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Industry equilibrium, uncertainty, and futures markets
InternationalJeumal of
ELSEVIER
International Journal of Industrial Organization 14 (1996) 53-70
Industrial Organization
Industry equilibrium, uncertainty, and futures markets Shoji Haruna Department of Economics, Okayama University, Tsushima-Naka 3-I-1, Okayama 700, Japan Accepted July 1994
Abstract
This paper studies competitive long-run industry equilibrium with uncertainty and futures trading. We also provide a long-run analysis of futures trading instead of the conventional short-run analysis. It is shown without using a general equilibrium model that, given uncertain demand and risk aversion, a bias (backwardation or contango) arises in long-run industry equilibrium. Given risk neutrality such a bias, however, disappears. We find that the occurrence of the bias depends not only on the existence of a risk premium, but also on the length of the period considered. Each risk-averse firm operates at a less efficient scale and, moreover, an expected spot price never accords with marginal cost. When the expected spot price is less than marginal cost, the risk-averse firm can secure profit by selling futures contracts more than the amount of its output. In the presence of production (supply) uncertainty a risk-neutral firm operates with excess capacity under additive risk, while the firm operates with proper capacity under multiplicative risk. The key lies in whether its average cost curve is shifted by production risk.
Keywords: Industry equilibrium; Uncertainty; Futures trading; Bias JEL classification: D41; D81; L l l
I. Introduction
A l t h o u g h the n u m b e r of firms in a competitive industry is fixed in the short run, the n u m b e r varies t h r o u g h entry and exit in the long run if there 0167-7187/96/15.00 (~ 1996 Elsevier Science B.V. All rights reserved .SSDI 0167-7187(94)00456-0
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are no entry and exit costs and no legal regulations on entry and exit. The optimal number of firms in the industry as well as the optimal output of each firm is determined in long-run industry equilibrium. The properties of such equilibrium without uncertainty have been considered and clarified. On the other hand, the conventional approach to competitive industry equilibrium with uncertainty is to dichotomize the uncertainty into the demand and supply sides. Given that firms are risk neutral, one can show different effects of uncertainty on the output price and level, depending on which side the uncertainty occurs. For example, the study by Beuthe et al. (1984) shows that if production (supply) is uncertain, the planned output of the firm is larger and the average equilibrium market price of the output is higher than those in the absence of uncertainty. As for the case of demand uncertainty but without production uncertainty, Sheshinski and Drrze (1976) and Lippman and McCall (1981) demonstrate that each firm's output is less than the minimum efficient scale that corresponds to average-cost minimizing output. Their discussions are, however, based on a special assumption that an industry demand curve is zero-elastic with respect to industry output. For a more general case with a downward-sloping industry demand curve, Pazner and Razin (1975) consider long-run industry equilibrium with output price (demand) uncertainty, and obtain a relationship between the equilibrium output of the firm and its attitude toward risk in the long run. Following Pazner and Razin, for example, Appelbaum and Katz (1986) and Chavas et al. (1988) have conducted comparative static analyses, i.e. the effects of output price uncertainty, input price, and fixed costs in long-run industry equilibrium, on the output of the individual firm and the number of firms in the industry. One of the standard theoretical results concerning futures trading is the separation theorem of hedging and production decisions; the firm's output level depends only on a non-stochastic futures price but is independent of its attitude toward risk and the probability distribution of the spot price (see, for example, Holthausen, 1979; Feder et al., 1980; and Anderson and Danthine, 1983); its output level is not affected by spot price uncertainty at all. Anderson and Danthine consider the problem of hedging (and speculation) and production decisions, and further causes of a bias, which is a difference between futures and expected spot prices in the presence of spot price and production uncertainty, using a general equilibrium model with a rational expectations hypothesis. 1 Hedging implies the sale of futures or selling output in a futures market and speculation is the sale of futures greater than output or buying futures. The existing literature is confined to
The bias is called backwardation(contango)when the futures price exceeds (is less than) the expected price. See Houthakker (1986) for backwardation.
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considering short-run futures trading and short-run futures market equilibrium. An exception is Ishii (1987). He conducts a comparative static analysis under long-run industry equilibrium with futures trading but does not mention the issue of whether the bias arises. The innovation of this paper is to explicate the problem of the bias (backwardation or contango) that might exist in the long run: the existing literature on this subject is also confined to short-run cases. The purpose of this paper is to consider long-run industry equilibrium with futures trading and uncertainty. We are concerned with how the long-run behavior of firms and a competitive industry with either demand or production uncertainty is affected by introducing futures contracts. 2 From another point of view, this paper provides a model for a long-run analysis of hedging: we discuss the hedging decision of the firm and the relationship between the futures and the expected spot prices in a long-run industry equilibrium. In Section 2 we investigate the properties of the long-run industry equilibrium with demand uncertainty and futures trading. The principal contribution is that the futures price is not an unbiased estimator of the random spot price, namely there certainly arises a bias, i.e. either backwardation or contango, in the futures market whenever firms are risk averse. In contrast, given risk neutrality, the futures price is equal to the expected spot price so that the bias vanishes. In the long run the risk-averse firm either hedges less than its entire output or speculates by selling futures contracts more than its output, whereas the risk-neutral firm hedges all of its entire output. It is found that whether the bias appears in the long-run equilibrium relates to both the existence of a risk premium and the length of the analytic period. We also show that the risk-averse firm is producing at less than efficient scale, where marginal cost falls short of average cost, and the expected spot price either exceeds or is less than marginal cost. When the expected spot price is less than marginal cost, the risk-averse firms can make a profit by positively taking advantage of futures contracts like a speculator and then can remain in the industry. The availability of futures trading makes a significant difference to the properties of the long-run industry equilibrium. In Section 3 we explore long-run industry equilibrium with production uncertainty incorporating a rational expectations hypothesis, and show that the firm's scale of operation is affected by the way risk is incorporated into production: a risk-neutral firm will produce more than capacity output, for which average cost is minimized, under additive risk, and just capacity output under multiplicative risk. The key is that the average cost curve of
2In this paper, however, we do not conduct comparative static analysis. See Ishii (1987) for that analysis in a similar model with demand uncertainty and futures contacts.
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the firm is shifted up by risk when it is additive, but not when it is multiplicative. Alternatively, the bias may not occur when production is random. Section 4 summarizes our conclusions.
2. The competitive industry and demand uncertainty We consider a competitive industry that produces a homogeneous commodity, which may be perishable. There are n identical firms in the industry with no barriers to entry and exit. In this section we explore the behavior of the competitive industry with demand uncertainty. The commodity is traded in both spot and futures markets, i.e. each firm has access to the futures market as well as the spot m a r k e t ) The futures market provides for producers a guaranteed price for their output and reduces some of the risk due to any fluctuation in the future price, with which they are faced. Hence, by use of futures trading (contracts) the firm can eliminate income risk. Oils and fats, gold, non-ferrous metals, and cotton thread are subject to demand risk, and futures markets for them are widely developed. Let us denote the output of each firm by qi. Suppose that the industry demand function facing the firms in the spot market is given by p = F(Q, a), where Q = ET=lq i = nqi is industry output, p spot price, and a a random variable with probability density function ~b(a). The function F(Q, a) is assumed to decrease in the total amount of output sold in the spot market; the spot price is decreasing in Q, i.e. dp/dQ = pQt < 0. We posit that F(Q, a) is an increasing function of a, and that F(0, a ) is positive for all a. Each firm can also join a futures market and use futures trading. When the quantity of firm i's futures trading (contracts) is denoted by h i, it is positive (negative) if the firm sells (purchases) futures. Since firms hedge in futures markets to eliminate income risk, they will generally supply or sell futures. On the other hand, speculators are located on the opposite side, i.e. on the demand side, in the markets, so that they will purchase (demand) futures. Sometimes, however, they may sell futures contracts. Anyway, the speculators trade only futures. T o make the analysis simple, we take their demand for futures to be given; that is, we do not explicitly treat the behavior of the speculators (see A n d e r s o n and Danthine, 1983, for their behavior). Then we assume that the d e m a n d function of the futures market takes the form of Ph = p h ( H ) , where Ph stands for futures price and H = ZT=1 h i = n h i is the total amount of
3 For a discussion of futures trading, see, for example, Holthausen (1979), Newbery and Stiglitz (1981), and A n d e r s o n and Dathine (1983).
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futures sold by firms in the futures m a r k e t . Futures price is decreasing in the total a m o u n t of hedging, d p h / d H = P'h < 0.4 W i t h the futures m a r k e t available the profits of each firm are 7ri = p ( q ~ - hi) + phh~ -- c ( q i ) ,
w h e r e c ( q i ) is the cost function with fixed costs such that c ( 0 ) = c o > 0 , d c ( q i ) / d q i = c'(qg) > 0, and c"(qi) > 0; we assume that a l t h o u g h the long-run total costs for each firm are zero at zero o u t p u t level, they rise to c o w h e n any p r o d u c t i o n at all is a t t e m p t e d . T h e average cost curve, c ( q i ) / q i is U - s h a p e d , and there exists unique o u t p u t q* > 0 such that c'(qg) - c ( q i ) / q i ~_ 0 as qi ~ q * , w h e r e q* is the o u t p u t for which average cost is m i n i m u m and is r e f e r r e d to as capacity output. T h e firm must d e t e r m i n e the a m o u n t s of o u t p u t and futures trading b e f o r e d e m a n d uncertainty is resolved. Its objective is n o w to c h o o s e t h e m so as to maximize the e x p e c t e d utility of profits, EU(Tri), where U(Tri) is a v o n N e u m a n n - M o r g e n s t e r n utility function with U ( 0 ) = 0, U ' ( ~ ' ) > 0 and U"(rr) ~< 0, and E is the expections operator. Maximization of the e x p e c t e d utility o f profits reflects the firm's attitude t o w a r d risk and its evaluation of profits u n d e r uncertainty. F o r simplicity, hereafter we suppress the subscript i. T h e p r o b l e m of the individual firm is to solve m a x EU(Tr) = E U [ p ( q q.h
- h ) + p h h - c(q)].
T h e first-order conditions for a m a x i m u m are given by 5 0EU(~-)
Oq
- E{U'(Tr)[p
= 0,
(1)
+ P h ) } = O.
(2)
- c'(q)]}
0EU(~r) Oh
= E(U'(Tr)(-p
F r o m these conditions we get Ph = c ' ( q ) ,
(3)
4 See, for example, Stein (1961, 1979), Danthine (1978), Newbery and Stiglitz (1981), and Anderson and Danthine (1983) for futures market equilibrium. Hereafter, we assume that all equilibrium solutions are unique. 5 For the risk-averse firm, the second-order conditions hold since [O2EU(Tr)/ Oq2][02EU(Tr)/ Oh2] _ [OzEUQr) / OqOh]2 = -c"(q)E[U '(Tr)]E[U"(Tr)(p - Ph)2] > 0. Although these conditions are not necessarily satisfied in the presence of risk neutrality, we assume that there is an interior solution.
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which is the separation theorem in the theory of futures trading. 6 In the absence of uncertainty, if there are fixed costs and they are assumed to rise when any production is attempted, firms will continue to enter (leave) the industry as long as they can make positive (negative) profits (see Kreps, 1990). The long-run industry equilibrium is not sustained in a situation where firms make negative profits at any output level. Therefore, all firms make zero profits in a long-run industry equilibrium. In contrast, firms must now decide whether to enter or exit before demand uncertainty is resolved. We take up the concept of reservation utility U R, and suppose an industry equilibrium condition: EU(Tr) = E U [ p ( q - h )
+ph h -- c(q)] = U R .
(4)
If the expected utility of each firm exceeds the reservation utility, it causes firms to enter the industry, while if the expected utility falls short of the reservation utility, firms leave the industry. In the long run the expected utility of each firm will be equated to U R, because the expected utility is a decreasing function of the number of firms. 7 For simplicity, we normalize the level of the reservation utility to zero, U R = 0, which is equivalent to the utility of not producing and not hedging (speculating), where q = 0 and h = 0. By analogy with the certainty case, the expected utility of each firm becomes equal to zero, not negative, in the long-run industry equilibrium since fixed costs are assumed not to arise before entry. Consider the firm's choice of output. From (1) we have q E [ U ' ( ~ ) p ] = q E [ U ' ( T r ) ] c ' ( q ) , and subtracting c ( q ) E U ' ( 1 r ) from both sides of this equation yields: E { U ' ( T r ) [ p q - c(q)]} = [ c ' ( q ) q - c(q)]EU'(1r).
Taking account of both E[U'('rr)]ph = E[U'(Tr)p], obtained from (2), and industry equilibrium condition (4), we get
6 F r o m (1) and (2) we obtain that if U"(~r) < 0, then q / > ( < ) h according as Ph <~( > ) E ( p ) , if U"(~') = 0, t h e n q = h according as Ph = E ( p ) . T h e s e results are the s a m e as those of Feder et al. (1980) and H o l t h a u s e n (1979). 7 Differentiating the left-hand side of (4) with respect to n yields:
d E U ( ~ ) / d n = E U ' ( ~ ' ) [ ( q - h )2p~ + h 2p'~l < O, b e c a u s e p ~ < 0 and p;, < 0.
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c(q) q
c'(q) =
E[U(Tr) - U'(rr)Ir] qEU'(Tr)
59
(5)
T h e n u m e r a t o r on the right-hand side of (5) is positive (negative) or zero, depending on whether the utility function is concave (convex) or linear. 8 Thus it follows that c ( q ) / q - c'(q)/> ( < )0, according to whether the firm is risk averse (risk-loving) or risk neutral. With a U-shaped average cost curve this leads to the following result: in the long-run competitive industry equilibrium with futures contracts, the firm produces where average cost is larger (less) than or equal to marginal cost, depending on whether it is risk averse (risk loving) or risk neutral. 9 We know from this that even if the firm has access to the futures market, there still exists excess capacity under risk aversion. 1° The firm's choice of output depends on its attitude toward risk exclusively, and the introduction of futures trading has no influence on its output choice. Thus it seems that its introduction is no inducement to mitigate the excess capacity of the risk-averse firm, although Holthausen (1979) and N e w b e r y and Stiglitz (1981) assert in their short-run models that the availability of futures markets induces a firm to increase output. 11 Consequently, their results do not carry over to the long-run analysis. Let us turn to examining the long-run industry equilibrium in some detail. We rewrite the industry equilibrium condition as follows: EU(~-) = U[E(~r) - p] = U R = O, where p ( > 0 or = 0) stands for the risk premium. We have s T h e m e t h o d of deriving (5) is essentially the same as in Pazner and Razin (1975). T h e sign of the n u m e r a t o r on the right-hand side of (5) depends crucially on the values of U(0) and U R as well as the firm's attitude toward risk. If assumptions U ( 0 ) = 0 a n d / o r U R = 0 are now relaxed, the results on that sign do not always accord with those derived in (5). For example, for U(0) = 0 and U R > 0 the sign of E[U(0r) - U'(~')~'] will be of either sign, or negative if the firm is risk averse or risk neutral. In contrast, for U R = 0 and U(0) > 0 the sign of E[U(zr) U'(~')Tr] will be positive if it is either risk averse or risk neutral. F u r t h e r m o r e , as pointed out by o n e of the referees, w h e n the utility function takes the form of U(~r) = - e -r~, with r > 0, for U R = 0, E [ U ( ~ - ) - U ( ~ ' ) z r ] will be of either sign because U ( 0 ) < 0 : namely, this result is different from that on the sign of E[U(~') - U'(zr)~r] in (5). 9 F r o m the fact that the sign of the n u m e r a t o r on the right-hand side of (5) d e p e n d s on the level of U(0) we find that the result does not always carry over to any affine transformation of the original utility function. ~0 This is the same as the result of Pazner and Razin (1975), but is different from that of Sheshinski and Dr6ze (1976) that excess capacity prevails u n d e r risk neutrality. This s e e m s to be due to their specific assumption that the r a n d o m industry d e m a n d is evenly divided a m o n g firms in the industry. 11 A n d e r s o n and D a n t h i n e (1983), however, show in their short-run model that the introduction of futures trading does not always cause the firm to increase expected output when production uncertainty is added.
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E(rr) = p.
(6)
Rewriting the e x p e c t e d profits gives: E(~') - p = ~ - - P h ) ( q -- h ) + c ' ( q ) q - c ( q ) - p, w h e r e fi = E ( p ) . W h e n e v e r the futures price is equal to the expected spot price, ~ = Ph, i.e. there is not a bias which is the difference b e t w e e n b o t h prices, the e q u a t i o n a b o v e is r e d u c e d to E ( ~ - ) - p = c ' ( q ) q - c ( q ) - p . P r o v i d e d that the firm is risk averse, E 0 r ) - p < 0 because c ( q ) - c ' ( q ) q > 0 a n d 0 > 0. It follows that, given risk aversion, E(Tr) - p is negative. T h u s a c o n t r a d i c t i o n arises in c o m p a r i n g (6) with the result E ( T r ) < p d r a w n u n d e r risk aversion. This involves that the industry equilibrium condition and the a s s u m p t i o n of unbiasedness that the futures price is an unbiased estimator of the e x p e c t e d spot price are inconsistent u n d e r risk aversion. 12 In the p r e s e n c e of a futures m a r k e t , in o r d e r for the industry equilibrium condition to be satisfied, it is necessary for the expected spot price not to coincide with the futures price, n a m e l y the futures price must be a biased estimator of the e x p e c t e d spot p r i c e ) 3 W h a t is the cause of the bias? T h e r e are two causes. First, the bias is c o n s i d e r e d to be incurred by the risk p r e m i u m and the level of operation. M o r e specifically, its cause can be ascribed to the risk p r e m i u m since the level of o p e r a t i o n is, in fact, d e p e n d e n t on the firm's attitude t o w a r d risk, which can also be r e p r e s e n t e d by the risk p r e m i u m . We find f r o m an e x a m i n a t i o n of e q u a t i o n (6) that its size also relies on its d e g r e e in the long run as well as in the short run; that size increases as the degree of the firm's risk aversion increases. A n d e r s o n and D a n t h i n e (1983) also m e n t i o n that the size o f the bias will d e p e n d on the degree of risk aversion of participants in the futures m a r k e t . In contrast, given risk neutrality, we obtain E ( T r ) = 0 since c ( q ) - c ' ( q ) q = 0 and p = 0, even if the futures price is an unbiased e s t i m a t o r of the expected spot price: the bias disappears w h e n the risk p r e m i u m is zero. This d e m o n s t r a t e s that, given risk neutrality, an industry equilibrium condition such as E U ( T r ) = 0 is consistent with the assumption ~2In his short-run equilibrium model with producers and speculators, Danthine (1978) demonstrates that the futures price is not an unbiased estimator of the expected spot price as long as the producers are risk averse and their initial position is riskier than the speculators'. Anderson and Danthine (1983) also describe that in the short run there is a bias if futures sales and purchases do not accord. 13When a differential between the futures and the expected spot prices exists, there is room for entry by speculators, although we do not explicitly introduce them; they could earn profits either by utilizing futures contracts if the futures price exceeds the expected spot price or by purchasing them if the futures price is lower than the expected spot price (see Anderson and Danthine, 1983, and Danthine 1978).
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of unbiasedness. Second, we cannot overlook the fact that the length of period considered plays a definite role in the incidence of the bias. 14 This is because even in the case of risk neutrality the bias may arise if the assumption U R = 0 is not satisfied. Therefore, our result addresses doubts about the validity of making such an assumption in the long run, although the assumption of unbiasedness is often made in the short run (see, for example, McKinnon, 1967, and Benninga et al., 1983). It should be borne in mind that not only the risk premium but also the length of the analytic period affects the existence of the bias. We are now in a position to examine the relationship between spot price and marginal cost. By Taylor's theorem and then taking expectations we have EU(Tr) = U(0) + U'(0)E(or) +
EIU"(ATr)(zr) 2]
0
where U ( 0 ) = 0. In the long-run industry equilibrium it follows that E ( 1 r ) = -E[U"(ATr)(rr)2]/2U'(O), so that E(~r) is positive or zero, according to whether the firm is risk averse or risk neutral. Since c ' ( q ) = P h , we have E ( ~ ) = (q - h ) ~ - c'(q)] + c ' ( q ) q - c(q). Combining these results yields: (q - h ) ~ - c'(q)] + c ' ( q ) q - c(q) >! 0
for U"(Arr) ~< 0 .
(7)
If the firm is risk neutral, we get p = c ' ( q ) because c ' ( q ) = c ( q ) / q , as shown previously. As can be seen from footnote 6, the expected spot price becomes equal to marginal and average costs and the futures price. Meanwhile, owing to c ( q ) - c ' ( q ) q > 0 for risk aversion, ( q - h ) ~ - c'(q)] must be positive so that s i g n ~ - c'(q)] = sign(q - h). Then, taking the result C'(Q)-~Ph into consideration, we obtain that <(>)Ph,
if and only if q < ( > ) h .
This result is summarized as follows. When the expected spot price is less than the futures price, i.e. there is contango, the firm will speculate by selling an amount greater than its output; when the expected spot price exceeds the futures price, i.e. there is backwardation, the firm will hedge less than its output. The result is the same as that in the short-run analysis (Feder et al., 1980, and Holthausen, 1979). More interestingly, given risk aversion, it does not hold that -fi=Ph,
if and only if q = h .
The reason is as follows: if it holds that c ' ( q ) - c ( q ) / q > 0 from (7), while, 14I am indebted to one of the referees for having pointed out this fact.
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as we have shown above, c ( q ) / q - c ' ( q ) > 0, then both results contradict each other. Thus, in the long-run industry equilibrium the risk-averse firm never hedges all of its output. This is different from the short-run result. In the long-run industry equilibrium the risk-neutral firm operates at the minimum (efficient) level of average cost, where average cost is equal to marginal cost, and the expected spot price becomes equal to marginal cost (and the futures price). In contrast, the risk-averse firm is producing at a less efficient scale, where average cost exceeds marginal cost. At this level of output the expected spot price does not equal marginal cost at all, i.e. that price is either higher or lower than marginal cost. This implies that there exists backwardation or contango, although we cannot specify whether backwardation or contango occurs. 15 When the expected spot price exceeds marginal cost, i.e. fi > c ' ( q ) = P h , the firm will hedge less than its output in the futures market to eliminate risk. The expected spot price will be, at least, higher than average cost. When the expected spot price is less than marginal cost, the firm will speculate by selling futures contracts in excess of its output to secure profit and increase the amount of futures contracts as contango diminishes. Therefore, when the expected spot price falls short of marginal cost, the firm can get profit by taking advantage of futures trading and remain in the industry, while without futures markets firms will carry on leaving the industry as long as expected price is less than marginal cost. In this case the number of firms in the industry with futures trading will be more than without it. We note that the availability of futures trading makes a significant difference to the properties of the long-run industry equilibrium.
3. The competitive industry and production uncertainty We deal with long-run industry equilibrium with production (or supply) uncertainty instead of demand uncertainty. In the real world it is observed that bad weather, and insect pests in the primary industry frequently break out, and the production of agricultural commodities such as wheat, soy bean, coffee, and sugar fluctuates significantly. Then the producers of these commodities are faced with production risk rather than demand risk. So, it would be rare for planned (ex ante) quantities of such commodities to accord with their actual (ex post) quantities. Incorporating supply risk,
15Dr6ze (1987) refers to the case with futures trading and mentions in his conclusion (p. 232) that "in an industry with uncertain future demand, one may expect the price on a forward market to fall below minimum average cost, if that price is to reflect expected marginal cost". His conclusion is, however, invalid in our model.
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therefore, seems reasonable. 16 Let q denote planned output and y(q, a) actual output, where a is a random variable with the probability function y ( a ) expressing disturbances affecting output. It is now assumed that Oy(q, a)/Oq = yq(q, a ) > 0 and by(q, a)/Oa = y~(q, a ) > O. Aggregate supply in the spot market is reduced to Qs = ny(q, a). We denote aggregate demand by Qd = Qd(P)" The spot price is determined by the marketclearing condition
ny(q, a) = Qd(P). The spot price thus becomes a random variable, the distribution of which is generated by the distribution of a, because risk is transferred to the spot price through the market-clearing condition. 17 The density function of the spot price relies on the structure of the spot market as well as the density function of output. We adopt the rational expectations hypothesis, which means that the equilibrium distribution of the spot price is known to every firm. In consequence, an individual firm in the industry is confronted with both spot price and production risks. To distinguish the current random spot price from that used in the previous section we express the former random spot price by t7. The maximization problem of the firm is max EU(~-) = EU[/7(y(q, a) - h) + ph h -- c(q)]. q,h
All symbols, except f o r / 7 and y(q, a) introduced in this section, are the same as in the preceding section. The first-order conditions are given by OEU(~) - - E { U ' ( ' n ' ) [ / 7 Or(q,oqa) Oq OEU(zr) -
-
Oh
-
E[U'(~')(-/7
+Pa)]
= O.
c'(q)]} =0 ,
(8)
(9)
Using (8) and (9) we have phE[U'('n')yq(q, a)] = c'(q)EU'(~r): the separation theorem fails to hold in the presence of production uncertainty (Anderson and Danthine, 1983).
16For an analysis of futures trading and production uncertainty, see, for instance, Newbery and Stiglitz (1981), Anderson and Danthine (1983), and Britto (1984). 17See Newbery and Stiglitz (1981), Anderson and Danthine (1983), and Britto (1984) for models that discuss spot or futures market equilibrium in the analysis of futures trading. See, for example, Newbery and Stiglitz, and Britto for analyses including (short-run) spot market equilibrium.
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Suppose that, as in the case of Section 2, the industry equilibrium condition is EU(Tr) = EU[/7(y(q, ~) - h) + ph h -- c(q)] = 0.
(4')
Firms must decide whether to enter or exit before a a n d / 7 are revealed. Consider the firm's behavior and the connection between the spot price and marginal cost in long-run industry equilibrium. In so doing, since y ( q , a ) is too general for analytical and practical purposes, suppose that y ( q , a ) takes two specified forms, as follows) 8 One is a multiplicatively separated form such as y ( q , a ) = aq, with E ( a ) = 1 and var(a) = 0-2, where var stands for variance. The other is an additively separated form such as y ( q , a ) = q + a, with E ( a ) = 0 and var(a) = 0.2. 3.1. Multiplicative risk When the relationship between planned output and risk is exhibited in a multiplicative way, (8) is rewritten as E { U ' ( ~ ) L ~ - c'(q)]} = 0 .
(8')
It follows from (8') that E [ U ' ( T r ) / T a ] q = E [ U ' ( ~ ) ] c ' ( q ) q . Subtracting E[U'(~r)]c(q) from both sides of this equation and taking (9) into consideration yields: E{U'(~r)[/7(aq - h) + ph h -- c(q)]} = E[U'(~r)][c'(q)q - c(q)]. F u r t h e r m o r e , taking (4') into consideration yields: E [ U ( ~ r ) - U'(Tr)Tr]= q E [ U ' ( T r ) ] [ c ~ )
c'(q)],
where E [ U ( T r ) - U'(~r)Tr]/> 0 according to the firm's attitude toward risk. H e n c e , in a long-run industry equilibrium with futures trading, if production risk is multiplicative, the firm operates at the point where average cost is larger than or equal to marginal cost, depending on whether it is risk averse or risk neutral; the risk-averse firm operates with excess capacity and the risk-neutral firm with proper capacity. The firm's scale of operation will be delimited by its attitude toward risk, as in the case of demand uncertainty. B e u t h e et al. (1984) have concluded that in the long-run industry equilibrium the risk-neutral firm chooses output where average cost is less than marginal cost, and hence less capacity prevails. This is at variance with our result. The difference is due to the following reason: following Beuthe et 18Newbery and Stiglitz (1981) also set up a model with special supply risk.
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al.'s notation, the connection between planned output q and realized output is expressed by a specific form as ~ = q + u ( q , e), with u , - qU~q > 0 , where e is a random variable, while in our multiplicative model such an assumption on u ( q , e) is not adopted. It was derived in Section 2 that the industry equilibrium condition contradicts the assumption of unbiasedness with respect to the futures price under risk aversion. Nevertheless, the issue of inconsistency does not emerge in both the multiplicative and additive risk cases. This issue might not arise when production and spot price uncertainty coexist. Let us now turn to examining the connection between the spot price and costs. By applying Taylor's theorem to (4') we get E(zr) = - E[U"(ATr)(Tr)2]/ 2U'(0), where 0 < A < 1. Given risk neutrality, this expression is rewritten as - c ( q ) / q = h(t~ --Ph)q -- COV(/~, a ) , where /~ = E(/7). Since c ' ( q ) = c ( q ) / q , we have /~ - e ' ( q ) = h ( l ~ - - P h ) / q -- COV(/7, a). The term -cov(/7, ~) can be interpreted as risk-bearing cost for the risk-neutral firm. Thus risk-bearing cost arises even for risk neutrality, different from the demand uncertainty case. When the demand curve in the spot market is downward sloping, [7 and a will move in opposite directions, so that cov(fi, a ) < 0. From (9) we get E ( ~ ) = / ~ = P h under risk neutrality so t h a t / ~ = c ' ( q ) - cov(/7, a ) = c ( q ) / q - cov(fi, a ) ; the expected spot price exceeds marginal cost by the amount of the risk-bearing cost, because cov(/7, a ) < 0 , and is higher than the minimum average cost. This outcome contrasts with that either under certainty or under demand uncertainty (Appelbaum and Lim, 1982), and is also in sharp contrast to that of Sheshinski and Dr6ze (1976) and Dr6ze (1987, ch. 11). It is an interesting result that the firm's output choice is not affected by risk at all, but the expected equilibrium spot price (and the futures price) is in excess of marginal cost as a result of risk appearance. For our model, however, if the firm is risk averse and furthermore if there is no bias, we have that E(cr) = qL~ + cov(/7, a ) - c ( q ) / q ] > 0. Since c ( q ) / q > c ' ( q ) under risk aversion, it follows that/~ > c ( q ) / q > c'(q); in the long-run industry equilibrium the expected spot price exceeds not only average and marginal costs but also that price without uncertainty. In short, the expected spot price becomes higher than marginal and average costs under both risk neutrality and risk aversion since risk-bearing cost exists.
3.2. A d d i t i v e risk
Consider the additive risk case of y ( q , a ) = q + a, with E(o0 = 0 and var(oz) = ~r2. Thus profits are reduced to 7r = / 7 ( q + a - h) + p h h - c(q), and (8) is replaced by E { U ' ( T r ) ~ - c'(q)]} = O.
(8")
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Combining this equation with (9) yields Ph c'(q), so that the separation t h e o r e m holds. Rewriting (8") and using industry equilibrium condition (4'), we have =
E[U(~)-
U'(u)u] = [c(q)-c'(q)qlEU'(~)
-E[U'(u)~a].
Provided that the firm is risk neutral, then E [ U ( T r ) - U'(Tr)~-] = 0, so that c ( q ) - c ' ( q ) q = E(/Ta). Since E(fia) = cov(jT, a ) < 0, we obtain that [c(q) cov(/7, a ) ] / q = c ' ( q ) . The average cost curve of the firm shifts upward by the a m o u n t of -cov(/~, a ) / q , but the marginal cost curve remains unchanged. Risk-bearing cost plays the same part as fixed costs. The firm thus determines planned output, q~, at which level average cost is less than marginal cost: it is producing more than capacity output. This is illustrated in Fig. 1. In contrast, the firm with certainty chooses output, qc, at which average cost is equal to marginal cost. Production risk leads the risk-neutral firm to produce more than capacity output ex ante; it behaves aggressively like a risk-lover. This result is counter-intuitive but a similar result is drawn by Beuthe et al. (1984). According to them, risk shifts both average and marginal cost curves up so that the firm operates with less capacity. T h e r e is, however, a difference in logic in deriving our and their results because the marginal cost curve is unchanged in our model, whereas it shifts in their
c'(q) c(q)/q- cov(~,ot)/q ~ _
i
~
c(q)/q
I ! I ! I
! I
1
0
qc qu~ Fig. 1. Additive risk case.
q
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model. The output decision of the risk-neutral firm is not independent of risk, unlike those under demand and multiplicative uncertainty. Its independence would not be violated by introducing futures trading. From the discussions in the previous and current subsections we note that whether risk is introduced multiplicatively or additively makes a difference to the output choice of the risk-neutral firm. Given risk aversion, since E [ U ( ~ ' ) U'(Tr)Tr] > 0, it follows that [c(q) - c ' ( q ) q ] E U ' ( q r ) > E[U'(~)/~a], where E[U'(~')/~a] = EU'(~') + cov(/7, a) + cov[U'(~r),/Ta]. Since cov[U'(Tr),/Ta] ~0, the term E[U'(Tr)/Ta] is of either sign. We cannot specify the firm's scale of operation under risk aversion. As mentioned in the preceding subsection, even if the firm is risk neutral, uncertainty affects its output behavior so that the expected spot price is higher than marginal cost. Let us examine if this result is valid in the presence of additive risk. In the same manner as in the above, using the industry equilibrium condition gives E(ar)= -E[_U"(Aar)(ar)2]/2U'(O). Provided that the firm is risk neutral, E(Tr) = 0 and/5 -- Ph from (9), SO that we obtain/~ - c ' ( q ) = c ( q ) / q - c'(q) - E(t~a)/q. Since c ( q ) / q - c ' ( q ) = E(fict)/ q, as indicated above, we have/~-- c'(q) > c ( q ) / q ; the expected spot price is equal to marginal cost. At a glance, this outcome seems to be the same as that under certainty but differs from that under multiplicative risk. The expected spot price moves up to the minimum level of [c(q) - cov(/~, a ) ] / q , and is higher than the certainty output price by the amount of average cost of risk bearing. With risk aversion it is ambiguous whether the industry equilibrium spot price becomes equal to or higher than marginal cost when we assume the unbiasedness of the futures market. Through the analyses in the two subsections it is important that the firm's scale of operation and the level of equilibrium spot price depend on the functional form of y ( q , a ) as well as its attitude toward risk. In particular the form of y ( q , a) has a great influence on the output decision of the risk-neutral firm. We have shown that, given risk neutrality, the equilibrium expected spot price exceeds the equilibrium price without uncertainty and futures trading. Nevertheless, it is difficult to determine whether this is due to incorporating futures contracts or a decrease in the number of firms in the industry, compared with the certainty case. The cause of high expected spot price consists of a decrease in that number if the availability of futures contracts does not increase industry demand for output by much.
4. Conclusion
We have considered competitive industry equilibrium with uncertainty and futures contracts. Long-run industry equilibria with either demand or supply uncertainty have been discussed.
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In the presence of d e m a n d uncertainty we have obtained the following results. First, with risk aversion the industry equilibrium condition is inconsistent with the assumption of unbiasedness about the futures price. T h e cause of this inconsistency will be attributed to both a risk p r e m i u m and the length of the period considered. W h e t h e r a bias arises depends on the length. This result suggests that it may be questionable to m a k e the assumption of unbiasedness in the long run. Although Anderson and D a n t h i n e (1983, p. 384) mention in their short-run (but general equilibrium) m o d e l that "...in a world of risk-averse individuals, bias will necessarily result if the plans of producers and processors are not perfectly in line with one a n o t h e r " , we have shown that there is also such a bias (backwardation or contango) for a long-run partial equilibrium model, which does not include speculators a n d / o r processors. The bias might not arise when production uncertainty exists. These results suggest that backwardation or contango occur m o r e easily in the presence of demand uncertainty rather than production uncertainty. The occurrence of the bias is closely related to a risk p r e m i u m , and the magnitude of the bias expands as the degree of the firm's risk aversion (its risk premium) increases. T a b l e 1 summarizes the relationship between the behavior of each firm in the long-run industry equilibrium and uncertainty, i.e. the properties of the long-run industry equilibrium under uncertainty. For example, the third row of the table shows that, given supply uncertainty and risk neutrality, when planned output and risk are expressed in multiplicative form, the firm operates with efficient capacity but the expected spot price exceeds marginal cost. In addition to the results shown in Table 1 the following results are obtained: (1) Given d e m a n d uncertainty, the firm will hedge less than its output w h e n the expected spot price exceeds marginal cost; the firm will speculate
Table 1 Uncertainty and the decisions of the firm
Demand uncertainty Supply uncertainty
Risk aversion Risk neutrality Risk neutrality
m.r.c? a.r.c,b
am.r.c. = multiplicative risk case. ba.r.c. = additive risk case.
Capacity l e v e l
Relationship between the expected spot price (e.s.p.) and marginal cost
Excess capacity
e.s.p. <>mc
Efficient capacity
e.s.p. = mc
Efficientcapacity Excesscapacity
e.s.p. >mc e.s.p. = mc
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by selling futures contracts in excess of its o u t p u t w h e n the e x p e c t e d spot price is less than marginal cost. (2) G i v e n supply uncertainty and that p r o d u c t i o n takes the multiplicative f o r m of risk and p l a n n e d o u t p u t , the expected spot price always exceeds m a r g i n a l cost, irrespective o f the firm's attitude t o w a r d risk. F u r t h e r extensions remain. O n e is to elaborate w h e t h e r a bias arises within a long-run general equilibrium m o d e l , because our discussion is limited to a long-run industry equilibrium approach. In o t h e r words, we have implicitly a s s u m e d that futures m a r k e t equilibrium holds. A n o t h e r extension is to construct a m o d e l incorporating a m o r e sophisticated rational expectations hypothesis. T h e discussion about welfare still remains.
Acknowledgements I a m grateful to H a j i m e Miyazaki, N o r m a n Ireland, H a j i m e Oniki, s e m i n a r participants at O s a k a University and, especially, two a n o n y m o u s referees for valuable c o m m e n t s and suggestions. I also wish to t h a n k T a k a o F u j i m o t o and Yasunori Ishii for c o m m e n t s on an earlier version o f this p a p e r . A n y remaining errors are m y own responsibility.
References Anderson, Ronald W. and Jean-Pierre Danthine, 1983, Hedger diversity in futures markets, Economic Journal 93, 370-389. Appelbaum, Elie and Eliakim Katz, 1986, Measures of risk aversion and comparative statics of industry equilibrium, American Economic Review 76, 524-529. Appelbaum, Elie and Chin Lira, 1982, Long-run industry equilibrium with uncertainty, Economics Letters 9, 139-145. Benninga, Simon, Rafaei Eldor and Itzhak Zilcha, 1983, Optimal hedging in the futures market under price uncertainty, Economics Letters 13, 141-145. Beuthe, Michel, Louis Eeckhoudt and Jean Lefoll, 1984, Production uncertainty and the market equilibrium of the competitive firm, European Economic Review 26, 1-11. Britto, Ronald, 1984, The simultaneous determination of spot and futures prices in a simple model with production risk, Quarterly Journal of Economics 98, 351-365. Chavas, Jean-Paul, Rulon D. Pope and Howard Leathers, 1988, Competitive industry equilibrium under uncertainty and free entry, Economic Inquiry 26, 331-344. Danthine, Jean-Pierre, 1978, Information, futures prices, and stabilizing speculation, Journal of Economic Theory 17, 79-98. Dr6ze, Jacques H., 1987, Essays on economic decisions under uncertainty (Cambridge University Press, Cambridge). Feder, Gershon, Richard E. Just and Andrew Schmitz, 1980, Futures markets and the theory of the firm under price uncertainty, Quarterly Journal of Economics 94, 317-328. Haruna, Shoji, 1988, Industry equilibrium with uncertainty and labor-managed firms, Economics Letters 26, 83-88.
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Holthausen, Duncan M., 1979, Hedging and the competitive firm under price uncertainty, American Economics Review 69, 989-995. Houthakker, Hendrick S., 1968, Normal backwardation, in: J. N. Wolfe, ed., Value, capital, and growth: Papers in honor of Sir John Hicks (Edinburgh University Press, Chicago, IL) 193-214. Ishii, Yasunori, 1987, Demand uncertainty and industry equilibrium in the presence of a forward contract market, Discussion Paper No.21, Yokohama City University. Ishii, Yasunori, 1989, Measures of risk aversion and comparative statics of industry equilibrium: Correction, American Economic Review 79, 285-286. Kreps, David M., 1990, A course in microeconomic theory (Princeton University Press, Princeton, NJ). Lippman, Steven A. and John J. McCall, 1981, Competitive production and increases in risk, American Economic Review 71, 207-211. McKinnon, Ronald I., 1967, Futures markets, buffer stocks, and income stability for primary problems, Journal of Political Economy 76, 844-861. Newbery, David M. G. and Josef E. Stiglitz, 1981, The theory of commodity price stabilization (Clarendon Press, Oxford). Pazner, Elisha A. and Assaf Razin, 1975, Industry equilibrium under random demand, European Economic Review 6, 387-395. Pratt, John W., 1964, Risk aversion in the small and in the large, Econometrica 32, 122-136. Sheshinski, Eytan and Jacues H. Dr6ze, 1976, Demand fluctuations, capacity utilization, and costs, American Economics Review 66, 731-742. Stein, Jerome L., 1961, The simultaneous determination of spot and futures prices, American Economic Review 51, 1012-1025. Stein, Jerome L., 1979, Spot, forward and futures, Research in Finance 1, 225-310. Varian, Hal R., 1993, Intermediate microeconomics: A modern approach, 3rd edn. (W.W. Norton & Company, New York).
ELSEVIER
International Journal of Industrial Organization 14 (1996) 53-70
Industrial Organization
Industry equilibrium, uncertainty, and futures markets Shoji Haruna Department of Economics, Okayama University, Tsushima-Naka 3-I-1, Okayama 700, Japan Accepted July 1994
Abstract
This paper studies competitive long-run industry equilibrium with uncertainty and futures trading. We also provide a long-run analysis of futures trading instead of the conventional short-run analysis. It is shown without using a general equilibrium model that, given uncertain demand and risk aversion, a bias (backwardation or contango) arises in long-run industry equilibrium. Given risk neutrality such a bias, however, disappears. We find that the occurrence of the bias depends not only on the existence of a risk premium, but also on the length of the period considered. Each risk-averse firm operates at a less efficient scale and, moreover, an expected spot price never accords with marginal cost. When the expected spot price is less than marginal cost, the risk-averse firm can secure profit by selling futures contracts more than the amount of its output. In the presence of production (supply) uncertainty a risk-neutral firm operates with excess capacity under additive risk, while the firm operates with proper capacity under multiplicative risk. The key lies in whether its average cost curve is shifted by production risk.
Keywords: Industry equilibrium; Uncertainty; Futures trading; Bias JEL classification: D41; D81; L l l
I. Introduction
A l t h o u g h the n u m b e r of firms in a competitive industry is fixed in the short run, the n u m b e r varies t h r o u g h entry and exit in the long run if there 0167-7187/96/15.00 (~ 1996 Elsevier Science B.V. All rights reserved .SSDI 0167-7187(94)00456-0
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are no entry and exit costs and no legal regulations on entry and exit. The optimal number of firms in the industry as well as the optimal output of each firm is determined in long-run industry equilibrium. The properties of such equilibrium without uncertainty have been considered and clarified. On the other hand, the conventional approach to competitive industry equilibrium with uncertainty is to dichotomize the uncertainty into the demand and supply sides. Given that firms are risk neutral, one can show different effects of uncertainty on the output price and level, depending on which side the uncertainty occurs. For example, the study by Beuthe et al. (1984) shows that if production (supply) is uncertain, the planned output of the firm is larger and the average equilibrium market price of the output is higher than those in the absence of uncertainty. As for the case of demand uncertainty but without production uncertainty, Sheshinski and Drrze (1976) and Lippman and McCall (1981) demonstrate that each firm's output is less than the minimum efficient scale that corresponds to average-cost minimizing output. Their discussions are, however, based on a special assumption that an industry demand curve is zero-elastic with respect to industry output. For a more general case with a downward-sloping industry demand curve, Pazner and Razin (1975) consider long-run industry equilibrium with output price (demand) uncertainty, and obtain a relationship between the equilibrium output of the firm and its attitude toward risk in the long run. Following Pazner and Razin, for example, Appelbaum and Katz (1986) and Chavas et al. (1988) have conducted comparative static analyses, i.e. the effects of output price uncertainty, input price, and fixed costs in long-run industry equilibrium, on the output of the individual firm and the number of firms in the industry. One of the standard theoretical results concerning futures trading is the separation theorem of hedging and production decisions; the firm's output level depends only on a non-stochastic futures price but is independent of its attitude toward risk and the probability distribution of the spot price (see, for example, Holthausen, 1979; Feder et al., 1980; and Anderson and Danthine, 1983); its output level is not affected by spot price uncertainty at all. Anderson and Danthine consider the problem of hedging (and speculation) and production decisions, and further causes of a bias, which is a difference between futures and expected spot prices in the presence of spot price and production uncertainty, using a general equilibrium model with a rational expectations hypothesis. 1 Hedging implies the sale of futures or selling output in a futures market and speculation is the sale of futures greater than output or buying futures. The existing literature is confined to
The bias is called backwardation(contango)when the futures price exceeds (is less than) the expected price. See Houthakker (1986) for backwardation.
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considering short-run futures trading and short-run futures market equilibrium. An exception is Ishii (1987). He conducts a comparative static analysis under long-run industry equilibrium with futures trading but does not mention the issue of whether the bias arises. The innovation of this paper is to explicate the problem of the bias (backwardation or contango) that might exist in the long run: the existing literature on this subject is also confined to short-run cases. The purpose of this paper is to consider long-run industry equilibrium with futures trading and uncertainty. We are concerned with how the long-run behavior of firms and a competitive industry with either demand or production uncertainty is affected by introducing futures contracts. 2 From another point of view, this paper provides a model for a long-run analysis of hedging: we discuss the hedging decision of the firm and the relationship between the futures and the expected spot prices in a long-run industry equilibrium. In Section 2 we investigate the properties of the long-run industry equilibrium with demand uncertainty and futures trading. The principal contribution is that the futures price is not an unbiased estimator of the random spot price, namely there certainly arises a bias, i.e. either backwardation or contango, in the futures market whenever firms are risk averse. In contrast, given risk neutrality, the futures price is equal to the expected spot price so that the bias vanishes. In the long run the risk-averse firm either hedges less than its entire output or speculates by selling futures contracts more than its output, whereas the risk-neutral firm hedges all of its entire output. It is found that whether the bias appears in the long-run equilibrium relates to both the existence of a risk premium and the length of the analytic period. We also show that the risk-averse firm is producing at less than efficient scale, where marginal cost falls short of average cost, and the expected spot price either exceeds or is less than marginal cost. When the expected spot price is less than marginal cost, the risk-averse firms can make a profit by positively taking advantage of futures contracts like a speculator and then can remain in the industry. The availability of futures trading makes a significant difference to the properties of the long-run industry equilibrium. In Section 3 we explore long-run industry equilibrium with production uncertainty incorporating a rational expectations hypothesis, and show that the firm's scale of operation is affected by the way risk is incorporated into production: a risk-neutral firm will produce more than capacity output, for which average cost is minimized, under additive risk, and just capacity output under multiplicative risk. The key is that the average cost curve of
2In this paper, however, we do not conduct comparative static analysis. See Ishii (1987) for that analysis in a similar model with demand uncertainty and futures contacts.
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the firm is shifted up by risk when it is additive, but not when it is multiplicative. Alternatively, the bias may not occur when production is random. Section 4 summarizes our conclusions.
2. The competitive industry and demand uncertainty We consider a competitive industry that produces a homogeneous commodity, which may be perishable. There are n identical firms in the industry with no barriers to entry and exit. In this section we explore the behavior of the competitive industry with demand uncertainty. The commodity is traded in both spot and futures markets, i.e. each firm has access to the futures market as well as the spot m a r k e t ) The futures market provides for producers a guaranteed price for their output and reduces some of the risk due to any fluctuation in the future price, with which they are faced. Hence, by use of futures trading (contracts) the firm can eliminate income risk. Oils and fats, gold, non-ferrous metals, and cotton thread are subject to demand risk, and futures markets for them are widely developed. Let us denote the output of each firm by qi. Suppose that the industry demand function facing the firms in the spot market is given by p = F(Q, a), where Q = ET=lq i = nqi is industry output, p spot price, and a a random variable with probability density function ~b(a). The function F(Q, a) is assumed to decrease in the total amount of output sold in the spot market; the spot price is decreasing in Q, i.e. dp/dQ = pQt < 0. We posit that F(Q, a) is an increasing function of a, and that F(0, a ) is positive for all a. Each firm can also join a futures market and use futures trading. When the quantity of firm i's futures trading (contracts) is denoted by h i, it is positive (negative) if the firm sells (purchases) futures. Since firms hedge in futures markets to eliminate income risk, they will generally supply or sell futures. On the other hand, speculators are located on the opposite side, i.e. on the demand side, in the markets, so that they will purchase (demand) futures. Sometimes, however, they may sell futures contracts. Anyway, the speculators trade only futures. T o make the analysis simple, we take their demand for futures to be given; that is, we do not explicitly treat the behavior of the speculators (see A n d e r s o n and Danthine, 1983, for their behavior). Then we assume that the d e m a n d function of the futures market takes the form of Ph = p h ( H ) , where Ph stands for futures price and H = ZT=1 h i = n h i is the total amount of
3 For a discussion of futures trading, see, for example, Holthausen (1979), Newbery and Stiglitz (1981), and A n d e r s o n and Dathine (1983).
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futures sold by firms in the futures m a r k e t . Futures price is decreasing in the total a m o u n t of hedging, d p h / d H = P'h < 0.4 W i t h the futures m a r k e t available the profits of each firm are 7ri = p ( q ~ - hi) + phh~ -- c ( q i ) ,
w h e r e c ( q i ) is the cost function with fixed costs such that c ( 0 ) = c o > 0 , d c ( q i ) / d q i = c'(qg) > 0, and c"(qi) > 0; we assume that a l t h o u g h the long-run total costs for each firm are zero at zero o u t p u t level, they rise to c o w h e n any p r o d u c t i o n at all is a t t e m p t e d . T h e average cost curve, c ( q i ) / q i is U - s h a p e d , and there exists unique o u t p u t q* > 0 such that c'(qg) - c ( q i ) / q i ~_ 0 as qi ~ q * , w h e r e q* is the o u t p u t for which average cost is m i n i m u m and is r e f e r r e d to as capacity output. T h e firm must d e t e r m i n e the a m o u n t s of o u t p u t and futures trading b e f o r e d e m a n d uncertainty is resolved. Its objective is n o w to c h o o s e t h e m so as to maximize the e x p e c t e d utility of profits, EU(Tri), where U(Tri) is a v o n N e u m a n n - M o r g e n s t e r n utility function with U ( 0 ) = 0, U ' ( ~ ' ) > 0 and U"(rr) ~< 0, and E is the expections operator. Maximization of the e x p e c t e d utility o f profits reflects the firm's attitude t o w a r d risk and its evaluation of profits u n d e r uncertainty. F o r simplicity, hereafter we suppress the subscript i. T h e p r o b l e m of the individual firm is to solve m a x EU(Tr) = E U [ p ( q q.h
- h ) + p h h - c(q)].
T h e first-order conditions for a m a x i m u m are given by 5 0EU(~-)
Oq
- E{U'(Tr)[p
= 0,
(1)
+ P h ) } = O.
(2)
- c'(q)]}
0EU(~r) Oh
= E(U'(Tr)(-p
F r o m these conditions we get Ph = c ' ( q ) ,
(3)
4 See, for example, Stein (1961, 1979), Danthine (1978), Newbery and Stiglitz (1981), and Anderson and Danthine (1983) for futures market equilibrium. Hereafter, we assume that all equilibrium solutions are unique. 5 For the risk-averse firm, the second-order conditions hold since [O2EU(Tr)/ Oq2][02EU(Tr)/ Oh2] _ [OzEUQr) / OqOh]2 = -c"(q)E[U '(Tr)]E[U"(Tr)(p - Ph)2] > 0. Although these conditions are not necessarily satisfied in the presence of risk neutrality, we assume that there is an interior solution.
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which is the separation theorem in the theory of futures trading. 6 In the absence of uncertainty, if there are fixed costs and they are assumed to rise when any production is attempted, firms will continue to enter (leave) the industry as long as they can make positive (negative) profits (see Kreps, 1990). The long-run industry equilibrium is not sustained in a situation where firms make negative profits at any output level. Therefore, all firms make zero profits in a long-run industry equilibrium. In contrast, firms must now decide whether to enter or exit before demand uncertainty is resolved. We take up the concept of reservation utility U R, and suppose an industry equilibrium condition: EU(Tr) = E U [ p ( q - h )
+ph h -- c(q)] = U R .
(4)
If the expected utility of each firm exceeds the reservation utility, it causes firms to enter the industry, while if the expected utility falls short of the reservation utility, firms leave the industry. In the long run the expected utility of each firm will be equated to U R, because the expected utility is a decreasing function of the number of firms. 7 For simplicity, we normalize the level of the reservation utility to zero, U R = 0, which is equivalent to the utility of not producing and not hedging (speculating), where q = 0 and h = 0. By analogy with the certainty case, the expected utility of each firm becomes equal to zero, not negative, in the long-run industry equilibrium since fixed costs are assumed not to arise before entry. Consider the firm's choice of output. From (1) we have q E [ U ' ( ~ ) p ] = q E [ U ' ( T r ) ] c ' ( q ) , and subtracting c ( q ) E U ' ( 1 r ) from both sides of this equation yields: E { U ' ( T r ) [ p q - c(q)]} = [ c ' ( q ) q - c(q)]EU'(1r).
Taking account of both E[U'('rr)]ph = E[U'(Tr)p], obtained from (2), and industry equilibrium condition (4), we get
6 F r o m (1) and (2) we obtain that if U"(~r) < 0, then q / > ( < ) h according as Ph <~( > ) E ( p ) , if U"(~') = 0, t h e n q = h according as Ph = E ( p ) . T h e s e results are the s a m e as those of Feder et al. (1980) and H o l t h a u s e n (1979). 7 Differentiating the left-hand side of (4) with respect to n yields:
d E U ( ~ ) / d n = E U ' ( ~ ' ) [ ( q - h )2p~ + h 2p'~l < O, b e c a u s e p ~ < 0 and p;, < 0.
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c(q) q
c'(q) =
E[U(Tr) - U'(rr)Ir] qEU'(Tr)
59
(5)
T h e n u m e r a t o r on the right-hand side of (5) is positive (negative) or zero, depending on whether the utility function is concave (convex) or linear. 8 Thus it follows that c ( q ) / q - c'(q)/> ( < )0, according to whether the firm is risk averse (risk-loving) or risk neutral. With a U-shaped average cost curve this leads to the following result: in the long-run competitive industry equilibrium with futures contracts, the firm produces where average cost is larger (less) than or equal to marginal cost, depending on whether it is risk averse (risk loving) or risk neutral. 9 We know from this that even if the firm has access to the futures market, there still exists excess capacity under risk aversion. 1° The firm's choice of output depends on its attitude toward risk exclusively, and the introduction of futures trading has no influence on its output choice. Thus it seems that its introduction is no inducement to mitigate the excess capacity of the risk-averse firm, although Holthausen (1979) and N e w b e r y and Stiglitz (1981) assert in their short-run models that the availability of futures markets induces a firm to increase output. 11 Consequently, their results do not carry over to the long-run analysis. Let us turn to examining the long-run industry equilibrium in some detail. We rewrite the industry equilibrium condition as follows: EU(~-) = U[E(~r) - p] = U R = O, where p ( > 0 or = 0) stands for the risk premium. We have s T h e m e t h o d of deriving (5) is essentially the same as in Pazner and Razin (1975). T h e sign of the n u m e r a t o r on the right-hand side of (5) depends crucially on the values of U(0) and U R as well as the firm's attitude toward risk. If assumptions U ( 0 ) = 0 a n d / o r U R = 0 are now relaxed, the results on that sign do not always accord with those derived in (5). For example, for U(0) = 0 and U R > 0 the sign of E[U(0r) - U'(~')~'] will be of either sign, or negative if the firm is risk averse or risk neutral. In contrast, for U R = 0 and U(0) > 0 the sign of E[U(zr) U'(~')Tr] will be positive if it is either risk averse or risk neutral. F u r t h e r m o r e , as pointed out by o n e of the referees, w h e n the utility function takes the form of U(~r) = - e -r~, with r > 0, for U R = 0, E [ U ( ~ - ) - U ( ~ ' ) z r ] will be of either sign because U ( 0 ) < 0 : namely, this result is different from that on the sign of E[U(~') - U'(zr)~r] in (5). 9 F r o m the fact that the sign of the n u m e r a t o r on the right-hand side of (5) d e p e n d s on the level of U(0) we find that the result does not always carry over to any affine transformation of the original utility function. ~0 This is the same as the result of Pazner and Razin (1975), but is different from that of Sheshinski and Dr6ze (1976) that excess capacity prevails u n d e r risk neutrality. This s e e m s to be due to their specific assumption that the r a n d o m industry d e m a n d is evenly divided a m o n g firms in the industry. 11 A n d e r s o n and D a n t h i n e (1983), however, show in their short-run model that the introduction of futures trading does not always cause the firm to increase expected output when production uncertainty is added.
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E(rr) = p.
(6)
Rewriting the e x p e c t e d profits gives: E(~') - p = ~ - - P h ) ( q -- h ) + c ' ( q ) q - c ( q ) - p, w h e r e fi = E ( p ) . W h e n e v e r the futures price is equal to the expected spot price, ~ = Ph, i.e. there is not a bias which is the difference b e t w e e n b o t h prices, the e q u a t i o n a b o v e is r e d u c e d to E ( ~ - ) - p = c ' ( q ) q - c ( q ) - p . P r o v i d e d that the firm is risk averse, E 0 r ) - p < 0 because c ( q ) - c ' ( q ) q > 0 a n d 0 > 0. It follows that, given risk aversion, E(Tr) - p is negative. T h u s a c o n t r a d i c t i o n arises in c o m p a r i n g (6) with the result E ( T r ) < p d r a w n u n d e r risk aversion. This involves that the industry equilibrium condition and the a s s u m p t i o n of unbiasedness that the futures price is an unbiased estimator of the e x p e c t e d spot price are inconsistent u n d e r risk aversion. 12 In the p r e s e n c e of a futures m a r k e t , in o r d e r for the industry equilibrium condition to be satisfied, it is necessary for the expected spot price not to coincide with the futures price, n a m e l y the futures price must be a biased estimator of the e x p e c t e d spot p r i c e ) 3 W h a t is the cause of the bias? T h e r e are two causes. First, the bias is c o n s i d e r e d to be incurred by the risk p r e m i u m and the level of operation. M o r e specifically, its cause can be ascribed to the risk p r e m i u m since the level of o p e r a t i o n is, in fact, d e p e n d e n t on the firm's attitude t o w a r d risk, which can also be r e p r e s e n t e d by the risk p r e m i u m . We find f r o m an e x a m i n a t i o n of e q u a t i o n (6) that its size also relies on its d e g r e e in the long run as well as in the short run; that size increases as the degree of the firm's risk aversion increases. A n d e r s o n and D a n t h i n e (1983) also m e n t i o n that the size o f the bias will d e p e n d on the degree of risk aversion of participants in the futures m a r k e t . In contrast, given risk neutrality, we obtain E ( T r ) = 0 since c ( q ) - c ' ( q ) q = 0 and p = 0, even if the futures price is an unbiased e s t i m a t o r of the expected spot price: the bias disappears w h e n the risk p r e m i u m is zero. This d e m o n s t r a t e s that, given risk neutrality, an industry equilibrium condition such as E U ( T r ) = 0 is consistent with the assumption ~2In his short-run equilibrium model with producers and speculators, Danthine (1978) demonstrates that the futures price is not an unbiased estimator of the expected spot price as long as the producers are risk averse and their initial position is riskier than the speculators'. Anderson and Danthine (1983) also describe that in the short run there is a bias if futures sales and purchases do not accord. 13When a differential between the futures and the expected spot prices exists, there is room for entry by speculators, although we do not explicitly introduce them; they could earn profits either by utilizing futures contracts if the futures price exceeds the expected spot price or by purchasing them if the futures price is lower than the expected spot price (see Anderson and Danthine, 1983, and Danthine 1978).
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of unbiasedness. Second, we cannot overlook the fact that the length of period considered plays a definite role in the incidence of the bias. 14 This is because even in the case of risk neutrality the bias may arise if the assumption U R = 0 is not satisfied. Therefore, our result addresses doubts about the validity of making such an assumption in the long run, although the assumption of unbiasedness is often made in the short run (see, for example, McKinnon, 1967, and Benninga et al., 1983). It should be borne in mind that not only the risk premium but also the length of the analytic period affects the existence of the bias. We are now in a position to examine the relationship between spot price and marginal cost. By Taylor's theorem and then taking expectations we have EU(Tr) = U(0) + U'(0)E(or) +
EIU"(ATr)(zr) 2]
0
where U ( 0 ) = 0. In the long-run industry equilibrium it follows that E ( 1 r ) = -E[U"(ATr)(rr)2]/2U'(O), so that E(~r) is positive or zero, according to whether the firm is risk averse or risk neutral. Since c ' ( q ) = P h , we have E ( ~ ) = (q - h ) ~ - c'(q)] + c ' ( q ) q - c(q). Combining these results yields: (q - h ) ~ - c'(q)] + c ' ( q ) q - c(q) >! 0
for U"(Arr) ~< 0 .
(7)
If the firm is risk neutral, we get p = c ' ( q ) because c ' ( q ) = c ( q ) / q , as shown previously. As can be seen from footnote 6, the expected spot price becomes equal to marginal and average costs and the futures price. Meanwhile, owing to c ( q ) - c ' ( q ) q > 0 for risk aversion, ( q - h ) ~ - c'(q)] must be positive so that s i g n ~ - c'(q)] = sign(q - h). Then, taking the result C'(Q)-~Ph into consideration, we obtain that <(>)Ph,
if and only if q < ( > ) h .
This result is summarized as follows. When the expected spot price is less than the futures price, i.e. there is contango, the firm will speculate by selling an amount greater than its output; when the expected spot price exceeds the futures price, i.e. there is backwardation, the firm will hedge less than its output. The result is the same as that in the short-run analysis (Feder et al., 1980, and Holthausen, 1979). More interestingly, given risk aversion, it does not hold that -fi=Ph,
if and only if q = h .
The reason is as follows: if it holds that c ' ( q ) - c ( q ) / q > 0 from (7), while, 14I am indebted to one of the referees for having pointed out this fact.
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as we have shown above, c ( q ) / q - c ' ( q ) > 0, then both results contradict each other. Thus, in the long-run industry equilibrium the risk-averse firm never hedges all of its output. This is different from the short-run result. In the long-run industry equilibrium the risk-neutral firm operates at the minimum (efficient) level of average cost, where average cost is equal to marginal cost, and the expected spot price becomes equal to marginal cost (and the futures price). In contrast, the risk-averse firm is producing at a less efficient scale, where average cost exceeds marginal cost. At this level of output the expected spot price does not equal marginal cost at all, i.e. that price is either higher or lower than marginal cost. This implies that there exists backwardation or contango, although we cannot specify whether backwardation or contango occurs. 15 When the expected spot price exceeds marginal cost, i.e. fi > c ' ( q ) = P h , the firm will hedge less than its output in the futures market to eliminate risk. The expected spot price will be, at least, higher than average cost. When the expected spot price is less than marginal cost, the firm will speculate by selling futures contracts in excess of its output to secure profit and increase the amount of futures contracts as contango diminishes. Therefore, when the expected spot price falls short of marginal cost, the firm can get profit by taking advantage of futures trading and remain in the industry, while without futures markets firms will carry on leaving the industry as long as expected price is less than marginal cost. In this case the number of firms in the industry with futures trading will be more than without it. We note that the availability of futures trading makes a significant difference to the properties of the long-run industry equilibrium.
3. The competitive industry and production uncertainty We deal with long-run industry equilibrium with production (or supply) uncertainty instead of demand uncertainty. In the real world it is observed that bad weather, and insect pests in the primary industry frequently break out, and the production of agricultural commodities such as wheat, soy bean, coffee, and sugar fluctuates significantly. Then the producers of these commodities are faced with production risk rather than demand risk. So, it would be rare for planned (ex ante) quantities of such commodities to accord with their actual (ex post) quantities. Incorporating supply risk,
15Dr6ze (1987) refers to the case with futures trading and mentions in his conclusion (p. 232) that "in an industry with uncertain future demand, one may expect the price on a forward market to fall below minimum average cost, if that price is to reflect expected marginal cost". His conclusion is, however, invalid in our model.
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therefore, seems reasonable. 16 Let q denote planned output and y(q, a) actual output, where a is a random variable with the probability function y ( a ) expressing disturbances affecting output. It is now assumed that Oy(q, a)/Oq = yq(q, a ) > 0 and by(q, a)/Oa = y~(q, a ) > O. Aggregate supply in the spot market is reduced to Qs = ny(q, a). We denote aggregate demand by Qd = Qd(P)" The spot price is determined by the marketclearing condition
ny(q, a) = Qd(P). The spot price thus becomes a random variable, the distribution of which is generated by the distribution of a, because risk is transferred to the spot price through the market-clearing condition. 17 The density function of the spot price relies on the structure of the spot market as well as the density function of output. We adopt the rational expectations hypothesis, which means that the equilibrium distribution of the spot price is known to every firm. In consequence, an individual firm in the industry is confronted with both spot price and production risks. To distinguish the current random spot price from that used in the previous section we express the former random spot price by t7. The maximization problem of the firm is max EU(~-) = EU[/7(y(q, a) - h) + ph h -- c(q)]. q,h
All symbols, except f o r / 7 and y(q, a) introduced in this section, are the same as in the preceding section. The first-order conditions are given by OEU(~) - - E { U ' ( ' n ' ) [ / 7 Or(q,oqa) Oq OEU(zr) -
-
Oh
-
E[U'(~')(-/7
+Pa)]
= O.
c'(q)]} =0 ,
(8)
(9)
Using (8) and (9) we have phE[U'('n')yq(q, a)] = c'(q)EU'(~r): the separation theorem fails to hold in the presence of production uncertainty (Anderson and Danthine, 1983).
16For an analysis of futures trading and production uncertainty, see, for instance, Newbery and Stiglitz (1981), Anderson and Danthine (1983), and Britto (1984). 17See Newbery and Stiglitz (1981), Anderson and Danthine (1983), and Britto (1984) for models that discuss spot or futures market equilibrium in the analysis of futures trading. See, for example, Newbery and Stiglitz, and Britto for analyses including (short-run) spot market equilibrium.
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Suppose that, as in the case of Section 2, the industry equilibrium condition is EU(Tr) = EU[/7(y(q, ~) - h) + ph h -- c(q)] = 0.
(4')
Firms must decide whether to enter or exit before a a n d / 7 are revealed. Consider the firm's behavior and the connection between the spot price and marginal cost in long-run industry equilibrium. In so doing, since y ( q , a ) is too general for analytical and practical purposes, suppose that y ( q , a ) takes two specified forms, as follows) 8 One is a multiplicatively separated form such as y ( q , a ) = aq, with E ( a ) = 1 and var(a) = 0-2, where var stands for variance. The other is an additively separated form such as y ( q , a ) = q + a, with E ( a ) = 0 and var(a) = 0.2. 3.1. Multiplicative risk When the relationship between planned output and risk is exhibited in a multiplicative way, (8) is rewritten as E { U ' ( ~ ) L ~ - c'(q)]} = 0 .
(8')
It follows from (8') that E [ U ' ( T r ) / T a ] q = E [ U ' ( ~ ) ] c ' ( q ) q . Subtracting E[U'(~r)]c(q) from both sides of this equation and taking (9) into consideration yields: E{U'(~r)[/7(aq - h) + ph h -- c(q)]} = E[U'(~r)][c'(q)q - c(q)]. F u r t h e r m o r e , taking (4') into consideration yields: E [ U ( ~ r ) - U'(Tr)Tr]= q E [ U ' ( T r ) ] [ c ~ )
c'(q)],
where E [ U ( T r ) - U'(~r)Tr]/> 0 according to the firm's attitude toward risk. H e n c e , in a long-run industry equilibrium with futures trading, if production risk is multiplicative, the firm operates at the point where average cost is larger than or equal to marginal cost, depending on whether it is risk averse or risk neutral; the risk-averse firm operates with excess capacity and the risk-neutral firm with proper capacity. The firm's scale of operation will be delimited by its attitude toward risk, as in the case of demand uncertainty. B e u t h e et al. (1984) have concluded that in the long-run industry equilibrium the risk-neutral firm chooses output where average cost is less than marginal cost, and hence less capacity prevails. This is at variance with our result. The difference is due to the following reason: following Beuthe et 18Newbery and Stiglitz (1981) also set up a model with special supply risk.
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al.'s notation, the connection between planned output q and realized output is expressed by a specific form as ~ = q + u ( q , e), with u , - qU~q > 0 , where e is a random variable, while in our multiplicative model such an assumption on u ( q , e) is not adopted. It was derived in Section 2 that the industry equilibrium condition contradicts the assumption of unbiasedness with respect to the futures price under risk aversion. Nevertheless, the issue of inconsistency does not emerge in both the multiplicative and additive risk cases. This issue might not arise when production and spot price uncertainty coexist. Let us now turn to examining the connection between the spot price and costs. By applying Taylor's theorem to (4') we get E(zr) = - E[U"(ATr)(Tr)2]/ 2U'(0), where 0 < A < 1. Given risk neutrality, this expression is rewritten as - c ( q ) / q = h(t~ --Ph)q -- COV(/~, a ) , where /~ = E(/7). Since c ' ( q ) = c ( q ) / q , we have /~ - e ' ( q ) = h ( l ~ - - P h ) / q -- COV(/7, a). The term -cov(/7, ~) can be interpreted as risk-bearing cost for the risk-neutral firm. Thus risk-bearing cost arises even for risk neutrality, different from the demand uncertainty case. When the demand curve in the spot market is downward sloping, [7 and a will move in opposite directions, so that cov(fi, a ) < 0. From (9) we get E ( ~ ) = / ~ = P h under risk neutrality so t h a t / ~ = c ' ( q ) - cov(/7, a ) = c ( q ) / q - cov(fi, a ) ; the expected spot price exceeds marginal cost by the amount of the risk-bearing cost, because cov(/7, a ) < 0 , and is higher than the minimum average cost. This outcome contrasts with that either under certainty or under demand uncertainty (Appelbaum and Lim, 1982), and is also in sharp contrast to that of Sheshinski and Dr6ze (1976) and Dr6ze (1987, ch. 11). It is an interesting result that the firm's output choice is not affected by risk at all, but the expected equilibrium spot price (and the futures price) is in excess of marginal cost as a result of risk appearance. For our model, however, if the firm is risk averse and furthermore if there is no bias, we have that E(cr) = qL~ + cov(/7, a ) - c ( q ) / q ] > 0. Since c ( q ) / q > c ' ( q ) under risk aversion, it follows that/~ > c ( q ) / q > c'(q); in the long-run industry equilibrium the expected spot price exceeds not only average and marginal costs but also that price without uncertainty. In short, the expected spot price becomes higher than marginal and average costs under both risk neutrality and risk aversion since risk-bearing cost exists.
3.2. A d d i t i v e risk
Consider the additive risk case of y ( q , a ) = q + a, with E(o0 = 0 and var(oz) = ~r2. Thus profits are reduced to 7r = / 7 ( q + a - h) + p h h - c(q), and (8) is replaced by E { U ' ( T r ) ~ - c'(q)]} = O.
(8")
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Combining this equation with (9) yields Ph c'(q), so that the separation t h e o r e m holds. Rewriting (8") and using industry equilibrium condition (4'), we have =
E[U(~)-
U'(u)u] = [c(q)-c'(q)qlEU'(~)
-E[U'(u)~a].
Provided that the firm is risk neutral, then E [ U ( T r ) - U'(Tr)~-] = 0, so that c ( q ) - c ' ( q ) q = E(/Ta). Since E(fia) = cov(jT, a ) < 0, we obtain that [c(q) cov(/7, a ) ] / q = c ' ( q ) . The average cost curve of the firm shifts upward by the a m o u n t of -cov(/~, a ) / q , but the marginal cost curve remains unchanged. Risk-bearing cost plays the same part as fixed costs. The firm thus determines planned output, q~, at which level average cost is less than marginal cost: it is producing more than capacity output. This is illustrated in Fig. 1. In contrast, the firm with certainty chooses output, qc, at which average cost is equal to marginal cost. Production risk leads the risk-neutral firm to produce more than capacity output ex ante; it behaves aggressively like a risk-lover. This result is counter-intuitive but a similar result is drawn by Beuthe et al. (1984). According to them, risk shifts both average and marginal cost curves up so that the firm operates with less capacity. T h e r e is, however, a difference in logic in deriving our and their results because the marginal cost curve is unchanged in our model, whereas it shifts in their
c'(q) c(q)/q- cov(~,ot)/q ~ _
i
~
c(q)/q
I ! I ! I
! I
1
0
qc qu~ Fig. 1. Additive risk case.
q
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model. The output decision of the risk-neutral firm is not independent of risk, unlike those under demand and multiplicative uncertainty. Its independence would not be violated by introducing futures trading. From the discussions in the previous and current subsections we note that whether risk is introduced multiplicatively or additively makes a difference to the output choice of the risk-neutral firm. Given risk aversion, since E [ U ( ~ ' ) U'(Tr)Tr] > 0, it follows that [c(q) - c ' ( q ) q ] E U ' ( q r ) > E[U'(~)/~a], where E[U'(~')/~a] = EU'(~') + cov(/7, a) + cov[U'(~r),/Ta]. Since cov[U'(Tr),/Ta] ~0, the term E[U'(Tr)/Ta] is of either sign. We cannot specify the firm's scale of operation under risk aversion. As mentioned in the preceding subsection, even if the firm is risk neutral, uncertainty affects its output behavior so that the expected spot price is higher than marginal cost. Let us examine if this result is valid in the presence of additive risk. In the same manner as in the above, using the industry equilibrium condition gives E(ar)= -E[_U"(Aar)(ar)2]/2U'(O). Provided that the firm is risk neutral, E(Tr) = 0 and/5 -- Ph from (9), SO that we obtain/~ - c ' ( q ) = c ( q ) / q - c'(q) - E(t~a)/q. Since c ( q ) / q - c ' ( q ) = E(fict)/ q, as indicated above, we have/~-- c'(q) > c ( q ) / q ; the expected spot price is equal to marginal cost. At a glance, this outcome seems to be the same as that under certainty but differs from that under multiplicative risk. The expected spot price moves up to the minimum level of [c(q) - cov(/~, a ) ] / q , and is higher than the certainty output price by the amount of average cost of risk bearing. With risk aversion it is ambiguous whether the industry equilibrium spot price becomes equal to or higher than marginal cost when we assume the unbiasedness of the futures market. Through the analyses in the two subsections it is important that the firm's scale of operation and the level of equilibrium spot price depend on the functional form of y ( q , a ) as well as its attitude toward risk. In particular the form of y ( q , a) has a great influence on the output decision of the risk-neutral firm. We have shown that, given risk neutrality, the equilibrium expected spot price exceeds the equilibrium price without uncertainty and futures trading. Nevertheless, it is difficult to determine whether this is due to incorporating futures contracts or a decrease in the number of firms in the industry, compared with the certainty case. The cause of high expected spot price consists of a decrease in that number if the availability of futures contracts does not increase industry demand for output by much.
4. Conclusion
We have considered competitive industry equilibrium with uncertainty and futures contracts. Long-run industry equilibria with either demand or supply uncertainty have been discussed.
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In the presence of d e m a n d uncertainty we have obtained the following results. First, with risk aversion the industry equilibrium condition is inconsistent with the assumption of unbiasedness about the futures price. T h e cause of this inconsistency will be attributed to both a risk p r e m i u m and the length of the period considered. W h e t h e r a bias arises depends on the length. This result suggests that it may be questionable to m a k e the assumption of unbiasedness in the long run. Although Anderson and D a n t h i n e (1983, p. 384) mention in their short-run (but general equilibrium) m o d e l that "...in a world of risk-averse individuals, bias will necessarily result if the plans of producers and processors are not perfectly in line with one a n o t h e r " , we have shown that there is also such a bias (backwardation or contango) for a long-run partial equilibrium model, which does not include speculators a n d / o r processors. The bias might not arise when production uncertainty exists. These results suggest that backwardation or contango occur m o r e easily in the presence of demand uncertainty rather than production uncertainty. The occurrence of the bias is closely related to a risk p r e m i u m , and the magnitude of the bias expands as the degree of the firm's risk aversion (its risk premium) increases. T a b l e 1 summarizes the relationship between the behavior of each firm in the long-run industry equilibrium and uncertainty, i.e. the properties of the long-run industry equilibrium under uncertainty. For example, the third row of the table shows that, given supply uncertainty and risk neutrality, when planned output and risk are expressed in multiplicative form, the firm operates with efficient capacity but the expected spot price exceeds marginal cost. In addition to the results shown in Table 1 the following results are obtained: (1) Given d e m a n d uncertainty, the firm will hedge less than its output w h e n the expected spot price exceeds marginal cost; the firm will speculate
Table 1 Uncertainty and the decisions of the firm
Demand uncertainty Supply uncertainty
Risk aversion Risk neutrality Risk neutrality
m.r.c? a.r.c,b
am.r.c. = multiplicative risk case. ba.r.c. = additive risk case.
Capacity l e v e l
Relationship between the expected spot price (e.s.p.) and marginal cost
Excess capacity
e.s.p. <>mc
Efficient capacity
e.s.p. = mc
Efficientcapacity Excesscapacity
e.s.p. >mc e.s.p. = mc
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by selling futures contracts in excess of its o u t p u t w h e n the e x p e c t e d spot price is less than marginal cost. (2) G i v e n supply uncertainty and that p r o d u c t i o n takes the multiplicative f o r m of risk and p l a n n e d o u t p u t , the expected spot price always exceeds m a r g i n a l cost, irrespective o f the firm's attitude t o w a r d risk. F u r t h e r extensions remain. O n e is to elaborate w h e t h e r a bias arises within a long-run general equilibrium m o d e l , because our discussion is limited to a long-run industry equilibrium approach. In o t h e r words, we have implicitly a s s u m e d that futures m a r k e t equilibrium holds. A n o t h e r extension is to construct a m o d e l incorporating a m o r e sophisticated rational expectations hypothesis. T h e discussion about welfare still remains.
Acknowledgements I a m grateful to H a j i m e Miyazaki, N o r m a n Ireland, H a j i m e Oniki, s e m i n a r participants at O s a k a University and, especially, two a n o n y m o u s referees for valuable c o m m e n t s and suggestions. I also wish to t h a n k T a k a o F u j i m o t o and Yasunori Ishii for c o m m e n t s on an earlier version o f this p a p e r . A n y remaining errors are m y own responsibility.
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