(1)
Suppose we are now at the start of year t = 0. We believe that the world will end in 20 years time. Whatever happens after year t = 19 is deemed to be irrelevant to human well-being. An initial stock of 100 (million tons, say) of a non-renewable resource exists. The problem is to decide how this stock is to be used over the 20 years.
The finite horizon social welfare function (SWF) is
with a fixed-size cake of resources, S . Utility – and so social welfare – is derived directly from consuming the resource. Once the cake is fully eaten, no further utility is possible. Note that because of our assumption that nothing matters after t = 19, the cake will be fully eaten by the end of period 19.
The utility at some point in time, U(R), depends on the amount consumed: U(R)  0R P(R)dR (2)
where P is the net price of the resource. We assume that the form of the resource demand function, P(R), is
P(R)  a ï€ bR (3)
where a = 50; b = 1; and r (discount rate) = 10%. We wish to maximise the sum of discounted utility over the specified time period by choosing an optimal extraction path. Using Excel Solver, find out the optimal time path of R.
[70 Marks]
(2)
Now suppose that the discount rate increases to 20%. How does your solution change? Explain intuitively. [20 + 10 = 30 Marks]
