rate 0; however, this is achieved by the optimal one-period hedge chosen at the beginning of period 2: Formally, it can be shown that the additional first-order condition due to this extra unbiased forward market can be derived from the other conditions.’

Our argument about the overhedging behavior (Theorem 2) which has been derived for a two-period model can be extended to any finite or infinite horizon model under some conditions. By using some value function argument one can extend the proof of Theorem 2 to such cases.

Let us, briefly, consider some policy implications of our results. A number of countries have instituted exchange rate guarantee programs for longer ranges of time relative to the existing forward markets. Our result shows that, except for the subsidy that might be involved in such a program, there is no effect upon the production decisions of the firms described here (assuming that unbiased forward markets are available).

Our result regarding the overhedging should be taken into account by the tax authorities in various countries. For example, in the USA for tax purposes one is considered as a speculator if one sells short in the forward market but is not long in the commodity (currency) itself. Our result shows that selling forward more than one’s output (or its value) may stem from hedging purposes in a many-period model.

It is traditional in this literature to compare the firm’s behavior under uncertainty with the ‘certainty equivalent’ case (see for example Sandmo, 1971; Leland, 1972). The classical result in the one-period model is that the behavior of the risk-neutral firm is identical to the certainty-equivalent case. Here we observe, in the multiperiod framework (with fixed and variable inputs), that this result does not hold. In the certainty-equivalent model the first-order conditions are: (Let Z2 = EZ,),

(A) ZIPIFK(KC, L:)+i?,P,F,(K’, L;)-r = 0,

03) ZIP,F,(KC, L;) = wl,

cc> Z,P,F,(K’, L;) = w2.

However, the optimality conditions for risk-neutral firms are,

<A’) E{Z,P,FK(K, E,)+PZ(e,)PJK(K &(e,))-r) = 0,

<B’) g,P,F,(K, L,) = ~1,

CC’> G(el)PPJK Mel)) = wz for all e, .

It is clear that these two sets of conditions yield different solutions since, for example, in (A’) Cov(?,(e,), F,(K, L,(e,))>O in general. Hence

t2P2ECE~(K &(e, ))I - w2 < 0. Some of the results in this paper hold without the assumption of constant

ITZHAK ZILCHA AND RAFAEL ELDOR 115

returns to scale technology. However, Theorems 1 and 3 do not hold without this assumption.

Appendix

Proof of Theorem 3: Assume that e{=Z, =EP, and e~(e,)=~,(e,)=E(P,)2, =e,). Note that X:(e,)= P2F(K*, L:(e,)) for all e, (see, for example, the proofofTheorem 2), hence

(Al) ir* = ic(K*, Lr, L:) = P,P,F(K*, L:)+P,P,F(K*, L:(e,))-rK*-w,L:-w,L:

=n*+XT(P,-Z,)+P,F(K*,Lt(e,)).(E,-Z,(e,)).

In particular, under the assumption of unbiasedness, ir* is riskier than n*, since Eir* = Erc* (see Rothschild and Stiglitz, 1970). From equation (7) we obtain that L.z(ei) is a nondecreasing function of e,. Also, note that equations (4) and (8) imply, when e{=t?,, that (19) holds.

By our assumption about the production function we know that for some constant 8,

and also

Therefore, Z* can be expressed as follows:

(A2) n*(ei) = Zi(PiF(K*, ~:)-X:)+~,(e,)P,F(K*, I;)-w,L:(e,)-w,L:-r

= PI(P,F(K*, ~:)-X:)+~,(e,)LT(e,)F,(K*, L:(el))+Z,(e,)P,F,(K*, Lz(e,))

-w’LZ(e,)-w,L,-rK*

= t,(P,F(K*, Lr)-X:)+L*Lz(e,)+const.

For some positive constant I*. We have used the fact that Ct(ei)FL(K*, L:(e,))= w2 =const. for all e, (see (9)). Using equations (19) and (AZ) we now derive:

(A3) Cov(L:(ei), U’(n*)) = $ Cov[n*(e,)+(Xr -PIF(K*, Lf))t,, U’(n*)]

= $ Cov(n*, v’(rr*)) < 0.

The last equality has been obtained due to (19). Now we use the fact that XT 2 P,F(K*, 15:) (see the proof of Theorem 2) to derive that

<A4) Lz(e,), aJ&u’(ir*)

EU’(ir*) ’ >I

First note that equation (A4) holds when W’ is linear since En* = E?*. When U’ is convex and XV’(X) concave, since ir* is riskier than K*, we find that

<A% EtY’(n*) < EU’(ir*),

<AfQ E,n*U’(n*) > Eic*U’(ir*),

116 Exporting firm and forward markets

which imply that equation (A4) holds because

and the fact (which follows from the monotonicity of ?r* in ei)

0 = Cov(P,, V’(77”)) 2 Cov(P,, cr’(ir*)).

NOW let us prove the assertion of the theorem. Let us rewrite condition (3). Since x* does not depend upon the realization of P,, for each given e, we can rewrite equation (3) as follows: For some positive constant q*,

(A7) E,[Z,P,FJK*, Lr)+q*Lf(e,)-r]U’(n*) = 0.

Now, using equation (A4) we obtain from equation (A7) that

(Ag) E,[PIPIFK(K*, t~)+q*L:(el)-r]Erv’(ff*) < 0.

Similarly, we obtain from equations (4) and (5),

(A9) E,(P,P,FL(K*, L:)-wI)EzU’(ir*) < 0,

(AlO) E,(7,(e,)P,F,(K*, L;(e,))-wl)E,U’(ir*) < 0.

Now consider the first-order conditions for K, t,, i, (which are necessary and sufficient conditions for optimality when X, ~0 and Xz SO). Since the maximand EU(ii) is concace in K, L, and L,, comparing conditions (AS), (A9). and (AlO) with equations (IO), (1 l), and (12) we derive, using the results in Theorem 1 and the assumption that F,,>O, that K*2E, L:>L, and Lz(e,)>L(e,) for all e,.

Notes

An exchange rate guarantee program is designed to afford exporters the benefits of a forward market where none exist. Countries which have established such programs and which also have forward markets for their currencies-among them France, Germany, and Japan-limit participation in the programs to commitments which cannot be covered by the forward markets (generally in excess of two years). In addition, some of these programs incorporate a distinct subsidy element. Let ei= EP, be the forward price for contracts between t =0 and r=2. X: is the corresponding forward sale in this market. Then the additional condition is:

E(ei -Z,)U’(n**) = 0

where R** = rr* + X:(e! - P2). This condition is implied from the unbiasedness assumption and conditions (6) and (7) since

Cov(P,, Ij’(77**)) = E,, Cov(e,(e,), r/‘(n**)ie,) = 0.

References

BENNINGA, S.R., R. ELDOR, AND I. ZILCHA, ‘Optimal International Hedging and Output Policies in Commodity and Currency Forward Markets,’ Journal of International Mane) and Finance, December 1985, 4: 537-552.

ETHIER, W., ‘International Trade and the Forward Exchange Market,’ American Economic Review, June 1973, 63: 494-503.

FEDER, G., R.E. JUST, AND S. SCHMITZ, ‘Futures Markets and the Theory of the Firm Under Price Uncertainty,’ Quarterly Journal of Economics, March 1980, 94: 317-328.

HOLTHAUSEN, D.M., ‘Hedging and the Competitive Firm under Price Uncertainty,’ American Economic Review, December 1979, 69: 989-995.

ITZHAK ZILCHA AND RAFAEL ELDOR 117

KAWAI, M., AND I. ZILCHA, ‘International Trade with Forward Futures Markets Under Exchange Rate and Price Uncertainty,’ Journal of international Economics, February 1986, 20: 83-98.

LELAND, H.E., ‘Theory of the Firm Facing Uncertain Demand,’ American Economic Review. June 1972, 62: 278-291.

ROTHSCHILD, M., AND J.E. STIGLITZ, ‘Increasing Risk: I. A Definition,’ Journal of Economic Theory, September 1970, 2: 225-243.

SANDMO, A., ‘On the Theory of the Competitive Firm Under Price Uncertainty,’ American Economic Review, March 1971, 61: 65-73.

STEWART, M., ‘Factor Price Uncertainty with Variable Proportions,’ American Economic Review, June 1978, 68: 468473.

TURNOVSKY, S., ‘Production Flexibility, Price Uncertainty and the Behavior of the Competitive Firm,’ International Economic Review, June 1973, 4: 395-413.

WRIGHT, B.D., ‘The Effects of Price Uncertainty on the Factor Choices of the Competitive Firm, Southern Economic Journal, October 1984, 5: 443-455.