Question

Assume that you own an exhaustible resore that is sold competitive,and the marginal cost of extraction at year t is given by:
Ct=5+0.5t+0.05t^2,
where t=0 at the beginning of 2010.it is also known that the interest rate is 7% per annum and the demand for the resource is:
Qt=100-Pt,
where Qt and Pt represent output level in,tons and price in year t respectively.
a)suppose that the price in year 2020 is anticipated to be 100, what were the prices,user costs and production level of the resource in 2015 and 2016?
b)if the market were monopolized in2017,what were the optimal production rule in that period?

Answer 1

Solution:

Marginal Cost of extraction, MC_t = 5 + 0.5t + 0.05t², where t = 0 at beginning of 2010

Interest rate, i = 7% p.a.

Demand for resource, Q_t = 100 - P_t

a) With t = 0 at beginning of 2010, in the year 2020, t = 2020 - 2010 = 10

In the year 2015, t = 2015 - 2010 = 5 and in the year 2016, t = 2016 - 2010 = 6

So, given that anticipated P₁₀ = 100, we have to find P₅, P₆, MC_5, MC_6, Q₅, and Q₆,

With i = 7% p.a., using the present value formula,

P₅ = P₁₀/(1 + i)^(10-5) = 100/(1.07)⁵ = 100/1.402 = 71.3 (approx)

And, P₆ = P₁₀/(1 + i)^(10-6) = 100/(1.07)⁴ = 100/1.31 = 76.3 (approx) [or simply, P₆ = P₅(1+r)]

MC₅ = 5 + 0.5*5 + 0.05(5)² = 8.75

MC₆ = 5 + 0.5*6 + 0.05(6)² = 9.8

Then, Q₅ = 100 - P₅ = 100 - 71.3 = 28.7

And, Q₆ = 100 - P₆ = 100 - 76.3 = 23.7

b) In case of monopoly, optimal production occurs where marginal cost of extraction must equal the marginal revenue of extraction of the resource.

Marginal cost is as given, now t = 2017 - 2010 = 7

So, MC₇ = 5 + 0.5*7 + 0.05(7)²

MC₇ = 5 + 3.5 + 2.45 = 10.95

Calculating marginal revenue:

Total revenue = Price*quantity = (100 - Qt)*Qt

TR = 100Qt - Qt²

Marginal revenue, MR = = 100 - 2*Qt

So, MR₇ = 100 - 2*Q₇

So, for optimality, we require MR₇ = MC₇

100 - 2*Q₇ = 10.95

So, optimal Q₇ = (100 - 10.95)/2 = 44.525

Thus, optimal production level in the period 2017 would be 44.525