The basic bilateral approach
Let xv > 0 be the level of care taken by the victim, and let xt > 0 be the level of care taken by the injurer. Then we assume the following probability of damage function:
Because the victim can now take care, the analysis becomes a little bit more complicated than the analysis of the unilateral care model. The marginal benefit of each party's action now depends on the action chosen by the other party. Thus, when choosing his action, each party must form a view about what the other party will do, because each player can act strategically under different institutional arrangements in response to what they believe the other player's best response is.
Let us assume that the injurer can take a level of care xt > 0, at a constant marginal cost of wi > 0, and the victim can take a level of care xv > 0 at a marginal cost of wv > 0. The probability of an accident occurring, given that care levels are xi and xv, is p(xi,xv) e [0,1]. Generalising this model to allow for the possibility that the level of harm could also depend on the level of care taken by each party, we have H(xilxv) = p(xi,xv) h(x,xv), where we assume:
- • dH < 0. If the injurer takes more care, the expected harm falls. d2 H
- • 2 > 0. There are diminishing marginal returns to providing more dx2
care by the injurer: more care reduces both the probability and harm from an accident, but the rate of reduction falls as more care is provided.
- • IlH < 0. If the victim takes more care, expected harm falls.
- 32 H
- • —— > 0. There are diminishing marginal returns to providing more 3xv
care by the victim: more care reduces the probability and harm from an accident, but the rate of reduction falls as more care is provided. 32 H
• > 0. For any level of care that is provided by the victim, as more
Эх;Эх„ J v у ,
care is provided by the injurer, the marginal effectiveness of the victim's care falls. And, conversely, for any level of care that is provided by the victim, as more care is provided by the injurer, the marginal effectiveness of the victim's care falls. Another way of saying this is - the levels of care taken by each of the players are substitutes: when the victim increases his amount of care, this reduces the marginal effectiveness of the injurer's care. To restore the marginal effectiveness to its original level, the injurer reduces his amount care. Conversely, if the injurer increases his level of care, the victim will reduce his level of care.
This final behavioural assumption need not always be true in reality - it will depend on the nature of the activity in question and the nature of the accident that is being analysed.
126.96.36.199 Efficient levels of care
Let us first characterise the efficient levels of care by both parties in this model. Recall that the efficient levels of care minimises the aggregate expected costs of care. The aggregate expected costs here are:
There will now be two first-order conditions instead of one. At the efficient care levels (x* > 0, x* > 0), these conditions are:
Again, these conditions are of the familiar form:
- 1. Social marginal cost of care by injurer = Expected social marginal reduction in damages due to care by injurer
- 2. Social marginal cost of care by victim = Expected social marginal reduction in damages due to care by victim