Question

Suppose that an injurer and victim have been in an accident. The level of precaution for each is given by x and y, respectively, and these must be non- negative real numbers. The ex ante probability of an accident was p(x, y) = 1/[x+y+1]. Note that for a probability function q(k) = 1/[k+1], the marginal probability is given by q′(k) = − 1/[k+1]2.

The constant marginal cost of care for the injurer and victim are w and z respectively, and the cost of an accident is given by A, which is 50,000. Suppose w = 5 and z = 10.
a) Find the socially optimal levels of precaution x* and y*.

Answer 1

The socially optimal levels of precaution are x* = 20 and y* = 40.

Efficiently balancing the marginal costs of care with the marginal reduction in accident probability yields the socially optimal precautionary levels of x* = 20 and y* = 40.

Step-by-step explanation:

To determine the socially optimal levels of precaution, we use the formula for the ex ante probability of an accident,
`\(p(x, y) = (1)/(x+y+1)\)`. The marginal probability is \(p_x' = -\frac{1}{(x+1)^2}\) for the injurer and
`\(p_y' = -(1)/((y+1)^2)\)` for the victim. Given the constant marginal costs of care (w = 5 for the injurer and z = 10 for the victim), the socially optimal levels of precaution occur when the marginal cost equals the marginal reduction in accident probability.

Setting (w =
`p_x'`) and (z =
`p_y'`) and solving for x and y, we find that (x* = 20) and (y* = 40). These values represent the socially optimal levels of precaution, ensuring that the marginal costs of precaution for both the injurer and the victim are equal to the marginal reduction in accident probability, leading to an efficient allocation of precautionary efforts.

Efficient allocation of precautionary efforts is a key concept in economic analysis, especially in the field of law and economics. It involves balancing the costs and benefits of precautionary measures to achieve an optimal level of risk reduction. This approach is crucial in understanding how legal and policy frameworks can incentivize individuals to take socially desirable levels of precaution to minimize the occurrence of accidents and associated costs.