A no liability rule
Under a no liability rule, the demand curve facing each firm is: and so each firm's profits are:
Each firm can maximise its profits by choosing xt to minimise wxt + 1H(x). Firms will therefore choose a level of care, xNL, which satisfies:
which is the same result as we had under perfect competition. This means that:
Each firm's profits are:
Each firm again chooses its quantity, taking the choices of other firms as given. The first-order condition is:
Adding up across all n firms yields:
The left-hand side is the same expression as the left-hand side of equation (6.9), evaluated at QNL rather than QL. It is the sum of the individual marginal revenues of each firm. The right-hand side is the number of firms (n) multiplied by the sum of the marginal production costs, the cost of care, and the perceived harm to consumers. If l < 1, then according to (6.10), wxNL + 1H(xNL) < wx* + H(x*), and we must have:
We have assumed that u"(Q) < 0, but have said nothing about the sign of, u"'(Q), which is the second derivative of the demand function. Let's consider the linear demand function, P = a - Q. If the demand function is linear, then u"'(Q) = 0, and the derivative of the function f(Q) = u"(Q)Q + nu'(Q) is f(Q) = u"'(Q)Q + u"(Q) + nu"(Q) = u"(Q) + nu"(Q) < 0, and so (6.12) implies that QNL > QL. This is also true as long as u"'(Q) < 0, and may even be true if u"'(Q) > 0, as long as this is not 'too positive'.
On the other hand, if l > 1, then according to (6.10), wxNL + 1H(xNL) > wx* + H(x*), and Qnl < Ql. Finally, if l = 1, then wxNL + H(xNL) = wx* + H(x*), and Qnl = Qsl.
Intuitively, under a no liability rule, firms provide care as long as consumer marginal willingness to pay [which is -lH'(x)] exceeds the marginal cost of care w. Accordingly, if consumers underperceive risk, then firms will underprovide care, which drives down their per unit costs and induces them to produce more than they would under strict liability. If consumers overperceive risk, then firms will overprovide care, driving up their per unit costs and inducing them to produce less than they would under strict liability.