Question

Let demand be given by Q = 150 - P + 2Y. This is the same for all problems of this type. Let r = 10%. Let Y = 50 across both periods. Let MC = 0. Let reserves = 200. Consider the basic two-period model. What is consumption of the resource in the present IF THE OWNER OF THE RESOURCE IS A MONOPOLIST?

94.29
		98.81
		101.19
		105.71
		none of the above

Answer 1

We have the following information

Q = 150 – P + 2Y; where Y = 50 for both the periods

So, Q = 150 – P + 100

Q = 250 – P

P = 250 – Q Where P is price and Q is the quantity

Marginal cost (MC) = 0

Total reserves = 200

Discount rate (r) = 10% or 0.1

The owner is Monopolist

In the case of indefinite stock, the amount of the resources consumed is determined by the profit maximizing condition of Marginal Revenue (MR) = MC.

Total Revenue (TR) = P × Q = 250Q – Q²

MR = ΔTR/ΔQ = 250 – 2Q

However, in the present case the stock is limited to 200. In a two-period model for an efficient allocation we need to equalize the marginal rent of period one and period two.

MR – MC(Q₁) = MR – MC(Q₂)

Now, the marginal profit of a given period has to be equal to the discounted marginal profit of the following period. This is called the “r-percent” rule or Hotelling’s rule and it is given by the following equation

MR – MC(Q₁) = [MR – MC(Q₂)]/(1 + r)

What the above equation says is that the marginal profit of a given period has to be r% higher than the marginal profit of the previous period. Solving the above

250 – 2Q₁ – 0 = [250 – 2Q₂ – 0]/(1 + 0.1)

250 – 2Q₁ = (250 – 2Q₂)/(1.1)

(250 – 2Q₁) = 250/1.1 – 2Q₂/1.1

Now the reserve constraint is

Q₁ + Q₂ = 200

We will replace Q₂ = 200 – Q₁

(250 – 2Q₁) = 250/1.1 – 2(200 – Q₁)/1.1

(250 – 2Q₁) = 227.27 – 400/1.1 + 2Q₁/1.1

(250 – 2Q₁) = 227.27 – 363.63 + 2Q₁/1.1

(250 – 2Q₁) = – 136.36 + 2Q₁/1.1

275 – 2.2Q₁ = – 150 + 2Q₁

4.2Q₁ = 425

Consumption of the Resources in the present: Q₁ = 101.19

Q₂ = 200 – Q₁

Consumption of the Resources in the next period: Q₂ = 98.80