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ECON 4910 Seminar exercise 3
Consider two firms with the following production structures
y1  f1 ( x1 , xR ), f11  0, f12  0
P  g ( x1 , xR ), g1  0, g 2  0
(1)
and
y2  f 2 ( x2 , P), f 21  0, f 22  0
(2)
where y1,y2 are outputs and P is a pollutant, and x1,x2,xR are the inputs. (The subscripts of
the derivatives point to the place of the argument in the functions.) The input xR is used
for purification. The firms are price-takers and the fixed product prices are p1,p2 and the
input prices q1,q2,qR.
1. Find the profit-maximising solution for both firms in the case of non-cooperation,
and give first Firm 1 the property right to discharge the pollutant and then give the
property right to Firm 2 to bloc any discharge. Comment on the difference
between the solutions. (Hint: set up the profit maximising problem according to
the property right regime. Look also at differences of profits for each firm in each
of the property rights regimes.)
2. In the case of Firm 1 having the property right to discharge pollution try to define
the total and the marginal damage function as a function of the amount of the
pollutant for Firm 2. (Hint: the damage to Firm 2 is loss of profit, set up an
expression for the difference between maximal profit for zero discharge of
pollutants and the profit with a given positive level of the pollutant.)
3. Assume that Firm 2 offers to subsidise a reduction of the discharge of pollutant
from Firm 1 with a fixed rate s1 per unit of reduction when this firm has the right
to pollute. How should Firm 2 set this rate? (Hint: Firm 2 knows that Firm 1 will
react by maximizing profit including the payment from Firm 1 of s1 per unit of
reduction of the pollutant from the profit-maximisation level found in problem 1.
Look at Firm1’s profit maximization problem and find x1, xR, P as functions of
the exogenous variables, find the sign of P / s1 , then look at Firm 2’s profit
2
maximisation problem using the amount of pollutant as a function of the subsidy
rate s1)
4. Explain how the solution to problem 3 is related to the Coase theorem. (Hint:
check Lecture 2.)
5. Compare the solution to problem 3 with the social optimal solution. Find the
optimal Pigou tax levied on discharge of the pollutant P in order to implement the
social solution within the competitive markets the firms operate in. (Hint: use the
fixed prices as the social unit value of products and inputs and maximise joint net
surplus from producing the goods. Then compute the profit maximizing market
conditions for use of the two types of resources of Firm 1 using a tax rate t on P
and the condition for the optimal use of the resource of Firm 2 taking the
pollution level as given. Compare the market conditions and the social solution to
find the optimal rate of t.)
6. Try to discuss the situation when Firm 2 has the right to a clean environment and
assume that Firm 1 offers to pay Firm 2 a fixed rate s2 per unit of increase in the
discharge of the pollutant for allowing Firm 1 to pollute. How should Firm 1 set
this rate? (Hint: Firm 1 knows that Firm 2 will react by maximizing profit
including the payment from Firm 1 of s2 per unit of the pollutant and calculate the
optimal amount P of pollutant to be tolerated. Firm 2 will then take this
relationship into consideration when maximizing his own profit and setting the
optimal rate of subsidy.)