ECOS3013 Tutorial 5: Solutions
Week 6, starting 4 September
Question 1: Net Present Value
This question is based on Kolstad Ch.6, Q1
A dam is proposed on a stretch of wild river, a river that is currently used for recreation. The dam will
generate electricity. The dam will have a useful life of 50 years, after which its reservoir will be full of
sediment and the dam will need to be removed. The following are characteristics of the dam:
• Initial Cost: $100,000,000
• Electricity produced: 100,000 MWh per year at $100/Mwh
• Cost of decommissioning dam: $10,000,000
• Value of recreation lost: $5,000,000 per year
Assume that annual costs and benefits start next year.
1. Government authorities are considering whether to approve the project. If the social discount rate
is 3% per year, is the dam a good idea?
2. Now suppose that authorities believe the dam will impose additional environmental damage species
living in the local area. If they require a potential Pareto improvement, what is the maximum level
of environmental damage that the authorities should tolerate (in present value terms)?
1
Question 2: Net Present Value
Suppose that firms emit a pure stock pollutant. That is, a tonne of emissions today causes a constant
amount of damage D each year, forever. They propose a short-term project that would emit one tonne
of pollution. The effect of the environmental damages would start next year.
1. If the discount rate is r, what is the present value of the damages?
2. Suppose that the annual environmental damage is D = $60 million, and the benefit of the project
is π = $2000 million of profit this year. Should the project go ahead if the social discount rate is
10%?
3. What discount rate would overturn the result in part 2?
Question 3: Social Discounting
1. Explain why the discount rate is important when measuring the damages caused by greenhouse gas
emissions.
2. Briefly discuss the arguments for and against using a high discount rate to measure damages caused
by climate change.
3. What discount rate should we use? Why is this a normative decision?
Question 4: Stock Pollution
Assume that the current stock of CO2 held in the atmosphere is s0 in Gt, and evolves according to:
st = δst−1 + et
where st is the stock at the end of period t. To simplify the analysis, consider 50-year increments, so that
t=0 is today, t=1 represents the next 50 years, t=2 the following 50 years and so on. The persistence
rate of CO2 in the atmosphere is δ (the amount remaining after 50 years). Assume that emissions are
constant at et every 50 years. The total damages over a 50 year period are:
Dt = 0.02s
2
t
where st is the end-of-period stock of CO2.
2
1. Write the stock of CO2 in the atmosphere at the end of period t = 1, 2, 3 as a function of s0 and
emissions in each period.
2. Find an expression for the marginal damage done in period t=3 from emissions released during
period 1, e1.
3. Suppose that s0=4000 and emissions each period are constant at et = 2000 for t = 1, 2, 3. Also
assume that δ = 0.8. What is the present value of damages caused by a small increase in emissions
in period 1? Assume that the social discount rate for a 50 year period is 0.6, and that marginal
damages are measured at the end of each period (so that we discount t=1 once).
4. How much should society as a whole be willing to pay to reduce emissions in period 1 by a small
amount, based on the damage that these emissions would do over 150 years?