The Stafford coal seam contains 25,000 tons of coal. It costs $100 per ton to extract the coal and deliver it to the market. (This is a constant marginal cost). The demand for coal can be represented by the following inverse demand equation:
P = 600 - 0.02*Q.
a. If you were only concerned with this period, what would the optimal consumption level be? What price would achieve this level of consumption? Graph this solution.
b. Now assume that there are two periods. Assume that there is no discounting, that the value of a dollar in period in two is the same as the value of a dollar in period 1. Graph the optimal allocation of coal across these two time periods (see pgs. 41-43 in the book for a guide). Also solve for the optimal quantity of coal in each period numerically.
c. In part b, what price would elicit the optimal level of coal consumption in period 1? Discuss what this price represents.
d. Now assume that there is discounting, that is that a dollar in period 2 is only worth $0.90 in period 1. Show graphically what this does to the optimal allocation of coal in periods 1 and 2. (You do not have to show this numerically.)
e. Now assume that the demand for coal in period 2 is double the demand for coal in period 1 (discounting still applies). Show graphically what this does to the optimal allocation of coal in periods 1 and 2.
Please see the attached file.
Marginal Cost $100.00
Demand (P) 600-.02Q
A) Total profit = (600 - .02Q)Q - 100Q Q P Profit
= 590Q - .02Q^2 500 590 245,000
d/dq = 500 -.04Q 1000 580 480,000
Q = 12,500 1500 570 705,000
2000 560 920,000
B) Let coal in period 1 = Q1 2500 550 1,125,000
3000 540 1,320,000
total profit = = ...
This Solution contains calculations to aid you in understanding the Solution to these questions.