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Technological advance and hedging decisions in unbiased futures markets
J ECO BUSN 1994; 46:215-225
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Technological Advance and Hedging Decisions in Unbiased Futures Markets Fathali Firoozi
This study investigates the effect of a productivity-improving technological advance on hedging behavior of producers in an environment where output price is uncertain and an unbiased futures market is available. A participation criterion that explicitly incorporates technology of production is stated as a minimum futures price required by a producer to participate in the futures market. It is shown that the effect of a technological advance on the minimum participation price is a function of the producer's attitude toward risk. The results indicate that a technological advance introduces an additional risk factor into the decision to assume risk by nonparticipation.
I. Introduction The theory of the firm under output price uncertainty has been studied by a number of authors, including Sandmo (1971), Leland (1972), and Batra and Ullah (1974). Subsequently, Holthausen (1979), Feder et al. (1980), and others extended the theory to an environment where there is an opportunity to hedge output price risk via a futures market. These latter studies have shown that when a futures market is present and the firm decides to participate, the level of optimal output is determined based on the prevailing futures price, as in the deterministic case. However, the firm may assume a futures position that may be different from its planned production level. Antonovitz and Roe (1986) and Dalai and Arshanapalli (1989), among others, have shown that the optimal position taken by the firm in a biased futures market depends on whether the futures price underestimates or overestimates the expected spot price (normal backwardation or contango, respectively). 1 The unsettled issue of futures market biasedness (normal baekwardation and contango) has been the subject of a number of studies with mixed results, e.g., Newbery and Stiglitz [(1981), p. 185], Anderson and Danthine (1983), and Williams University of Texas, San Antonio, Texas. Address reprint requests to Fathali Firoozi, Division of Economics and Finance, University of Texas, San Antonio, Texas 78249. ~In the case of participation, the difference between the adopted futures position and planned outpout constitutes the speculative component of the position, which may be positive or negative. Journal of Economics and Business © 1994 Temple University
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i l,ir~)~ [(1986), p. 84]. In a comprehensive study, Newbery and Stiglitz [(1)81), p. 185; suggest that for many commodity futures the hypothesis of zero bias is reasonable. Several studies, including Holthausen [(1979), p. 990) and Newbery and Stiglitz [(1981), p. 185], have shown that when the futures market is unbiased, the optimal futures position adopted by the firm amounts to hedging the entire output with a zero speculative component. O ' H a r a (1985) incorporated technology of production explicitly and showed that the presence of an unbiased futures market is not sufficient, even for a risk-averse producer, to participate in the futures market. O ' H a r a assumed that the decision to hedge output price risk is determined by the attitude toward price risk, which is only partially determined by the attitude toward profit risk reflected on the producer's utility function U(Tr). The other portion of attitude toward price risk is determined by the technology of production reflected on the profit function 7r(P). Given unbiased futures markets, O ' H a r a ' s participation criterion is that the producer assumes a futures position to hedge the entire output if the producer is averse toward price risk; otherwise, the producer assumes risk by nonparticipation. 3 A deficiency associated with O ' H a r a ' s participation criterion is that the perceived future spot price distribution does not play any role in the criterion. 4 In the present study we utilize a general participation criterion that incorporates technology of production, attitude toward profit risk, and spot price distribution. The criterion is that the producer participates in the futures market to hedge output price risk if maximum utility derived from participation exceeds maximum expected utility derived from nonparticipation. We proceed to show that the participation criterion can be stated in terms of a minimum participation price P,II. If the observed futures price P~ is below P~, the producer assumes risk by nonparticipation. The determinants of P,~ will be specified. Second, we evaluate the effects of a productivity-improving technological advance on decision by a producer regarding participation in the futures market. Third, we show that in response to a technological advance, whether the number of participating hedgers in the aggregate increases or declines is determined by producers' attitude toward profit risk. Finally, we examine some implications of the aggregate hedging response for the futures price~
2The assumption of unbiased futures has been adopted in a number of studies, including McKinnon (1967), Binniga et al. (1984), and O'Hara (1985). 3To elaborate, define the function V(P) = U(Tr(P)). It is clear that aversion toward profit risk (U~, < 0) is not sufficient for aversion toward price risk (Vpp < 0). The shape of function V is determined by the shapes of both functions U and ~r, where the shape of rr is determined by technology of production. O'Hara [(1985), Proposition 1] shows that when the production technology X = F(L) has the usual concave shape (FLL < 0), the profit function 7r(P) is convex or linear (Itm, > 0). In O'Hara [(1985), Proposition 2] itis shown that the firm participates in the unbiased futures (Vpp < 0) if the condition - U ~ / U ~ > (1/rrp)rrrp/rr p holds. It is clear that the usual risk-aversion condition r = -U~,,/U,~ > 0 (concave utility, U~ < 0) is not sufficient for participation in the futures market (Gp < o3. 4It should be noted that the O'Hara model investigated Hicks' conjecture regarding normal backwardation. When hedgers are the only participants in forward markets, O'Hara shows that technology can cause a producer to sell output forward (long hedging) without buying input forward (short hedging). In aggregate, O'Hara concludes, such an imbalance in initial unbiased forward markets can lead to normal baekwardation (Hicks' conjecture).
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II. Environment The single-input deterministic production of a producer is defined by X = A F ( L ) , where X is the level of production, A > 0 is a multiplicative productivity index,5 L is the level of input employed, and F is a function with the usual production characteristics of increasing and strict concavity, FL - - d F / d L > 0 and FLL = d 2 F / d L 2 < 0. The producer buys the input needs L in a competitive market at a given rate w. The uncertainty regarding the spot price of the output is reflected on a given density function g ( P ) with the expected value/z, E[P] --/~. The producer has an opportunity to participate in an unbiased futures market with the prevailing futures price p0 =/z. The producer's utility (U) is an increasing function of profits (zr) defined by U = U(1r), U' > 0. The function U satisfies the von Neumann-Morgenstern axioms [DeGroot (1970)] so that optimal decision under uncertainty (nonparticipation) is derived from maximization of expected utility. Risk aversion, neutrality, and preference are reflected, respectively, by U" < 0, U" = 0, and U" > 0. The profit function is defined by 7r = P X - wL = P A F ( L ) - wL. A s indicated in the last section, in the case of participation in the unbiased futures market, the producer hedges the entire output with zero speculative component. Hence, optimal employment and output in the case of participation are derived from the maximization of 7r = P ° A F ( L ) - wL. In the case of nonparticipation, optimal employment and output are derived from the maximization of E[U(~')].
III. Participation C r i t e r i o n The producer participates in the futures market if maximum utility derived from participation is greater than maximum expected utility derived from nonparticipation. 6 In this section we demonstrate that this criterion can be stated as a minimum futures price pO required by the producer to participate. If the prevailing futures price p0 is below pO, the producer chooses to assume risk by nonparticipation. We will characterize the determinants of the minimum participation price pO. If the producer chooses to assume risk by nonparticipation, the optimal factor employment is derived from the maximization of E[U(~)] with respect to L, where E[U(~r)] = E { U [ P A F ( L ) - wL]}. The first-order condition is7 E[ P U ' ] A F L = E[U']w.
(1)
5The parameter A can also be interpreted as the index of any scheme of output expansion at fixed input. 6A fundamental difference between the present participation criterion and the criterions applied by Antonovitz and Roe (1986) and Dalai and Arshanapalli (1989) is the dependence property of the present criterion in unbiased environments (p0 =/x), i.e., the present criterion is sensitive to parametric changes such as a technological advance or a change is spot price distribution. The participation decisions emerging from the criterions applied in the above studies are independent of such parametric changes. The dependence property of the present criterion will be shown at the end of this section and in the next section. 7If the producer is risk-averse, the strict concavity of production is sufficient but not necessary for the maximization. If the producer is risk-neutral, the strict concavity is both sufficient and necessary for the maximization as in the certainty case. These results are shown by Batra and Ullah (1974).
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F FirooJ Utilizing the definition h = P A F r ...... w. the first-order condition can be written a~ E [ h U ' ] = O.
(2~
Let L* be the solution to (1) or (2). The maximum expected utility derived from nonparticipation is denoted by (in the continuous case) U* = max E[U(Tr)] = m a x f p U [ P A F ( L )
- w L ] g ( P ) dP,
(3)
where the maximization is over L. Equivalently U * = E[U(Tr*)], where ~r*= P A F ( L * ) - wL*. The nonparticipation optimal output is X* = A F ( L * ) .
If the producer participates in the market at the futures price p0, the optimal hedge (output) is derived from the maximization of ~-= P ° A F ( L ) - wL. The first-order condition P°AFL - w = 0 is solved to derive the optimal factor employment L ° as a function of p08: L ° = L°(P°),
d L ° / d P ° > 0.
(4)
Utilizing (4), the participation optimal profit, utility, and output are defined as functions of p0: 71"O =
(5)
pOAF( L °) - wL o = 7r°(p°),
(6)
U ° = U(,n -°) = U ° ( p ° ) , X ° = A F ( L °) = X ° ( p ° ) , where it follows from (4) that dX°/dP ° = ( OX°/c)L°)(dL°/dP°)
= A F L ( d L ° / d P °) > O.
Differentiating U ° and utilizing the participation first-order condition P°AFL - w = 0 lead to o u O / o p o = U ' [ d ~ - ° / d p o] = U'[ A F ( L °) + (P°AFL - w ) ( d L O / d p ° ) ] = U ' X ° > 0.
(7)
It is clear that the participation maximum utility U ° is a monotonically increasing function of the futures price p0. The participation decision is now derived from a comparison of (3) and (6). The producer is indifferent toward participation at a futures price pO and nonparticipation if the corresponding maximum expected utilities are equal. Thus, p0 is the solution to u°(e~) = u*,
(8)
8Differentiatingthe first-o~er condition P°AFL - w = 0 yi~h~s P°AFL L( dL° / dP ° ) + A F L = O. The solution is d L ° / d P ° ffi " [ F L / ( P ° F L L ) ] > O, where the sign is due to the production concavity, ELL <~0.
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where U ° and U* are specified in (6) and (3), respectively. Because p 0 > pO implies U°(P °) > U°(P° ) = U*, the producer chooses to participate in the futures market when p0 > pO. The hedging function is then given by H = X ° if p0 > pO and H = 0 if p0 < pm0, where X ° is the participation optimal output. For a futures price p0
IV. Technology
and Participation
We now evaluate the effect of a productivity-improving technological advance on the producers' decision regarding participation in the futures market. A technological advance is reflected by a rise in the productivity index A of the production function. We evaluate the effect of a rise in A on the minimum participation price po. Therefore, the objective is to evaluate the sign of dP°s/dA. The following result shows that the sign of d P ° / d A for a marginal producer is completely determined by the producer's attitude toward profit risk, i.e., the shape of the utility function U(~r). Proposition. The minimum participation price P ° o f a marginal producer 11 responds to a productivity-improving technological advance as follows: (i) Pm° falls if the producer is risk-averse. (ii) pO rises if the producer is risk-loving. (iii) pO remains unchanged if the producer is risk-neutral.
9A better name for/Do would be the inferior of the participation price set or the superior of the nonpartidpation price set1°The present participation criterion is sufficientlygeneral so that it covers any shape that the distribution g(P) may assume. Note that the computationof U* ha 0) involvesthe expectationof the random variable U(w) = U[PAF(L ) - wL]. Because U and F are general nonlinear functions,assurehag a specificshape for the distributionof P (e,g., the normal distr~ution) does not produce a particular result regardingthe distributionof U0r). u GLventhe futures price p0, we considera producerwhose initial decisionregardingparticipation is close to the borderline. For such a "marginalproducer",(i) the initial deviation IP,° - p0j is sufficiently small so that Pm ° ffi p0 is an approximation and (ii) the initial deviation [pO _ pe"I is smaller than the mal'~mnalchange IdP°l that emerges from a technologicaladvance(dA), Clearly,a producer for whom initiallyP,~ ffipu is a marginalproducer.
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i tqroozi
paoov Evaluate dP~/dA utilizing (8), U°I,P,¢,I) - U*, when the futures price ~Y'~ .... /*) is at the minimum participation level, p0 = P,II = /x. Differentiation of (8) with respect to A yields (aU°/c)A) + ( aU" / aP~,l)( dP,~,i/dA } = dU* /dA. Solving for d P ° / d A leads to
dP°m/dA = [1/( OU°/OP°m)]{(dU*/d,4) - (OU°/OA)}.
(9)
It is shown in (7) that OU°/OP2 > 0. It follows that the sign of dP°m/dA is determined by the sign of the term in braces in (9), which is denoted by T:
T = {(dU*/dA) - (OU°/,gA)}. It is shown in the Appendix that the sign of T is completely determined by the producer's attitude toward risk as follows: (i) If the producer is risk-averse, then T < 0 and d P ° / d A < O. (ii) If the producer is risk-loving, then T > 0 and d P ° / d A > O. (iii) If the producer is risk-neutral, then T --- 0 and d P ° / d A = O.
[]
Given the futures price p0 and spot price distribution, the proposition demonstrates the likely changes in the number of hedging participants in the futures market emerging from a technological advance: 1. In the case where producers are risk-averse, initial nonparticipation decisions by marginal producers (when initial pO is at or marginally above p0) are reversed to participation as pO falls. Hence, a likely outcome is that the number of participating hedgers will rise. 2. If producers are risk seekers, initial participation decisions by marginal producers (when initial Pm° is at or marginally below p0) are reversed to nonparticipation as pO rises. In this case, a likely outcome is that the number of participating heders will fall. An intuitive explanation follows. A productivity-improving technological advance leads to higher levels of optimal employment and output in both participation and nonparticipation cases. Given the futures price and spot price distribution, higher employment and output will raise the level of risk assumed under nonparticipation. When producers are risk-averse, this additional nonparticipation risk factor introduced by the technological advance may be sufficient to induce some producers to switch from initial nonparticipation to participation. This outcome is formally shown in the proposition by a fall in pO of risk-averse marginal producers. When producers are risk seekers, the additional risk factor under nonparticipation may induce some producers to switch from initial participation to nonparticipation. This outcome is established in the proposition by a rise in po of risk-seeking marginal producers.
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V. Discussion Studies on commodity futures can be divided into two groups: 1. Those focusing on behavior of an individual particpant where futures price and spot price distribution are given exogenously, e.g., Holthausen (1979), Feder et al. (1980), O'Hara (1985), Antonovitz and Roe (1986), and Dalai and Arshanapalli (1989). 2. Those concentrating on market equilibrium (partial and general) with endogenous prices and identification of relations between spot and futures prices, e.g., Newbery and Stiglitz (1981), Anderson and Danthine (1983), Turnovsky (1983), and Stein (1986). The equilibrium studies typically involve specifications of supply and demand sides and evaluate variations in equilibrium prices or quantities. The studies in group 1, on the other hand, evaluate variations in one side of the market. Therefore, the studies in group 1 represent short-term evaluations in the sense that long-term responses by market prices (changes in spot price distribution and futures price) are not fed back into the evaluations. However, short-term evaluations identify underlying microeconomic behavior, which is essential in long-term (equilibrium) studies. Clearly, the present study is a short-term evaluation that concentrates on hedging behavior of producers in response to a technological advance. Some aggregate implications, however, can be drawn in certain ad hoc contexts. We proceed with an example. Following the market assumptions and terminology of O'Hara (1985), suppose the futures market under study is composed of hedgers only. Long contracts (selling output forward) are entered by producers in the primary industry that experiences technological advance, and short contracts (buying input forward) are entered by producers in another industry that experiences no change. As an example, consider a forward wheat market in which long and short contracts are entered by farmers and bakers, respectively. Suppose a technological advance occurs in wheat production only. Assuming farmers are risk-averse (U" < 0), the proposition in the last section shows that the minimum participation price falls and a likely outcome is that the number of participating farmers will rise. In addition, as optimal participation output rises due to the technological advance, the volume of each long contract will also rise. It is clear that a combined effect will be a fall in the futures price. If the expected future spot price remains unchanged, as in O'Hara (1985), normal backwardation emerges in the futures market. 12 If farmers are risk seekers (U" > 0), however, the propostion shows that a technological advance leads to a rise in the minimum participation price and a likely fall in the number of participating farmers. In this case, the effect of a technological advance on the futures price is ambiguous (because the volume of each long contract rises and the number of participating farmers may fall). In the case where farmers are risk-neutral (U" = 0), a technological advance produces no chang e in the number of participating farmers. However, the rise in volume of long contracts leads to a fall in the futures price unambiguously.
12Clearly, should the futures market remain unbiased, a fall in the mean of the future spot price distribution follows.
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J;. Firoozi The proposition and the above discussion can be utilized to evaluate the case where technological progress takes place on behalf of the competitors of a producer only. Assuming that the competitors are risk-averse, both the number of producers selling output forward and volume of each contract rise. Hence, the futures price falls and the producer who does not experience the technological progress may switch from initial participation to nonparticipation (this happens if the futures price falls below the producer's minimum participation price).
Vl. Conclusions A producer participates in an unbiased futures market to hedge output price risk if maximum utility derived from participation exceeds maximum expected utility derived from nonparticipation. Such a participation criterion can be stated as a minimum participation price pO. A producer participates in the futures market if the prevailing futures price exceeds pO. We have specified the determinants of po. The criterion is a generalization of the O'Hara (1985) criterion in the sense that the present criterion incorporates the future spot price distribution as well as the attitude toward profit risk and technology of production. We have evaluated the effects of a technological advance on futures market participation by producers. The results indicate that a productivity-improving technological advance introduces an additional risk factor into the decision to assume risk by nonparticipation: 1. In the case where producers are risk-averse, a technological advance leads to a fall in the minimum participation price of marginal producers and a likely increase in the number of participating long hedgers. If the futures market is composed of hedgers only, a fall in the futures price follows. 2. In the case where producers are risk seekers, a technological advance leads to a rise in the minimum participation price of marginal producers and a likely fall in the number of participating long hedgers. Because the volume of each long contract rises, the net effect on the futures price in this case is ambiguous. 3. If producers are risk-neutral, no change in the number of participating long hedgers is expected to emerge from a technological advance. However, the futures price is expected to fall in this case due to a rise in volume of each long contract.
Appendix In this Appendix we show the relation between the sign of T = {(dU*/dA) (aU°/OA)} and the producer attitude toward profit risk. The relation was utilized in the proof for the proposition in the text. The results show that, for a marginal risk-averse producer facing a technological advance, the rise in maximum expected utility derived from nonparticipation (dU*/dA) is smaller thari the rise in maximum utility derived from participation (OU°/aA). For a risk-loving producer, dU*/dA > OU°/ OA, and for a risk-neutral producer, dU*/dA = OU°/ OA.
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Following the proof in the text, we first evaluate the second term in T. Note that (5) and (6) imply c~U°/ OA = U'( ~r°)[ d~r°/dA] = U'(~ro)[pOX° + P ° A F L ( d L ° / d A ) - w(dL°/dA)]
= U'(~r°)[ tzX ° + (P°AFL - w)(dL°/dA)] = izU'(~r°)X ° > O, (A.1) where the last equality is due to the participation first-order condition Pm° AFc - w -- 0 and U'(~r °) is U' evaluated at Ir °. In evaluating the first term in T, it follows from (3) that d U * / d A = ( d / d A ) E [ U ( ~ ) ] = E[U'(d~r/dA)],
(A.2)
where 7r = P A F ( L ) - wL and L satisfies the nonparti¢ipation first-order condition (1). Utilizing the definition for h, the evaluation of d~r/dA leads to dcr/dA = P F ( L ) + h ( d L / d A ) .
(A.3)
Differentiating the first-order condition (1) with respect to A yields
E[ PU" (d'rr//dA)]AFL + E[ PU']FL + E[ PU'IAFLr.(dL/dA) = E[U" (d~'/dA)]w.
(A.4)
Substitution of (A.3) into (A.4) and rearranging terms lead to E[PU"F(L)(PAFr
- w)] + E[PU']FL
1
+
E[U"(P,'WL - w ) 2 + P U ' A F L L ] ( d L I d A ) = 0.
(A.5)
Solving (A.5) for d L / d A with a use of the definition for h yields the constant d L / d A = - E [ P U " h F ( L ) + PU'FL]/E[U~h 2 + PU'AFL~],
(A.6)
where all the right-hand side terms are evaluated using the iaitial nonparticipation optimal indues. Because d L / d A is the constant defined in (A.6), substitution of (A.3) into (A.2) yields dU*/dA = E[PU'IF(L) + E[hU'I(dL/dA).
An application of'the first-order condition (2)shows that d U * / d A = E [ P U ' ] F ( L ) = E [ P U ' O r ' ) ] X * > 0.
Substitution of (A.I) and (A.7) into T yields i
T = E[PU'(~'*)]X* - ~ U ' ( ~ ° ) X °.
(A.7)
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! !:irooz Note that E[ PU'] = txE[U'] ~ covt P, U'I. It follows that T - - ix{E[U'(rr*)]X *
//'lrr~)X~'} ~ - X * c o v ( P , U ' )
Recall that the equality (8) can be written as U{J(Tr t~) = E[U(Tr*)]. Differentiating both sides shows that U'(~r~q = E[U'(vr*)]. Therefore, T can be written as T = t x U ' ( r r % ( X * - X °) + X * cov(P, U').
(A.8)
We now evaluate T in the following three cases. 1. The producer is risk-averse. In this case U'(~r) > 0 and U"(~r) < 0, i.e., U is a strictly increasing and concave function of ~r. It follows from d z r / d P > 0 that 8 U ' / S P = U " ( d T r / d P ) < 0, implying c o v ( P , U ' ) < 0. This establishes the negativity of the second term in (A.8). To show the negativity of the first term, note that X* is the optimal output under uncertainty (nonparticipation) and X ° is the optimal output under certainty (participation), where the certainty price p0 is equal to the expected uncertain price ~. A number of studies, including Sandmo [(1971), p. 66], Batra and Ullah [(1974), p. 541], and Hey [1979), p. 129], have shown that X* < X ° for a risk-averse producer. It follows that the first term in (A.8) is also negative. Therefore, T < 0. 2. The producer is risk-loving. In this case U" > 0 and coy(P, U') > 0, which shows the positivity of the second term in T. The studies mentioned in case 1 above have also shown that under risk preference, X* > X °. Therefore, T>0. 3. The producer is risk-neutral. In this case U" = 0 and U' is a constant. Hence, cov(P, U ' ) = 0. The studies mentioned in case 1 above have shown that under risk-neutrality, X* = X °. It follows that T = 0.
The author thanks two anonymousreferees for helpful comments.
References Anderson, R. W., and Danthine, J. 1983. Hedger diversity in futures markets. Economic Journal 93:370-389. Antonovitz, F., and Roe, T. 1986. Effects of expected cash and futures prices on hedging and production. Journal of Futures Markets 6:187-205. Batra, R. N., and Ullah, A. 1974. Competitive firm and the theory of input demand under price uncertainty. Journal of Political Economy 82:537-548. Benniga, S., Eldor, R., and Zilcha, I. 1984, The optimal hedge ratio in unbiased futures markets. Journal of Futures Markets 4:155-160. Dalai, A. J., and Arshanapalli, B. 1989. Effects of expected cash and futures prices on hedging and production: Comments and extensions. Journal of Futures Markets 4:337-345. DeGroot, M. H. 1970. Optimal Statistical Decisions. New York: McGraw-Hill. Feder, G., Just, R., and Schmitz, A. 1980. Futures markets and the theory of the firm under price uncertainty. Quarterly Journal of Economics 94:317-328.
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Hey, J. D. 1979. UncertaintY in Microeconomics. New York University Press. Holthausen, D. M. 1979. Hedging and the competitive firm under price uncertainty, American Economic Review 5:989-995. Leland, H. 1972. theory of the firm facing uncertain demand. American Economic Review 62:278-291. McKinnon, R. I. 1967. Futures markets, buffer stocks, and income stability for primary producers. Journal of Political Economy 75:844-861. Newbery, D., and Stiglitz, J. E. 1981. The Theory of Commodity Price Stabilization. New York: Oxford University Press. O'Hara, M. 1985. Technology and hedging behavior: A proof of Hicks' conjecture. American Economic Review 75:1186-1190. Sandmo, A. 1971. On the theory of the competitive firm under price uncertainty. American Economic Review 61:65-73. Stein, J. L 1986, The Economics of Futures Markets. New York: Basil Blackwell. Turnovsky, S. 1983. The determination of spot and futures prices with storable commodities. Econometrica 51:1363-1387. Williams, J. 1986. The Economic Function of Futures Markets. New York: Cambridge University Press.
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Technological Advance and Hedging Decisions in Unbiased Futures Markets Fathali Firoozi
This study investigates the effect of a productivity-improving technological advance on hedging behavior of producers in an environment where output price is uncertain and an unbiased futures market is available. A participation criterion that explicitly incorporates technology of production is stated as a minimum futures price required by a producer to participate in the futures market. It is shown that the effect of a technological advance on the minimum participation price is a function of the producer's attitude toward risk. The results indicate that a technological advance introduces an additional risk factor into the decision to assume risk by nonparticipation.
I. Introduction The theory of the firm under output price uncertainty has been studied by a number of authors, including Sandmo (1971), Leland (1972), and Batra and Ullah (1974). Subsequently, Holthausen (1979), Feder et al. (1980), and others extended the theory to an environment where there is an opportunity to hedge output price risk via a futures market. These latter studies have shown that when a futures market is present and the firm decides to participate, the level of optimal output is determined based on the prevailing futures price, as in the deterministic case. However, the firm may assume a futures position that may be different from its planned production level. Antonovitz and Roe (1986) and Dalai and Arshanapalli (1989), among others, have shown that the optimal position taken by the firm in a biased futures market depends on whether the futures price underestimates or overestimates the expected spot price (normal backwardation or contango, respectively). 1 The unsettled issue of futures market biasedness (normal baekwardation and contango) has been the subject of a number of studies with mixed results, e.g., Newbery and Stiglitz [(1981), p. 185], Anderson and Danthine (1983), and Williams University of Texas, San Antonio, Texas. Address reprint requests to Fathali Firoozi, Division of Economics and Finance, University of Texas, San Antonio, Texas 78249. ~In the case of participation, the difference between the adopted futures position and planned outpout constitutes the speculative component of the position, which may be positive or negative. Journal of Economics and Business © 1994 Temple University
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i l,ir~)~ [(1986), p. 84]. In a comprehensive study, Newbery and Stiglitz [(1)81), p. 185; suggest that for many commodity futures the hypothesis of zero bias is reasonable. Several studies, including Holthausen [(1979), p. 990) and Newbery and Stiglitz [(1981), p. 185], have shown that when the futures market is unbiased, the optimal futures position adopted by the firm amounts to hedging the entire output with a zero speculative component. O ' H a r a (1985) incorporated technology of production explicitly and showed that the presence of an unbiased futures market is not sufficient, even for a risk-averse producer, to participate in the futures market. O ' H a r a assumed that the decision to hedge output price risk is determined by the attitude toward price risk, which is only partially determined by the attitude toward profit risk reflected on the producer's utility function U(Tr). The other portion of attitude toward price risk is determined by the technology of production reflected on the profit function 7r(P). Given unbiased futures markets, O ' H a r a ' s participation criterion is that the producer assumes a futures position to hedge the entire output if the producer is averse toward price risk; otherwise, the producer assumes risk by nonparticipation. 3 A deficiency associated with O ' H a r a ' s participation criterion is that the perceived future spot price distribution does not play any role in the criterion. 4 In the present study we utilize a general participation criterion that incorporates technology of production, attitude toward profit risk, and spot price distribution. The criterion is that the producer participates in the futures market to hedge output price risk if maximum utility derived from participation exceeds maximum expected utility derived from nonparticipation. We proceed to show that the participation criterion can be stated in terms of a minimum participation price P,II. If the observed futures price P~ is below P~, the producer assumes risk by nonparticipation. The determinants of P,~ will be specified. Second, we evaluate the effects of a productivity-improving technological advance on decision by a producer regarding participation in the futures market. Third, we show that in response to a technological advance, whether the number of participating hedgers in the aggregate increases or declines is determined by producers' attitude toward profit risk. Finally, we examine some implications of the aggregate hedging response for the futures price~
2The assumption of unbiased futures has been adopted in a number of studies, including McKinnon (1967), Binniga et al. (1984), and O'Hara (1985). 3To elaborate, define the function V(P) = U(Tr(P)). It is clear that aversion toward profit risk (U~, < 0) is not sufficient for aversion toward price risk (Vpp < 0). The shape of function V is determined by the shapes of both functions U and ~r, where the shape of rr is determined by technology of production. O'Hara [(1985), Proposition 1] shows that when the production technology X = F(L) has the usual concave shape (FLL < 0), the profit function 7r(P) is convex or linear (Itm, > 0). In O'Hara [(1985), Proposition 2] itis shown that the firm participates in the unbiased futures (Vpp < 0) if the condition - U ~ / U ~ > (1/rrp)rrrp/rr p holds. It is clear that the usual risk-aversion condition r = -U~,,/U,~ > 0 (concave utility, U~ < 0) is not sufficient for participation in the futures market (Gp < o3. 4It should be noted that the O'Hara model investigated Hicks' conjecture regarding normal backwardation. When hedgers are the only participants in forward markets, O'Hara shows that technology can cause a producer to sell output forward (long hedging) without buying input forward (short hedging). In aggregate, O'Hara concludes, such an imbalance in initial unbiased forward markets can lead to normal baekwardation (Hicks' conjecture).
Technological Advance and Hedging Decisions
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II. Environment The single-input deterministic production of a producer is defined by X = A F ( L ) , where X is the level of production, A > 0 is a multiplicative productivity index,5 L is the level of input employed, and F is a function with the usual production characteristics of increasing and strict concavity, FL - - d F / d L > 0 and FLL = d 2 F / d L 2 < 0. The producer buys the input needs L in a competitive market at a given rate w. The uncertainty regarding the spot price of the output is reflected on a given density function g ( P ) with the expected value/z, E[P] --/~. The producer has an opportunity to participate in an unbiased futures market with the prevailing futures price p0 =/z. The producer's utility (U) is an increasing function of profits (zr) defined by U = U(1r), U' > 0. The function U satisfies the von Neumann-Morgenstern axioms [DeGroot (1970)] so that optimal decision under uncertainty (nonparticipation) is derived from maximization of expected utility. Risk aversion, neutrality, and preference are reflected, respectively, by U" < 0, U" = 0, and U" > 0. The profit function is defined by 7r = P X - wL = P A F ( L ) - wL. A s indicated in the last section, in the case of participation in the unbiased futures market, the producer hedges the entire output with zero speculative component. Hence, optimal employment and output in the case of participation are derived from the maximization of 7r = P ° A F ( L ) - wL. In the case of nonparticipation, optimal employment and output are derived from the maximization of E[U(~')].
III. Participation C r i t e r i o n The producer participates in the futures market if maximum utility derived from participation is greater than maximum expected utility derived from nonparticipation. 6 In this section we demonstrate that this criterion can be stated as a minimum futures price pO required by the producer to participate. If the prevailing futures price p0 is below pO, the producer chooses to assume risk by nonparticipation. We will characterize the determinants of the minimum participation price pO. If the producer chooses to assume risk by nonparticipation, the optimal factor employment is derived from the maximization of E[U(~)] with respect to L, where E[U(~r)] = E { U [ P A F ( L ) - wL]}. The first-order condition is7 E[ P U ' ] A F L = E[U']w.
(1)
5The parameter A can also be interpreted as the index of any scheme of output expansion at fixed input. 6A fundamental difference between the present participation criterion and the criterions applied by Antonovitz and Roe (1986) and Dalai and Arshanapalli (1989) is the dependence property of the present criterion in unbiased environments (p0 =/x), i.e., the present criterion is sensitive to parametric changes such as a technological advance or a change is spot price distribution. The participation decisions emerging from the criterions applied in the above studies are independent of such parametric changes. The dependence property of the present criterion will be shown at the end of this section and in the next section. 7If the producer is risk-averse, the strict concavity of production is sufficient but not necessary for the maximization. If the producer is risk-neutral, the strict concavity is both sufficient and necessary for the maximization as in the certainty case. These results are shown by Batra and Ullah (1974).
218
F FirooJ Utilizing the definition h = P A F r ...... w. the first-order condition can be written a~ E [ h U ' ] = O.
(2~
Let L* be the solution to (1) or (2). The maximum expected utility derived from nonparticipation is denoted by (in the continuous case) U* = max E[U(Tr)] = m a x f p U [ P A F ( L )
- w L ] g ( P ) dP,
(3)
where the maximization is over L. Equivalently U * = E[U(Tr*)], where ~r*= P A F ( L * ) - wL*. The nonparticipation optimal output is X* = A F ( L * ) .
If the producer participates in the market at the futures price p0, the optimal hedge (output) is derived from the maximization of ~-= P ° A F ( L ) - wL. The first-order condition P°AFL - w = 0 is solved to derive the optimal factor employment L ° as a function of p08: L ° = L°(P°),
d L ° / d P ° > 0.
(4)
Utilizing (4), the participation optimal profit, utility, and output are defined as functions of p0: 71"O =
(5)
pOAF( L °) - wL o = 7r°(p°),
(6)
U ° = U(,n -°) = U ° ( p ° ) , X ° = A F ( L °) = X ° ( p ° ) , where it follows from (4) that dX°/dP ° = ( OX°/c)L°)(dL°/dP°)
= A F L ( d L ° / d P °) > O.
Differentiating U ° and utilizing the participation first-order condition P°AFL - w = 0 lead to o u O / o p o = U ' [ d ~ - ° / d p o] = U'[ A F ( L °) + (P°AFL - w ) ( d L O / d p ° ) ] = U ' X ° > 0.
(7)
It is clear that the participation maximum utility U ° is a monotonically increasing function of the futures price p0. The participation decision is now derived from a comparison of (3) and (6). The producer is indifferent toward participation at a futures price pO and nonparticipation if the corresponding maximum expected utilities are equal. Thus, p0 is the solution to u°(e~) = u*,
(8)
8Differentiatingthe first-o~er condition P°AFL - w = 0 yi~h~s P°AFL L( dL° / dP ° ) + A F L = O. The solution is d L ° / d P ° ffi " [ F L / ( P ° F L L ) ] > O, where the sign is due to the production concavity, ELL <~0.
Technological Advance and Hedging Decisions
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where U ° and U* are specified in (6) and (3), respectively. Because p 0 > pO implies U°(P °) > U°(P° ) = U*, the producer chooses to participate in the futures market when p0 > pO. The hedging function is then given by H = X ° if p0 > pO and H = 0 if p0 < pm0, where X ° is the participation optimal output. For a futures price p0
IV. Technology
and Participation
We now evaluate the effect of a productivity-improving technological advance on the producers' decision regarding participation in the futures market. A technological advance is reflected by a rise in the productivity index A of the production function. We evaluate the effect of a rise in A on the minimum participation price po. Therefore, the objective is to evaluate the sign of dP°s/dA. The following result shows that the sign of d P ° / d A for a marginal producer is completely determined by the producer's attitude toward profit risk, i.e., the shape of the utility function U(~r). Proposition. The minimum participation price P ° o f a marginal producer 11 responds to a productivity-improving technological advance as follows: (i) Pm° falls if the producer is risk-averse. (ii) pO rises if the producer is risk-loving. (iii) pO remains unchanged if the producer is risk-neutral.
9A better name for/Do would be the inferior of the participation price set or the superior of the nonpartidpation price set1°The present participation criterion is sufficientlygeneral so that it covers any shape that the distribution g(P) may assume. Note that the computationof U* ha 0) involvesthe expectationof the random variable U(w) = U[PAF(L ) - wL]. Because U and F are general nonlinear functions,assurehag a specificshape for the distributionof P (e,g., the normal distr~ution) does not produce a particular result regardingthe distributionof U0r). u GLventhe futures price p0, we considera producerwhose initial decisionregardingparticipation is close to the borderline. For such a "marginalproducer",(i) the initial deviation IP,° - p0j is sufficiently small so that Pm ° ffi p0 is an approximation and (ii) the initial deviation [pO _ pe"I is smaller than the mal'~mnalchange IdP°l that emerges from a technologicaladvance(dA), Clearly,a producer for whom initiallyP,~ ffipu is a marginalproducer.
2211
i tqroozi
paoov Evaluate dP~/dA utilizing (8), U°I,P,¢,I) - U*, when the futures price ~Y'~ .... /*) is at the minimum participation level, p0 = P,II = /x. Differentiation of (8) with respect to A yields (aU°/c)A) + ( aU" / aP~,l)( dP,~,i/dA } = dU* /dA. Solving for d P ° / d A leads to
dP°m/dA = [1/( OU°/OP°m)]{(dU*/d,4) - (OU°/OA)}.
(9)
It is shown in (7) that OU°/OP2 > 0. It follows that the sign of dP°m/dA is determined by the sign of the term in braces in (9), which is denoted by T:
T = {(dU*/dA) - (OU°/,gA)}. It is shown in the Appendix that the sign of T is completely determined by the producer's attitude toward risk as follows: (i) If the producer is risk-averse, then T < 0 and d P ° / d A < O. (ii) If the producer is risk-loving, then T > 0 and d P ° / d A > O. (iii) If the producer is risk-neutral, then T --- 0 and d P ° / d A = O.
[]
Given the futures price p0 and spot price distribution, the proposition demonstrates the likely changes in the number of hedging participants in the futures market emerging from a technological advance: 1. In the case where producers are risk-averse, initial nonparticipation decisions by marginal producers (when initial pO is at or marginally above p0) are reversed to participation as pO falls. Hence, a likely outcome is that the number of participating hedgers will rise. 2. If producers are risk seekers, initial participation decisions by marginal producers (when initial Pm° is at or marginally below p0) are reversed to nonparticipation as pO rises. In this case, a likely outcome is that the number of participating heders will fall. An intuitive explanation follows. A productivity-improving technological advance leads to higher levels of optimal employment and output in both participation and nonparticipation cases. Given the futures price and spot price distribution, higher employment and output will raise the level of risk assumed under nonparticipation. When producers are risk-averse, this additional nonparticipation risk factor introduced by the technological advance may be sufficient to induce some producers to switch from initial nonparticipation to participation. This outcome is formally shown in the proposition by a fall in pO of risk-averse marginal producers. When producers are risk seekers, the additional risk factor under nonparticipation may induce some producers to switch from initial participation to nonparticipation. This outcome is established in the proposition by a rise in po of risk-seeking marginal producers.
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V. Discussion Studies on commodity futures can be divided into two groups: 1. Those focusing on behavior of an individual particpant where futures price and spot price distribution are given exogenously, e.g., Holthausen (1979), Feder et al. (1980), O'Hara (1985), Antonovitz and Roe (1986), and Dalai and Arshanapalli (1989). 2. Those concentrating on market equilibrium (partial and general) with endogenous prices and identification of relations between spot and futures prices, e.g., Newbery and Stiglitz (1981), Anderson and Danthine (1983), Turnovsky (1983), and Stein (1986). The equilibrium studies typically involve specifications of supply and demand sides and evaluate variations in equilibrium prices or quantities. The studies in group 1, on the other hand, evaluate variations in one side of the market. Therefore, the studies in group 1 represent short-term evaluations in the sense that long-term responses by market prices (changes in spot price distribution and futures price) are not fed back into the evaluations. However, short-term evaluations identify underlying microeconomic behavior, which is essential in long-term (equilibrium) studies. Clearly, the present study is a short-term evaluation that concentrates on hedging behavior of producers in response to a technological advance. Some aggregate implications, however, can be drawn in certain ad hoc contexts. We proceed with an example. Following the market assumptions and terminology of O'Hara (1985), suppose the futures market under study is composed of hedgers only. Long contracts (selling output forward) are entered by producers in the primary industry that experiences technological advance, and short contracts (buying input forward) are entered by producers in another industry that experiences no change. As an example, consider a forward wheat market in which long and short contracts are entered by farmers and bakers, respectively. Suppose a technological advance occurs in wheat production only. Assuming farmers are risk-averse (U" < 0), the proposition in the last section shows that the minimum participation price falls and a likely outcome is that the number of participating farmers will rise. In addition, as optimal participation output rises due to the technological advance, the volume of each long contract will also rise. It is clear that a combined effect will be a fall in the futures price. If the expected future spot price remains unchanged, as in O'Hara (1985), normal backwardation emerges in the futures market. 12 If farmers are risk seekers (U" > 0), however, the propostion shows that a technological advance leads to a rise in the minimum participation price and a likely fall in the number of participating farmers. In this case, the effect of a technological advance on the futures price is ambiguous (because the volume of each long contract rises and the number of participating farmers may fall). In the case where farmers are risk-neutral (U" = 0), a technological advance produces no chang e in the number of participating farmers. However, the rise in volume of long contracts leads to a fall in the futures price unambiguously.
12Clearly, should the futures market remain unbiased, a fall in the mean of the future spot price distribution follows.
222
J;. Firoozi The proposition and the above discussion can be utilized to evaluate the case where technological progress takes place on behalf of the competitors of a producer only. Assuming that the competitors are risk-averse, both the number of producers selling output forward and volume of each contract rise. Hence, the futures price falls and the producer who does not experience the technological progress may switch from initial participation to nonparticipation (this happens if the futures price falls below the producer's minimum participation price).
Vl. Conclusions A producer participates in an unbiased futures market to hedge output price risk if maximum utility derived from participation exceeds maximum expected utility derived from nonparticipation. Such a participation criterion can be stated as a minimum participation price pO. A producer participates in the futures market if the prevailing futures price exceeds pO. We have specified the determinants of po. The criterion is a generalization of the O'Hara (1985) criterion in the sense that the present criterion incorporates the future spot price distribution as well as the attitude toward profit risk and technology of production. We have evaluated the effects of a technological advance on futures market participation by producers. The results indicate that a productivity-improving technological advance introduces an additional risk factor into the decision to assume risk by nonparticipation: 1. In the case where producers are risk-averse, a technological advance leads to a fall in the minimum participation price of marginal producers and a likely increase in the number of participating long hedgers. If the futures market is composed of hedgers only, a fall in the futures price follows. 2. In the case where producers are risk seekers, a technological advance leads to a rise in the minimum participation price of marginal producers and a likely fall in the number of participating long hedgers. Because the volume of each long contract rises, the net effect on the futures price in this case is ambiguous. 3. If producers are risk-neutral, no change in the number of participating long hedgers is expected to emerge from a technological advance. However, the futures price is expected to fall in this case due to a rise in volume of each long contract.
Appendix In this Appendix we show the relation between the sign of T = {(dU*/dA) (aU°/OA)} and the producer attitude toward profit risk. The relation was utilized in the proof for the proposition in the text. The results show that, for a marginal risk-averse producer facing a technological advance, the rise in maximum expected utility derived from nonparticipation (dU*/dA) is smaller thari the rise in maximum utility derived from participation (OU°/aA). For a risk-loving producer, dU*/dA > OU°/ OA, and for a risk-neutral producer, dU*/dA = OU°/ OA.
Technological Advance and Hedging Decisions
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Following the proof in the text, we first evaluate the second term in T. Note that (5) and (6) imply c~U°/ OA = U'( ~r°)[ d~r°/dA] = U'(~ro)[pOX° + P ° A F L ( d L ° / d A ) - w(dL°/dA)]
= U'(~r°)[ tzX ° + (P°AFL - w)(dL°/dA)] = izU'(~r°)X ° > O, (A.1) where the last equality is due to the participation first-order condition Pm° AFc - w -- 0 and U'(~r °) is U' evaluated at Ir °. In evaluating the first term in T, it follows from (3) that d U * / d A = ( d / d A ) E [ U ( ~ ) ] = E[U'(d~r/dA)],
(A.2)
where 7r = P A F ( L ) - wL and L satisfies the nonparti¢ipation first-order condition (1). Utilizing the definition for h, the evaluation of d~r/dA leads to dcr/dA = P F ( L ) + h ( d L / d A ) .
(A.3)
Differentiating the first-order condition (1) with respect to A yields
E[ PU" (d'rr//dA)]AFL + E[ PU']FL + E[ PU'IAFLr.(dL/dA) = E[U" (d~'/dA)]w.
(A.4)
Substitution of (A.3) into (A.4) and rearranging terms lead to E[PU"F(L)(PAFr
- w)] + E[PU']FL
1
+
E[U"(P,'WL - w ) 2 + P U ' A F L L ] ( d L I d A ) = 0.
(A.5)
Solving (A.5) for d L / d A with a use of the definition for h yields the constant d L / d A = - E [ P U " h F ( L ) + PU'FL]/E[U~h 2 + PU'AFL~],
(A.6)
where all the right-hand side terms are evaluated using the iaitial nonparticipation optimal indues. Because d L / d A is the constant defined in (A.6), substitution of (A.3) into (A.2) yields dU*/dA = E[PU'IF(L) + E[hU'I(dL/dA).
An application of'the first-order condition (2)shows that d U * / d A = E [ P U ' ] F ( L ) = E [ P U ' O r ' ) ] X * > 0.
Substitution of (A.I) and (A.7) into T yields i
T = E[PU'(~'*)]X* - ~ U ' ( ~ ° ) X °.
(A.7)
224
! !:irooz Note that E[ PU'] = txE[U'] ~ covt P, U'I. It follows that T - - ix{E[U'(rr*)]X *
//'lrr~)X~'} ~ - X * c o v ( P , U ' )
Recall that the equality (8) can be written as U{J(Tr t~) = E[U(Tr*)]. Differentiating both sides shows that U'(~r~q = E[U'(vr*)]. Therefore, T can be written as T = t x U ' ( r r % ( X * - X °) + X * cov(P, U').
(A.8)
We now evaluate T in the following three cases. 1. The producer is risk-averse. In this case U'(~r) > 0 and U"(~r) < 0, i.e., U is a strictly increasing and concave function of ~r. It follows from d z r / d P > 0 that 8 U ' / S P = U " ( d T r / d P ) < 0, implying c o v ( P , U ' ) < 0. This establishes the negativity of the second term in (A.8). To show the negativity of the first term, note that X* is the optimal output under uncertainty (nonparticipation) and X ° is the optimal output under certainty (participation), where the certainty price p0 is equal to the expected uncertain price ~. A number of studies, including Sandmo [(1971), p. 66], Batra and Ullah [(1974), p. 541], and Hey [1979), p. 129], have shown that X* < X ° for a risk-averse producer. It follows that the first term in (A.8) is also negative. Therefore, T < 0. 2. The producer is risk-loving. In this case U" > 0 and coy(P, U') > 0, which shows the positivity of the second term in T. The studies mentioned in case 1 above have also shown that under risk preference, X* > X °. Therefore, T>0. 3. The producer is risk-neutral. In this case U" = 0 and U' is a constant. Hence, cov(P, U ' ) = 0. The studies mentioned in case 1 above have shown that under risk-neutrality, X* = X °. It follows that T = 0.
The author thanks two anonymousreferees for helpful comments.
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