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Imperfect job risk information and optimal workmen's compensation benefits
Journal
of Public
Economics
14 (1980) 319-337.
Q North-Holland
Publishing
IMPERFECT JOB RISK INFORMATION OPTIMAL WORKMEN’S COMPENSATION
Company
AND BENEFITS
W. Kip VISCUSl* Northwestern
University,
Ecunston,IL 60201,U.S.4
Received June 1978, revised version
received November
1979
If workers have imprecise, but unbiased risk perceptions, they will accept a lower wage and less of a marginal wage reduction in response to an increase in injury compensation or a reduction in risk. Market outcomes will provide too little compensation and too great a job risk, as compared with the perfect information case. The optimal workmen’s compensation policy will provide a supra-optimal benefit level in an effort to create incentives for safety. The signaling function of market-provided injury compensation will eliminate the bias in worker priors and increase their precision, eliminating the market failure in some instances.
1, Introduction Compensation for job risks may be received in the form of ex urrte wage compensation, ex post compensation after an injury has occurred, or manipulation of some other job attribute valued by the worker. If one considers the partial equilibrium context in which there is a competitive wage schedule in which a market wage rate is associated with each possible level of risk and mix of job rewards, the decisions by workers and firms in an otherwise competitive market will lead to an efficient level of risk, wages. and ex post injury compensation.’ This result and the analysis in this paper abstract from more fundamental, general equilibrium concerns such as the determination of the equilibrium wage schedule and the decisions by firms to introduce jobs with a different level of risk into the set of possible job options available to workers. The implications of research in this vein do not necessarily bolster the findings of more traditional compensating wage differential analyses since decisions by firms that influence the available job quality mix often lead to inefficient outcomes.* *Helpful comments were provided by two referees, one which I now address in section 4. ‘See, for example, the discussion in Oi (1973a, 1974) and analysis for product markets appears in Oi (1973b). 2The analvses of Lancaster 11975) and Dr6ze and Ha.een for product quality, although many of the results generalize firm.
of whom Thaler
raised
the signaling
issue,
and Rosen (1976). A parallel
(1978) consider these issues primarily to the choice of work qualiiy by thk
320
WK.
Viscusi, Imperfect job risk
infirmutiun
Other deviations from the competitive norm, which pertain more specifically to the particular employment relationship, include (a) actuarially unfair market insurance, (b) moral hazard problems, (c) imperfect worker information, and (d) externalities to society at large.3 The primary focus of this paper is on the impact of imperfect worker information on the welfare implications of decisions by workers and firms and on the optimal policy choice regarding the level of workmen’s compensation benefits. Two kinds of imperfections can be distinguished. Workers may have biased mean values of their prior assessments of the chance of an accident. This is the familiar instance in which firms will provide too little ex posr compensation and too high a level of risk if workers systematically underassess the risks they face.” The second informational shortcoming is that even if workers’ priors are unbiased, they rarely possess full information regarding the actual level of risk faced. As additional information is acquired, worker priors will be updated. My principal focus will not be on the possibility of initial bias in workers’ judgements but on the implications of imprecise prior beliefs and the influence of worker learning. The subsequent findings regardin g the implications of adaptive worker behavior are based on the uncontroversial assumption that workers often do not have perfect information regarding the level of risks they face. Indeed, there is considerable evidence that workers learn about job risks through their on-the-job experiences and quit if these experiences are sufficiently unfavorable.’ The absence of perfect prior information and the potential for learning by experience are the two essential ingredients of the subsequent analysis. In contrast, analyses of biased worker perceptions hinge on assumptions regarding both the direction of the bias and the responsiveness of the there is some experimental perceptions to changes in the risk. 6 Although evidence indicating that individuals tend to underestimate very small probabilities, workers have a greater interest in learning about the level of the risk than do individuals facing hypothetical choices. Moreover, the average injury risk facing workers is fairly sizeable since it exceeds lo-’ annually. Even worker death risks, which average about 10e4 annually for the average American worker, command compensating wage differentials that correspond
‘For a survey of these imperfections, see Zeckhauser and Nichols (1978). 4Thls result was derived by Diamond (1977) for iob risks in an analvsis that closelv uarallels Spence’s (1977) treatment of product quality. Other-discussions of biased perceptions i%lude Oi (1973) and Viscusi (1979a). 5Fuller description of the supporting empirical results is provided in Viscusi (1979a). ‘ln particular, primary attention has focused on the situation in which workers underestimate the level of risk and revise their prior beliefs less than proportionately with respect to an increase in safety. See, especially, Spence (1977) and Diamond (1977).
WK. Viscusi, Imperfect job risk infbrmation
321
to an implicit value of life on the order of $2-3 million.7 These results are bolstered by evidence that there is widespread awareness of many job hazards, as individuals’ job risk perceptions are strongly correlated with objective measures of the industry injury rate. While none of these findings imply that workers’ probabilistic beliefs are unbiased, these substantial market impacts provide evidence that risks are not completely ignored by workers. The economic foundations for workmen’s compensation and other regulations could clearly be strengthened if they did not hinge on an assumption that probabilistic beliefs are slanted in a particular direction but instead derived from the absence of complete information prior to incurring the risk. The impact of imperfections in worker information on individual responses to the level of safety at the workplace and the amount of ex post injury compensation provided will be assessed in section 2. These responses will influence the incentives for the firm to provide an optimal level of risk and compensation. The results in section 3 imply that even if workers’ prior assessments are accurate, the imprecision of workers’ judgements will result in too high a level of risk and a suboptimal level of injury compensation. These shortcomings can be partially alleviated by a workmen’s compensation system, such as that presented in section 3.3. Government imposition of compensation levels will sacrifice the informational content that discretionary choices of compensation would have had. The potentially important signaling function of injury compensation is addressed in section 4.
2. Individual job choice under uncertainty 2.1. Notation
and analytic framework
A pivotal determinant of enterprise decisions is the responsiveness of workers to different actions taken by the firm. Here we will investigate the behavior of a representative worker and the effect of imperfect information on the attractiveness of different job characteristics. The analysis throughout assumes that individuals assess subjective probabilities, update these probabilities in conventional Bayesian fashion, and undertake actions to maximize their expected utility. The implications of this analysis will be used subsequently in assessing decisions made by the firm, given this pattern of individual action.
‘These estimates are in 1979 prices based on the representative samples of workers in Viscusi (1979a) and in current unpublished work by the author. Estimates by Smith (1976) are similar. The value of life for workers in high risk jobs is, as one would expect, considerably lower. Thaler and Rosen’s estimates for workers facing annual death risks of 10-j imply a value of life of roughly S400.000 in current prices.
WK. Viscusi, Imperj>ct job risk
322
i:Jorrnation
Consider a two-period situation in which a worker must choose between a potentially hazardous job with an assessed prior probability p of remaining uninjured and a safe job alternative offering a wage w~.~ If the worker is not injured on the job, he receives a wage rate w, and if he is injured he receives ex post compensation s. For simplicity, I will restrict myself to the income maintenance function of workmen’s compensation rather than get involved in medical insurance issues. Being injured also may lower the marginal utility of consumption x.~ To allow for this possibility and to reflect the assumption that the worker would rather not be injured for any given level of consumption, I will adopt a state-dependent utility function format in which Ur is the utility function when the worker is healthy and U” is the utility function when ill, so that
U’(x)>
P(x)
u: zu;>o, and
where subscripts indicate partial derivatives.” The formation and revision of workers’ prior beliefs will be specified in fairly unrestrictive terms. The assessed prior probability p of remaining uninjured is assumed to be an increasing function of the actual probability q, where c'p/Zq>O and (:2p/2q2~0. The actual injury probability may not be observable, but q will affect the firm’s accident record and other information that can be used by the worker in forming his beliefs prior to work on the job. The enterprise characteristics that form the intervening link between p and the unobservable q will not be detailed explicitly. Section 4 addresses this issue more formally, analyzing the role of s as a signal for q. The precision of the worker’s prior will be denoted by 7. where higher values of 3’ correspond to greater information. The extreme case of perfect information can be viewed as the limiting case of tight priors as Y-+X. The parameter y will be given a more explicit interpretation in the discussion of beta distributions below. Although individuals start the risky job with imperfect information, they
‘Alternatively the alternative job could offer a lottery with known probabilities. The probability of an accident is assumed to be independent of worker actions. This case is considered in Viscusi (1979b). ‘Medical care expenditures may bc quite effective in reducing pain and in increasing the probability that the worker returns to the healthy state. These issues are considered within a medical insurance context by Zeckhauser (1970). “State-dependent utility functions have been used in a variety of studies, includmg Zeckhauser (1970),Spence (1977), Diamond (1977), and Vlscusi (1979a).
WK.
Viscusi,
Imperfect
job
risk informution
323
can acquire information on the job by direct experience of adverse outcomes, and observation of others’ injury exobservation of job characteristics, periences. For simplicity I will restrict the analysis to the effect of injuries on Let p(m,n) represent the assessed probability of the worker’s perceptions.” no accident after m successful job risk lottery outcomes at the firm and n unsuccessful outcomes (accidents). The subjective probability of not being injured in period 2 after not being injured in the initial period is assumed to be updated in standard Bayesian fashion where
The focus here will be on the lowest initial wage-injury compensation package that will attract the worker to the job in the initial period. In such situations the worker will always quit after an unfavorable outcome so unfavorable updating of the worker’s prior need not be considered.i3 The worker’s objective is to maximize the discounted expected utility from the income he receives in each period. where h is the discount factor equal to the reciprocal of 1 plus the interest rate.” For the worker to be indifferent between the two jobs in the initial period, the value of the indirect utility function Vfor the two jobs must be equal, or
V(w, P, s,Y I= I/b,, 40, % 1,
(1)
where the index ;j of the sharpness of the worker’s prior is infinite if the worker is perfectly informed. The discounted expected value of the uncertain job is given by I/(w,p,s,;‘)=pc”(~)+(l
-p)v2(s)+bp[p(l,O)U’(\1!)
+ (1 -p(ltO))UZ(s)]
f h(1 -p)U’(Wo).
(2)
The first two terms on the right-hand side of eq. (2) represent the first period reward. All subsequent terms are weighted by the discount factor since they pertain to period 2. The first period 2 term consists of the probability of a successful first period outcome multiplied by the expected reward in period 2 after a successful job experience in period 1, while the
“An analysis of experiences of other workers and other work characteristics that may influence one’s probabilistic perceptions is presented in Viscusi (1979a). The results are quite similar to those generated here. r3A proof that the reservation wage rate will always lead to worker quitting is provided in Viscusi (1979a). I41 assume that workers do not transfer resources across periods. Relaxation of this assumption would make consumption rn period 1 a choice variable, further complicating the model.
324
WK. Viscusi, Imperject job risk information
final term consists of the probability of an accident in period 1 followed by the value for work on the alternative job after the individual has switched positions. The value of the safe job alternative is defined analogously and is given by V(y), l,O, Co)=U’(w,)(l
+b).
Since the properties of the utility function are unaffected by adding a constant to their value. I will define my utility metric by setting U’(W,) equal to zero. The condition that the worker be indifferent between the two jobs, which is given by eq. (1) simplifies to the requirement that 0=pcr’+(1-p)u~+~l)~p(1,0)u~+(1-p(1,0))u2~, where the arguments before.
2.2. Determinants
of the utility
ofthe
reservation
functions
wage
(3)
are M’for U’ and s for U2, as
level
The analysis in the following section will indicate that a fundamental determinant of enterprises’ choice of the level of safety and injury compensation is how the reservation wage required by workers is affected by the level of s, the looseness of their prior assessments, and the level of the true job risk. The true probability that the worker will not suffer a job injury will be indicated by q. By implicit differentiation of the indirect utility function for the risky job, one can readily determine the effect of marginal changes in s and q. In particular, one obtains the result that
(‘M:
-= 2s
V -2= VW
v:C(l-P)+hP(l-P(l,o))l vxPU
+
b(LO))l
’
(4)
and L;w
-= r?q
V __A= VW
_ i
Gil.‘--D.‘)+b
fP-
%(I, 0) (U’-U2) ip
where both ?w/c?s and c?w/8q are negative.
~cp(l,o)uL+(lp(l,o))c’~, [
31
:U:[Ip(l+bP(l,o)]}-‘,
(5)
325
WK. Viscusi, Imperjhct job risk informtrtiorl
In order to analyze the nature of the reservation responses in greater detail, I will use a specific class of prior probability distributions. Beta distributions will be used since this distribution is quite flexible and can assume a variety of skewed and symmetric shapes. Moreover, as Raiffa and this distribution is ideally suited to Schlaifer (1961) have emphasized, to normal Bernoulli-type processes such as this and is far superior approximations. Letting the distribution fl(p,l;) be reparameterized in terms of the mean p of the prior, where 0~ p< 1, and its precision y, where 0 < y < a, one has the posterior probability p(m, n) of a successful job outcome after m successes and II failures (accidents) given by
v+m
p(m,n)= y+m+n' Individuals with prior probabilistic perceptions p(p, y) in effect act as if their prior experiences consisted of 7 trials of which a fraction p were successful outcomes. The development below will also utilize the cross partials and second derivatives of these wage response terms. Letting p’ denote dp/dq and using the beta distribution values for p(l,O), one has
~=~~[p’(~+1-yb+2ybp)l>0 ??qC% ~~Cpb+1+byp+b)l
’
a%
-p)l >.
s=
-~$b+ 1-w-P+$p(l
and, after a considerably
’
U:[p(v+lfbyp+b)] more complicated
calculation,
a value of
a2W b-0 a4
2
The wage reduction from a marginal increase in safety q or compensation s diminishes as the other variable is increased. Moreover, reservation wages decline with the levels of 4 and s, but they do so at a decreasing rate. 2.3. The role of the precision
of one’s initial judgements
The primary matter of concern is how provision of safety and compensation. The the worker’s prior assessed probability of the learning process and, more specifically,
worker learning affects the firm’s focus consequently will not be on an accident but on the nature of on the precision y of the worker’s
initial beliefs. As one would values are updated less, as c7p(l,O)
P-l
2Y-~--(:,+
1)’
expect,
tighter
priors
associated
with larger
3
Inspection of the expression for V(w,p,s, 7) in eq. (2) indicates that for the marginal worker, only the greater upward revision of loose priors is of consequence. The downward revision after an unfavorable outcome does not enter since the worker leaves the hazardous job after an unsuccessful period I outcome. Consequently, loose priors (i.e. low 7 values ) serve to boost the value of p(l,O), raising the attractiveness of the uncertain job provided that U’ exceeds U2.t” One might expect that workers who viewed the job on more favorable terms in period 2 would require a lower wage to accept the position and would place a lower value on marginal increases in safety or additional ex post compensation. wage. Upon implicit Consider first the impact of :’ on the reservation differentiation of V(w, p, s, y), one obtains
as expected. The lower posterior probabilities of a successful outcome for workers with high values of y raise the wage they require. Loose priors enhance the attractiveness of the job. Upon differentiation of eq. (6) with respect to s and q, we get
(p-lW,Z
d2W
i?$s
(y+ i)‘U:[l+
and, after considerable (35
+/+q
-
bp(l,O)]
’
simplification,
(U’ - U2)bp’(l+ ___.II_
b)
L$~~+l+ybp+b]*
The value of ?2ti~j3yds is clearly
’
negative
since pc 1. Since it will be shown
15For the optimal compensation levels derived in sections 3 and 4, this relationship will always hold. A notable exception is the case in which accidents involve only monetary losses that are fully compensated, so that U’ = U* and ?MJ/+~ is zero.
WK. Viscusi, Imperfect job risk information
327
that the uninjured state is preferred, U’ > CT’, d’~1~/8ydq likewise is negative. Increased precision of one’s intial judgements makes one willing to incur a greater marginal wage reduction in response to an increase in y or s, while looser priors diminish the attractiveness of such changes. An important issue for the subsequent analysis is how worker responses are affected by the absence of perfect information. With perfect information, the value of the employment path would be
where U’(w,) is set at zero as before. The effect of s and q on the reservation wage will be denoted by ?w*lZs and i;w*/Zq, where
i?w* --Yz
-V”
s
_-V,z(l-q)(l+hq)_-u2(1-q)
2s
v;
2w*
-T/y*_
Qqu + bq)
uh
(7)
and ---=-----34 v;,
-[(U’-U2)(1+2bq)+bUZ] uh(l+bq)
-~
(8)
The analysis below will focus primarily on unbiased priors in which p=q. If this condition holds, the value of &*/irs with perfect prior information is simply the limiting value of i3w,i?s as y-‘nc! and when p is replaced by q. Consequently, any decrease i; ; represents a departure from the perfect information situation. The negative value for ?2~~l/?$s implies that ?N/?s will be more negative for perfectly informed workers than for individuals with imprecise initial information. This relation also holds if p>q.16 To assess the impact of imperfect information on i?wiiq, one also must ascertain the magnitude of p’. With perfect information, one’s accurate prior beliefs increase proportionately with the probability of no accident, or p’ equals 1. If the worker’s prior judgements can be characterized similarly for the imperfect information situation (i.e. p =q and p’= l), the value of &:*,i$ with perfect information will equal the limiting value of h/2q as y-)x. Since higher values of 1 make &,/?q more negative. the value of i+~/icl with imperfect information will be larger (or less negative). Upon comparison of the magnitudes of c’w/c?q and c!w*/iq, it can be shown that so long as
16The larger response for &v*/i% can be verified by comparing eqs. (4) and (8). The result does not require that p(l,O) be updated using the beta distribution. Any updated prior with p(l,O)>p will suflice.
328
l4X. Viscusi, Impwfect
joh risk inforfnntiofl
0 < p’ 5 1 and p 2 4, ?w/C:q will be larger with imperfect information.” If p’ > 1, the imperfectly informed worker overestimates the marginal increases in safety so that Zw/&l may become more negative with imperfect information, though this need not be the case.18 The analysis below will assume that workers do not overestimate the value of p’ sufficiently to lead to a reversal in the otherwise positive effect of imperfect information on Sw/aq.
3. Imperfect information, 3.1. Worker
uncertainty
enterprise decisions, and workmen’s compensation and enterprise
behaoior
Consider a firm that selects the level of q and s to minimize its costs for a given level of output. The costs C per worker consist of the wage costs W, P.Y post compensation costs at a level [( 1 -q)s]/q if insurance is purchased on an actuarially fair basis, and the cost h(q) of providing a safe environment, where h’ > 0 and h” > 0. For equally costly technologies associated with the same risk and p values for workers, the firm will always prefer the uncertain technology with the lower 9 since ?w/Sy is negative. There will consequently be a bias toward new technologies that, are not fully understood. In the discussion below, we will assume that the firm is selecting from a class of technologies with the same 7 but differing q. After analyzing the optimal decision. the effect of 7 on s and 4 will be assessed. Alternatively, the model could be framed in terms of the firm picking from some frontier of best available technologies where for each level of 4 there is an associated value of y(q). The conclusions below are reinforced if ~‘20 and are altered substantially if 1~‘<0. In effect, we will be making the neutral assumption that 7’ =O. The enterprise’s objective is to
subject to the constraint that it attracts conditions for an optimum are that
workers
to the firm. The first-order
17This result also does not require that priors belong to the beta family, but rather all that is needed is that p(l,O)>O. “Whether this reversal takes place depends on the relative magnitudes of U’ and Zl’ as well as the various risk parameters.
WK. Viscusi, Imperfect job risk information
329
and
where dw/ds and &/8q are given by eqs. (4) and (5) for the representative worker.” The second-order conditions that C,,>O and C,, >O are both satisfied since i’w/?? and F2w/?q2 are positive, as is h”. To ensure that the solution is a relative optimum, the value of C,,C,,(C,,)‘, which will be denoted by H, is assumed to be positive. To assess the impact of y, totally differentiate the first order conditions, yielding
r
Solving
11
for the effect of 7 on s and q,
and
&l -=al,
1 - U’,,Wqy+ W,? Wqs-7 1 4 i [
H
I
.
Using the wage results from section 2 and the positive signs assumed for H and h”, it is clear that both Zs/C:y and iq/Gy will be positive if the positive wsq term exceeds l/q2. This relationship can be verified using eq. (6) above.” Greater precision of workers’ priors will raise the optimal no-accident probability q and raise the level of ex post compensation. In contrast, dimly understood technologies associated with loose priors create a bias toward inadequate compensation and too great a level of risk. To assess the overall effect of imperfect information, one would like to ascertain the effect of biased perceptions coupled with imprecise priors. First, consider the outcomes with perfect information. The first-order conditions (9) “In effect, the firm is a price-taker in the product market and a wage-taker in the labor market. The risk choice by the firm does not alter the available work quality options for workers. This assumption is stronger than that in D&e and Hagen (1978). Z°Clearly if workers overestimate the risk by too great an amount so that q is close to zero and p is near 1, this inequality may not hold. For the inequality to hold, let psq, p’( 1, and u; 2 u:.
330
WK.
Viscusi. Impeyfert
job risk injbrmation
and (10) are unchanged except that dw*jZs and ?w*/c?q now replace C:w/(?.s and c7w!:r?q.The optimal s, for given q, has the most convenient interpretation since, after substituting from eq. (7) one obtains the familiar result that iJi = li,!.” If injuries are equivalent to monetary losses, t’x post compensation will equalize the absolute utility levels in each state. If an injury diminishes the marginal utility of consumption, equalization of the marginal utilities of consumption will require a lower level of consumption after an accident. The values of C?M./?Sand ?w/dq will each be less negative than in the perfect information case if p 2 q and p’ 5 1. Assuming these conditions hold, consider the choices implied by eqs. (9) and (10). For fixed values of q, and consequently fixed values of (1 -q)/q, a larger (less negative) value of the W, function will lead to a lower optimal level of s. Similarly, for fixed values of s, it can be shown that a less responsive u’~ term will lead to a lower optimal q level. Coupling of the standard assumptions concerning misperceptions in static models with the adaptive behavior when learning takes place simply reinforces the biases in the risk and ex post compensation levels. The more novel portion of this result is that even with unbiased and perfectly responsive prior probabilistic beliefs (i.e. p=q and p’= 1 ), the subsequent updating of these perceptions will lead to types of market shortcomings generally associated with misperceptions in static contexts.
3.2. Elf&t of’exogenous
compensution
lerels
Suppose that the level of s is not a choice variable for the enterprise but instead is imposed on a nondiscretionary basis through a workmen’s compensation program. Let a fraction f of benefits be merit-rated, where Ozf 5 1, and the remainder of the insurance program costs be financed by a lump-sum tax on enterprises. Large firms often self-insure or are experiencerated and have values of ,f close to 1, while smaller enterprises have f values near 0 under the U.S. workmen’s compensation system.*’ The first-order condition for an optimizing firm selecting safety level q will be
“This is a standard result when actuarially fair insurance IS available. See, for example, Arrow (1971). I2 Ratings for smaller enterprises are based largely on the average industry risk, which is affected bery little by the accidents at a particular firm. Only IS percent of all enterprises are experience-rated to -any degree, but smce these are the larger firms. over 85 percent of all workmen’s compensation premiums are paid by such firms. See pp. 3637 of the National Commission on State Workmen’s Compensation Laws (1973).
WK. Viscusi, Impetfect
job risk irfornmtion
331
The primary matter of interest is the effect of s on the level of q selected by the firm, since this relationship defines the firm’s reaction to the policy choice variable. Totally differentiating eq. (1 I), one obtains
dq-
- wys+.i?q* ---
(12)
ds - wyq+ h” + 2s/y3
The denominator is simply C,,, which is positive. From section 2 we know that wqSis also positive. If there is no merit rating so that f equals zero, dqlds will be negative. Higher workmen’s compensation levels reduce safety in this instance by enhancing the attractiveness of hazardous jobs to workers, while providing no marginal safety incentives for the firm. In the situation of complete merit rating, one obtains the opposite result if qs is U’ is smaller than l/q*. As was indicated in section 3.1, this inequality always satisfied for the types of prior beliefs being considered. For some critical merit-rating fraction ,f ** there will be no marginal effect of s on q. Abovef**, dq/ds will be positive. and below that critical value, dq/ds will be negative.
3.3. Optimal
workmen’s
compensation
hizne$ts
If enterprise incentives were not affected by the workmen’s compensation policy, the optimal compensation level would be determined by ascertaining the value of s that would maximize the worker’s expected utility if he were perfectly informed, where the insurance plan must break even on an actuarial basis. In such instances the level of s would be set to equate the marginal utility in each health state. The absolute level of welfare will be less when the worker is injured if, as was assumed here, an injury reduces the marginal utility derived from any given level of income. If an injury results only in a loss of worker income, the optimal s will equalize both the absolute and marginal utility levels in each state. Since the full information competitive outcome also satisfies this optimal insurance requirement, ideally one would like to set the level of s and q in a manner that replicated this outcome. While the government can manipulate the level of s, it influences q only indirectly through the incentives for safety created by a higher S. More specifically. the shape of the enterprise response function q(s) will be dictated by eq. (12) above. The policy objective is to minimize the costs per unit of output subject to the constraint that worker utility I’* based on the actual job risk probabilities is at some constant level. As before, the wage reductions accruing to the firm from higher levels of s will be dependent upon worker perceptions rather than the actual risks involved.
332
WK. Viscusi, Imperfrct
More specifically,
the policy objective
minL=w+h(q)+----~
41-d 4
leading
to the first-order
conditions
job risk injbrmution
is to
_3v*
I’ for optimal
insurance,23
An optimizing lirm will set q so that eq. (11) is satisfied, resulting in a value of zero for the bracketed coefficient of dq/ds. The optimal workmen’s compensation condition can be rewritten as
(13) This condition differs from eq. (9) for an unregulated market by the addition of the final term. If the bracketed expression is positive, the marginal productivity of compensation s will be above that in a market allocation so that it will be optimal for the policy to increase s above market levels. The first component term is given by
av*
aw
-=~qu:(l+bq)+(l-q)C’,Z(l+bq).
as
If workers were perfectly informed initially, dw/& would assume the value for ~w*/c% in eq. (7), implying that dV*/as would be zero. There would be no rationale in terms of the direct effect of s on utility to warrant additional compensation of fully informed workers. However, the imperfect information case is associated with a suboptimal s level so that aV*/i% will be positive. The second component of the bracketed ;1 coefficient is dqaV* ---=z ds dq
dq
(U’-V)(l
.
+2bq)+hU’+$J:(q+hq2)
Since the level of safety q is too low in the unregulated dV*/aq will be positive when assessed at the market
I market, values
the value of of q and s.
2’For simplicity, the total enterprise costs omit the merit rating fraction f and any lump sum component involved. It is assumed that the workmen’s compensation plan must break even so that regardless of the merit-rated and lump-sum tax share, total costs are unchanged. The value off is reflected in the enterpripe’s reaction to s, through dq/ds.
WK. Viscusi, Imperfect job
risk
in$?rmution
333
Consequently, the sign of this expression hinges on that of dq/ds, which can be positive, negative, or zero. In the case of relatively strong merit rating (i.e. f>f**), higher values of s will not exert any negative effect on worker welfare through its effect on q. Since ?V*/?s is positive, the beneficial effect of additional compensation on worker well-being will always lead to a positive value of the final term in eq. (13), so that the optimal level for s will exceed that in the market. If merit rating is relatively weak (i.e. S
4. Ex post compensation
as a signal for risk
Since the level of q is not observable to workers, other forms of information must be used in making inferences about the level of safety. Here I will focus on the level of ex post compensation as a possible signal for the underlying q characterizing the firm’s decisions.25 More specifically, let the worker’s prior assessed probability of no accident be r(s) so that the expected utility B of the employment path on the uncertain job is P(w, Y(S), f)= r(s)U’ + (I+ br(s)
rb)W’
Yr(s)+
1
-q-(1/‘-u*)tu*
24Given the utility-taking hypothesis adopted here, an additional fine on accidents that is reimbursed to workers will have no effect. Other forms of penalties and safety standards may be useful, but these are beyond the scope of the analysis here. “The analysis here parallels that of Spence (1977), with the principal modification being the introduction of learning into the model.
334
Since v is independent of y. the value of ?W/iq for the signaling case is zero.26 Letting 7 denote the precision of worker priors with signaling, the effect of s is now enhanced by its influence on worker perceptions so that
U;r[r+ The optimizing
ybr + 1 + h]
firm will select s and q so that (14)
and
Since C,, is positive, s will serve as a signal for q in equilibrium, implying that r(s)=q in the expression for C~$/SS.~~ The utilization of s as a signal for y will affect the precision of the worker’s initial judgements as well. I will view the presence of the signal as equivalent to additional observations regarding outcomes of the stochastic job risk process so that, in effect ^J is increased to a value r>r. I will assume that the value of y is independent of the particular s selected so that the mean of the prior and not its informational content is influenced by s. Compensation for injuries is generally provided at a fixed level across broad classes of workers at the firm or for the enterprise as a whole. If all jobs at the firm pose the same risk and workers are aware of the signaling relationship, s will provide perfect information. This important limiting case in which ~~-+x. will be discussed further below. If these conditions do not hold, s will result in unbiased priors for the firm-specific risk but would not resolve all uncertainties. Even workers on representative jobs with safety probability q will update their priors because they are not aware that their initial judgements were correct. “1 will continue to focus on a hypothetical worker. Variegated preferences are an essenttal as noted in Spence (1977). Such diversity can be prerequisite for a signaling equihbrtum, achieved by letting the individual be of class i. where there are different utility functions for different classes. 27The requirements for an effective signal are dtscussed in greater detail by Spence (1974. 1977). In a signaling equilibrtum. he observes that prices cannot be a stgnal for product quahty since there is no cost to raising prtces. The same result pertains to wages for the model here.
WK.
Viscusi,
Imperfect
job
335
risk informrrtion
Although the signaling equilibrium will not generally be efficient, in the case in which the only losses involved are monetary equivalents, Spence (1977) has shown that guarantees produce efficient outcomes when consumer products are subject to failure. To assess whether this result generalizes to the job safety case with adaptive behavior, let the monetary loss be 0 so that the utility function takes the form U2(x)= LJ’(x -fl). In the perfect information situation, s is selected so that Vi = U$ or s=8. For these utility functions, this result in turn implies that U’ =U2, which both equal zero since the value of the job alternative was arbitrarily set at this value. Substitution of these utility values into the expression for (7w*/(!q in eq. (8) implies that aw*/dq equals zero. Consequently, with perfect information, the two first-order conditions will be
and h'=
s/q’.
(15)
Equation (15) is identical to that in the signaling case described above. For fixed levels of s, q will be set as in the perfect information situation. The signaling equilibrium will lead to efficient outcomes for both q and s if setting 8 equal to s would satisfy eq. (14). Noting that this level of compensation would have the aforementioned effects on utilities and marginal utilities, for the signaling equilibrium in which r(s)=q, eq. (14) reduces to y+ 1 +
$q
(16)
y+1+“g7q+b=1.
This equality will not be satisfied since b>O with finite interest rates. The difficulty derives from the updating of initially unbiased prior assessments. However, as was indicated above, the utilization of s as a signal for q not only affects the mean of workers’ priors, but it also provides additional information tantamount to observing additional trials on the uncertain job. In the limiting case in which the signals provide completely precise information (i.e. as *;+a), eq. (16) will be satisfied. The signaling equilibrium will be efficient if all losses are financial and if signals provide complete information to the worker.28 Market allocations for ex post compensation consequently serve a signaling function that eliminates the bias in prior perceptions and increases the sharpness of the prior, thus diminishing the aforementioned difficulties 28This limiting analysis.
result
in effect reduces
the model
to a job market
analog
of Spence’s
(1977)
336
WK.
Viscusi,
Imperfrct
job
risk iTformation
deriving from adaptive worker behavior. A workmen’s compensation program that imposes mandatory levels of s eliminates both forms of information content of discretionary s values. The suppression of this information transmitted through a market signaling process can potentially undermine the efficiency of market allocations so that even an optimally designed workmen’s compensation policy may lower worker welfare.29 In short, the optimal workmen’s compensation system may require no interference with outcomes in an unregulated market.
5. Conclusion The presence of learning and adaptive worker behavior will lead market outcomes to provide too little er post compensation and too great a level of risk even if workers’ priors are unbiased. If a workmen’s compensation program is merit-rated to a sufficient degree, higher mandated levels of compensation for injuries will provide additional incentives for safety as well as greater compensation for injured workers. The imposition of uniform injury compensation levels may, however, have an unintended impact. Prior to government intervention, injury compensation levels would have served as a signal for the underlying risk level, thus enhancing the degree of prior information and leading to unbiased priors in a signaling equilibrium. Levels of compensation specified by the government will eliminate this information transfer, making the desirability of intervention unclear. ‘“Whether welfare is actually lowered depends on a variety of influences about which very little is known, such as the availabihty of other job risk signals. the precision of worker prrors with and wrthout the e.x post compensation signal, and the bras in priors in the absence of signaling.
References Arrow. Kenneth, 1971, Essays in the theory of rusk bearing (Markham, Chicago). Diamond. Peter. 1977. insurance theoretic aspects of workers’ compensation, in: Natural resources. uncertainty. and general equilibrium systems (Academic Press. New York) 67-89. Drere. Jacques H. and Kare P. Hagen, 1978, Choice of product quahty: Equihbrrum and efficiency, Econometrica 46, 493-514. Lancaster, Kelvin, 1975, Socially optimal product differentiation, American Economic Review 65, 567- 585. National Commission on State Workmen’s Compensation Laws, 1973, Compendium on Workmen’s Compensation (U.S. Government Printing Office, Washington). Oi, Walter, 1973a, An essay on workmen’s compensation and industrial safety. in: Supplemental studies for the National Commission on State Workmen’s Compensation Laws (U.S. Government Printing Office. Washington) 41-106. of Economics and Oi. Walter, 1973b, The economtcs of product safety, Bell Journal Management Science, 3-28. Oi, Walter, 1974, On the economics of industrial safety, Law and Contemporary Problems 38, 669 699.
WK.
Viscusi,
Imperject
job
risk information
337
Raiffa, Howard and Robert Schlaifer, 1961, Applied statistical deciston theory (M.I.T. Press, Cambridge). Smith, Robert, 1976, The Occupattonal Safety and Health Act: Its goals and achievements (American Enterprise Institute, Washington). Spence, Michael, 1974, Market Signaling: Informational Transfer in Hiring and Related Screening Processes (Harvard University Press, Cambridge). Spence, Michael, 1977, Consumer misperception, product failure, and product Ilability, Review of Economic Studies 44, 561L572. Thaler, Richard and Sherwin Rosen, 1976. The value of saving a life: Evidence from the labor market, in N. Terleckyz, ed., Household production and consumption (NBER, New York) 265-298. Viscusi, W. Kip, 1979a, Employment hazards: An investigation of market performance (Harvard University Press, Cambridge). Viscusi. W. Kip. 1979b, The impact of Occupational Safety and Health Regulation, Bell Journal of Economics 10, 117-140. Zeckhauser, Rtchard J., 1970, Medical insurance: A case study of the tradeoff between risk spreading and appropriate incenttves, Journal of Economic Theory 2, 1@26. 197X, The Occupational Safety and Health Zeckhauser, Richard and Albert Ntchols, Administration: An overview, The Study on Federal Regulation of the Senate Committee on Governmental Affairs (U.S. Government Printing Office, Washington).
of Public
Economics
14 (1980) 319-337.
Q North-Holland
Publishing
IMPERFECT JOB RISK INFORMATION OPTIMAL WORKMEN’S COMPENSATION
Company
AND BENEFITS
W. Kip VISCUSl* Northwestern
University,
Ecunston,IL 60201,U.S.4
Received June 1978, revised version
received November
1979
If workers have imprecise, but unbiased risk perceptions, they will accept a lower wage and less of a marginal wage reduction in response to an increase in injury compensation or a reduction in risk. Market outcomes will provide too little compensation and too great a job risk, as compared with the perfect information case. The optimal workmen’s compensation policy will provide a supra-optimal benefit level in an effort to create incentives for safety. The signaling function of market-provided injury compensation will eliminate the bias in worker priors and increase their precision, eliminating the market failure in some instances.
1, Introduction Compensation for job risks may be received in the form of ex urrte wage compensation, ex post compensation after an injury has occurred, or manipulation of some other job attribute valued by the worker. If one considers the partial equilibrium context in which there is a competitive wage schedule in which a market wage rate is associated with each possible level of risk and mix of job rewards, the decisions by workers and firms in an otherwise competitive market will lead to an efficient level of risk, wages. and ex post injury compensation.’ This result and the analysis in this paper abstract from more fundamental, general equilibrium concerns such as the determination of the equilibrium wage schedule and the decisions by firms to introduce jobs with a different level of risk into the set of possible job options available to workers. The implications of research in this vein do not necessarily bolster the findings of more traditional compensating wage differential analyses since decisions by firms that influence the available job quality mix often lead to inefficient outcomes.* *Helpful comments were provided by two referees, one which I now address in section 4. ‘See, for example, the discussion in Oi (1973a, 1974) and analysis for product markets appears in Oi (1973b). 2The analvses of Lancaster 11975) and Dr6ze and Ha.een for product quality, although many of the results generalize firm.
of whom Thaler
raised
the signaling
issue,
and Rosen (1976). A parallel
(1978) consider these issues primarily to the choice of work qualiiy by thk
320
WK.
Viscusi, Imperfect job risk
infirmutiun
Other deviations from the competitive norm, which pertain more specifically to the particular employment relationship, include (a) actuarially unfair market insurance, (b) moral hazard problems, (c) imperfect worker information, and (d) externalities to society at large.3 The primary focus of this paper is on the impact of imperfect worker information on the welfare implications of decisions by workers and firms and on the optimal policy choice regarding the level of workmen’s compensation benefits. Two kinds of imperfections can be distinguished. Workers may have biased mean values of their prior assessments of the chance of an accident. This is the familiar instance in which firms will provide too little ex posr compensation and too high a level of risk if workers systematically underassess the risks they face.” The second informational shortcoming is that even if workers’ priors are unbiased, they rarely possess full information regarding the actual level of risk faced. As additional information is acquired, worker priors will be updated. My principal focus will not be on the possibility of initial bias in workers’ judgements but on the implications of imprecise prior beliefs and the influence of worker learning. The subsequent findings regardin g the implications of adaptive worker behavior are based on the uncontroversial assumption that workers often do not have perfect information regarding the level of risks they face. Indeed, there is considerable evidence that workers learn about job risks through their on-the-job experiences and quit if these experiences are sufficiently unfavorable.’ The absence of perfect prior information and the potential for learning by experience are the two essential ingredients of the subsequent analysis. In contrast, analyses of biased worker perceptions hinge on assumptions regarding both the direction of the bias and the responsiveness of the there is some experimental perceptions to changes in the risk. 6 Although evidence indicating that individuals tend to underestimate very small probabilities, workers have a greater interest in learning about the level of the risk than do individuals facing hypothetical choices. Moreover, the average injury risk facing workers is fairly sizeable since it exceeds lo-’ annually. Even worker death risks, which average about 10e4 annually for the average American worker, command compensating wage differentials that correspond
‘For a survey of these imperfections, see Zeckhauser and Nichols (1978). 4Thls result was derived by Diamond (1977) for iob risks in an analvsis that closelv uarallels Spence’s (1977) treatment of product quality. Other-discussions of biased perceptions i%lude Oi (1973) and Viscusi (1979a). 5Fuller description of the supporting empirical results is provided in Viscusi (1979a). ‘ln particular, primary attention has focused on the situation in which workers underestimate the level of risk and revise their prior beliefs less than proportionately with respect to an increase in safety. See, especially, Spence (1977) and Diamond (1977).
WK. Viscusi, Imperfect job risk infbrmation
321
to an implicit value of life on the order of $2-3 million.7 These results are bolstered by evidence that there is widespread awareness of many job hazards, as individuals’ job risk perceptions are strongly correlated with objective measures of the industry injury rate. While none of these findings imply that workers’ probabilistic beliefs are unbiased, these substantial market impacts provide evidence that risks are not completely ignored by workers. The economic foundations for workmen’s compensation and other regulations could clearly be strengthened if they did not hinge on an assumption that probabilistic beliefs are slanted in a particular direction but instead derived from the absence of complete information prior to incurring the risk. The impact of imperfections in worker information on individual responses to the level of safety at the workplace and the amount of ex post injury compensation provided will be assessed in section 2. These responses will influence the incentives for the firm to provide an optimal level of risk and compensation. The results in section 3 imply that even if workers’ prior assessments are accurate, the imprecision of workers’ judgements will result in too high a level of risk and a suboptimal level of injury compensation. These shortcomings can be partially alleviated by a workmen’s compensation system, such as that presented in section 3.3. Government imposition of compensation levels will sacrifice the informational content that discretionary choices of compensation would have had. The potentially important signaling function of injury compensation is addressed in section 4.
2. Individual job choice under uncertainty 2.1. Notation
and analytic framework
A pivotal determinant of enterprise decisions is the responsiveness of workers to different actions taken by the firm. Here we will investigate the behavior of a representative worker and the effect of imperfect information on the attractiveness of different job characteristics. The analysis throughout assumes that individuals assess subjective probabilities, update these probabilities in conventional Bayesian fashion, and undertake actions to maximize their expected utility. The implications of this analysis will be used subsequently in assessing decisions made by the firm, given this pattern of individual action.
‘These estimates are in 1979 prices based on the representative samples of workers in Viscusi (1979a) and in current unpublished work by the author. Estimates by Smith (1976) are similar. The value of life for workers in high risk jobs is, as one would expect, considerably lower. Thaler and Rosen’s estimates for workers facing annual death risks of 10-j imply a value of life of roughly S400.000 in current prices.
WK. Viscusi, Imperj>ct job risk
322
i:Jorrnation
Consider a two-period situation in which a worker must choose between a potentially hazardous job with an assessed prior probability p of remaining uninjured and a safe job alternative offering a wage w~.~ If the worker is not injured on the job, he receives a wage rate w, and if he is injured he receives ex post compensation s. For simplicity, I will restrict myself to the income maintenance function of workmen’s compensation rather than get involved in medical insurance issues. Being injured also may lower the marginal utility of consumption x.~ To allow for this possibility and to reflect the assumption that the worker would rather not be injured for any given level of consumption, I will adopt a state-dependent utility function format in which Ur is the utility function when the worker is healthy and U” is the utility function when ill, so that
U’(x)>
P(x)
u: zu;>o, and
where subscripts indicate partial derivatives.” The formation and revision of workers’ prior beliefs will be specified in fairly unrestrictive terms. The assessed prior probability p of remaining uninjured is assumed to be an increasing function of the actual probability q, where c'p/Zq>O and (:2p/2q2~0. The actual injury probability may not be observable, but q will affect the firm’s accident record and other information that can be used by the worker in forming his beliefs prior to work on the job. The enterprise characteristics that form the intervening link between p and the unobservable q will not be detailed explicitly. Section 4 addresses this issue more formally, analyzing the role of s as a signal for q. The precision of the worker’s prior will be denoted by 7. where higher values of 3’ correspond to greater information. The extreme case of perfect information can be viewed as the limiting case of tight priors as Y-+X. The parameter y will be given a more explicit interpretation in the discussion of beta distributions below. Although individuals start the risky job with imperfect information, they
‘Alternatively the alternative job could offer a lottery with known probabilities. The probability of an accident is assumed to be independent of worker actions. This case is considered in Viscusi (1979b). ‘Medical care expenditures may bc quite effective in reducing pain and in increasing the probability that the worker returns to the healthy state. These issues are considered within a medical insurance context by Zeckhauser (1970). “State-dependent utility functions have been used in a variety of studies, includmg Zeckhauser (1970),Spence (1977), Diamond (1977), and Vlscusi (1979a).
WK.
Viscusi,
Imperfect
job
risk informution
323
can acquire information on the job by direct experience of adverse outcomes, and observation of others’ injury exobservation of job characteristics, periences. For simplicity I will restrict the analysis to the effect of injuries on Let p(m,n) represent the assessed probability of the worker’s perceptions.” no accident after m successful job risk lottery outcomes at the firm and n unsuccessful outcomes (accidents). The subjective probability of not being injured in period 2 after not being injured in the initial period is assumed to be updated in standard Bayesian fashion where
The focus here will be on the lowest initial wage-injury compensation package that will attract the worker to the job in the initial period. In such situations the worker will always quit after an unfavorable outcome so unfavorable updating of the worker’s prior need not be considered.i3 The worker’s objective is to maximize the discounted expected utility from the income he receives in each period. where h is the discount factor equal to the reciprocal of 1 plus the interest rate.” For the worker to be indifferent between the two jobs in the initial period, the value of the indirect utility function Vfor the two jobs must be equal, or
V(w, P, s,Y I= I/b,, 40, % 1,
(1)
where the index ;j of the sharpness of the worker’s prior is infinite if the worker is perfectly informed. The discounted expected value of the uncertain job is given by I/(w,p,s,;‘)=pc”(~)+(l
-p)v2(s)+bp[p(l,O)U’(\1!)
+ (1 -p(ltO))UZ(s)]
f h(1 -p)U’(Wo).
(2)
The first two terms on the right-hand side of eq. (2) represent the first period reward. All subsequent terms are weighted by the discount factor since they pertain to period 2. The first period 2 term consists of the probability of a successful first period outcome multiplied by the expected reward in period 2 after a successful job experience in period 1, while the
“An analysis of experiences of other workers and other work characteristics that may influence one’s probabilistic perceptions is presented in Viscusi (1979a). The results are quite similar to those generated here. r3A proof that the reservation wage rate will always lead to worker quitting is provided in Viscusi (1979a). I41 assume that workers do not transfer resources across periods. Relaxation of this assumption would make consumption rn period 1 a choice variable, further complicating the model.
324
WK. Viscusi, Imperject job risk information
final term consists of the probability of an accident in period 1 followed by the value for work on the alternative job after the individual has switched positions. The value of the safe job alternative is defined analogously and is given by V(y), l,O, Co)=U’(w,)(l
+b).
Since the properties of the utility function are unaffected by adding a constant to their value. I will define my utility metric by setting U’(W,) equal to zero. The condition that the worker be indifferent between the two jobs, which is given by eq. (1) simplifies to the requirement that 0=pcr’+(1-p)u~+~l)~p(1,0)u~+(1-p(1,0))u2~, where the arguments before.
2.2. Determinants
of the utility
ofthe
reservation
functions
wage
(3)
are M’for U’ and s for U2, as
level
The analysis in the following section will indicate that a fundamental determinant of enterprises’ choice of the level of safety and injury compensation is how the reservation wage required by workers is affected by the level of s, the looseness of their prior assessments, and the level of the true job risk. The true probability that the worker will not suffer a job injury will be indicated by q. By implicit differentiation of the indirect utility function for the risky job, one can readily determine the effect of marginal changes in s and q. In particular, one obtains the result that
(‘M:
-= 2s
V -2= VW
v:C(l-P)+hP(l-P(l,o))l vxPU
+
b(LO))l
’
(4)
and L;w
-= r?q
V __A= VW
_ i
Gil.‘--D.‘)+b
fP-
%(I, 0) (U’-U2) ip
where both ?w/c?s and c?w/8q are negative.
~cp(l,o)uL+(lp(l,o))c’~, [
31
:U:[Ip(l+bP(l,o)]}-‘,
(5)
325
WK. Viscusi, Imperjhct job risk informtrtiorl
In order to analyze the nature of the reservation responses in greater detail, I will use a specific class of prior probability distributions. Beta distributions will be used since this distribution is quite flexible and can assume a variety of skewed and symmetric shapes. Moreover, as Raiffa and this distribution is ideally suited to Schlaifer (1961) have emphasized, to normal Bernoulli-type processes such as this and is far superior approximations. Letting the distribution fl(p,l;) be reparameterized in terms of the mean p of the prior, where 0~ p< 1, and its precision y, where 0 < y < a, one has the posterior probability p(m, n) of a successful job outcome after m successes and II failures (accidents) given by
v+m
p(m,n)= y+m+n' Individuals with prior probabilistic perceptions p(p, y) in effect act as if their prior experiences consisted of 7 trials of which a fraction p were successful outcomes. The development below will also utilize the cross partials and second derivatives of these wage response terms. Letting p’ denote dp/dq and using the beta distribution values for p(l,O), one has
~=~~[p’(~+1-yb+2ybp)l>0 ??qC% ~~Cpb+1+byp+b)l
’
a%
-p)l >.
s=
-~$b+ 1-w-P+$p(l
and, after a considerably
’
U:[p(v+lfbyp+b)] more complicated
calculation,
a value of
a2W b-0 a4
2
The wage reduction from a marginal increase in safety q or compensation s diminishes as the other variable is increased. Moreover, reservation wages decline with the levels of 4 and s, but they do so at a decreasing rate. 2.3. The role of the precision
of one’s initial judgements
The primary matter of concern is how provision of safety and compensation. The the worker’s prior assessed probability of the learning process and, more specifically,
worker learning affects the firm’s focus consequently will not be on an accident but on the nature of on the precision y of the worker’s
initial beliefs. As one would values are updated less, as c7p(l,O)
P-l
2Y-~--(:,+
1)’
expect,
tighter
priors
associated
with larger
3
Inspection of the expression for V(w,p,s, 7) in eq. (2) indicates that for the marginal worker, only the greater upward revision of loose priors is of consequence. The downward revision after an unfavorable outcome does not enter since the worker leaves the hazardous job after an unsuccessful period I outcome. Consequently, loose priors (i.e. low 7 values ) serve to boost the value of p(l,O), raising the attractiveness of the uncertain job provided that U’ exceeds U2.t” One might expect that workers who viewed the job on more favorable terms in period 2 would require a lower wage to accept the position and would place a lower value on marginal increases in safety or additional ex post compensation. wage. Upon implicit Consider first the impact of :’ on the reservation differentiation of V(w, p, s, y), one obtains
as expected. The lower posterior probabilities of a successful outcome for workers with high values of y raise the wage they require. Loose priors enhance the attractiveness of the job. Upon differentiation of eq. (6) with respect to s and q, we get
(p-lW,Z
d2W
i?$s
(y+ i)‘U:[l+
and, after considerable (35
+/+q
-
bp(l,O)]
’
simplification,
(U’ - U2)bp’(l+ ___.II_
b)
L$~~+l+ybp+b]*
The value of ?2ti~j3yds is clearly
’
negative
since pc 1. Since it will be shown
15For the optimal compensation levels derived in sections 3 and 4, this relationship will always hold. A notable exception is the case in which accidents involve only monetary losses that are fully compensated, so that U’ = U* and ?MJ/+~ is zero.
WK. Viscusi, Imperfect job risk information
327
that the uninjured state is preferred, U’ > CT’, d’~1~/8ydq likewise is negative. Increased precision of one’s intial judgements makes one willing to incur a greater marginal wage reduction in response to an increase in y or s, while looser priors diminish the attractiveness of such changes. An important issue for the subsequent analysis is how worker responses are affected by the absence of perfect information. With perfect information, the value of the employment path would be
where U’(w,) is set at zero as before. The effect of s and q on the reservation wage will be denoted by ?w*lZs and i;w*/Zq, where
i?w* --Yz
-V”
s
_-V,z(l-q)(l+hq)_-u2(1-q)
2s
v;
2w*
-T/y*_
Qqu + bq)
uh
(7)
and ---=-----34 v;,
-[(U’-U2)(1+2bq)+bUZ] uh(l+bq)
-~
(8)
The analysis below will focus primarily on unbiased priors in which p=q. If this condition holds, the value of &*/irs with perfect prior information is simply the limiting value of i3w,i?s as y-‘nc! and when p is replaced by q. Consequently, any decrease i; ; represents a departure from the perfect information situation. The negative value for ?2~~l/?$s implies that ?N/?s will be more negative for perfectly informed workers than for individuals with imprecise initial information. This relation also holds if p>q.16 To assess the impact of imperfect information on i?wiiq, one also must ascertain the magnitude of p’. With perfect information, one’s accurate prior beliefs increase proportionately with the probability of no accident, or p’ equals 1. If the worker’s prior judgements can be characterized similarly for the imperfect information situation (i.e. p =q and p’= l), the value of &:*,i$ with perfect information will equal the limiting value of h/2q as y-)x. Since higher values of 1 make &,/?q more negative. the value of i+~/icl with imperfect information will be larger (or less negative). Upon comparison of the magnitudes of c’w/c?q and c!w*/iq, it can be shown that so long as
16The larger response for &v*/i% can be verified by comparing eqs. (4) and (8). The result does not require that p(l,O) be updated using the beta distribution. Any updated prior with p(l,O)>p will suflice.
328
l4X. Viscusi, Impwfect
joh risk inforfnntiofl
0 < p’ 5 1 and p 2 4, ?w/C:q will be larger with imperfect information.” If p’ > 1, the imperfectly informed worker overestimates the marginal increases in safety so that Zw/&l may become more negative with imperfect information, though this need not be the case.18 The analysis below will assume that workers do not overestimate the value of p’ sufficiently to lead to a reversal in the otherwise positive effect of imperfect information on Sw/aq.
3. Imperfect information, 3.1. Worker
uncertainty
enterprise decisions, and workmen’s compensation and enterprise
behaoior
Consider a firm that selects the level of q and s to minimize its costs for a given level of output. The costs C per worker consist of the wage costs W, P.Y post compensation costs at a level [( 1 -q)s]/q if insurance is purchased on an actuarially fair basis, and the cost h(q) of providing a safe environment, where h’ > 0 and h” > 0. For equally costly technologies associated with the same risk and p values for workers, the firm will always prefer the uncertain technology with the lower 9 since ?w/Sy is negative. There will consequently be a bias toward new technologies that, are not fully understood. In the discussion below, we will assume that the firm is selecting from a class of technologies with the same 7 but differing q. After analyzing the optimal decision. the effect of 7 on s and 4 will be assessed. Alternatively, the model could be framed in terms of the firm picking from some frontier of best available technologies where for each level of 4 there is an associated value of y(q). The conclusions below are reinforced if ~‘20 and are altered substantially if 1~‘<0. In effect, we will be making the neutral assumption that 7’ =O. The enterprise’s objective is to
subject to the constraint that it attracts conditions for an optimum are that
workers
to the firm. The first-order
17This result also does not require that priors belong to the beta family, but rather all that is needed is that p(l,O)>O. “Whether this reversal takes place depends on the relative magnitudes of U’ and Zl’ as well as the various risk parameters.
WK. Viscusi, Imperfect job risk information
329
and
where dw/ds and &/8q are given by eqs. (4) and (5) for the representative worker.” The second-order conditions that C,,>O and C,, >O are both satisfied since i’w/?? and F2w/?q2 are positive, as is h”. To ensure that the solution is a relative optimum, the value of C,,C,,(C,,)‘, which will be denoted by H, is assumed to be positive. To assess the impact of y, totally differentiate the first order conditions, yielding
r
Solving
11
for the effect of 7 on s and q,
and
&l -=al,
1 - U’,,Wqy+ W,? Wqs-7 1 4 i [
H
I
.
Using the wage results from section 2 and the positive signs assumed for H and h”, it is clear that both Zs/C:y and iq/Gy will be positive if the positive wsq term exceeds l/q2. This relationship can be verified using eq. (6) above.” Greater precision of workers’ priors will raise the optimal no-accident probability q and raise the level of ex post compensation. In contrast, dimly understood technologies associated with loose priors create a bias toward inadequate compensation and too great a level of risk. To assess the overall effect of imperfect information, one would like to ascertain the effect of biased perceptions coupled with imprecise priors. First, consider the outcomes with perfect information. The first-order conditions (9) “In effect, the firm is a price-taker in the product market and a wage-taker in the labor market. The risk choice by the firm does not alter the available work quality options for workers. This assumption is stronger than that in D&e and Hagen (1978). Z°Clearly if workers overestimate the risk by too great an amount so that q is close to zero and p is near 1, this inequality may not hold. For the inequality to hold, let psq, p’( 1, and u; 2 u:.
330
WK.
Viscusi. Impeyfert
job risk injbrmation
and (10) are unchanged except that dw*jZs and ?w*/c?q now replace C:w/(?.s and c7w!:r?q.The optimal s, for given q, has the most convenient interpretation since, after substituting from eq. (7) one obtains the familiar result that iJi = li,!.” If injuries are equivalent to monetary losses, t’x post compensation will equalize the absolute utility levels in each state. If an injury diminishes the marginal utility of consumption, equalization of the marginal utilities of consumption will require a lower level of consumption after an accident. The values of C?M./?Sand ?w/dq will each be less negative than in the perfect information case if p 2 q and p’ 5 1. Assuming these conditions hold, consider the choices implied by eqs. (9) and (10). For fixed values of q, and consequently fixed values of (1 -q)/q, a larger (less negative) value of the W, function will lead to a lower optimal level of s. Similarly, for fixed values of s, it can be shown that a less responsive u’~ term will lead to a lower optimal q level. Coupling of the standard assumptions concerning misperceptions in static models with the adaptive behavior when learning takes place simply reinforces the biases in the risk and ex post compensation levels. The more novel portion of this result is that even with unbiased and perfectly responsive prior probabilistic beliefs (i.e. p=q and p’= 1 ), the subsequent updating of these perceptions will lead to types of market shortcomings generally associated with misperceptions in static contexts.
3.2. Elf&t of’exogenous
compensution
lerels
Suppose that the level of s is not a choice variable for the enterprise but instead is imposed on a nondiscretionary basis through a workmen’s compensation program. Let a fraction f of benefits be merit-rated, where Ozf 5 1, and the remainder of the insurance program costs be financed by a lump-sum tax on enterprises. Large firms often self-insure or are experiencerated and have values of ,f close to 1, while smaller enterprises have f values near 0 under the U.S. workmen’s compensation system.*’ The first-order condition for an optimizing firm selecting safety level q will be
“This is a standard result when actuarially fair insurance IS available. See, for example, Arrow (1971). I2 Ratings for smaller enterprises are based largely on the average industry risk, which is affected bery little by the accidents at a particular firm. Only IS percent of all enterprises are experience-rated to -any degree, but smce these are the larger firms. over 85 percent of all workmen’s compensation premiums are paid by such firms. See pp. 3637 of the National Commission on State Workmen’s Compensation Laws (1973).
WK. Viscusi, Impetfect
job risk irfornmtion
331
The primary matter of interest is the effect of s on the level of q selected by the firm, since this relationship defines the firm’s reaction to the policy choice variable. Totally differentiating eq. (1 I), one obtains
dq-
- wys+.i?q* ---
(12)
ds - wyq+ h” + 2s/y3
The denominator is simply C,,, which is positive. From section 2 we know that wqSis also positive. If there is no merit rating so that f equals zero, dqlds will be negative. Higher workmen’s compensation levels reduce safety in this instance by enhancing the attractiveness of hazardous jobs to workers, while providing no marginal safety incentives for the firm. In the situation of complete merit rating, one obtains the opposite result if qs is U’ is smaller than l/q*. As was indicated in section 3.1, this inequality always satisfied for the types of prior beliefs being considered. For some critical merit-rating fraction ,f ** there will be no marginal effect of s on q. Abovef**, dq/ds will be positive. and below that critical value, dq/ds will be negative.
3.3. Optimal
workmen’s
compensation
hizne$ts
If enterprise incentives were not affected by the workmen’s compensation policy, the optimal compensation level would be determined by ascertaining the value of s that would maximize the worker’s expected utility if he were perfectly informed, where the insurance plan must break even on an actuarial basis. In such instances the level of s would be set to equate the marginal utility in each health state. The absolute level of welfare will be less when the worker is injured if, as was assumed here, an injury reduces the marginal utility derived from any given level of income. If an injury results only in a loss of worker income, the optimal s will equalize both the absolute and marginal utility levels in each state. Since the full information competitive outcome also satisfies this optimal insurance requirement, ideally one would like to set the level of s and q in a manner that replicated this outcome. While the government can manipulate the level of s, it influences q only indirectly through the incentives for safety created by a higher S. More specifically. the shape of the enterprise response function q(s) will be dictated by eq. (12) above. The policy objective is to minimize the costs per unit of output subject to the constraint that worker utility I’* based on the actual job risk probabilities is at some constant level. As before, the wage reductions accruing to the firm from higher levels of s will be dependent upon worker perceptions rather than the actual risks involved.
332
WK. Viscusi, Imperfrct
More specifically,
the policy objective
minL=w+h(q)+----~
41-d 4
leading
to the first-order
conditions
job risk injbrmution
is to
_3v*
I’ for optimal
insurance,23
An optimizing lirm will set q so that eq. (11) is satisfied, resulting in a value of zero for the bracketed coefficient of dq/ds. The optimal workmen’s compensation condition can be rewritten as
(13) This condition differs from eq. (9) for an unregulated market by the addition of the final term. If the bracketed expression is positive, the marginal productivity of compensation s will be above that in a market allocation so that it will be optimal for the policy to increase s above market levels. The first component term is given by
av*
aw
-=~qu:(l+bq)+(l-q)C’,Z(l+bq).
as
If workers were perfectly informed initially, dw/& would assume the value for ~w*/c% in eq. (7), implying that dV*/as would be zero. There would be no rationale in terms of the direct effect of s on utility to warrant additional compensation of fully informed workers. However, the imperfect information case is associated with a suboptimal s level so that aV*/i% will be positive. The second component of the bracketed ;1 coefficient is dqaV* ---=z ds dq
dq
(U’-V)(l
.
+2bq)+hU’+$J:(q+hq2)
Since the level of safety q is too low in the unregulated dV*/aq will be positive when assessed at the market
I market, values
the value of of q and s.
2’For simplicity, the total enterprise costs omit the merit rating fraction f and any lump sum component involved. It is assumed that the workmen’s compensation plan must break even so that regardless of the merit-rated and lump-sum tax share, total costs are unchanged. The value off is reflected in the enterpripe’s reaction to s, through dq/ds.
WK. Viscusi, Imperfect job
risk
in$?rmution
333
Consequently, the sign of this expression hinges on that of dq/ds, which can be positive, negative, or zero. In the case of relatively strong merit rating (i.e. f>f**), higher values of s will not exert any negative effect on worker welfare through its effect on q. Since ?V*/?s is positive, the beneficial effect of additional compensation on worker well-being will always lead to a positive value of the final term in eq. (13), so that the optimal level for s will exceed that in the market. If merit rating is relatively weak (i.e. S
4. Ex post compensation
as a signal for risk
Since the level of q is not observable to workers, other forms of information must be used in making inferences about the level of safety. Here I will focus on the level of ex post compensation as a possible signal for the underlying q characterizing the firm’s decisions.25 More specifically, let the worker’s prior assessed probability of no accident be r(s) so that the expected utility B of the employment path on the uncertain job is P(w, Y(S), f)= r(s)U’ + (I+ br(s)
rb)W’
Yr(s)+
1
-q-(1/‘-u*)tu*
24Given the utility-taking hypothesis adopted here, an additional fine on accidents that is reimbursed to workers will have no effect. Other forms of penalties and safety standards may be useful, but these are beyond the scope of the analysis here. “The analysis here parallels that of Spence (1977), with the principal modification being the introduction of learning into the model.
334
Since v is independent of y. the value of ?W/iq for the signaling case is zero.26 Letting 7 denote the precision of worker priors with signaling, the effect of s is now enhanced by its influence on worker perceptions so that
U;r[r+ The optimizing
ybr + 1 + h]
firm will select s and q so that (14)
and
Since C,, is positive, s will serve as a signal for q in equilibrium, implying that r(s)=q in the expression for C~$/SS.~~ The utilization of s as a signal for y will affect the precision of the worker’s initial judgements as well. I will view the presence of the signal as equivalent to additional observations regarding outcomes of the stochastic job risk process so that, in effect ^J is increased to a value r>r. I will assume that the value of y is independent of the particular s selected so that the mean of the prior and not its informational content is influenced by s. Compensation for injuries is generally provided at a fixed level across broad classes of workers at the firm or for the enterprise as a whole. If all jobs at the firm pose the same risk and workers are aware of the signaling relationship, s will provide perfect information. This important limiting case in which ~~-+x. will be discussed further below. If these conditions do not hold, s will result in unbiased priors for the firm-specific risk but would not resolve all uncertainties. Even workers on representative jobs with safety probability q will update their priors because they are not aware that their initial judgements were correct. “1 will continue to focus on a hypothetical worker. Variegated preferences are an essenttal as noted in Spence (1977). Such diversity can be prerequisite for a signaling equihbrtum, achieved by letting the individual be of class i. where there are different utility functions for different classes. 27The requirements for an effective signal are dtscussed in greater detail by Spence (1974. 1977). In a signaling equilibrtum. he observes that prices cannot be a stgnal for product quahty since there is no cost to raising prtces. The same result pertains to wages for the model here.
WK.
Viscusi,
Imperfect
job
335
risk informrrtion
Although the signaling equilibrium will not generally be efficient, in the case in which the only losses involved are monetary equivalents, Spence (1977) has shown that guarantees produce efficient outcomes when consumer products are subject to failure. To assess whether this result generalizes to the job safety case with adaptive behavior, let the monetary loss be 0 so that the utility function takes the form U2(x)= LJ’(x -fl). In the perfect information situation, s is selected so that Vi = U$ or s=8. For these utility functions, this result in turn implies that U’ =U2, which both equal zero since the value of the job alternative was arbitrarily set at this value. Substitution of these utility values into the expression for (7w*/(!q in eq. (8) implies that aw*/dq equals zero. Consequently, with perfect information, the two first-order conditions will be
and h'=
s/q’.
(15)
Equation (15) is identical to that in the signaling case described above. For fixed levels of s, q will be set as in the perfect information situation. The signaling equilibrium will lead to efficient outcomes for both q and s if setting 8 equal to s would satisfy eq. (14). Noting that this level of compensation would have the aforementioned effects on utilities and marginal utilities, for the signaling equilibrium in which r(s)=q, eq. (14) reduces to y+ 1 +
$q
(16)
y+1+“g7q+b=1.
This equality will not be satisfied since b>O with finite interest rates. The difficulty derives from the updating of initially unbiased prior assessments. However, as was indicated above, the utilization of s as a signal for q not only affects the mean of workers’ priors, but it also provides additional information tantamount to observing additional trials on the uncertain job. In the limiting case in which the signals provide completely precise information (i.e. as *;+a), eq. (16) will be satisfied. The signaling equilibrium will be efficient if all losses are financial and if signals provide complete information to the worker.28 Market allocations for ex post compensation consequently serve a signaling function that eliminates the bias in prior perceptions and increases the sharpness of the prior, thus diminishing the aforementioned difficulties 28This limiting analysis.
result
in effect reduces
the model
to a job market
analog
of Spence’s
(1977)
336
WK.
Viscusi,
Imperfrct
job
risk iTformation
deriving from adaptive worker behavior. A workmen’s compensation program that imposes mandatory levels of s eliminates both forms of information content of discretionary s values. The suppression of this information transmitted through a market signaling process can potentially undermine the efficiency of market allocations so that even an optimally designed workmen’s compensation policy may lower worker welfare.29 In short, the optimal workmen’s compensation system may require no interference with outcomes in an unregulated market.
5. Conclusion The presence of learning and adaptive worker behavior will lead market outcomes to provide too little er post compensation and too great a level of risk even if workers’ priors are unbiased. If a workmen’s compensation program is merit-rated to a sufficient degree, higher mandated levels of compensation for injuries will provide additional incentives for safety as well as greater compensation for injured workers. The imposition of uniform injury compensation levels may, however, have an unintended impact. Prior to government intervention, injury compensation levels would have served as a signal for the underlying risk level, thus enhancing the degree of prior information and leading to unbiased priors in a signaling equilibrium. Levels of compensation specified by the government will eliminate this information transfer, making the desirability of intervention unclear. ‘“Whether welfare is actually lowered depends on a variety of influences about which very little is known, such as the availabihty of other job risk signals. the precision of worker prrors with and wrthout the e.x post compensation signal, and the bras in priors in the absence of signaling.
References Arrow. Kenneth, 1971, Essays in the theory of rusk bearing (Markham, Chicago). Diamond. Peter. 1977. insurance theoretic aspects of workers’ compensation, in: Natural resources. uncertainty. and general equilibrium systems (Academic Press. New York) 67-89. Drere. Jacques H. and Kare P. Hagen, 1978, Choice of product quahty: Equihbrrum and efficiency, Econometrica 46, 493-514. Lancaster, Kelvin, 1975, Socially optimal product differentiation, American Economic Review 65, 567- 585. National Commission on State Workmen’s Compensation Laws, 1973, Compendium on Workmen’s Compensation (U.S. Government Printing Office, Washington). Oi, Walter, 1973a, An essay on workmen’s compensation and industrial safety. in: Supplemental studies for the National Commission on State Workmen’s Compensation Laws (U.S. Government Printing Office. Washington) 41-106. of Economics and Oi. Walter, 1973b, The economtcs of product safety, Bell Journal Management Science, 3-28. Oi, Walter, 1974, On the economics of industrial safety, Law and Contemporary Problems 38, 669 699.
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Imperject
job
risk information
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Raiffa, Howard and Robert Schlaifer, 1961, Applied statistical deciston theory (M.I.T. Press, Cambridge). Smith, Robert, 1976, The Occupattonal Safety and Health Act: Its goals and achievements (American Enterprise Institute, Washington). Spence, Michael, 1974, Market Signaling: Informational Transfer in Hiring and Related Screening Processes (Harvard University Press, Cambridge). Spence, Michael, 1977, Consumer misperception, product failure, and product Ilability, Review of Economic Studies 44, 561L572. Thaler, Richard and Sherwin Rosen, 1976. The value of saving a life: Evidence from the labor market, in N. Terleckyz, ed., Household production and consumption (NBER, New York) 265-298. Viscusi, W. Kip, 1979a, Employment hazards: An investigation of market performance (Harvard University Press, Cambridge). Viscusi. W. Kip. 1979b, The impact of Occupational Safety and Health Regulation, Bell Journal of Economics 10, 117-140. Zeckhauser, Rtchard J., 1970, Medical insurance: A case study of the tradeoff between risk spreading and appropriate incenttves, Journal of Economic Theory 2, 1@26. 197X, The Occupational Safety and Health Zeckhauser, Richard and Albert Ntchols, Administration: An overview, The Study on Federal Regulation of the Senate Committee on Governmental Affairs (U.S. Government Printing Office, Washington).