European Journal of Political Economy 9 (1993) 141-146. North-Holland

Further results on comparative statics under uncertainty

Yaffa Machnes*

Bar-llan University, Ramat-Gan, Israel and ESSEC, France

Accepted for publication June 1992

This note develops conditions for stochastically dominant shifts in a random variable to have a qualitative identifiable effect on a decision variable in two cases: Leland’s two-period consump- tion saving model and Sandmo’s model of a firm under uncertainty. In both models, the risk is additive. While the literature focused on mean-preserving spreads, we derive qualitative conclusions in a different group of cases.

1. Introduction

The purpose of this note is to provide a variation of the impact of a stochastic change of the random variable as presented in two different decision making problems: consumption-saving and production.

Since Leland (1968) and Sandmo (1970), it has been recognized that under convexity of the marginal utility function, uncertainty about future income will reduce current consumption and will increase current saving. The consumption-saving model was recently analysed by Kimball (1990) who looked at the effect of changes in the distribution of the random income as described by Rothschild and Stiglitz (1970).

In section 3 we replace the classical mean-preserving shift of the variance by the stochastic orders introduced by Hanoch and Levy (1969) and Hadar and Russell (1969). We utilize the approach of Cheng, Magi11 and Shafer (1987) to familiar decision making models while adopting the common assumptions about the preferences of the decision taker.

In the context of a firm which decides about output before uncertain demand is resolved, we add, in section 4, another source of randomness involved with the fixed costs of the firm. We thus extend a well known model which deals with comparative static analysis to a case of multiple sources of randomness.

Correspondence to: Y. Machnes, Department of Economics, Bar-Ban University, 52900 Ramat-Gan, Israel.

*The author thanks Christian Gollier and an anonymous referee for their remarks on an earlier version.

01762680/93/$06.00 0 1993-Elsevier Science Publishers B.V. All rights reserved

142 Y. Macht~es, Compuratice statics

2. Common notation and assumptions

We discuss two different models presented in the literature. X denotes the decision variable, and our agent maximizes his expected utility. In the two different models, the utility function is denoted by I/. In section 4, the cost function is denoted by C and the revenue by R. These three functions are well behaved ~ continuous and twice differentiable in their arguments. We assume that maximization is obtained by having an internal solution. U, C and R have positive first derivatives. R-C and U are assumed to be concave.

A decision which is consistent with the expected utility rule was identified by Hanoch and Levy. Without recourse to the shapes of individuals’ utility functions, they applied the following general rule of First Degree Stochastic Dominance (FSD). It is set out in terms of monetary outcomes and their probabilities and not in terms of utilities.

Let F, and F, denote cumulative probability distribution functions. Any random income A is preferable to random income B if FA(t) sFs(f),

for all values of f (and a strict inequality holds for some value t); i.e., if the cumulative probability distribution of B lies to the left of that of A.

A private case of FSD is the case where the random income is shifted by a constant d. It is called first degree risk deterministic transformation. In this case it holds true that F,.,( t + d) = F,(t).

It follows from FSD that the two cumulative probability distributions do not intersect. If we make the usual assumptions of risk aversion, we can discriminate between projects by applying the Second Degree Stochastic Dominance (SSD) criterion.

The random income A is preferable to the random income B for all risk- averters if the cumulative difference between F, and F, is non negative over the entire domain of t. If the risk aversion criterion is fulfilled, the expected utility of income A is higher than the expected utility of income B.

3. The effect of uncertainty on saving decisions

In a two-period model, Leland (1968) and Sandmo (1970) show how a mean-preserving change of the income’s variance affects savings. Let us present their two-period model.

X is consumption in the present.

p- 1 is the interest rate which is known.

W,Z are the present values of the consumer’s alternative permanent random income. F and G denote the cumulative distribution functions. Let us denote the distributions W-F. Z-G. The shift from W to Z may be caused by a change in social security benefits or tax regulations.

Y. Machnes, Comparative statics 143

The consumer chooses the optimal level of consumption X, i.e.,

Max E U[X, p( W- X)], and Max E U[X, p(Z-X)] x w x z

Let us denote aU( )/3X = U,; aU( )/a[~( W-X)] = U,;

au,la[P(w-x)l=~,,; and ~3U,/a[p( W -X)] = Uz2.

X, denotes the optimal consumption when the random income is w and Xc denotes the optimal consumption when the random income is Z.

Proposition 1. For every utility function that satisfies pU,,~U,,, if F dominates G by the First Degree Stochastic Dominance (FSD) rule then X,2X,.

Proof: In order to compare the optimal choices under the two alternate cumulative distributions functions F and G we solve the two equations E&U, -pU,)=O and E,(U, -pU,)=O.

In order to utilize the FSD criterion, we look for an increasing function. The function (U, -pU,) increases with W at X=X, and thus also fulfills

the FSD criterion, since F dominates G for every non-decreasing function at the point Xr,

EVJ/,-PU,)~~E(U,-PU,). W z

Cheng, Magi11 and Shafer (1987) proved that the last inequality assures XF 2 x,. q

Leland (1968) and Sandmo (1970) showed that a positive third derivative of the utility function indicates a precautionary saving motive - that is, that uncertainty about future income will decrease consumption in the first period. Thus, the sign of the third derivative of the utility function governs the presence or absence of a precautionary saving. As consumption in the first period is known with certainty, the convexity of marginal utility from consumption in the second period plays the main role in decision making under uncertainty.

Let us denote the third derivatives:

U 122=~~12/xPw-m ~,n=~~*,l~rP(w-ml.

We assume convexity of marginal utility from consumption in the second period.

144 Y. Machnes, Comparatke statics

Proposition 2. For every utility function that satisfies U,22<=pU,ZZ and pUZ2 2 U,, ij’F dominates G hy the Second Degree Stochastic Dominance rule (SSD), the consumer chooses X, 2 X,.

Proof: (V, -pU,) increases and is concave in w so as F is SSD to G, for every non-decreasing concave function E,( U 1 - p U,) 5 E,( U 1 - p U 2)r and x,2x,. 0

Propositions 1 and 2 generalize the conclusions of Leland and Sandmo to a larger group of changes in the random income distributions. Our generali- zation is applicable to financial and insurance markets. Due to tax regula- tions, many insurance contracts may fit the SSD ordering. Although the insurer gets a premium which is higher than the expected loss, the insured in most countries can share their expenses with the government through the tax system.

4. Production and the distribution of fixed costs

Let us generalize the model presented by Sandmo (1971) and Leland (1972). The output X is chosen before demand is realized. A firm faces an uncertain revenue function and has the usual variable costs C(X) plus fixed random costs B,+D. R(X) is the revenue and y is a positive random variable. We assume that output is chosen so as to maximize the expected utility of profits

max E U[yR(X)-C(X)-B,-D]. X Y.BO

(2)

X is the decision variable and B, +D is called ‘background risk’ in the literature.

Sandmo (1971) and Katz, Paroush and Kahana (1982) (hereafter KPK) assumed certainty of costs and show how, in the case of random demand, certain changes in the fixed costs influence the optimal output X.

We generalize the model to the case of uncertain fixed costs. Firms are uncertain considering B,. Natural disasters or tire can risk the whole plant. In our paper we deal with the influence of changes in the distribution of the background risk B, on production decisions. We compare the optimal output X under two different distributions: B, and B,. In order to keep the traditional denotion of stochastic dominance, we look at the value of -B,. The firm desires more of -B. We denote the cumulative distributions by G and F so that -B,-G and -B,-F.

We first generalize the results of Sandmo (1971) Leland (1972) and of KPK to the case of multiple sources of randomness. In the case in which a

Y. Machnes, Comparative statics 145

firm faces a constant reduction of costs we show that a deterministic change of the distribution of B, so that B,= Bo- D (and F( - B, + D) = G( - Be)) will increase the optimal output under the same assumptions about the utility function made by the former authors.

Proposition 3. If F and G fulfill first degree deterministic transformation, then a firm whose absolute risk aversion decreases will produce more under F than under G.

Proof Assuming y and B, are independent random variables, Levy and Sarnat (1971) have shown that when F is FSD to G, yR(X)- C(X)- B, is FSD to yR(X) - C(X) - B, - D.

We solve the equation:

E [yR’(X)-C’X)]U’[yR(X)-C(X)- Bo-D] =O. (3) Y.B

The derivative of the left side of eq. (3) with respect to D is

-E [yR’(X) - C’(X)] U”[ 1. (4) v.B

Following the proofs of Sandmo and KPK (in their Lemma 3) we can see that the derivative in (4) is negative under the assumption of decreasing absolute risk aversion. It has been proven that by decreasing the random expenditures by a fixed sum, the firm increases the quantity of X. 0

The stochastic change of the distribution of B, as described in Proposition 3 is very special and is equivalent to receiving a fixed subsidy, independent of output or exogenous losses occurring due to the randomness of Bo.

For the more general FSD changes of B, we have the following result:

Proposition 4. If F is FSD to G, a firm whose utility function fuljlls U”‘s 0, will produce less under B, than under B,.

Proof. Since U” decreases with y, we know that for every value of B

cov [(?R’ - C’), U”] 5 0

E(yR’-C’)U”sE(yR’-C’)EU” V V

As Sandmo showed, for the initial optimal X, EyR’(X) > C’(X). Since U” 50 we learn that E,(yR’- C’)U” <O so E(yR’-c’) U’ decreases for every value of B. 0

We do not have opposite results, neither for the case of U”‘>O nor for the case of decreasing absolute risk aversion.

We conclude from Propositions 3 and 4 that every risk averse firm whose utility function fulfills U”‘j0 will decrease its output in the case in which the

146 I: Machnes, Comparative statics

distribution of B is more favorable to the firm. Our conclusions generalize the results of Sandmo who presented a model with certain fixed costs. Further generalizations involve more assumptions about the fourth derivative of the utility functions and are outside the scope of this note.

5. Conclusion

We have shown how risk averse decision takers react to an exogenous change in a random variable. In the two models presented, the random variable is additive and statistically independent of other variables. We showed how the value of the third derivative of the utility function determines the sign of the change of the decision variable.

The aim of several public projects is to reduce the uncertainty that consumers and firms face. Governments introduce several social security plans like health and unemployment insurance. A variety of projects that reduce losses due to natural disasters and investments in safety are public goods in most countries.

Applications of our note can be found by studying the effect of public policies that decrease the risks that individuals and firms face. We have shown how specific changes in the distribution of income can affect consumption, and how public investments that decrease the risk of natural losses can affect the output produced.

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