Offsetting behavior and the benefits of safety regulations.

Hause, John C.

I. INTRODUCTION

Many regulatory, safety, and health policies are adopted to reduce harm to potential victims from accidents and other harmful events. Economists now widely recognize attenuation and even reversal of the direct policy effect on expected harm may occur because of offsetting behavior (OB) by potential victims as they reduce care in response to the policy. When policy makers ignore OB where it is significant, the predicted policy effect will be overstated.

Peltzman's (1975) study of automobile safety regulation is apparently the first to argue that these regulations would induce OB by the victim (driver), thereby increasing the probability of accidents, and perhaps even increasing expected accident loss. Since Peltzman's (1975) initial empirical work, there is now a vast empirical literature on OB and accidents, dealing primarily with traffic, workplace, and consumer product accidents. (1) Despite ongoing controversy about the pervasiveness and magnitude of the OB effect, its existence is indisputable. Motorists do drive slower on icy roads and faster with better road lighting or wider lanes (Wilde 1994; Assum et al. 1999; Noland forthcoming; Noland and Oh 2003). Antilock brakes lead to closer following in traffic (Sagberg et al. 1997). Bicycle helmets have not yielded the reduction in head injuries that had been forecast. (2) Furthermore, OB is relevant not only for the study of accidents but also for understanding the effect of health policies on lifestyle-dependent disease and mortality (e.g., Wilde 1994). More generally, OB is potentially relevant in any application where adopting a policy changes the victim's payoffs in a way that reduces the marginal value of his own accident avoidance expenditures. Despite accumulating evidence on the empirical relevance of OB, none of the theoretical literature has provided a model determining formal conditions under which dominant or partial OB occurs, much less the magnitude of the OB effect on expected accident loss. (3)

II. A SIMPLE MODEL OF OB

The following stylized and simplified model of OB has two main components. The first is the "production function" of expected accidental loss:" (4)

(1) A(x,y) [equivalent to] [pi](y)L(x),

where A(x, y) is the (monetary equivalent) value of a potential risk-neutral victim's expected loss from an accident, x summarizes the level of safety regulation (modeled as expenditure for convenience) and y represents the monetary equivalent of victim accident avoidance behavior. [pi](y) is the probability of the accident occurring, and L(x) is the monetary equivalent loss to the victim if an accident occurs. Assume that [pi](y) and L(x) are nonnegative, strictly decreasing smooth convex functions defined on x, y [greater than or equal to] 0, so that [A.sub.y], [A.sub.x] < 0; [A.sub.yy], [A.sub.xx] > 0. (5) Define y(x) > 0 as the accident victim's best response for all values of x we consider.

The second component of the OB model is the behavioral assumption that a (risk neutral) victim chooses avoidance expenditures y to maximize expected consumption

(2) E(C) = I - [A(x,y) + y],

where I is total income. The objective function (2) describes the victim's trade-off between using y to reduce the "bad" of an expected accident loss or buying other market goods. It is equivalent to minimizing the sum of the expected accident loss and accident avoidance expenditure. (6) The maximization of (2) implies that (7)

y' = -(L'[pi]'/L[pi]").

Initially set x = 0 (no policy has been adopted) so that y = y(0) and the expected accident loss is [pi][y(0)]L(0). Let a new safety policy be adopted with permanent flow expenditures [x.sup.1] > 0.

DEFINITION 1. Victim OB occurs from adopting the policy if

[pi][y([x.sup.1])]L([x.sup.1]) > [pi][y(0)]L([x.sup.1]),

that is, if the direct effect of the policy in reducing expected loss is attenuated by the induced change in victim behavior.

The inequality defining OB is satisfied because the slope of the victim's best response function is negative so that the induced decrease in y increases [pi]. Proposition 1 immediately follows.

PROPOSITION 1. Positive public safety (or regulatory) expenditures x always induce OB by the victim in this model of expected accident loss.

Proof. Follows because y' > 0 implies that y(x) > y(0) [for all] x > 0, and Definition 1 of OB is satisfied.

An intuitive interpretation is that the multiplicative separability of A(x, y) and assumptions on [pi](y) and L(x) imply that x and y are technological substitutes in reducing A.

Specifically, [A.sub.xy] > 0, so an increase in x reduces the marginal product of y in reducing A. (8) Hence, given the victim's objective, an increase in x lowers the marginal product of y expenditure, inducing a decline in y.

When victim OB is present, we make the following qualitative distinction.

DEFINITION 2. Victim OB is dominant if it more than completely offsets the reduction in expected accident loss from the direct effect of the policy, that is, if the net effect of adopting the policy actually increases the initial expected accident loss. Victim OB is partial if it less than completely offsets the reduction in expected accident loss from the direct effect of the policy, that is, if the net effect of adopting the policy decreases the initial expected accident loss, but by less than the direct effect.

The following proposition describes the technology required for dominant OB to occur. A numerical example is given that provides useful intuition for the more quantitative results derived in the next two sections.

PROPOSITION 2. If an increase in x induces dominant OB behavior by the victim, then public regulatory/safety expenditure x must be an inferior factor in the production of expected loss mitigation.

Proof. By construction we have OB so that y([x.sup.1]) > y(0). The OB is dominant if A[[x.sup.1], y([x.sup.1])] > A[0, y(0)]. By definition, a factor of production used in producing a good is inferior if higher output of the good uses less of the factor. The expected loss A(x,y) is a "bad," so -A is a good. It follows that when more x is used A increases (a bad) and there is less of the good -A. These definitions immediately imply that if an increase in x induces dominant OB, then x must be an inferior factor in the production of the good -A. (9)

EXAMPLE 1. This example shows the simple geometric intuition for the comparative statics of dominant OB. (10) The highest curve in Figure 1 graphs the expected victim loss when x = 0, A(0, y), for 125 [less than or equal to] y [less than or equal to] 250. Discrete increases of x to [x.sup.m] and [x.sup.h] shift down and flatten this curve because A(x, y) decreases with x and because the change in the marginal product of the victim's y expenditure in reducing A, decreases with x. (11) The victim's optimal y(x) for each curve is indicated by the heavy dots in Figure 1. At a given y(x) each of the curves has the same slope, -1, from the victim's FOC. Hence, y(x) must be a decreasing function of x because of the flattening of the curves as x decreases, as Figure 1 clearly shows. Furthermore, Figure 1 shows dominant OB for this example, because A[x, y(x)] increases as x increases. But the victim's preferences are better satisfied as x increases. The undesirable increase in A[x, y(x)] is more than offset by the decrease of the victim's accident reduction expenditures y, as Table 1 shows.

[FIGURE 1 OMITTED]

III. DOMINANT OB

To determine conditions assuring dominant OB in this model, I decompose the marginal net effect of x into the direct effect of x and the indirect OB effect of x on y. Define A(x) = A [x, y(x)] and take the total derivative to obtain (12)

(3) [dA(x)/dx] = [A.sub.x]{1 - [([A.sub.xy]/[A.sub.yy])/ ([A.sub.x]/[A.sub.y])]},

where ([A.sub.xy]/[A.sub.yy])/([A.sub.x]/[A.sub.y]) is the victim's marginal OB and measures the fraction by which the direct marginal effect of x on A is reduced by victim OB. It has a simple geometric interpretation as the ratio of the slope of the victim's best response function (in this case, the slope of a constant marginal product function [A.sub.y] = K to the slope of the iso-loss curve A(x, y) = M. If I assume marginal OB is greater than 1 for 0 [less than or equal to] x [less than or equal to] [x.sup.*], then this condition is sufficient to ensure OB is dominant for the safety policy [x.sup.*]. Similarly, if marginal OB is less than 1, OB is partial for the safety policy [x.sup.*].

PROPOSITION 3. For risk-neutral victims, a necessary and sufficient condition for dominant marginal OB, that is, dA(x)/dx > 0 is that x be an inferior factor of production.

Proof. By assumption, [A.sub.x], [A.sub.y] < 0 and [A.sub.xy], [A.sub.yy] > 0. If ([A.sub.xy]/[A.sub.yy])/([A.sub.x]/[A.sub.y]) > 1, the inequality can be rewritten as ([A.sub.y][A.sub.yx] - [A.sub.x][A.sub.yy]) < 0. Dividing by [A.sup.2.sub.y] then implies

[[differential]([A.sub.x]/[A.sub.y])/[differential]y] < 0.

This inequality implies that the marginal rate of substitution dx/dy along constant expected accident loss curves decreases as y increases (holding x constant), which implies that x is an inferior factor of production in reducing expected accident loss (see Bear 1965; Ferguson and Saving 1969). (13) QED

Although this result is conceptually simple, it may be difficult to verify empirically. A much sharper result is obtained by substituting (1) into (3). Then we have

(4) dA(x)/dx [equivalent to] [pi]L'[1 - ([[pi]'.sup.2]/[pi][pi]")].

From (4), [[pi]'.sup.2]/[pi][pi]" is the fractional reduction of the marginal direct effect of x due to OB, which depends on y, but not on x.

PROPOSITION 4. If log[[pi](y)] is a decreasing concave function, OB is dominant. If log[[pi](y)] is a decreasing convex function, OB is partial.

Proof. Because [pi]' < 0, log[[pi](y)] is a decreasing function of y. If [pi](y) is log concave, then

[d.sup.2]log[[pi](y)]/d[y.sup.2] = ([pi][pi]" - [[pi]'.sup.2])/[[pi].sup.2] < 0.

Multiplying this inequality by [[pi]'.sup.2]/[pi][pi]" yields

1 - ([[pi]'.sup.2]/[pi][pi]") < 0,

which from (4) and the fact that L' < 0 implies that dA(x)/dx < 0, the condition for OB to be dominant. A similar proof shows that if [pi](y) is log convex, OB is partial. (14)

Proposition 4 provides a surprisingly simple formal criterion for determining whether the OB effect will more than offset the direct safety policy effect on expected accident loss. It may be important in empirical work, because testing for evidence whether log[[pi](y)] is concave or convex is a simple criterion.

IV. THE PARAMETRIC FAMILY OF [pi](y) WITH CONSTANT MARGINAL OB

Theory

This section obtains my main quantitative results on the size of the OB effect. I determine the parametric family of [pi](y) for which the marginal OB effect is a constant. This result is important because it provides further insight into the properties of [pi](y) determining whether the OB effect is large or small. This parametric model of [pi](y) should also be useful in empirical efforts to estimate and test the size of the OB effect.

The identity (4) decomposes the net marginal effect of the safety policy expenditures x on expected accident loss into the direct effect and the OB effect. From (4), if [pi](y) satisfies

[[pi]'.sup.2]/[pi][pi]" = c,

(where c > 0 is a constant) it follows that the marginal OB effect is a constant proportion, c, of the direct effect. It turns out that the reciprocal of the marginal OB effect,

(5) [pi][pi]"/[[pi]'.sup.2]

is the natural concept for describing properties of [pi](y) determining the size of the OB effect. I therefore solve the equivalent problem of finding the family of [pi](y) for which (5) is a positive constant that yields the same result.

DEFINITION 3. The coefficient of diminishing returns (CDR) for the accident probability function [pi](y) is

(6) [delta](y) [equivalent to] ([pi][pi]"/[[pi]'.sup.2]),

where [delta](y) is the reciprocal of the marginal OB effect. (15)

DEFINITION 4. Diminishing returns are weak (WDR), moderate (MDR), or strong (SDR), depending on whether 0 < [delta](y) < 1, [delta](y) = 1, or [delta](y) > 1, respectively.

Hence, WDR and dominant OB are equivalent; as are SDR and partial OB. Further, [pi](y) with constant CDR are the solutions to the differential equation (6), with [delta] > 0, which yields the following proposition.

PROPOSITION 5. The CDR [delta] is constant for the following [pi](y) functions. (1) For WDR (0 < [delta] < 1, [[pi].sub.w](y) [equivalent to] a[([mu] - y).sup.k], a positive power function, with [delta] [equivalent to] = (k - 1)/k; [for all] k > 1, a,[mu] > 0, and a restricted so that [[pi].sub.w](0) < 1, 0 [less than or equal to] y [less than or equal to] [mu]. (2) For MDR ([delta] = 1), [[pi].sub.m](y) [equivalent to] a[e.sup.-[bar.ky]], a negative exponential function, with 0 < a < 1, k > 0, y [greater than or equal to] 0. (3) For SDR ([delta] > 1), [[pi].sub.s](y) [equivalent to] a[([mu] + y).sup.k], a negative power function, with [delta] [equivalent to] (k - 1)/k, [for all] k < 0, a,[mu], < 0, and a restricted so that [[pi].sub.s](0) < 1, y [greater than or equal to] 0.

Proposition 5. is potentially important for determining the empirical significance of OB. It shows that the estimated power function parameter k in this model summarizes the magnitude of the OB effect, because k/(k - 1), [for all] k < 0 and [for all] k > 1 is the proportion by which OB reduces the marginal direct effect of x on A(x, y).

The results in Proposition 5 are summarized in Table 2. Table 2 shows that at the critical value [delta] = 1, where the direct effect of an increase in x is precisely canceled by the OB effect, [pi](y) is the negative exponential function. For all [delta] > 1, the OB effect only partially offsets the direct effect of x, and [pi](y) is a left-translated negative power function with k < 0. For 0 < [delta] < 1, the OB effect is dominant and more than offsets the direct effect of x. In this case, [pi](y) is a right-translated positive power function with k > 1.

The family of constant [delta] functions [pi](y) described in Proposition 5 can be uniquely partitioned so that in each partition member every [pi](y) has the same initial value [pi](0), and the same initial slope [pi]'(0). Each partition member contains a unique [pi](y) with a constant [delta], [for all] [delta] > 0. Comparing the properties of [pi](y)'s with different [delta]'s is greatly simplified by restricting the comparisons to [pi](y)'s belonging to the same partition member.

Figure 2 illustrates the three different types of [pi](y). They have [delta] = 0.5 (dominant OB), [delta] = 1 (complete OB), and [delta] = 1.5 (partial OB) for the partition member with [pi](0) = 0.1, [pi]' (0) - 0.00008. The figure provides geometric intuition on how properties of [pi](y) vary With [delta]. By construction, all elements of a partition member intersect only once, at y = 0.

[FIGURE 2 OMITTED]

Figure 2 shows that as the victim's accident reduction expenditures y increase, [pi](y) tapers off more rapidly the larger the [delta] (i.e., the stronger are diminishing returns). This property explains why the parameter [delta] is defined as the coefficient of diminishing returns. The simple inverse relation between the magnitude of the marginal OB effect (1/[delta]) and the strength of diminishing returns [delta] greatly simplifies the analysis of models in which the expected accident loss function, A(x,y), is multiplicatively separable.

A distinctive feature of [pi](y) with WDR (dominant OB) is that [pi](y) actually attains 0 (at y = [mu]), where [pi](y) is tangent to the y-axis. The other types of [pi](y) only approach the y-axis asymptotically. This distinction has little empirical relevance, because the model implies the victim's OB censors outcomes with y > y(0). However Proposition 3 implies that log[[pi](y)] is a decreasing concave (convex) function if OB is dominant (partial), which may allow empirical testing.

Figure 2 implies the surprising result that for [pi](y)'s in the same partition, those with a larger OB effect (i.e., [pi](y)'s with smaller [delta]) are technologically superior in reducing A(x,y) to those with a smaller OB effect. Proposition 5 formulates this result more precisely.

PROPOSITION 6. (1) Less victim expenditure y is required to attain any specified accident probability [[pi].sup.*] = [pi](y) < [pi](0) the smaller [delta]. (2) Furthermore, for any specified level of regulatory expenditure x, the victim equilibrium accident probability [pi][y(x)] increases monotonically with [delta]. (16) (3) Finally, the equilibrium value of [pi][y(x)]L(x) + y(x), which the victim seeks to minimize, increases monotonically with [delta].

Discussion. Part (1) is just a verbal description of the nonintersection of the elements of [pi](y) except at y = 0, as illustrated in Figure 2. Parts 2 and 3 are obtained by straightforward but tedious comparative statics calculations of the victim's first-order conditions as k changes. (17)

Net Effects of Safety Policy Expenditures if OB Is Ignored

The effect on expected loss of adopting a policy with expenditure x relative to not adopting it is

{[pi][y(x)]L(x)/[pi][y(0)]L(0)}

which takes into account the OB effect increasing n by the victim's reducing y. A prediction based just on the direct policy effect ignores OB and treats the initial probability [pi][y(0)] as a constant independent of x, yielding the prediction

{[pi][y(0)]L(x)/[pi][y(0)]L(0)} = L(x)/L(0),

which for x > 0 always exaggerates how much the policy will reduce expected accident loss.

This section summarizes a few illustrative calculations indicating the prediction error from ignoring OB. Numerical results are extended to the case of victims with constant relative risk aversion (CRRA). Three discrete levels of regulatory safety expenditures are considered, none, moderate, and high, denoted as x = 0, [x.sup.m], [x.sup.h] associated with losses L(0) = 100,000, L([x.sup.m]) = 50,000, L([x.sup.h]) = 25,000 when an accident occurs. The corresponding victim accident avoidance expenditures that optimize their objective function are y(0), y([x.sup.m]), y([x.sup.h]).

The entries in the two panels of Table 3 measure the expected accident loss for the two levels of [x.sup.i] relative to the expected loss when x = 0 (i.e., before the policy is adopted) as

[pi][y([x.sup.i])L([x.sup.i])]/[pi][y(0)L(0)], i = m, h.

The three rows in each panel correspond to [pi](y)'s with [delta] = 0.5, 1.0, 1.5, respectively. The four columns in each panel correspond to behavior of victims with CRRA parameter [rho] = 0, 0.5, 1.0, 2.0. In the following calculations, potential victims choose y to maximize expected utility defined as

[(1 - [rho]).sup.-1]{[pi](y)[[I - L(x) - y].sup.1-[rho]] + [1 - [pi](y)][(I - y).sup.1-[rho]]}; [rho] > 0, [rho] [not equal to] 1

{[pi](y)ln[I - L(x) - y] + [1 - [pi](y)]ln(I - y)}; [rho] = 1,

where [rho] is the CRRA parameter. For risk-neutral victims ([rho] = 0), maximizing expected utility and minimizing [pi](y)L(x) + y yield the same optimal y. For the calculations in this part, I = 120,000. For [delta] = 0.5, [pi](y) = 0.00000016[(y - 250).sup.2]). For [delta] = 1, [pi](y) = 0.01[e.sup.-0.008y]. For [delta] = 1.5, [pi](y) = 625 [(y + 250).sup.-2]. (18)

The direct policy effect ratios (which ignore the OB effect) of increasing x from 0 to [x.sup.m] corresponding to panel 1 would be the same constant, L([x.sup.m])/L(0) = 0.5 for all entries in the panel. Similarly, the direct effect of increasing x from 0 to [x.sup.h] corresponding to panel 2 would be the constant, L([x.sup.h])/L(0) = 0.25.

All the entries in the panels are much larger than the corresponding direct effects, indicating large OB effects in these examples. Entries greater than 1 indicate dominant OB, where the net effect of the safety policy increases the expected accident loss.

The first column in the two panels reveals the behavior of risk-neutral victims ([rho] = 0) as [delta] and x vary. It confirms how OB decreases as [delta] increases and that the expected accident loss increases with x when [delta] > 1. The net expected loss is at least 58% larger than the predicted expected loss if the OB effect is ignored. Of course, the prediction error from ignoring OB will be relatively minor for sufficiently large [delta] (say, [delta] > 5). But the empirical work has not yet been done to determine the relevant range of 5 and the implied size of the OB effect for real world applications.

As the risk-aversion parameter p increases, the entries in all rows of Table 3 increase. Thus higher risk aversion is associated with relatively larger OB. This result follows from the decreasing marginal importance of risk aversion as the size of the loss, L(x), decreases (from an increase in x). (19)

In the risk-neutral case the ratio,

[pi][y([x.sup.i])]L([x.sup.i])/[pi][y(0)]L(0)

decreases as [delta] increases, given x and [rho].

Table 3 shows for [rho] > 0, [delta] = 1 is no longer the critical condition for precisely offsetting OB, and [delta] = 1 now leads to dominant OB. Indeed, the last row in both panels shows the OB effect is dominant (>1) for sufficiently large [rho] when [delta] = 1.5, even though [delta] = 1.5 always leads to partial OB for risk-neutral victims.

These calculations clearly show how risk aversion increases the size of marginal OB effects. It follows regulatory policy that ignores OB further exaggerates the expected accident loss reduction likely to occur for victims with significant risk aversion.

V. WELFARE IMPLICATIONS OF OB

The welfare implications of OB depend critically on whether the value of the reduction in victim accident avoidance expenditures y is excluded or included as a social gain. Here I briefly discuss this issue, assuming that the safety regulator understands victim OB and takes into account in choosing the optimal x (if the safety policy is adopted).

Consider two alternative regulatory criteria. Choose x to minimize: (20)

(7) [W.sub.a][x,y(x)] = [pi][y(x)]L(x) + x [W.sub.b][x,y(x)] = [pi][y(x)]L(x) + x + y(x).

Both [W.sub.a], [W.sub.b] value the victim's expected accident loss, [pi]L, the same as the victim. Both policy objectives also recognize a trade-off in values between reducing [pi]L and the opportunity costs of achieving it. However, [W.sub.a] assigns no policy value to the victim's reduction of y, whereas [W.sub.b] includes it.

The first-order condition for these two policy objectives are:

(8) d[W.sub.a]/dx = 1 + d{[pi][y(x)]L(x)}/dx = 0

d[W.sub.b]/dx = 1 + d{[pi][y(x)L(x))/dx + dy(x)/dx = 0.

The discussion is facilitated by considering the [W.sub.i]; i = a, b, that have a unique interior minimum satisfying the first-order conditions in (8) and the second-order sufficient condition [d.sup.2][W.sub.i]/d[x.sup.2] > 0, 0 [less than or equal to] x [less than or equal to] [x.sup.i]. These assumptions ensure that d[W.sub.i]/dx < 0, 0 [less than or equal to] x < [x.sup.i]. I also assume that the marginal OB effect is partial over the relevant domain 0 [less than or equal to] x [less than or equal to] [x.sup.i] (so d{[pi][y(x)L(x)]}/dx < 0) or else OB is dominant over the relevant domain (so d{[pi][y(x)L(x)]}/ dx > 0).

When the OB effect is partial, [x.sup.b] > [x.sup.a], and therefore regulatory criterion [W.sub.b] leads to a greater reduction of expected accidental harm than criterion [W.sub.a]. (21) When the marginal OB effect is dominant, [x.sup.a] = 0 (i.e., the regulators don't adopt a safety policy). This conclusion follows because d[W.sub.a]/dx > 0, x = 0 if the OB effect is dominant. Thus only a corner solution exists in this case when the regulatory criterion is [W.sub.a]. Dominant marginal OB is consistent with the regulatory criterion [W.sub.b] if increases in x induce a larger decrease in y. This follows, because if dy(x)/dx < -1, d{[pi][y(x)L(x)]}/dx > 0 to satisfy the first-order condition, d[W.sub.b]/dx = 0.

In summary, using criterion [W.sub.a], a safety regulator will want to increase x as long as its marginal cost, 1, is less than the marginal benefit of reducing [pi]L. However, victim OB reduces the size of this marginal benefit. With criterion [W.sub.b.], the regulator will want to increase x as long as its marginal cost is less than the net marginal benefit from reducing [pi]L + y. With this criterion it makes no difference how the marginal benefit is distributed between reductions of [pi]L and y, which is why [pi]L can increase if y decreases by more than the increase in x.

Minimizing [W.sub.b] is a plausible social welfare criterion in some applications. (22) However, the political economy of safety regulation makes [W.sub.a] far more likely than [W.sub.b], as the regulatory cost-benefit criterion, even if the regulators fully understand OB and how it affects final outcome. It seems unlikely that regulators consider any reduction in y induced by their policy a social gain. y is not part of the regulator's budget, much less of the regulator's mandate. (23) Annual regulatory reports usually say little about the relationship between regulatory policy expenditures and changes in accident rates and expected accident losses; they say nothing about victim costs of accident avoidance or changes in them. The main reason that safety regulators do not treat the value of the reduction of y as a social gain is doubtlessly because a decrease in y directly increases the accident rate [pi]. The tenure of a safety regulator boasting the results from a policy that increases expected accident losses, but decreases accident avoidance costs to victims by even more can probably be measured in microseconds.

Regardless of whether the OB cost saving of y is included as a benefit, a competent analysis of the effects of a safety policy on expected accident loss and accident rates must take account of OB effects when they are large.

VI. CONCLUSIONS AND EXTENSIONS

The potential importance of OB on the net effect of safety regulation policies is ultimately an empirical question. This article has several important implications for the empirical study of safety policies and their effects. Specifying [pi](y) as a constant [delta] function provides a natural parametric micro-model for studying the size of OB effects. If data permit sufficiently precise estimation of the power parameter k, the marginal OB effect is obtained from 1/[delta] = k/k - 1. For example, if k < -1, OB reduces the direct safety policy effect between 50% and 100%. Even if data are too crude to estimate k, testing whether [pi](y) is decreasing and log-concave or log-convex is equivalent to testing whether OB is dominant or partial.

In this production function model of expected accident loss, the victim's reduction of y is the mechanism through which OB takes place. Whether the reduction of y is considered a benefit or not, it is important to look more carefully for empirical evidence of victims reducing y in response to the adoption of safety policies. In applications where y and changes in y can be directly observed, proxied, or reasonably inferred, this information could supply strong support for the OB story.

The illustrative calculations showed that increasing the risk averse taste parameter [rho] for a given safety technology parameter [delta] increases the size of the OB effect. Further work is required to determine how far one can disentangle the effects of [delta] and [rho] on OB.

The production function approach of this article, with its emphasis on [delta], is not limited to studying the effect of regulatory and safety policies. For example, an improvement in medical technology that reduces the consequences of severe injury may induce OB by potential victims, by reducing the marginal productivity of their accident avoidance expenditures.

APPENDIX: ANALYSIS OF OB WITH OTHER MODELS OF A(x,y)

The article specified the expected accident loss has the form A(x,y) [equivalent to] [pi](y)L(x). This is a plausible form for Peltzman's (1975) original study of automobile safety regulation, but is inappropriate for some applications. (24) This appendix maintains the assumption that the victim chooses y to minimize A(x,y) + y, and imposes the following assumptions.

Assumptions on the Expected Accident Loss A(x,y)

1. A(x,y) is a smooth, strictly decreasing function of x and y, that is, [A.sub.x], [A.sub.y] > 0, [there exist]x, y [greater than or equal to] 0.

2. A(x,y) is subject to diminishing returns from both x and y, that is, [A.sub.xx], [A.sub.yy] > 0.

3. x and y are technological substitutes in reducing A(x,y), that is, [A.sub.xy] > 0, an increase in x reduces the marginal product of y in reducing A (x,y).

4. The iso-loss curves, A(x,y) = K, are convex.

5. The value of the potential accident victim's best response to safety policy expenditures y(0) > 0.

If, contrary to assumption (3), x and y are technological complements (25) {A.sub.xy] < 0 in reducing A(x,y), then augmenting behavior is present instead of OB. This follows because the derivative of the victim's best response, y'(x) > 0. Although augmenting behavior may be relevant in some applications, it is ignored in this study.

With these assumptions, one again obtains relation (3) from the main text, that is,

dA(x)/dx = [A.sub.x]{1 - ([A.sub.xy]/[A.sub.yy])/([A.sub.x]/[A.sub.y])]}.

Hence the effect of OB precisely cancels the direct effect of an increase in x if

[([A.sub.xy]/[A.sub.yy])/([A.sub.x]/[A.sub.y])] = 1

This PDE condition is satisfied if the level curves of A (x,y) are vertical displacements of y = F(x), where F(0) > 0 F'(x) < 0, F"(x) > 0. (26) The PDE

[([A.sub.xy]/[A.sub.yy])/([A.sub.x]/[A.sub.y])] = K

with K > 0 can be solved by separation of variables, A(x,y) = F(x)G(y); F',G' < 0; F", G" > 0, this A(x,y) satisfies the five assumptions. The coefficient of diminishing returns, [delta], can be defined for G(y), and [delta] = 1/K.

The specification in the main text A(x,y) [equivalent to] [pi](y)L(x) is a special case, where the separability arises in a natural way. An important alternative specification is A(x,y) [equivalent to] [pi](x,y)L, where regulatory x and victim y accident reduction expenditures reduce the accident probability, and the loss L (or its expected value, conditional on an accident occurring) is constant. The five assumptions on A(x,y) are inherited by [pi](x,y). If the accident probability is multiplicatively separable so that [pi](x,y) = [alpha](x)[beta](y); [alpha]',[beta]' < 0; [alpha]", [beta]" > 0. [delta] is now defined for [beta](y). It again quantifies the strength of diminishing returns to the victim's y expenditures, and its reciprocal is the OB effect magnitude. This separable form is the most plausible accident probability model for many applications. It captures the idea that for all levels of regulatory safety expenditures, an increase in victim accident avoidance expenditures y leads to the same percentage reduction in [pi].

REFERENCES

Adams, John. Risk. London: University College London Press, 1995.

Allen, R. G. D. Mathematical Analysis for Economists. New York: St. Martin's Press, 1938.

Assum, Terje, Torkel Bjornskau, Stein Fosser, and Fridulv Sagberg. "Risk Compensation--the Case of Road Lighting." Accident Analysis and Prevention, 31, 1999, 545-53.

Barnes, Julian E. "A Bicycling Mystery: Head Injuries Piling Up." New York Times, August 29, 2001, 1.

Bear, D. V. T. "Inferior Factors and the Theory of the Firm." Journal of Political Economy, 73, 1965, 287-89.

Brown, John Prather. "Toward an Economic Theory of Liability." Journal of Legal Studies, 2, 1973, 323-50.

Calabresi, Guido. The Costs of Accidents. New Haven, CT: Yale University Press, 1970.

Dulisse, Brian. "Methodological Issues in Testing the Hypothesis of Risk Compensation." Accident Analysis and Prevention, 29, 1997, 285-92.

Ferguson, Charles E., and Thomas R. Saving, "Long-Run Scale Adjustments of a Perfectly Competitive Firm and Industry." American Economic Review, 59, 1969, 774-83.

Landes, William M., and Richard A. Posner. The Economic Structure of Tort Law. Cambridge, MA: Harvard University Press, 1987.

Noland, Robert B. "Traffic Fatalities and Injuries: the Effect of Changes in Infrastructure and Other Trends." Accident Analysis and Prevention (forthcoming).

Noland, Robert B., and Lyoong Oh. "The Effect of Infrastructure and Demographic on Traffic-related Fatalities and Crashes: A Case Study of Illinois County-level Data." Working Paper, Centre for Transport Studies, Imperial College, 2003.

Peltzman, Sam. "The Effects of Automobile Safety Regulation." Journal of Political Economy, 83, 1975, 677-725.

Sagberg, Fridulv, Stein Fosser, and Inger-Anne F. Saetermo. "An Investigation of Behavioural Adaptation to Airbags and Antilock Brakes among Taxi Drivers." Accident Analysis and Prevention, 29, 1997, 293-302.

Shavell, Steven. "Strict Liability vs. Negligence." Journal of Legal Studies, 9, 1980, 1-25.

--. Economic Analysis of Accident Law. Cambridge, MA: Harvard University Press, 1987.

Viscusi, W. Kip. Fatal Tradeoffs. New York: Oxford University Press, 1992.

Wilde, Gerald J. S. Target Risk. Toronto: PDE Publications, 1994.

(1.) See, for example, Wilde (1994), Adams, (1995), Viscusi (1992), and Dulisse (1997) for discussion of some of the findings and controversies.

(2.) See, for example, Barnes (2001), who reports head injury rates from bicycle accidents have increased by 10% despite the much wider use of bicycle helmets. Some safety analysts think this partly reflects riskier behavior by the victims.

(3.) Peltzman (1975) and Viscusi (1992) conclude that theoretical conditions for dominant OB have not been established. Wilde's (1994) risk homeostasis theory implies that risk to an individual will be unchanged by external factors unless they alter the individual's target level of risk. But the model is not intended to measure the size of the change in expected accident loss if the target level changes.

(4.) The expected accident loss function A(x,y) was essentially expressed as [pi](y)L(x) in Peltzman (1975), which leads to less clutter in the formal discussion than the widely used alternative A(x,y) [equivalent to] [pi](x,y)L (with L constant) in Brown (1973), Viscusi (1992), and Landes and Posner (1987). The relevance of the model does not depend on assuming [pi] is determined by the victim and L by the safety policy. See the appendix, which shows that essentially the same results are obtained for A(x,y) [equivalent to] [pi](x,y)L if [pi](x,y) = [alpha](x)[beta](y).

(5.) These assumptions imply the iso-loss curves A(x,y) = K are convex.

(6.) The assumption that the potential victim chooses y to minimize A(x,y) + y or to optimize some equivalent measure is the most widely employed formal hypothesis on victim behavior in the safety regulation literature, for example, Peltzman (1975), Viscusi (1992), and in the law and economics literature on tort liability, for example, Brown (1973), Landes and Posner (1987), and Shavell (1980, 1987). The analysis abstracts from imperfect information of the victim about A(x,y) or about the x chosen by the regulator, and ignores learning. Given the theoretical focus, it further abstracts from the regulator's objective (except in section IV) and possible strategic behavior. And it ignores the possibility of positive utility from taking risks.

(7.) By implicit differentiation of [pi]'(y)L(x) + 1 = 0, the first-order condition for maximizing (2).

(8.) Because A is a "bad," [A.sub.xy] [equivalent to] [pi]'L' > 0 implies that x and y are technological substitutes, that is, an increase in x decreases the marginal product of y. This reverses the sign of the definition of input substitutes for normal products.

(9.) This definitional argument does not prove the existence of an expected accident loss function A(x, y) [equivalent to] [pi](y)L(x) that will satisfy the victim's specified objective and possess the assumed properties of [pi](y) and L(x). It just shows that if strong OB is implied by the model, then x must be an inferior factor. An analytic proof that such solutions exist, with x an inferior factor is given in section III.

(10.) In this example of dominant OB, A(x,y) [equivalent to] [pi](y)L(x) [equivalent to] 0.00000016[(250 - y).sup.2]L(x) for three levels of x, 0 < [x.sup.m] < [x.sup.h], and L(0) = 100,000, L([x.sup.m]) = 50,000, L([x.sup.h]) = 25,000.

(11.) In this specific example, the increase in x decreases L(x), which lowers and flattens the top curve. The explanation in the text for this geometric behavior is more general.

(12.) dA/dx [equivalent to] [A.sub.x] + [A.sub.y]y' [equivalent to] [A.sub.x]{1 + [y'/([A.sub.x]/[A.sub.y])]} [equivalent to] [A.sub.x]{1 - ([A.sub.xy]/[A.sub.yy])/([A.sub.x]/[A.sub.y])]}.

(13.) This result makes no use of the multiplicative separability of A(x,y), because it only requires that [A.sub.xy] > 0, that is, that x and y are substitutes in production.

(14.) Log concavity of [pi](y) also implies strong OB in Peltzman's (1975) slightly more complicated model. In it, potential accident victims also make expenditures c to reduce L(x, c). The expected accident loss is therefore A(x,y,c) = [pi](y)L(x,c), and the victim chooses y and c to minimize [pi](y)L(x, c) + y + c. The multiplicative separability of [pi]L implies the first-order condition for minimizing with respect to y, [pi]'L(x, c) + 1 = 0 is of the same form, whether L contains the endogenous variable c or not. Suppose initially the regulatory variable x = 0, and victim optimization leads to a loss of L(0, [c.sup.0]) if the accident occurs. With regulation, x' > 0, the optimized loss is L([x.sup.r], [c.sup.r]). The only requirement for the log concavity of [pi](y) to ensure that expected loss increases with [x.sup.r] is that L([x.sup.r], [c.sup.r]) < L(0, [c.sup.0]), that is, x and c cannot be supersubstitutes such that an increase in x leads to a reduction in c that actually causes a net increase in L.

(15.) This definition of [delta](y) is formally identical to the reciprocal of the own-price elasticity of substitution for homogenous functions, [[sigma].sub.yy]. See Allen (1938), where for homogenous of degree one production functions f(x,y), the partial own-price elasticity of substitution [[sigma].sub.yy] = ([f.sub.y][f.sub.y])/(f[f.sub.yy]). In my definition of the CDR [delta] = ([pi][[pi].sub.yy])/([[pi].sub.y][[pi].sub.y]), the formal reciprocal of [[sigma].sub.yy]. Of course, the class of functions for applying these definitions are quite different.

(16.) Suppose a safety regulator understands how OB affects the net outcome and wants to minimize [pi](y)L(x)+x. Part 2 of Proposition 5 shows that the minimum value of the optimized function [pi](y)L(x) + x decreases as [delta] decreases even though the marginal offset effect fraction, [[delta].sup.-1] increases. This outcome reflects the technological superiority of [pi](y)'s with smaller [delta]'s when they are elements of the same partition member.

(17.) In these calculations, a and [mu] are treated as functions of k so that the [pi](y)'s all belong to the same partition member.

(18.) Maple is used to calculate the victim's optimal y(0), y([x.sup.m]), y([x.sup.h]), corresponding to the losses L(0) = 100,000, L([x.sup.m]) = 50,000, L([x.sup.h]) = 25,000.

(19.) For any given L(x), [pi][y(x)] decreases as [rho] increases, because of greater risk aversion. But [pi][y(x)]/ [pi][y(0)] increases because the marginal importance of risk aversion is greater for L(0) than for L([x.sup.i]); i = m, h.

(20.) Although the victim actually chooses y(x), the regulator's knowledge of the victim's behavior assumed here enables the regulator to ensure this selection of y through the appropriate choice of x.

(21.) This follows because d[W.sub.b]/dx = d[W.sub.a]/dx + dy(x)/ dx. Because dy(x)/dx < 0 and at [x.sup.a], d[W.sub.a]/dx = 0, d[W.sub.b]/ dx < 0 at [x.sup.a]. Therefore [x.sup.b] > [x.sup.a].

(22.) The criterion of minimizing A(x,y) + x + y or some equivalent was introduced informally into the law and economics theoretical tort liability literature on bilateral accidents by Calabresi (1970), and formalized by Brown (1973), Landes and Posner (1987), and Shavell (1980, 1987).

(23.) Suppose y primarily reflects the victim's time cost, or the victim's psychological distaste for the effort in taking accident avoidance. In that case, a regulator might assign no value to the reduction in y by the victim, while understanding how OB constrains the policy outcome.

(24.) See discussion of Peltzman's slightly more elaborate model in note 14.

(25.) For complements, assume that if and y are increased by a factor t (so that the expected accident loss is A(tx, ty); [A.sub.t] < 0, [A.sub.tt] > 0.

(26.) The conditions on F satisfy the assumption on A(x,y), that the level curves of A(x,y) are decreasing convex curves. The general solution of the PDE [A.sub.xy][A.sub.yy]/ [A.sub.x][A.sub.y] = 1 is A(x, y) = G[y - F(x)], with G > 0, G' < 0, G" > 0.

JOHN C. HAUSE, This study was initiated at the Stigler CSES, University of Chicago, and I am indebted to its support. Norman Meyers's comments greatly improved the general results in the appendix. Tom Saving, Finis Welch, Larry Sjaastad, Yoram Barzel, Gary Becker, Michael Grossman, Anders Klevmarken, Tom Muench, Walter Oi, Sam Peltzman, Warren Sanderson, and several workshop and seminar participants at Chicago, Columbia, Hamburg, Imperial College Centre of Transport Studies, Institute of Transport Economics (Oslo), Kiel, London School of Economics, Minnesota, Rochester, SUNY Stony Brook, Texas A&M, and Uppsala provided substantial comments on earlier drafts. Lee Hause provided significant editorial improvements of countless revisions. Li Xu efficiently generated the computer graphics. I am solely accountable for conclusions, lapses, and errors.

Hause: Professor Emeritus, Dept. of Economics, State University of New York, Stony Brook, NY 11794-4384. Phone 1-212-489-7384, E-mail jhause@notes. cc.sunysb.edu

TABLE 1
Example of Dominant Offsetting Behavior

 Victim Victim [pi] [pi][y(x)]
 Loss Care Probability [y(x)] L(x) +
x L(x) y(x) [pi][y(x)] L(x) y(x)

0 100,000 218.75 0.000156 15.625 234.38
[x.sup.m] 50,000 187.50 0.000625 31.250 218.75
[x.sup.h] 25,000 125.00 0.002500 62.500 187.50

TABLE 2
Accident Probability Functions [pi](y) with [delta] Constant

[pi](y) Domain of [delta] Extent of OB

a[([mu] - y).sup.k] 0 < [delta] < 1 Dominant

a[e.sup.-ky] [delta] = 1 Complete

a([mu] + y) [delta] = 1 Partial

[pi](y) Domain of Victim Care Parameter Domain

a[([mu] - y).sup.k] [mu] [greater than a > 0, k > 1, [mu] > 0
 or equal to] y [greater
 than or equal to] 0

a[e.sup.-ky] y [greater than a > 0, k > 0
 or equal to] 0

a([mu] + y) y [greater than a > 0, k < 0, [mu] > 0
 or equal to] 0

[pi](y) Sign of Change in A

a[([mu] - y).sup.k] +

a[e.sup.-ky] 0

a([mu] + y) -

TABLE 3
Ratio of Expected Accident Losses for CRRA
Victims When Regulatory Expenditures
Reduce L(x)

 [rho] = 0 [rho] = 0.5 [rho] = 1.0 [rho] = 2.0

{[pi][y([x.sup.m])]L([x.sup.m])}/{[pi][y(0)L(0)]}

[delta] = 0.5 2.00 3.10 5.60 24.90
[delta] = 1.0 1.00 1.26 1.70 3.50
[delta] = 1.5 0.79 0.90 1.10 1.83

{[pi][y([x.sup.h])]L([x.sup.h])}/{[pi][y(0)L(0)]}

[delta] = 0.5 4.00 7.20 14.80 92.10
[delta] = 1.0 1.00 1.34 1.90 4.80
[delta] = 1.5 0.63 0.70 0.90 1.79