5. Consider a non-renewable resource owned by firms that are profit-maximizing a

Question

5. Consider a non-renewable resource owned by firms that are profit-maximizing and price-takers. There are two periods, now and later. The demand curve in each period (t = 1, 2) is Qt = 200 - 2Pt. The stock of the resource is 20 units. Extraction costs are approximately zero. The interest rate is 7 percent. (a) A market equilibrium requires identifying price and quantity at all times. What are the four variables for which we need to find numerical values to find the equilibrium? (b) Assume production costs are zero. What is the algebraic formula for the Hotelling Rule in this case? (c) Assume production involves a constant marginal cost of $c per unit of the resource extracted. What is the algebraic formula for the Hotelling Rule in this case? (d) Why is the formula in your answer to (b) different from that in your answer to (c)? Explain the economic logic underlying the Hotelling Rule? (e) Assume a production cost of $c per unit of the resource extracted, as in (c). The owners of the resource maximize the discounted present value of their profit from the resource, acting as a price taker. The amount they will choose to extract in each period and the market price at which they sell their output in each period are governed by four equations. Write down those four equations. (f) Assume that c = $3. Solve for the equilibrium values of P1, P2, Q1 and Q2. (g) In this solution, does price rise at the rate of the interest rate? If not, why not? (h) In this solution, does the quantity extracted decline over time? (i) What is the discounted present value of the profit over the two periods? (j) Without doing the specific calculations, how would you expect the solution to change if the interest rate was 12%? (k) With a pollution externality, Pigou demonstrated that the presence of a harmful real externality leads private decision making by a profit-maximizing polluter to generate an outcome that is not socially optimal. Is that also the case with the inter-temporal externality associated with extraction of this nonrenewable resource – is the profit maximizing solution you obtained in part (f) socially optimal? Explain the reason for your answer.

Explanation / Answer

a).

Consider the given problem, here there are 2 periods and we need to find out the “P” and “Q” for each period, => we need to find out, “P1”, “P2”, “Q1” and “Q2”.

b).

Assume that the production cost is “0”, “MC=0”, so the algebraic form of “Hoteling rule”, is given by, “P2/P1 = 1+r”, => “P2/P1=1.07”.

c).

Now, suppose that the “MC=c”, so now the algebraic form of the Hoteling rule is given by,

“(P2-c)/(P1-c) = 1+r”, => “P2-c = 1.07*(P1-c)”.

d).

as we can see here that the stock of resource is fixed, => there is an opportunity cost if we consume more today. So, according to this rule at the socially optimum rate of extraction the rate of growth of the unit price or marginal profit is equal to “1+r”.

So, in “b”, the MC=0, so the “price” itself be the “unit profit” but in “c”, “MC=c” so the unit profit be “Pi-c” for the “ith” period.

e).

the required 4 equation are given below,

1). The demand curve, “Qt = 200 - 2Pt”.

2). The supply curve , “P=c”

3). Total Stock, Q=Q1+Q2=20.

4). The rate of interest rate, “r”.

f).

So, according to the “hoteling rule” (P2-c) = 1.07*(P1-c), => P2 = 1.07*(P1-c) + c.

=> P2 = 1.07*P1- 1.07*c + c, => P2 = 1.07*P1- 0.07*c.

Now, form the total stock we get, “Q1+Q2=20, 200 – 2*P1 + 200 – 2*P2 = 20.

=> 200 – 2*P1 + 200 – 2*(1.07*P1- 0.07*c) = 20

=> 200 – 2*P1 + 200 – 2.14*P1- 0.14*c = 20

=> 400 – 20 – 4.14*P1- 0.14*c = 0

=> 380 - 0.14*c = 4.14*P1, => (380 - 0.14*c)/4.14 = P1

=> P1 = (380 – 0.14*3)/4.14 = 91.68

=>Q1 = 200 – 2*P1 = 200 – 183.37 = 16.63, => Q2 = 3.37.

=> P2=1.07*P1- 0.07*c=97.88.

So, the value of the 4 variable are, “P1=91.68”, “Q1=16.63”, “P2=97.88” and “Q2=3.37”.

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