# Numerical Exam Questions for Environmental Economics

Spring 2006

Krister Hjalte

Question 2. (29/3 1998)

The inverse demand function for a non-renewable resource is Pt = a- bRt, where Pt is the market price and Rt the extraction in period t. The total gross benefit from extracting this resource can be written as an integral

The extraction cost Ct= cRt, where c is a constant. Total available amount of the resource is denoted by S. From a social point of view we want to maximise the net benefits from extracting this resource subject to the general constraint that the sum of extraction in different time periods must equal the total available amount. Start by formulating an expression for the net benefits. Your problem is then to estimate how much that will be extracted in each period under the following assumptions

a= 8

b= 0.4

c= 2

r= discount rate

S = 30

T= 0,1,2 (three periods)

Use the Lagrange multiplier method for the maximisation and denote the multiplier by (.

a) How much is extracted in each of the three periods if the discount rate is 10%? What is the value of ( and give an economic interpretation of it. b) How will your answer in a) change if the discount rate r is 0 in stead? c) If the discount rate r is 20%?

d) How much of the total stock must be available for the extraction not to give rise to an economic intertemporal allocation problem? In what way does the assumption of different discount rates influence your answer to this? Explain carefully. e) Imagine that the resource would have been a renewable resource, say fish or mammals. In what principle way will the analysis of an intertemporal welfare maximisation harvesting policy change the optimality conditions?

Question 1. (29/3 2000)

An important issue focuses on the consequences of uncertainty for the efficiency loss arising from use of various instruments when wrong information is used. In situations with certainty where decision makers know the pollution abatement cost and the pollution damage functions and accordingly can determine efficient abatement levels, market based instruments are in general the best means of achieving such targets. However, this may not be true with uncertainty of costs and damage functions. In the following example the true marginal abatement cost function, MAB can be written as MAB = 1 000 - 3M, where M is the level of pollution flow. The marginal damage function is MD = 5M. The pollution regulating authority could use either a pollution tax or an emission permit. However, the authority underestimates the marginal abatement cost function and thinks it is MAB = 450 - 4M in stead. Which pollution instrument should you recommend to use? Motivate your answer carefully.

If the cost of information for reducing the uncertainty of the marginal abatement cost function is estimated at 30 000 money units, what is your recommendation of instrument now?

Question 1 (5/5 2000)

Assume that the relationship between the growth of a fish population and the population stock can be expressed as G(S)= 4S- 0.1S2 , where the G is the growth in tons and S is the stock in thousands of tons. The constant price per ton is 100 money units. The total harvesting cost expressed in terms of population stock is estimated to TC = 8000 - 200S.

a) What are the maximum sustainable yield (harvest) and the corresponding population stock?

b) Compute the sustainable yield and the corresponding population stock assuming an open access solution. c) Compute the efficient profit maximisation sustainable yield and corresponding population stock.

Question 2. (5/5 2000)

Two firms can control emissions at the following marginal costs: MC1 = 200 q1, MC2 = 100q2, where q1 and q2 are, respectively, the amount of emissions reduced by the first and the second firms. Assume that with no control at all, each firm would be emitting 20 units of emissions or a total of 40 units for both firms.

a) Estimate the cost-effective allocation...

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