Consider a unilateral care accident model in which the probability of an acciden

Question

Consider a unilateral care accident model in which the probability of an accident is given by p(x) = e^-x where x is the level of injurer care, and e is the base of the natural logarithm. Let D be the dollar amount of damages suffered by the victim in the event of an accident, and let c be the unit cost of care for the injurer. Write down the expression for total accident costs (the injurer's cost of care plus expected damages and solve for the optimal level of care, x as a function of c and D (Remember that In(e^x) = x.) Show that x * is increasing in D and decreasing in c Consider the basic unilateral accident model where p(x) is the probability of an accident, p 0, x is the injurer's spending on care; and D is the victim's fixed damages. Under a strict liability rule, the injurer faces liability, L. equal to D. and hence chooses efficient care of x * Suppose that the injurer has limited assets, A to spend on liability. That is, L lessthanorequlato A. How does this constraint affect the injurer's care choice when A

Explanation / Answer

1)

a) p(x) = e-x

Total accidents costs = cx + expected damages

                               =cx + p(x).D = cx + e-x . D

min (cx + e-x . D) . differentiating we get

c -De-x =0

c=De-x

c/D = e-x

c/D=1/ex

ex = D/C

taking log on both sides

x* = ln(D/C) = Ln d - ln c

b) dx/dD = 1/d >0 .......increasing

dx/dc=-1/c<0...decreasing

c) x* = ln(500/10) = 3.91

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