Efficiency when goods are potentially harmful
Consider a market for a good that is potentially harmful to consumers. Suppose any positive level of care can be chosen by the injurer, which in this case is the firm that is selling the good to the consumers, who are also victims in this setup. We will assume that all firms are identical. Let the total expected harm to consumers be equal to nqH(x) where q is the quantity produced by each firm, x is the level of care taken by each firm, and H(x) is the expected harm per unit of the good that is consumed. Let each firm's cost function be equal to C(q). Finally, let each firm's per unit cost of care be equal to w. Total welfare is given by:
Welfare maximisation now requires three conditions. First, for a given number of (identical) firms, it must not be possible to increase welfare by having each firm produce more. This means that:
Equation (6.1) states that consumption benefits from consuming the last unit of the good must equal the full marginal costs of consumption, where those costs now include the marginal expected harm that occurs as a result of consuming the good, as well as the cost of care. Notice that this condition implies that consumers should consume less of the good than they would if the good was harmless.
Second, for a given quantity, it must not be possible to increase welfare by changing the number of firms. This means that:
Equation (6.2) states that the marginal consumption benefits must also equal average expected costs, where these costs again include the expected harm that occurs as a result of consuming the good. Since the consumer equates marginal benefits with price P, welfare maximisation again requires price = marginal cost = average cost.
Finally, efficiency requires that it must not be possible to increase welfare by altering the level of care:
Equation (6.3) states that for the last unit of care that is provided, the reduction in expected harm (which is the marginal benefit of care) should be equal to the marginal cost providing that last unit of care.