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Optimal loss reduction and increases in risk aversion.

Optimal Loss Reduction and Increases in Risk Aversion

ABSTRACT

The impact of increases in risk aversion on the optimal level of loss reduction is analyzed in models in which the magnitude of the prospective loss is uncertain. The results are sensitive to the specification of the source of the uncertainty.

Introduction

Dionne and Eeckhoudt (1985) show that a more risk averse individual will always choose a higher level of loss reduction (activity which reduces the magnitude of a potential loss) but may not always choose a higher level of loss prevention (activity which reduces the probability of loss).(1) However, following Ehrlich and Becker (1972), they consider only a simple two state model in which the severity of the possible loss is fixed. A crucial property of a two state model is that the productivity of loss reduction is not subject to uncertainty. This article extends the previous analysis by examining the impact of increases in risk aversion on optimal loss reduction within a more general model which allows the productivity of loss reduction to be uncertain. It is shown that the results for the simple model used by Dionne and Eeckhoudt do not carry over to more general models.

It is assumed that uncertainly surrounding the loss reduction decision can arise form either of two distinct sources. In the first case, the severity of the potential loss is random, but there is no uncertainty about the effectiveness of loss reduction measures in reducing this loss. In the second case, the size of the potential loss is nonrandom, but the effectiveness of any loss reduction measure depends on conditions which happen to prevail at the time a loss actually occurs. In both cases the productivity of loss reduction is uncertain since the actual size of the loss avoided is unknown when the level of loss reduction is chosen. It is shown that an increase in risk aversion unambiguously increases the optimal level of loss reduction in the first case but not in the second case. Hence the results are critically dependent on the specification of the source of the uncertainty.(2)

This contrast in the results for the two alternative specifications has a simple explanation. If uncertainty arises because the potential loss is random, then an increase in loss reduction activity also reduces the variance of the (conditional) loss. Hence, a more risk averse individual chooses a less risky position by selecting a higher level of loss reduction.(3) On the other hand, if the effectiveness of loss reduction measures is uncertain, then an increase in loss reduction activity increases the variance of the (conditional) loss. Consequently, a more risk averse individual may actually prefer a lower level of loss reduction because the risk increasing character of this activity lowers its value to the individual.

Uncertainty about the Potential Loss

Consider an individual with an initial wealth, W, which is subject to the risk of partial loss. The individual can undertake loss reduction, activity which reduces the size of the actual loss but which leaves the probability of a loss unchanged. In this section it is assumed that the effectiveness of this loss reduction is nonrandom. The magnitude of the loss is written as L(1 - h(x)), where L is a positive random variable representing the size of the potential loss and h(x) represents the proportion of the loss avoided by the loss reducing activity, x. It is assumed that there are diminishing returns to loss reduction (h'(x) > 0, h"(x) < 0) and that h(0) = ) and h( ) = 1. At the time x is chosen, the marginal productivity of x in reducing the loss, Lh'(x), is random. The cost of loss reduction activities is represented by a convex cost function, C(x).

The objective is to maximize

(1 - p)U(W - C(x)) + pE[U(W - L(1 - h(x)) - C(x))] (1) where U is a strictly concave utility function and p is the probability that a loss occurs.(4) The first order condition for the optimal level of loss reduction is

(1 - p)U'(A) . - C'(x) + pE[U'(B)(Lh'(x) - C'(x))] = 0 (2) where A and B denote final wealth in the no loss and loss states, respectively.

Equation (2) can be written as(5)

(1)The terms "self insurance" and "self protection" are used in the economics literature to refer to "loss reduction" and "loss prevention," respectively.

(2)It is also possible to show that the impact on the optimal level of loss prevention of an increase in risk aversion is ambiguous in the more general model used in this article. Hence, the ambiguity obtained by Dionne and Eeckhoudt within the simple model carries over to the more general case.

(3)The use of variance as a measure of the riskiness of a random variable is quite restrictive. An increase in riskiness can be defined more generally in terms of a "mean preserving spread" of the probability distribution [see Rothschild and Stiglitz (1970)]. Nevertheless, for simplicity, variance is used to indicate riskiness in this article.

(4)If L is nonrandom, the model used by Dionne and Eeckhoudt is obtained.

(5)The covariance of any two random variables Z(1) and Z(2) can be written as Cov (Z(1),Z(2)) = E[Z(1)Z(2)] - E[Z(1)]E[Z(2)].

-((1 - p)U'(A) + pEU'(B)) . C' + pEU'(B)ELh' + pCov(U'(B),Lh') = 0. Dividing by EU'(B) and rearranging yields

pELh' + pCov(U'(B),Lh')/EU'(B) = ((1 - p)U'(A) + pEU'(B)) . C'/EU'(B). Since Cov(U'(B),Lh') is positive(6) and ((1 - p)U'(A) + pEU'(B))/EU'(B) is less than one,(7) it follows that pELh' < C'.

Thus, a risk averse individual pursues loss reducing activity to the point where the expected marginal product of loss reduction is less than its marginal cost. This reflects the fact that an increase in loss reduction not only reduces the expected loss but also reduces the variance of the loss. The risk averse individual values this reduction in variance in addition to the reduction in the expected loss.

Uncertainty about the Effectiveness of Loss Reduction

Rather than assuming that the effectiveness of loss reduction is completely known beforehand, it may be more realistic in some situations to assume that the effectiveness of loss reduction measures depends on conditions which happen to prevail at the time of loss. For example, the effectiveness of fire retardant material may depend on the temperature and wind conditions which prevail when a fire occurs. For simplicity, let L be nonrandom and let the damage reduction function be given by eh(x), where E is a positive random variable (0 <= e <= 1) representing the effect of conditions which prevail if a loss occurs.(8) The function h(x) represents the proportion of the loss avoided under the most favorable possible conditions, e = 1. The actual loss, should it occur, is L(1 - eh(x)). At the time x is chosen, the marginal productivity of loss reduction, Leh'(x), is uncertain. In this specification, an increase in loss reduction reduces the expected loss but increases the variance of the prospective loss.

The optimal level of loss reduction is determined by

(1 - p)U'(A) . - C'(x) + pE[U'(B)(Leh'(x) - C'(x))] = 0. (3)

(6)Cov (U'(B),Lh') is

E[(U'(B) - EU'(B))(Lh' - ELh')]

= E[U'(B)(Lh' - ELh')] + E[U'(B)]E[Lh' - ELh']. Since the second term on the right hand side is equal to zero, this can be written as

E[U'(B)(Lh' - ELh')] + U'(/B)E[Lh' - ELh']

= E[(U'(B) - U'(/B))(Lh' - ELh')] where /B = W - EL(1 - h) - C. Since larger values of L correspond to smaller values of wealth, when U is concave the terms in parentheses in the expectation on the right hand side always have the same signs which changes once at L = EL, making the expectation positive.

(7)((1 - p)U'(A) + pEU'(B))/EU'(B) is less than one since A > B implies U'(A) < U'(B) by the assumption that U is concave.

(8) If L is also assumed to be random, the impact of an increase in risk aversion is ambiguous in general. Since the result for the simple specification used in this section is similarly ambiguous, it is used to demonstrate that the effect of increased risk aversion depends critically on the source of the uncertainty. Rewriting condition (3), dividing by EU'(B), and rearranging gives

pLEeh' + pCov(U'(B),Leh')/EU'(B) =

((1 - p)U'(A) + pEU'(B)) . C'/EU'(B). Since Cov(U'(B),Leh') is negative and (1 - p)U'(A) + pEU'(B))/EU'(B) is less than one, it follows that pLEeh' <> C'.(9) This ambiguity reflects the fact that an increase in loss reduction not only decreases the expected loss but also increases the variance of the loss.

Increases in Risk Aversion

Consider a more risk averse individual whose utility function V can be represented by a strictly concave transformation of U [see Pratt (1964)]. That is, V = k(U), where k' > 0 and k" < 0.

When the potential loss is uncertain, the optimum for V is determined by

(1 - p)k'(U(A))U'(A). - C'(x) + pE[k'(U(B))U'(B)(Lh'(x) - C'(x))] = 0.

(4)

To compare the optimal choices for U and V the left hand side of (4) can be evaluated at the optimal level of x for U.

The left hand side of (4) can be written as

(1 - p)k'(U(A))U'(A). - C'(x*)

+ p caret L the integration of 0 k'(U(B))U'(B)(Lh'(x*) - C'(x*))f(L)dL

+ p infinity the integration of caret L k'(U(B))U'(B)(Lh'(x*) - C'(x*))f(L)dL (5) where caret L is defined by caret Lh'(x*) - C'(x*) = 0, x* is the optimal level of loss reduction for U, and f is the density function of L. Multiplying the first order condition for U, equation (2), by the constant k'(U(caret B)) yields

(1 - p)k'(U(caret B))U'(A) . - C'(x*)

+ pk'(U(caret B)) caret L the integration of 0 U'(B)(Lh'(x*) - C'(x*))f(L)dL

+ pk'(U(caret B)) infinity the integration of caret L U'(B)(Lh'(x*) - C'(x*))f(L)dL = 0 (6) where caret B is defined by W - caret L(1 - h(x*)) - C(x*). The value of expression (5) can be compared with the value of expression (6). Consider the first terms in both (5) and (6). Since U(A) > U(caret B), it follows that k'(U(A)) < k'(U(caret B)). Multiplying this latter inequality by the negative factor -(1 - p)C'(x*) implies that the first term in (5) is greater than the first term in (6). Consider the second terms in the two expressions. Note that larger values of L correspond to smaller values of wealth. For L < caret L, U(B) > U(caret B) and hence k'(U(B)) < k'(U(caret B)). Multiplying this last inequality by (Lh'(x*) - C'(x*)) < 0, one finds that the second term in (5) exceeds the second term in (6). Finally, consider the third terms in both expressions. For L > caret L, U(B) < U(caret B) and k'(U(B)) > k'(U(caret B)). Since (Lh'(x*) - C'(x*)) > 0, the third term in (5) also exceeds the corresponding term in (6). Hence, expression (5) is greater than (6) and consequently must be positive. It follows that the optimal value of loss reduction for V must exceed x* in order to satisfy condition (4). A more risk averse decision maker will always select a higher level of loss reduction.

(9) Cov (U'(B),Lepsilonh') can be written as

E[(U'(B) - U'(/B))(Lepsilonh' - LEepsilonh')] where /B = W - L(1 - Eepsilonh) - C. Since larger values of epsilon correspond to larger values of wealth, the terms in parentheses in this expectation always have opposite signg when U is concave, making the expectation negative.

However, when the effectiveness of loss reduction is uncertain, the impact of an increase in risk aversion is ambiguous.

The first order condition for an optimum for V requires

(1 - p)k'(U(A))U'(A) . - C'(x) + pE[k'(U(B))U'(B)(Lepsilonh'(x) - C'(x))] = 0. (7) Define caret epsilon by Lcaret epsilonh'(x**) - C'(x**) = 0, where x** is the optimum value of self insurance for individual with utility U. The left hand side of (7) can be written as

(1 - p)k'(U(A))U'(A) . - C'(x**) + p the integration of k'(U(B))U'(B)(L epsilon h'(x**) - C'(x**))g(epsilon) d epsilon as upper limit approaches caret epsilon and lower limit approaches 0 + p the integration of k'(U(B))U'(B)(L epsilon h'(x**) - C'(x**))g(epsilon) where g is the density function of epsilon. This expression can be compared with

(1 - p)k'(U(bar B))U'(A). -C'(x**) + pk'(U(bar B)) the integration of U'(B)(L epsilon h'(x**) - C'(x**))g (epsilon)d epsilon as upper limit approaches caret E and lower limit approaches 0
+ pk'(U(bar B)) the integration of U'(B)(L epsilon h'(x**) - C'(x**))g
(epsilon)d epsilon = 0 as upper limit approaches 1 and lower limit approaches ca
ret E (9)


where bar B denotes W - L(1 - caret epsilon(x**)) - C(x**). As in the previous case, the first term in (8) is greater than the first term in (9). Next, consider the second terms, noting that larger values of epsilon correspond to larger values of wealth. For epsilon < caret epsilon, U(B) < U(bar B) and hence k'(U(B)) > k'(bar B)). Multiplying by (L epsilon h'(x**) - C'(x**)) < 0 implies that the second term in (8) is less than the second term in on the other hand, if epsilon > caret epsilon, then U(B) > U(bar B) and k'(U(B)) < k'(U(bar B)). Multiplying by (L epsilon h'(x**) - C(x**)) > 0, one finds that the third term in (8) is greater than that in (9). Consequently, the expression in (8) does not necessarily exceed the expression in (9).

Therefore, when the productivity of self insurance is uncertain, the more risk averse individual does not necessarily choose a higher level of self insurance. The reason for this result is the variance increasing character of loss reduction activity.

Conclusion

This article considers the effect of increased risk aversion on the decision to undertake loss reduction activities when the magnitude of the prospective loss, and hence the productivity of loss reduction, is uncertain. Uncertainty is assumed to arise from two contrasting sources. In the first model, uncertainty is present because the potential loss is random. In the second model, uncertainty arises because the effectiveness of loss reduction measures varies in a random manner. It is shown that increased risk aversion always leads to increased loss prevention in the first model but may not lead to greater loss prevention in the second model. Hence, the result of Dionne and Eeckhoudt (1985) for the two state model does not, in general, carry over to more complex models.
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Author:Hiebert, L. Dean
Publication:Journal of Risk and Insurance
Date:Jun 1, 1989
Words:2567
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