Question

A nonrenewable resource stock of 200 units

Two periods

Demand in the first period (period 0) is p 0 =300-q 0

Demand in the second period (period 1) is p 1 =400-2q

The marginal extraction cost is zero.

Competitive industry, profit maximizing firms

1

a. Draw the 2-period graph for this case. On your graph, label the values of all vertical intercepts for the two demand curves (1 pt.)

b. State the value of the quantity that would be extracted in the first period if the discount rate is zero. Show your work.

(2 pts)

a. Quantity extracted in the first period: ____

Now suppose the discount rate is 10%. For each question below, show your work.

c. What would be the quantity extracted in the first period? (2 pts.)

Quantity extracted in the first period: ____

1 d. What is the price of the resource in the first period (t=0)? (1 pt)

Price in the first period (period 0) =______

e. What is the price of the resource in the second period (t=1)? (1 pt)

Price in the second period (period 1) =______

f. What is the marginal user cost of extraction in the first period? Explain your answer in 1 sentence. (2 pts.)

Marginal user cost in the first period: ______

Answer 1

a).

Consider the given problem here the demand function of the two periods are given below.

=> P0 = 300 – q0, for period 1.

=> P1 = 400 – 2*q1, for period 2, the following fig shows the demand curve for “period 1” and “period 2”.

Demand for period 1 Demand for period 2

Here “D0” and “D1” are the demand curve for “period 1” and “period 2” respectively.

b).

Let’s assume that the discount rate is zero implied “r=0”, => the “present value of net marginal benefit” in both period are given below.

=> PVNMB0 = (MB-MC) = 300 – q0, of period 1.

=> PVNMB1 = (MB-MC)/1+r = 400 – 2*q1, of period 2.

Now, at the equilibrium “PVNMB0” should be equal to “PVNMB1”.

=> 300 – q0 = 400 – 2*q1, where “q0+q1=200”, => q1=200-q0”.

=> 300 – q0 = 400 – 2*(200-q0) => 300 – q0 = 400 – 400 + 2*q0 = 2*q0.

=> q0 = 300/3 = 100, => q0 = q1 = 100.

So, the quantity extracted in “period 1” is “100 units”.

c).

Let’s assume that the discount rate is zero implied “r=0.1”, => the “present value of net marginal benefit” in both period are given below.

=> PVNMB0 = (MB-MC) = 300 – q0, of period 1.

=> PVNMB1 = (MB-MC)/1+r = (400 – 2*q1)/1.1 = 363.64 – 1.82*q1, of period 2.

Now, at the equilibrium “PVNMB0” should be equal to “PVNMB1”.

=> 300 – q0 = 364 – 1.82*q1, where “q0+q1=200”, => q1=200-q0”.

=> 300 – q0 = 364 – 1.82*(200-q0) => 300 – q0 = 364 – 364 + 1.82*q0 = 1.82*q0.

=> q0 = 300/2.82 = 106.38, => q0 = 106.38, q1 = 93.62.

So, the quantity extracted in “period 1” is “106.38 units”.

d).

So, the price of resources in period1 is “P0 = 300 – q0 = 300 – 106.38 = 193.62”, => “P0 = $193.62”.