Consider a firm with the production function x=f(l)=l a , where <0 and emissions are proportional

Consider a firm with the production function x=f(l)=l^a, where α<0 and emissions are proportional to x so that e=d•x. Suppose as well that the firm has a separable abatement cost function AC(e)=β(ê−e).

(a) Derive the overall cost function C(x,e) and examine its properties.

(b) Suppose instead that the emissions are given by e=d•l. Derive the overall cost function for this case, and compare its properties to part (a).

(c) Suppose instead that the production function is x=f(l)= γ•l.
Derive the overall cost functions for the e=d•x and e=d•l cases.

(d) Consider the production function in exercise 5.2, with e=d•x and separable abatement cost function AC(e)=β(ê−e). Derive the overall cost function C(x,e) and determine its properties. 

(e) Repeat part (d) using e=d•l₂.

Data from exercise 5.2

Consider a firm whose technology satisfies assumption 5.2. The cost function is c(x)=c·x²/2. The output price is fixed and denoted by: p. Emissions are proportional to output according to e=α·x. The firm is subject to an emission tax τ.