Final answer:
The optimal levels of precaution for a potential injurer and victim can be found by setting the marginal benefit of increased care (reduction in expected accident costs) equal to the marginal cost of care. Given the probability of accident p(x, y) and the specific costs for injurer w and victim z, as well as the cost of an accident A, we solve for x and y that equate marginal benefits to marginal costs.
Step-by-step explanation:
To determine the optimal levels of precaution for a potential injurer and victim (denoted by x and y, respectively), we use the principle that marginal benefits (MB) should equal marginal costs (MC). The probability of an accident is modelled by the function p(x, y) = 1/[(x+1)(y+1)]. The marginal changes in probability with respect to x and y are given by px(x,y) = -1/[x+1]2[y+1] and py(x,y) = -1/[x+1][y+1]2, and the cost of an accident is A = 2500. Given constant marginal costs of care w = 5 for the injurer and z = 10 for the victim, we look for levels of x and y where the marginal cost of increasing care by one unit equals the marginal benefit in terms of the reduction in the expected cost of accidents.
The expected benefit of increasing precaution by one unit is the reduction in the expected cost of accidents, which can be calculated as the decrease in probability of an accident times the cost of an accident A. Therefore, the marginal benefit for the injurer is A × px and for the victim, it is A × py. Equating these to the marginal costs, we get the equations:
- MB for injurer: A × px = w
- MB for victim: A × py = z
By solving these equations, we find the values for x and y that optimize the total costs considering both prevention efforts and the cost of accidents. The actual calculation would involve setting up these equations and solving for x and y.