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1. In this problem, we determine the marginal abatement cost function starting from the general

cost function c(q,e) = }((q-e)2 +q).

3.5. PROBLEMS

41

a. First we find TBAU. Write the firms profit maximizing function. Take the derivative with

respect to e and q. Solve the two first order conditions for e and q as a function of p, the

price of the good. Substitute these values back into the objective function and simplify.

This gives TBAU

b. Next we find the restricted solution in which e = r. Write the firms profit maximizing

function. Take the derivative with respect to q. Solve the first order condition for q as a

function of p and r. Substitute this value back into the objective function and simplify.

This gives (g(x),x).

c. Now determine C(x) = TBAU – #),r). This is the abatement cost function.

d. Now determine the marginal abatement cost function -C”. What are business as usual

emissions?

2. Using the definition of the abatement cost function and the firms profit maximizing problem

we have

C(z) = TBAU – (q(T),z) = TEAU – (pa(T) – ckq(T), ),

where (2) is the profit maximizing output given that the firm is restricted to emit z units of

pollution.

a. Take the derivative of C(2) with respect to I using the chain rule. Show that this

derivative simplifies to be

b. To interpret the result in part a, use the explicit cost function cq, e) = (-e)2 + q).

First determine de Now substitute the constraint q() = PI Now substitute in the

constraint e = r. You now have an expression for S. How does this compare to C’ you

found in part d of problem 1? Explain.

3. Let firm 1 have abatement cost function C1 (11) = }(0 11), where of course 11 is firm l’s

of pollution. Det

atical formula the marginal abatemen

function. Sketch a graph of the marginal abatement cost function. What is the interpretation

of ;? How much does it cost to reduce emissions to zero?

4. Let firm 1 have abatement cost C. (21) = (01 – 112 and let firm 2 have abatement cost

C2(12) = (02 – 12). Given a level of pollution r, determine the cost effective emissions of

each firm. Now determine the aggregate abatement cost function C(2). What kind of function

with respect to r is it?

5. Consider two firms. The marginal abatement costs for the firms are given by MAC1 =1-11

and MAC2 = 1 – 2.12. Sketch the marginal abatement cost functions. Then sketch the

aggregate marginal abatement cost function. Finally, determine the algebraic formula for

the aggregate marginal abatement cost function. (Remember, the aggregate function is the

horizontal addition of the individual functions).

6. Consider a problem with 2 firms and 2 citizens. The marginal abatement costs for the firms

are given by MAC = 1 – 11 and MAC2 = 1 – 2.12. Pollution is uniformly mixed. Marginal

damages are given by MD = I and MD2 = 2c where I = I1 + 12 is the total emissions of

pollution. Determine the efficient quantity of pollution mathematically and graphically.

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Explanation & Answer:

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Pollution Models

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