Output decisions of firms under uncertainty: some micro-theoretic analysis.

Guru-Gharana, Kishor Kumar ; Rahman, Matiur ; Parayitam, Satyanarayana 等

INTRODUCTION

Traditionally, risk-return models have been used in portfolio theory. The theory of the firm under uncertainty has not been analyzed with the risk-return approach except for few works such as the mean-standard deviation model of Hawawini (1978). Expected utility approach seems to be popular in this field of study notwithstanding the intuitive appeal of the risk-return dichotomy in the business world.

This paper is a theoretical re-exploration of various risk-return models to the output decision making of a competitive firm facing uncertainty in the product price. Our attempt here is to find if different results do arise under risk-return vs. expected utility approaches. The results of the risk-return models in this paper are compared to those of the expected utility literature. One important difference in the results is that the risk-return approach requires some restrictions on the cost function of the firm whenever the assumption of decreasing absolute risk aversion is required in the expected utility approach in order to obtain deterministic comparative statics results.

The remainder of the paper is structured as follows. Section 2 presents a brief review of some related literature. Section 3 is on the output decisions of a competitive firm under product-price uncertainty within mean-general risk model, and mean-standard deviation model. Section 4 concludes by comparing the major findings of this paper with the corresponding results in the expected utility approach.

LITERATURE REVIEW

Mills (1959) studies a monopolist under uncertainty and is one of the earliest works introducing uncertainty in microeconomics. But Mills' work suffers from serious limitations, such as, assumption of risk neutrality. Zabel (1970) is a generalization of Mills' article in some respects. Zabel uses multiplicative form of uncertainty instead of additive separability of uncertainty. Sandmo (1971) and Baron (1970) deal with a competitive firm facing an uncertain price of its product. These two articles are complementary to each other. The marginal impact of changing the distribution of price is studied by Sandmo but not by Baron, while Baron studies the effect of an increase in risk aversion which Sandmo ignores. Sandmo finds that the overall-impact of uncertainty is to reduce output assuming that the marginal cost is rising. However, Sandmo himself could not determine the sign of the marginal impact of uncertainty (for example the effect of a mean-preserving spread). But this problem was later taken up by Ishi (1977) who showed that nondecreasing absolute risk aversion is a sufficient condition for the marginal impact to be in the same direction as the overall impact.

Baron proves that, "... optimal output is a nondecreasing function of the firm's index (Arrow-Pratt measure) of risk aversion." Moreover, an increase in fixed cost decreases output for decreasing absolute risk aversion. Baron also concludes that if risk aversion is prevalent, as seems reasonable, prices are higher and outputs are lower than if firms were indifferent to risk. Finally, Baron finds that under uncertainty it is possible for the short run supply function of the risk averse firm to have a negative slope. This is a result which cannot occur in deterministic microeconomic theory. Baron (1971) demonstrates that for an imperfectly competitive firm under uncertainty the strategies of offering a quantity or changing a fixed price yield different results. Leland (1972) considers three alternatives behavioral modes under uncertainty, and claims that the result of Baron (1970) and Sandmo (1971) are special cases of his more general results. Lim (1980) addresses the question of ranking these behavioral modes under risk neutrality.

Batra and Ullah (1974) follow Sandmo heavily except that they use a production function and adopt an input approach instead of an output approach. As shown by Hartman (1975,1976), the Batra and Ullah paper suffers from a partial approach in deriving their conclusions about the overall impact of uncertainty on input demands considering both inputs simultaneously. Batra-Ullah also assume concavity of the production function and decreasing absolute risk aversion to show that the marginal impact of uncertainty in terms of mean-preserving spread is to reduce input demands. Hartman (1975) criticizes the partial approach adopted by Batra-Ullah and in his 1976 article, Hartman also relaxes the assumption that all inputs are chosen before the product price is observed. He claims that the results of Batra-Ullah and Sandmo are rather sensitive to that particular assumption. Korkie (1975) comments that Leland's conclusion is the result of the assumption called the principle of increasing uncertainty. Blair (1974) also discusses some implications of random input prices on the theory of the firm.

Similar to the objective function of the present paper involving mean and risk Arzac (1976) allows for substitution between expected profit and safety, and proposes the following objective:

Maximize x + g([alpha]), g' < 0, g'' < 0., where [alpha] is the probability that profit falls below a disaster level.

This criterion satisfies the continuity axiom but only its linear form satisfies the independence axiom and is compatible with utility theory (Markowitz (1959) and Arzac (1976)). Arzac also applies the safety-first approaches to the theory of the firm under uncertainty and concludes the followings:

a) The overall impact of uncertainty is to lower output.

b) Maximizing the certainty equivalent profit has the same comparative statics properties as the certainty model.

c) If suitable empirical evidence on the firm's past responses to changes in the profit tax rate and in lump sum taxes and subsidies are available, then an almost complete discrimination among the alternative criteria can be made.

A survey of stochastic dominance principle which is used in comparative statics analyses of this paper is found in Levy (1992). Examples of various applications of this principle in investment decision making are available in Levy and Robinson (1998) ,Kim (1998) and Kjetsaa and Kieff (2003). Empirical works on this principle are reported in Porter and Gaumnitz (1972) and Barret and Donald (2003). Gotoh and Konno (2000) study relationship between Third Degree Stochastic Dominance and Mean-Risk Analyses.

Output Decisions of a Competitive Firm Under Uncertainty

In this paper we study the output decision making of a competitive firm facing uncertainty in the product price, assuming that the firm has a subjective probability distribution for the product price.

Assumptions

i) The firm must choose the volume of output (denoted by x) prior to the sales date when the market price (denoted by p) becomes known.

ii) The firm's beliefs about the sales price can be summarized by a subjective probability distribution function F(p) with density function f(p) and mean p. Since the firm is unable to influence this distribution function, the basic assumption is that the firm is a price taker in a probabilistic sense.

iii) We are concerned here with the short run decisions of the firm; the fixed cost (denoted by b) appears in the total cost function.

iv) We assume that the firm's objective function is given by V = m - [lambda]r, where m is the mean and r is the risk of profit [pi]. Profit is given by [pi] = px - c(x) - b. The parameter [lambda] > 0 is the relative weight given to risk component and can be interpreted as the rate of substitution between return and risk and also as a measure of risk-aversion of the agent. For a risk-neutral firm [lambda] = 0.

v) The variable cost function c(x) is assumed to have the following properties: c (0) = 0, c'(x) > 0, and c''(x) [greater than or equal to] 0.

That is, we assume that marginal cost (denoted by MC) is a positive and nondecreasing function of the output level.

The assumptions about the risk function r will be different for different models and therefore will be stated at the appropriate places.

MEAN-GENERAL RISK MODEL

The Risk Function and the Objective Function

In the general risk framework risk r is given by

r = [[integral].sup.t.sub.-c(x)-b] [psi](t - [pi])g([pi]).sup.d[pi], (1)

Where g([pi]) is the probability density function of profits, and t is the target value of profit. This risk measure implies that agents associate risk only with outcomes below a target value (Markowitz (1952, 1959), Mao (1970), Fishburn (1977 and 1984), and Holthausen (1976 and 1981)).

The function [psi] is assumed to have the following properties:

[psi](0) = 0, [psi]'( ) > 0 for [pi] < t, and [psi]"( ) [greater than or equal to] 0.

The lower limit -c(x)-b is the minimum profit level for a given choice of output x, and occurs when p is zero. Changing variables from [pi] to p in equation (1) we obtain

r = [[integral].sup.[??].sub.0] [psi][t - px + c(x) + b]f(p)dp, (2)

where [??] = t+c(x)+b/x. For lack of another name, we can call [??] as the "target" price or the price which would give revenue equal to target profit and total cost.

The objective function to be maximized is given by

V = [bar.p]x - c(x) - b - [lambda] [[integral].sup.[??].sub.0] [psi][t - px + c(x) + b] f (p)dp. (3)

It can easily be shown that the necessary and sufficient condition for an interior maximum, is [bar.p] > c'(0), which we assume below.

First Order Condition

The first order condition for maximization is

Vx = [bar.p] - c'(x) - [lambda][[integral].sup.[??].sub.0] [psi]'( )[c'(x) - p]f (p)dp = 0 (4)

Solving (4) we can get the optimal value of x in terms of the parameters as

x = h([bar.p],t,b,[lambda]). (5)

The form of the function h is determined by the cost function c(x) and the risk function [psi]. From the first order condition it is proved below that the expected level of price exceeds the marginal cost at the optimum output level.

Proposition (1) if [x.sup.*] denotes the optimum output level, then [bar.p] > c'([x.sup.*]).

Proof: From the first order condition we have

[bar.p] - c' = [lambda] [[integral].sup.[??].sub.0] [psi]'(t - [pi])(c' - p) f (p)dp, (6)

Where [pi] = px - c(x) - b.

Case (A): c' [greater than or equal to] [??]. In this case, the integral in equation (6) is clearly positive because [psi]' ( ) > 0. Therefore, proposition (1) immediately follows.

Case (B): c' < [??]. In this case we prove proposition (1) by the method of contradiction. The integral in equation (6) can now be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

If [??] and [??] are such that [pi](0) < [??] < [pi](c'), and [pi](c') < [??] < [pi]([??]), then applying the mean value theorem, we obtain from equation (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Since [psi]" [greater than or equal to] 0 is assumed, we have from equation (8)

I = [greater than or equal to] [psi]" [t - [pi](c')] [[integral].sup.p.sub.0] (c' - p) f (p)dp. (9)

From the definition of the expected value [bar.p], we have

[bar.p] - C' = [[integral].sup.[??].sub.0] (p - c') f (p) dp + [[integral].sup.[infinity].sub.[??]] (p - c') f (p) dp, (10)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The sign of the integral on the right hand side of equation (11) follows from our assumption c' < [??]. Now if proposition (1) is violated, that is, if [bar.p] - c' [less than or equal to] 0, then from (11) we obtain

[[integral].sup.[??].sub.0] (p - c') f (p) dp < 0, (12)

or equivalently,

[[integral].sup.[??].sub.0] (c'-p)f (p)> 0. (13)

On the other hand, if [bar.p] - c' [less than or equal to] 0, then from equation (6) we have

I [less than or equal to] 0, (14)

Which along with equation (8) implies

[[integral].sup.[??].sub.0] (c'- p) f (p) dp [less than or equal to] 0. (15)

But inequalities (13) and (15) contradict each other. Therefore, it follows that we must have [bar.p] - c' > 0. This proves proposition (1).

One implication of proposition (1) is that the uncertainty output will be less than the corresponding certainty output. Sandmo (1971) calls this result the "overall impact of uncertainty." This result is illustrated in figure (2), where MC is the marginal cost curve.

In figure (1), [x.sup.*.sub.c] is the optimum output level for the corresponding certainty case (where price equals marginal cost), and [x.sup.*.sub.u] is the optimum output level for the uncertainty case (where [bar.p] > c').

[FIGURE 1 OMITTED]

Considering equation (6), another implication of proposition (1), which we use in some comparative statics results, is that

[[integral].sup.p.sub.0] [psi]'(t - [pi])(c' - p) f (p)dp > 0. (16)

Second Order Condition

The second order condition for an interior maximum is given by

Vxx = - c"[1 + [lambda] [[integral].sup.[??].sub.0] [psi] 'f (p)dp] - [lambda] [[integral].sup.[??].sub.0] [psi]"[(c' - p).sup.2] f (p)dp (17)

Our assumption c" > 0, [psi]" [greater than or equal to] 0, and [psi]"(0) [greater than or equal to] 0 are sufficient for the second order condition to be satisfied.

Comparative Statistics

To find the effects of a change in a parameter i (where i can be one of the parameters [bar.p], t, b, and [lambda]) we proceed in the following way:

The first order condition can be written as

Vx[h([bar.p],t,b,[lambda]),[bar.p],t,b,[lambda]] = 0. (18)

Differentiating with respect to i, we obtain

[V.sub.xi] + [V.sub.xx][h.sub.i] = 0. (19)

That is,

[h.sub.i] = - V x i/V xx. (19')

Since Vxx < 0 from the second order condition, the sign of [h.sub.i] is the same as the sign of Vxi. Hence in the following, we will determine the sign of [h.sub.i] from that of Vxi.

(a) Change in Risk Aversion.

From equation (4), the derivative Vx[lambda] is given by

Vx[lambda] = - [[integral].sup.[??].sub.0] [psi]'(c' - p f (p)dp < 0 (20)

Inequality (16) implies the sign of inequality in (20).

Thus, an increase in risk aversion indicated by a rise in [lambda], reduces the optimal output.

(b) Change in Fixed Cost.

From equation (4) we have

Vxb = - [lambda] [[integral].sup.[??].sub.0] [psi]''(c' - p) f (p) dp - [lambda]/x [psi] '(0)(c' - [??]) f ([??]). (21)

From equation (21) we infer

Vxb < 0 if c' [greater than or equal to] [??] = t + c + b/x. (21')

That is, if marginal cost is at least as large as t + c + b/x, then an increase in fixed cost reduces the output level. However, the condition in (21') is only a sufficient condition but not necessary. If c' is less than t + c + b/x, then the second term in (21) is positive but the sign of the first term is ambiguous.

(c) Change in Expected Price.

In order to study the effect of a change in the expected price, we will transform p to [p.sup.*] = p + k, where k is the shift parameter. Then, we have the objective function as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Where [[??].sup.*] = t + c + b/x - k. Therefore, the first order condition is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Equation (23) gives x as a function of k. As discussed above, the sign of [h.sub.k] will be the same as that of [V.sup.*.sub.xk], and will show the direction of the effects of a change in the expected price on the optimal output.

We have from equation (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Where [??] = t + c + b/x as above. The last term contains [??] because [[??].sup.*] = [??] when p = t + c + B/x - k.

From equation (24) we get the following results:

If c' [greater than or equal to] t + c + b/x, then [V.sup.*.sub.xk] > 0 (24')

In other words, a sufficient condition for the supply function of the competitive firm to be positively sloped with respect to the expected price is given by the inequality c' [greater than or equal to] t + c + b/x.

This condition reduces to marginal cost greater than or equal to average cost when the target value t equals zero. Holthausen (1981) and Crosby, et al. (1985) conclude after the review of many empirical studies that the target level is frequently equal to zero. Also, the marginal cost is at least as large as average cost if average cost is everywhere nondecreasing. However, the sufficient condition in the comparative statics results for t equal to zero require only that the optimal output should not lie in the decreasing section of the average cost curve.

(d) Change in the Target Value.

The comparative statics analysis with respect to a change in the target value is of special interest because we know that, other things remaining the same, the measure of risk increases as the target value increases. Thus, an increase in the target value can be considered as one way of increasing the measure of risk. However, the derivative [V.sub.xt] is found to be identical to [V.sub.xb] and is therefore of ambiguous sign. Again, the inequality t + c + b/x is a sufficient condition for an increase in the target value (and consequently in risk) to reduce the optimal level of output.

It may be interesting to note that, out of the four parameters [bar.p], b, t, and [lambda], only the parameter [lambda] gives an unambiguous comparative statics result. In the case of the other three parameters, the inequality c' [greater than or equal to] t + c + b/x provides a sufficient condition for a determinate comparative statics result.

In the remainder of this paper we will study further comparative statics effects related to some changes in the distribution of p, and the tax rate.

(e.) A First Degree Stochastic Dominance (FSD) Shift.

Here we investigate the effects of a FSD shift which will make new distribution dominating. It is proved below that such a shift will increase the optimal output level if the inequality c' [greater than or equal to] t + c + b/x is satisfied.

Let us denote the initial distribution of p by [f.sup.1] and the new distribution by [f.sup.2]. We want to show that, if [f.sup.2] FSD [f.sup.1], then the optimal output corresponding to [f.sup.2] is larger than the optimal output corresponding to [f.sup.1]. Let [V.sup.1] and [V.sup.2] denote the objective function corresponding to [f.sup.1] and [f.sup.2], respectively. Let [x.sup.1] denote the optimal output corresponding to [f.sup.1]. Then, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25')

Where [p.sup-1] and [p.sup.-2] are the means of p corresponding to [f.sup.1] and [f.sup.2], respectively. Denoting [psi]'(t - [pi]) [c'([x.sup.1]) - p] by [phi] (p) we have from (25) and (25').

([v.sup.2]x - [v.sup.1]x)|[sub.x=x] 1 = ([p.sup.-2] - [p.sup.-1] + [lambda][[integral].sup.p.sub.0] [PHI] (p)[[f.sup.1] - [f.sup.2]]dp. (26)

We know that the first term on the right hand side of (26) is positive. Let us investigate the sign of the second term. We have, [PHI](p) = [psi]'(t - [pi])(c'-p). Since we assume [psi]' [greater than or equal to] 0, it immediately follows that [PHI] (p) is positive in the integral in equation (26) if c' is at least as large as [??]. Therefore, the left hand side in (26) is also positive if c' [greater than or equal to] [??]. Now, differentiating [PHI] (p) we obtain

[PHI]' = -[psi]'(t - [pi])-x[psi]"(c'-p) < 0, (27)

Since [psi]" [greater than or equal to] 0 is our basic assumption. Thus, [PHI] is a decreasing function. Considering (26) and (27), and the results of stochastic dominance as derived in the article by Hadar and Russell in Balch, et al. (1974), it follows that the output increases as a result of a FSD shift which makes the new distribution dominating, given that c' is at least as large as t + c + b/x.

(f) Second Degree Stochastic Dominance (SSD) Shift.

We now show that the inequality c' [greater than or equal to] t + c + b/x, and the assumption [psi]'" [greater than or equal to] 0 are sufficient for a SSD shift, which makes the new distribution dominating, to increase output. To show this result, we need to demonstrate that the function [PHI](p) in (26) is convex.

We have

[phi]"(p) = 2x[psi]" + [psi]'"[x.sup.2] (c'-p) [greater than or equal to] 0, (28)

Since [psi]" [greater than or equal to] 0, [psi]'" [greater than or equal to] 0, and c' [greater than or equal to] [??]. Therefore, considering (26) and the results of stochastic dominance found in Hadar and Russell's article (1969 ,1971 and 1974) it follows that the output increases when the distribution is made dominating in the SSD sense.

(g) Change in Per Unit Tax.

Let us denote the per unit tax by [delta]. Then the net profit denoted by [??] is

[??] = px - c(x) - b - [delta]x. (29)

The objective function is

V = [bar.p]x - c(x) - b - [delta]x - [lambda] [[integral].sup.[??].sub.0] [psi] (t - [??]) f (p) dp, (30)

Where [??] = t + c + b/x + [delta] = [??] + [delta]. We show below that a rise in the per unit tax reduces output if c' is at least as large as t + c + b/x,

From equation (30) we obtain the first order condition as

[V.sub.x] = [bar.p] - c' - [delta] - [lambda] [[integral].sup.[??].sub.0][psi]'(t - [??])(c' - p + [delta]) f (p)dp = 0. (31)

Differentiating [V.sub.x] with respect to [delta] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Therefore, it is evident from (32) that c' [greater than or equal to] t + c + b/x is a sufficient condition for [V.sub.xt] < 0. In other words, given the inequality c' [greater than or equal to] [??], a rise in per unit tax reduces output.

In the following section, we study the mean-standard deviation model where risk is defined as the standard deviation of the profit level.

Mean Standard Deviation Model

Let the standard deviation of price be denoted by o and that of profit by [[sigma].sub.[pi]]. Then we have

[[sigma].sub.[pi]] = x[sigma]. (33)

The objective function in this framework is

V = [bar.p]x - c(x) -b - [lambda]x[sigma]. (34)

Shut-Down Condition

Here, we have

Vx|x = 0 = [bar.p] - c'(0) - [lambda][sigma]. (35)

Assuming that V is concave in x (which in this model requires c" > 0), we find the necessary and sufficient condition for a positive output as

[bar.p] > c'(0) + [lambda][sigma]. (36)

As discussed earlier, the condition for a positive output level in the risk formulation involving a target value is that [bar.p] exceeds c'(0). Thus, we see that the condition for a positive output level in the mean standard deviation model is more stringent than that in the model with the risk function defined in terms of the target level of profit.

In the subsequent paragraphs we derive the first and second order conditions for an interior maximum, assuming that inequality (36) is satisfied.

First Order Condition

From equation (34), we have the first order condition as

Vx = [bar.p] - c'(x)- [lambda][sigma] = 0. (37)

Solving for x we obtain

x = [c'.sup.-1]([bar.p] - [lambda][sigma]) = H([bar.p] - [lambda][sigma]). (38)

Equation (37) also gives

[bar.p] = c'(x) + [lambda][sigma] > c'(x). (39)

Thus, in this model we can explicitly express output as a function of a linear combination of mean and standard deviation of price. Moreover, the function h in equation (38) is the inverse of the marginal cost function with the linear combination of mean and risk as its argument. Equation (39) shows that, in this model too, the overall impact of uncertainty is to reduce output.

Second Order Condition

We have in this model

[V.sub.xx] = -c". (40)

Therefore, c" > 0 is necessary and sufficient for the second order condition to be satisfied in this model. In other risk formulations increasing marginal cost is not a necessary condition for an interior maximum.

Comparative Statistics

(a) Change in the Expected Price.

If we transform p to [p.sup.*] = p + k, k > 0, and denote the objective function corresponding to [p.sup.*] by [v.sup.*], then we have

[V.sup.*] = ([bar.p] + k)x - c(x)- b - [lambda]x[sigma]. (41)

The first order condition is

[V.sub.x.sup.*] = [bar.p] + k - c'-[lambda][sigma]=0. (42)

Differentiating with respect to k, we obtain

[V.sup.*.sub.vk] = 1 > 0. (43)

Therefore, the supply function with respect to the expected price is upward sloping in this model without any additional assumption, whereas in other risk formulations we could only give a sufficient condition (c' [greater than or equal to] t + c + b/x) for such a result.

Notably, the results of the mean-standard deviation model are closer to those of the corresponding deterministic model than to the results of other risk formulations. This is evidently true in the case of the positive slope of the supply function which requires only that the marginal cost is upward sloping. We will find that the same is true for other comparative statics results, provided they are applicable to the certainty case.

(b) Change in the Risk Aversion Measure.

The derivative Vx[lambda] is given by

Vx[lambda] = -[sigma] < 0. (44)

Hence, as in other models, an increase in risk aversion decreases output.

(c) Change in Fixed Cost.

The solution for the optimal output as in equation (38) clearly shows that a change in the fixed cost has no effect on the optimal output in this model. Again, we see that the result of the mean-standard deviation model is similar to that of the corresponding deterministic model.

Next, we consider a mean-preserving spread which changes the variance (and consequently the risk or the standard deviation) leaving the mean of the random variable unaltered. This is also a special type of SSD shift where the mean of the two distributions is the same.

(d) Mean-Preserving Spread (MPS).

We have from equation (37)

Vx[sigma] = -[lambda] < 0. (45)

Thus, a MPS transformation of p will reduce the optimal output level for a risk averse firm, and will have no effect for a risk neutral firm.

(e) Change in Per Unit Tax.

If per unit tax is denoted by [sigma], then we have the objective function as

V = [bar.p]x-c(x)-b-[delta]x-[lambda]x[sigma]. (46)

The first order condition is

Vx = [bar.p] - c'-[lambda][sigma]-[delta]=0. (47)

Differentiating with respect to [delta] we obtain

Vx[delta]=-1< 0. (48)

Thus, an increase per unit tax reduces output without any further assumption, whereas in other risk formulations we could provide only a sufficient condition for this result.

(f) Change in Profit Tax.

Let [theta] be the full loss offset profit tax rate. Then, the net profit is given by

[n.sup.*]=(1-[theta])[px - c(x) - b]. (49)

The objective function is

[V.sup.*] = (1 - [theta])[[bar.p]x - c( x) - b - [lambda]x[sigma]], (50)

Since the standard deviation of [[pi].sup.*] is (1-[theta])x[delta]. The first order condition is

[V.sub.x.sup.*] = (1-[theta])([bar.p]-c'-[lambda][sigma]] = 0. (51)

Solving for the optimal output we have

x = [c'.sup.-1][[bar.p] - [lambda][sigma]], (52)

which is similar to equation (37). Equation (52) clearly shows that a change in the profit tax rate of the full loss-offset type does not affect output.

CONCLUSIONS

This paper concludes as follows:

(a) A positive slope of the supply function with respect to expected price is obtained in expected utility models by assuming decreasing absolute risk aversion as observed in Baron (1970). In our mean-risk models with a target value, a positive slope of the supply curve is obtained given the sufficient condition c' [greater than or equal to] t + c + b/x (which reduces to marginal cost greater than or equal to average cost when t equals zero). On the other hand, in the mean-standard deviation model, this result holds without any further assumption.

(b) The effect of increased risk aversion on output is negative in mean-risk model, which is also true in expected utility models (see for example Baron (1970)).

(c) In the general risk formulation, the inequality c' [greater than or equal to] t + c + b/x is a sufficient condition for an increase in fixed cost to reduce output, whereas in expected utility models the assumption of decreasing absolute risk aversion is a sufficient condition for such a result. Thus, the sufficient condition of the mean-risk model involves the behavior of the cost function, whereas the sufficient condition of the expected utility model involves the risk aversion attitude of the firm. In the mean-standard deviation model fixed cost does not have any effect on the optimal output.

Thus, we see that different models can give significantly different results. For example, suppose c' < t + c + b/x holds for a firm, then the standard deviation model predicts zero effect, whereas the general risk model and the expected utility model give an ambiguous result. Similarly, if c' = t + c + b/x holds for a firm, then the standard deviation model predicts zero effect of fixed cost on output, the general risk model predicts a negative result and the expected utility model gives ambiguous results if decreasing absolute risk aversion is not assumed.

(d) In the case of a mean-preserving spread, a negative effect on output is obtained in the mean-general risk model assuming c' [greater than or equal to] t + c + b/x, in the mean-standard deviation model without any further assumption, and in the expected utility model assuming non-increasing absolute risk aversion (see, for example, Sandmo (1971)).

(e) A negative effect of an increase in per unit tax is obtained in the mean-risk model involving a target value by assuming c' [greater than or equal to] t + c + b/x, in the mean-standard deviation model without any further condition, and in the expected utility in model by assuming non-increasing absolute risk aversion (see, for example, Sandmo (1971)).

(f) The effect of a proportional tax in the expected utility model of Sandmo (1971) depends on the monotonicity property of the measure of relative risk aversion. Our mean-general risk model gives an ambiguous result. A change in the profit tax rate [theta] has no effect on output in the mean-standard deviation model.

APPENDIX

In parts A and B of the appendix, we provide a summary of the main results. The sign (+) means a positive effect on output, the sign (-) means a negative effect on output, and (0) means no effect on output. The assumption of decreasing absolute risk aversion is denoted by DARA. To state further, the mean-standard deviation model has results almost identical to the deterministic model when price equals with probability of 1.To note again, a mean-preserving spread reduces output in the mean-standard deviation model without further assumption, whereas in the expected utility model the assumption of nonincreasing absolute risk aversion gives such a result. On the other hand, in the case of models involving a target value the inequality is a sufficient condition for this result.

Summary of Results: Part A

Model                 Change in          Change in

                      Overall Price      Expected Price
                      of Uncertainty

General Risk Model           -           if c'[greater than
                                           or equal to]
                                            t + c + b/x

Standard                     -                   +
Deviation
Model

Expected                     -                   +
Utility                                       if DARA
Model

Deterministic               Not                  +
Model with              Applicable
Price

Model                   Change in             Change in

                        Fixed Cost            Per Unit Tax

General Risk Model    if c'[greater than    if c'[greater than
                        or equal to]          or equal to]
                         t + c + b/x           t + c + b/x

Standard                      0                     -
Deviation
Model

Expected                      -                     -
Utility                    if DARA           if nonincreasing
Model                                         absolute risk
                                                 aversion

Deterministic                 0                     -
Model with
Price

Model                      Change in

                         Profit Tax at
                          [theta] =0

General Risk Model         Ambigous

Standard                       0
Deviation
Model

Expected                       -
Utility               if decreasing relative
Model                    risk aversion

Deterministic                  0
Model with
Price

Summary of Results: Part B

Model         Change in

                   Risk                Target
                 Aversion               Value

General              -                    -
Risk Model                       if c'[greater than
                                    or equal to]
                                     t + c + b/x

Standard             -              Not Applicable
Deviation
Model

Expected             -              Not Applicable
Utility
Model

Model

                  FSD Shift               SSD Shift

General               +                       +
Risk Model    if c'[greater than    if [psi]"' [greater
                or equal to]              than or
                 t + c + b/x            equal to] 0,
                                             and
                                     if c'[greater than
                                        or equal to]
                                         t + c + b/x

Standard              +                 Not available
Deviation       Not available
Model

Expected        Not Available           Not Available
Utility
Model

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Kishor Kumar Guru-Gharana, McNeese State University

Matiur Rahman, McNeese State University

Satyanarayana Parayitam, University of Massachusetts Dartmouth