A NOTE ON ADJUSTMENT TO PRODUCTION UNCERTAINTY AND THE THEORY OF THE FIRM

LAURENCE BOOTH*

This paper analyzes the impact of production uncertainty on the firm ’s optimal output decision. lf uncertainty is introduced by an additive risk variable, then short-run optimal output is un- changed, but the owner-manuger s expected utility can change causing long-run output effects. If uncertainty is introduced by a multiplicative random variable, then short-run output can change as well.

I. INTRODUCTION

The pioneering work of Sandmo [ 197 11 and Leland [ 19721 was followed by a spate of papers on the theory of the firm under uncertainty. Research on this topic waned, however, as contributions by Nielsen [1975] and Hite [1978] made economists aware that a Pareto efficient stock market made firms behave in a manner analogous to that under certainty. More recently, the evident incompleteness, and by implication Pareto inefficiency, of stock markets has led to renewed interest, as research by Booth [1983] showed how the two strands of research could be integrated. Recent studies have also focused on technological considerations. The original contributions of Hart- man [1973], lbrnovsky [1973] and Epstein [ 19781 examined the implications of ex-post production changes, that is, the ability of firms to meet production targets after the uncertainty in demand has been resolved. In contrast, Ratti and Ullah [1976], Ingene and Yu [1981] and MacMinn and Holtman [1983] considered uncertainty in the production process itself. More recently, in this journal, Flacco and Kroetch [ 19861 analyzed a model combining ex-post production flexibility with production uncertainty.

This latter research is especially interesting, since the authors claim to find that not just the risk-averse but also the risk-neutral and risk-seeking firm will produce less under uncertainty than under certainty, a result that conflicts with much of the previous research. Flacco and Kroetch’s main result is that “the sign of the marginal risk premium of the firm at an optimum of ex ante output is not affected by the non-linearity of risk preferences.”’ However, this result is not correct. Instead, with additive uncertainty short- run output is unchanged, whereas multiplicative uncertainty effects the stan- dard result that risk-averse firms produce less and risk-preferring firms pro- duce more than under certainty.

* Professor of Finance and Economics, Faculty of Management, University of Toronto. 1. Flacco and Kroctch [ 1986, p. 4861.

616 Economic Inquiry Vd. XXVIII, July 1990, 616-621

BOOTH: PRODUCTION UNCERTAINTY 617

II. MODEL FORMULATION

We use the same framework as Flacco and Kroetch. The firm has to determine its target output level (x) which is presold at a fixed price (f). It has some monopoly power in its market, and so is faced with a downward sloping demand curve. Its non-stochastic cost structure is comprised of vari- able costs (C(x)) and fixed costs (b). The firm is faced with output uncer- tainty; consequently actual output may deviate from its target level. Any excess production is sold (at a price (d)), whereas insufficient production can be made up only by incurring incremental costs (h) .

An intuitive motivation for the problem is readily at hand. The firm can be thought of as contracting before actual production for the sale of its target output, or as hedging the price uncertainty by means of futures contracts. What the firm cannot hedge is its output uncertainty. Hence, shortfalls must be made up by market purchases and excess amounts of output disposed of by sales. In this case, with a certain price a competitive firm would have P=d=X, since shortfalls and excess output can be corrected by transactions in the secondary market which must exist.’ However, given monopoly power, one can legitimately argue that the excess output would have to be dumped (d<P) or that, conversely, the firm would have to incur overtime and other incidental costs to meet its obligations, since alternative market sources of output are not available (h>P) . The problem is thus an interesting one that differs from the typical production flexibility problem reviewed by Epstein. 3

The model is formally written as

max E[U(K)] X

where U i s the firm’s derived shareholder welfare function4, E(.) is the ex- pectation operator and K the uncertain level of profits. The variable K is defined a s

(z-x) X iff z-x<O (z-x) d iff z-x>O K = f x - c(x) - h +

where if z-x<O, output is below target and the firm incurs additional costs, (z-x)X, because of overtime and incremental costs of X. If z-x>O, then the excess output is dumped to yield (z-x)d in extra revenues.

2. The problem then becomes similar to the one analyzed by Ratti and Ullah [ 19761, whcrc the perfectly competitive firm faces output uncertainty, because the amounts of capital and labour actually used deviate from those intended.

3. Note that although Flacco and Kroetch talk extensively o f adjustment to output uncertainty, there is none in their model. This adjustment requires a second period control variable such as labour. The optimal first period control variable, usually capital, is then found in the normal way by using recursive dynamic programming. In the Flacco and Kroetch model *adjustment” is achieved by simply imposing random costs to make up for output shortfalls.

4. Owner’s utility function for an individual owner-nianagcr.

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It is important a t this point to note that the way uncertainty is introduced into the model will affect the model's results. Our initial assumption is the one used by Flacco and Kroetch, namely that uncertainty is introduced as an additive random variable, Z=X+E, where E is a mean zero random variable reflecting the deviation of actual output from target. This means that what- ever the level of output chosen, the distribution of uncertain output remains fixed. That is, at a target output level of 1,000 units, the uncertain output would be the same at plus or minus ten units, as at a target level of 100 units. This is a simplistic assumption that may or may not be reasonable. Clearly, however, by assuming additive uncertainty Flacco and Kroetch elim- inate uncertainty from the optimization problem, since the marginal change in risk resulting from output changes is zero!

If we substitute the uncertainty assumption Z=X+E, into the profit function, we have

where the term in brackets is not affected by the optimization problem. Optimal output is found by substituting (3) into (1) and differentiating:

E[ll'(lr)][P+P'(x)X-~'(x)]=O . (4)

Since E[U'(x)] is a constant, we have the standard certainty result that mar- ginal revenue (P+P'(x)x) should equal marginal cost (c'(x)) at the optimal target level of output.

This result differs from that found by Flacco and Kroetch, since they failed to substitute their uncertainty assumptions into (2). Hence, they differentiated (1) after substitutin in equation (2) to obtain a marginal risk premium term that does not exist. The result also conflicts with standard uncertainty re- sults, because the firm acts as if it is operating under certainty.

The reason for our result is straightforward. The firm presells all its target output, which it produces at a certain cost. Profit uncertainty is only intro- duced as deviations of actual from target output, which produce unexpected cash expenses to boost output when there are shortfalls in output, and unex- pected surpluses when there is excess output. In both cases, however, because uncertainty is introduced as an additive random variable, the profit uncer-

9

5. Their first order condition (equation 4, p. 489) is

[P+P(x)r-c'(x)-h]E~[ U'(x) ]+[P+P'(x)r -~' (x) -d jE2[ U'(x)]

where El and E2 are the expectations operators for values of z below and above the target output levcls respectively. Clearly, the terms X and d only enter the optimality condition because they failed to recognize that, with their uncertainty assumption, (z-X)=E and is a 'constant' unaffected by the optimum output target, x .

BOOTH: PRODUCTION UNCERTAINTY 619

tainty is independent of the target output level. Hence, the uncertainty does not change the firm’s optimal output decision.

This is not to say that the uncertainty is irrelevant. Although the firm’s output level is unchanged, the owner’s expected utility will differ from that under certainty. To see this, write the owner’s expected level of utility at the optimal output level as

Ei[U(n* + E A)] + E2 [U(X* + d ~ ) ] ( 5 )

where El is the expectation over shortfalls in output E(E<O), E2 the expec- tation over excess output ( E > 0) and n* the optimum level of profit.

If we introduce shift variables (a and p) and define a new mean-zero random variable, E* = a + p E, we can define the Rothchild-Stiglitz [1970] mean-preserving spread by substituting the new random variable into (5) and observing the constraint da/dp=O. Making this substitution and differentiat- ing subject to the constraint gives

Ei[u’(rc)k &I+&( U’(n)d~)=o, (6 )

which shows that the effect of increasing uncertainty (and implicitly from going from certainty to uncertainty) on the owner-manager’s expected utility is ambiguous.

If secondary markets exist, so that d=h=P and shortfalls cost a s much a s windfalls produce, then (6) collapses to E[u’(x)PE], and only if investors are risk neutral, so that u’(-) is a constant, will uncertainty have no effect. To see the effect of uncertainty more clearly, we can expand the expectation to get

E[u‘(sr)P~]=E[u’(n)]E[~]~+cov[u’( A ) , E ~ ] . (7)

Since E[E]=O, (7) collapses to the familiar covariance function (cov(.,.)) a s the determinant of the effect of an increase in risk.

Risk attitudes determine how output uncertainty (E) affects profits (IT) and the owner/shareholder’s marginal utility. For shortfalls in output (E<O) , profit falls; and for risk-averse individuals, marginal utility increases. Conversely, with excess output (E>O), profit increases and marginal utility falls. Hence, cov[ U’(~),E]P<O for a risk-averse owner/shareholder, and increased uncer- tainty causes utility losses.

In the risk-averse case, uncertainty acts like an increase in a fixed cost; it does not change the optimal output decision, but owner/shareholders are worse off as a result. If our short-run model is extended to include entry and exit from the industry, the fact that the owners of capital feel worse off as a result of the increased uncertainty will cause some firms to leave the

620 ECONOMIC INQUIRY

industry. Hence, the output uncertainty, like any other fixed cost, will have long-run resource implications.

In the general case of differential costs and benefits where X>P>d, an expansion of ( 5 ) gives

6

where El and COVI(.) are respectively the expectations and covariance oper- ators defined over E<O, and E2 and cov2(.) are the same defined over E>O. Equation (10) is ambiguous, since there are two effects at work. First, even though E is subject to a mean-preserving spread, expected profits can change depending on the values of X and d. For example, in the extreme case where excess output is valueless since it is a unique product (d=O), expected profits will fall because E2[~]<0. This expected profit effect is captured in the first two terms of (10). The second two terms capture the pure uncertainty effect and with risk aversion are both negative.

If the uncertainty assumption is modified from additive uncertainty 8 Z=X+E, to multiplicative uncertainty, z=x( 1+~) , then the results change again. In this case, the optimality condition is

7

E[ C/'( K)( P+P'(x)X-C'(X)]+E~ [ U'(IC)E]X+EZ[ V( K ) E ] d (12)

where the first term is the optimality condition for additive uncertainty, and the next two terms are the uncertainty effects, which are identical to equation (6). Hence, the impact of multiplicative uncertainty on the optimal output decision is identical to the impact of additive uncertainty on expected utility described above. To recap, in the secondary markets case where d=h=P, the last two terms in (12) become E[U'(K)EP] which is unambiguously negative with risk aversion. Hence, optimal output falls. In the general case, where h>P>d, optimal output can increase or decrease, depending on the parameters h and d, because of the expected profit effect.

6. If we use a value function instead of an expected utility function, then the increased uncertainty, in a risk averse market, would cause the firm's market value to fall. This in turn would cause exit from the industry when market value falls below replacemcnt cost.

If the output uncertainty is normally distributed or U(.) is quadratic, then Rubinstein [1976] showed that (7) can be reduced to -Acov(x,~P), where A is the hall-Arrow measure of absolutc risk aversion. In our case, since output is the only source of uncertainty cov(a,eP)>O, and because risk aversion implies A>O the comparative static is unambiguously negative.

7. Note with normality or a quadratic form, equation (10) can be simplified to obtain thc partial absolute risk aversion coefficients. These reflect the individual's attitude towards risk above ( 0 0 ) and below (E<O) the mean. If d=O or U(.) is constant above the mean, then only 'downside" risk matters and semi-variance is the appropriate measure of risk.

8. The uncertain profits are

whcre the increased cost, x ~ h , and benefits, xcd, are scale dependent and thus entcr into the first ordcr conditions.

BOOTH: PRODUCTION UNCERTAINTY 62 1

I I I. CONCLUSIONS

This paper corrects and extends the analysis of output uncertainty pre- sented by Flacco and Kroetch. We show that in the case of additive uncer- tainty, their adjustment to output uncertainty just introduces random profits that are not affected by the firm’s target level of output. Hence, the optimal output decision is unaffected by this form of uncertainty. However, the un- certainty does affect the owner-manager’s expected utility, which will cause entry and/or exit from the industry and long-run output effects. In contrast, with multiplicative uncertainty the uncertain profits become scale dependent. Hence, they will affect the target level of output, since increased output means more uncertainty. However, because of the asymmetric valuation of shortfalls versus excess output, along with the uncertainty effect there is an expected profit effect, which renders the change in output from the certainty level ambiguous. Only when secondary markets exist such that d=h=P is it unambiguous that expected utility falls in the additive uncertainty case and output falls in the multiplicative uncertainty case.

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