By choice of units, 1 unit of pollution is generated for each unit of X produced. We call this the base level of pollution and denote it by B. Producers have access to an abatement technology however, which for simplicity we assume uses only good X as an input. For a given base level of pollution B, the amount of pollution abated, A, is given by the function 1A(xa,B), where xa is the amount of resources allocated to abatement. We will treat l as a parameter that may be affected by technological change. Pollution emissions are then given by B minus A, or:
where 9 = xg/x is the fraction of X output devoted to abatement, and a(9) ° A(9,1). We assume there is no abatement without inputs, and that it is not possible to fully abate all pollution: i.e. a(0) = 0 and 1a(1) < 1. Note our specification implies increasing marginal abatement costs since, for a given level of base pollution, there are diminishing returns to abatement activity.
We can now specify the equilibrium conditions for the production side of the economy. We assume the government uses pollution emission taxes (which are endogenous) to reduce pollution. Given the pollution tax t, the profits px for a firm producing X are given by revenue, less production costs, pollution taxes, and abatement costs:
Firms will jointly choose gross output (x) and their abatement fraction 9 to maximize profits. Define
Because of constant returns to scale, the output of an individual firm is indeterminate, but for any level of output, the first order condition for the choice of 9 implies
where 9′ > 0. As one would expect, abatement activity is increasing in the level of the pollution tax.
With free entry, firms will enter each industry until profits are zero. Using (2.4), we have for the X industry
We assume both industries are active, and hence (2.7) and (2.8) determine factor prices w and r as functions of p . Factor prices in turn determine the unit input coefficients for each sector. For example, by Shepherd’s Lemma, the unit labor requirement in X is given by cw ° dcx/dw, etc. The full employment conditions then determine outputs:
where, as noted before, x denotes gross output of X. Net output of X (that remaining for consumption and/or export) is xn = x – xa = x(1 – 9).