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Environmental and Resource Economics

Mathematical analysis

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Question 1
Suppose there are two plants that emit a pollutant. Plant 1 can abate emissions by the level
A1 at costs C1 (A1) = 3. A1 3 . Plant 2 can abate emissions by the level A2 at costs C2 (A2)
= 1 +A2. You have the objective to reduce total emissions by A = A1+ A2, with A = 1, at the
lowest total costs C = C1 + C2.
1. Formulate this as a constrained optimasation problem and write down the lagrangian.
2. Compute the marginal abatement costs from both plants.
3. Compute the cost minimal of abatement plant one and plant two
Question 2
Assume a fish stock S growth with a rate G(S) = 2 S (1-S) [tons], and the social discount rate
is ρ=0.04. Further assume that extraction costs are C(R) = 0.25 R2 [€], and marginal utility to
be p(R) = 0.6-R [€/ton]. Determine:
( a) the socially optimal extraction rate in the long-run equilibrium;
(b) the laissez-faire extraction rate and price if marginal utility is equal to the market price;
(c) the Pigovian tax that leads to socially optimal extraction.

Question 3
Assume there are 20 polluting firms. Of those, 10 are type-A firms with abatement costs CA
(MA) = 10 – 2 MA + 0 .1 MA 2 [€]; without regulation, they emit MA = 10 [t] each. The 10
type-B firms each emit MB = 30 [t] without regulation, and have abatement costs CB(MB) =
22.5 – 1.5 MB + 0.025 MB 2 [€]. An emissions trading system is introduced to reduce total
emissions down to M = 150 [t].
1. Determine the marginal abatement costs for both firm types. • How much do they emit if
the certificate price would be p = 0.5 [€/t]? [MA = 7.5 and MB = 20]
2. How much do all firms emit in total if they face an arbitrary certificate price p?
3. What would be the market equilibrium certificate price if there is a total number of
certificates for 150 [t]? [p = 1]
Question 4
Suppose a city wants to provide a total area of new green space A=2.125km2
. There are two
quarters in the city, where parks with area x or y, respectively, can be supplied at different
costs C1(x)=7.x1.2 M€/km2
, and C2(y)=5.0y1.4 M€/km2
What is the possibility of to provide a total area A=x+y?
Question 5
Suppose a production function q=f (x,y) =x0.4 y0.2 , where q is output, and x,y are production
factors. Suppose the unit price of factor x is 1€, and the unit price of factor y is 2€.
(a) Write down the cost minimization problem to determine the cost function C(q).
(b) Write down the Lagrangean
(c) Determine inputs x/y

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