**Environmental and Resource Economics Mathematical analysis**

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