D IFFERENT T ORT M ODELS *
5.3. Stochastic variation of the harm
In the foregoing sections we have described a deterministic model, that is a model in which the harm is exactly determined either exogenously, h, or endogenously as a function of the level of precaution, h(x) or h(z). In this section we introduce a stochastic variation of the harm for each of the four models, and discuss possible variations of the conclusions reached so far.
As we are assuming risk-neutrality, the increase in the risk borne by the injurer does not affect per se the outcome.
We find that stocasticity does not change the conclusions reached for the first model, because the injurer’s expectation will be the same. In contrast, under the other three models, the level of the harm depends on the injurer’s precaution. In the deterministic model, when the injurer decreases his level of precaution, and the harm becomes too great, he will pay just the threshold. In the stochastic model this sharp effects fades, because when the injurer decreases his level of precaution, he will not be able to pay damages if the harm is high, but he will pay for them if the harm is low.
Therefore, there is an intermediate zone between paying the harm and paying only the threshold; in this zone the injurer pays sometimes the harm (if low) and sometimes the threshold (if the harm is high). This fact might change the results: the injurer might take an inefficient level of precaution, but different from the one the deterministic model would
17 In this case, as st=s*, then tmin becomes equal to h(z*)+z*/p(s*), which is clearly not satisfied by t=h(z*), The injurer takes optimal precaution-reducing precaution but still opts for no magnitude-reducing precaution.
predict.
First (stochastic) model: probability model
With respect to the first model nothing changes if we introduce stochasticity. In fact, the harm is independent from the level of precaution and the injurer, instead of paying a determined harm, pays an expected harm.
We introduce a stochastic variation for h between a lower limit, h, and an upper limit, h; π(h) will be the density function for h, and E(h) the expected harm.
(21) =
∫
hh
hdh h h
E( ) π( ) .
The injurer’s expenses function is p(x)E(h)+x, and it is minimized by x*, the optimal level of precaution18. In the presence of a threshold, t<h, the expected harm becomes:
(22) E(t) (h)hdh (h)tdh E(h)
t
h
h
t
<
+
=
∫
π∫
π .19The injurer’s expenses function is p(x)E(t)+x, and it is minimized by xt, a lower than optimal level of precaution, as he expects to pay lower damages for any level of precaution20. Therefore, as in the deterministic model, if the threshold is lower than (some realizations of) the harm, the injurer does not fully internalize the harm that he can cause and hence takes less than optimal precaution.
Second (stochastic) model: magnitude model
In the magnitude model the harm is a function of the level of precaution, h(x). In order to capture the stochastic component of the harm, we make a simplifying assumption: we assume that nature randomly select a magnitude function, hi(x), among all possible magnitude functions, with probability πi, (
∑
πi =1), and we assume that all possible magnitude functions have the same characteristics of being decreasing in x at a decreasing rate and that they do not intersect. As the core of our argument does depend on the presence of a stochastic variation and not on the number of different possible harm functions, it will not change if we consider the simplest case of only two magnitude functions, h(x)<h(x), for any x, occurring respectively with probability π and π , (π +π =1).
18 As under the deterministic model x* is optimal because is yields the minimum in the social cost function, p(x)E(h)+x.
19 Obviously, if t is lower than the lower limit, the first addendum disappears, and the injurer always pay t.
20 This point can be easily verified by confronting the first order conditions for E(h) and for E(t).
The difference with respect to the deterministic model, Exp. (5), is that in this case we can single out three cases instead of only two: J(x) occurs when the injurer can always compensate the victim for the harm, t≥h(x); Jt(x) occurs when the injurer cannot pay full compensation for any harm, t≤h(x); in addition to those, there is an intermediate case, Jm(x), occurring when the injurer can pay the harm only if the lower harm results, h(x)<t<h(x).
The injurer fully internalizes the harm in J(x) and his level of precaution will be the optimal level of precaution x*, which minimizes his expenditure function, as in the deterministic model. Similarly, Jt(x) is minimized by x=0, as in the deterministic model. The only difference is therefore the second case. Jm(x) is clearly convex21, and is minimized by a level of precaution, denoted by xm, which is lower than optimal, xm<x*22.
After confronting J(x*), Jt(0) and Jm(xm), the injurer will select the level of precaution which yields the lowest cost. Therefore, the feasible outcomes are not just two as in the deterministic model, but three: x*, x=0, and xm. In other words, the outcome can be optimal precaution, no precaution, or underprecaution.
There are however two criteria to be used to limit the range of possible outcomes:
(i) If t >h(0), then Jm(xm)<Jt(0)23, otherwise either Jm(xm)<Jt(0) or Jm(xm)>Jt(0) can result;
(ii) If t<h(x*), then Jm(xm)<J(x*)24, otherwise either Jm(xm)<J(x*) or Jm(xm)>J(x*) can result.
Therefore, we can predict some of the outcomes, by just looking at the level of the threshold. By combining the two former results we obtain:
(a) If t<h(0) and t<h(x*), the outcome might be either x=0 or xm; (b) If h(0)<t<h(x*), the outcome will be xm;
(c) If h(x*)<t<h(0), the outcome could be any of x=0, xm, and x*.
21 It is worth recalling that h”(x)>0.
22 By confronting the first order conditions for J(x) and Jm(x) we can verify this result. For J(x) we have
π h”(x)>0, necessarily also x*>xm.
23 In fact in this case: Jt(0)>Jm(0)>Jm(xm).
24 In fact in this case: J(x*)>Jm(x*)>Jm(xm).
(d) If t>h(0)and t>h(x*), the outcome could be either xm or x*.
More precise answer to the question of what level of precaution that the injurer will choose may be obtained by adopting a more refined model for the stochastic variation of the harm. The point we want to make here is that the stochastic variation might sometimes change the conclusions of our model. However, a little stochastic variation around the expected value of the harm is not likely to affect our conclusions.
Third (stochastic) model: joint probability-magnitude model
In the third model, the injurer can reduce the probability of an accident as well as the magnitude of the harm. Nevertheless, the same kind of reasoning as before can be applied here also.
The four criteria described above ought to be rephrased as follows:
(a) If t<h(0) and t<h(x*), the outcome could be either xt or xm; (b) If h(0)<t<h(x*), the outcome will be xm;
(c) If h(x*)<t<h(0), the outcome could be any of xt, xm, and x*.
(d) If t>h(0)and t>h(x*), the outcome could be either xm or x*.
Fourth (stochastic) model: separate (3-dimensional) probability-magnitude model The fourth model results as a combination of the first two; therefore, the same considerations already made can be applied here.