The bilateral care model in a market setting
We now extend the previous analysis to encompass a market setting.
Consider two perfectly competitive industries, denoted by i and v. Firms
in industry i cause harm to firms in industry v, but firms in both industries can take actions to reduce expected harm to firms in industry v. To
simplify the analysis, we make a series of assumptions:
- • Consumers of i and v regard the goods as neither substitutes nor complements. The benefits of consumption of each good are identical, so that the industries have the same demand curves, Qt = Qv ^ u'(Qt): = u'(Qv) .
- • Firms in each industry have identical constant marginal costs of ci = cv = c .
- • As a result of the constant marginal cost assumption, we can treat firms in each industry as a single representative price taking firm. The representative firm in industry i produces Qi units of output, and the firm in industry v produces Qv units of output.
- • The marginal costs of care in each industry are identical, so that wi = wv = w.
- • The expected per unit harm function has the following symmetry property: дН = дй ^ X = xv. (For example, this condition is satisfied by the constant elasticity of expected harm function if a = в = 0.5.)
Under these conditions, total welfare is:
The situation we have in mind is similar to that discussed in Chapter 4, with the added assumption that beer-producing firms can reduce harm by decontaminating the water before using it in beer production. This decontamination process is costly. The per unit expected costs to the beer industry of water pollution are still H (xi, xv). The total harm of pollution rises with the quantity produced in each industry. For a given level of total output of industry v (say Qv units), then the total harm that the industry faces could increase if either (or all) of the following occur:
- • Firms in industry i industry each choose a lower level of care (e.g. less decontamination);
- • Firms in industry i choose a low level of care;
- • Firms in industry i produce more output; or
- • Firms in industry v produce more beer.
The efficiency conditions are now:
The first two conditions in (5.1) and (5.2) relate to the output levels. Given optimal levels of per unit care, the marginal consumption benefit of the last unit of output in industry i should be equal to the full marginal cost, where the marginal cost depends on the production cost, the per unit costs of care, and the marginal cost of harm to firms in industry v. Similarly, the marginal consumption benefit of the last unit of output in industry v should be equal to the sum of the marginal production cost, the per unit costs of care, and the marginal cost of harm to firms in industry v. Note that under our assumptions, we have:
which implies that Q* = Q*.
Figure 5.4.1 Efficiency in the bilateral care model in a competitive market setting
The last two conditions in (5.3) and (5.4) relate to care, and simply state that care should be undertaken by firms in each industry up to the point where marginal cost equals marginal benefit, noting that for each industry marginal benefit depends on output levels in the other industry. Note that under our assumptions, we have:
which, together with the assumptions on H and the fact that Q* = Q*, imply that x* = x* = x*.
Since the demand curves in each industry are Pi = u' (Qi) and Pv = u' (Qv), and are equal to each other, the efficiency conditions imply that at the efficient allocation, the efficient prices in each industry must also be identical:
The efficient outcome is shown in Figure 5.4.1.