Comparative statics and risk aversion

Comparative statics and risk aversion

Economics Letters 5 (1980) 125-128 0 North-Holland Publishing Company COMPARATIVE STATICS AND RISK AVERSION Steinar EKERN Norwegian School of Econo...

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Economics Letters 5 (1980) 125-128 0 North-Holland Publishing Company

COMPARATIVE

STATICS AND RISK AVERSION

Steinar EKERN Norwegian School of Economics

and Business Administration,

N-5000 Bergen, Norway

Received 28 May 1980

Assuming risk aversion, comparative available. Three propositions facilitate ges in some structural parameter.

statics properties of ‘new’ economic models may be readily signing the optimal response of a control variable to chan-

1. Introduction Assuming a risk averse expected utility decision maker, this paper considers a basic uncertainty model with an outcome function depending on a control variable, a single random variable, and a structural parameter. By imposing various conditions on the outcome function and on the risk aversion functions, three propositions about the comparative statics of the general model are developed. These propositions provide a shortcut to the implications of ‘new’ economic models possessing the characteristics of the basic model. The paper supplements the sign determination rules of Diamond and Stiglitz (1974), and of Hadar and Russel(l978).

2. Model The following notation

will be used (with subscripts indicating

(Y: a control variable, with (Y*being an optimal value, 0 : a random variable, with distribution function F(B), fl : a structural parameter, y = G(cY, 0, 0) : an outcome function, r/(j) : a utility function, with U, > 0 and U,,, < 0, 0 = ECU(y)} = Je U(G(q p, 0)) dF(B): expected utility, Rb) f -l&,/U,, : the absolute risk aversion function, r(j) = -ylJ,,/U,, : the relative risk aversion function, sgn ( . ) : the Signum function, h : an arbitrary positive constant. 125

derivatives):

S. Ekern / Comparativestaticsand risk aversion

126

Under suitable regularity conditions, the first-order condition 0, = E {U,,G,} = 0 determines a unique and finite optimal value cr* = 01~(0) of the control variable, which satisfies the second-order condition a7,, = [email protected],,,Gi + U,G,,} < 0.

3. Comparative

statics analysis

The comparative statics analysis consists of examining ferentiation in the first-order condition, da*ldfl=

[email protected],,G&i

+ UYG,,)I&

dol*(P)/dfl. By implicit dif-

,

which has the sign of the expectation term by the second-order The following initial assumptions are made: A.1. A.2. A.3. A.4. A.5.

(1) condition.

There exists a unique value 8, such that G, changes sign at 6. Ge is non-zero and uniformly signed. R is strictly monotonous in y. Gp equals zero. G,, equals zero.

U, > 0 and 0, = 0 require G, to change sign in 0. By A.1, there is exactly one such sign change in G,. Note that A.1 does not require G, to be either continuous or monotonous in 8, and G, is not necessarily zero at 8. Let G,(0 10 < 6) be the valaue of G, evaluated at some 0 < 6. Often in economics A.2 will follow naturally. A.3 rules out any utility function U whose absolute risk aversion is neither strictly decreasing, constant, nor strictly increasing over its entire domain. A.4 and A.5 imply additive separability. Proposition 1. The effect on the optimal value (Y* of the control variable from changes in the structural parameter /3, is given by

sgn[da/dfil

= sgn(Ge) sgn(Gp) sgn(G,@

IO< e^))sgn[R(y + h) - RO)l

(2)

under assumptions A. I, A.2, A.3, A.4 and A.5. Any outcome function

of the general form

~(a, 0, 0 ) = OH(o) + K(P) + ~(0)

(3)

is consistent with assumptions A.1, A.2, A.4 and AS. Hence Proposition 1 relates the absolute risk aversion to outcome functions exhibiting multiplicative uncertainty and additive separability in the control variable and the parameter. The independence assumption A.5 may appear as rather restrictive. It may be relaxed, at the cost of replacing a complete sign determination with a partial sign determination.

S. Ekern / Comparative statics and risk aversion

A.6. G,p is non-zero

and uniformly

signed.

A. 7. sgn(G,p) = sgn(G0) sgn(Gp) sgn(G,(B 10 < 6)) sgn [RO, t h) - I]

A.8. sgn(r - 1) = sgn(-G

127

.

+ G,Gp/G,,).

Assumptions A.7 and A.8 hardly convey much intuitive economic insight at this level of generality and should be considered as formal conditions to be checked out for each specific model. Proposition

(i) w(da*/@)

2. 7;cIeeffect on the optimal 1y*from changes in /3 is given by (ha)

= s&G,&

under assurnpttons A.l, A.2, A.3, A.#, A.6, and A. 7,

(ii) sgn(da* /d/3) = sgn(G,J

sgn( 1 - r)

(4b)

under assumptions A. 6 and A. 8.

When the parameter /3 affects the outcome through a multiplicative function rather than through an additive one, the relative risk aversion function replaces the absolute risk aversion function as the key ingredient. The following new assumptions are called for: A.9. The outcome function is given by y = G(cll, fl,O) = H(cw,O)[email protected]). A. IO. r is strictly monotonous in y. Proposition 3. The effect on the optimal value LY*of the control variable from changes in the structural parameter /3, is given by

sgn[da*/@l

= sgn(Ho) sgn(H,(O IO < e^)) sgn(K) sgn(Kp) sgn[rO + h) -r(j)]

(5)

under assumptions A.1, A.2, A.9, and A.lO.

Proofs of the three propositions above are given in Ekern (1979) which is available on request. Basically, the proofs are derived by mimicking the procedures of Arrow (1971). 4. Concluding

remarks

The three propositions of this paper may significantly reduce the effort required for signing the optimal response of a single control variable to changes in structural parameters. It is noteworthy how properties of the absolute risk aversion function are relevant when the outcome is additively separable in the control variable and the parameter, whereas a multiplicatively separable outcome function ties with properties of the relative risk aversion function. Ekern (1979) shows how the propositions may be applied to some popular models.

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S. Ekern / Comparative

statics and risk aversion

References Arrow, K.J., 1971, Essays in the theory of risk-bearing (Markham, Chicago, IL). Diamond, P.A. and J.E. Stiglitz, 1974, Journal of Economic Theory 8, 337-360. Ekern, S., 1979, Comparative statics and risk aversion, Discussion Paper 16R/78, revised June 1979 (Norwegian School of Economics and Business Administration, Bergen). Hadar, J. and W.R. Russel, 1978, Applications in economic theory and analysis, in: G.A. Whitmore and M.C. Findlay, eds., Stochastic dominance (DC. Heath, Lexington, NC).