Suppose a nonrenewable resource can be extracted over two periods. Both periods are characterized by MB...

Suppose a nonrenewable resource can be extracted over two periods. Both periods are characterized by MB = 250 - .25Q and MC = 50 + .25Q and there are 350 units total available for extraction. Assume the discount rate is .15 (or 15%).

Identify the socially optimal rates of extraction for period 1 and period 2.
Identify and define the user cost associated with the extraction of this resource.
Identify the socially optimal price of the resource for period 1 and period 2 and the user cost for period 1 and period 2.

Solutions

Expert Solution

We are given with MB = 250 - .25Q and MC = 50 + .25Q and in total 350 units we have with discount rate of 15%.

a) If we put MC = MB , 250 - .25Q = 50 + .25Q,we get Q = 400 which is greater than 350 units. so discount rate comes into picture. We need to equalize present value of period 1 and period 2 as follows and Q₁ + Q₂ = 350.

PV(MB₁ - MC₁) = PV(MB₂ - MC₂) 1 and 2 are denoting periods.

250 - .25Q₁ - 50 - .25Q₁ = (250 - .25Q₂ - 50 - .25Q₂)/1.15¹ , 200 - 0.5Q₁ = (200 - 0.5Q₂)/1.15 by solving and putting Q₂ = 350 - Q₁ , we get 230 - 0.575Q₁ = 25 + 0.5Q₁ , Q1 = 205/1.075 = 190.69 , Q2 = 350 - Q1 = 350 - 190.69 = 159.31

b) User cost is the cost or expense which incurred due to depletion of resources. As the resources are non renewable, they deplete each time we do extraction. The user cost of extraction increases as there are less resources present as compare to each previous period. User cost can also be defined by difference between price of resource and marginal cost of extracting the resource.

c) Optimal price of the resource is calculated by putting units extracted in marginal benefit function for each period.

P1 = 250 - 0.25Q1 = 250 - 0.25*190.69 = P1 = 202.32 similarly, P2 = 250 - 0.25Q2 = 250 - 0.25*159.31 = P2 = 210.17

Price have been increased as user cost increase with time.

User cost is defined in above part and can be calculated as P - MC

For period 1, UC1 = P1 - MC1 = 202.32 - (50 + 0.25*190.69) = UC1 = 104.64

For period 2, UC2 = P2 - MC2 = 210.17 - (50 + 0.25*159.31) = UC2 = 120.34

You can notice that extraction cost or user cost has been increased from period 1 to period 2 as there are less resources present and extracting resources demands more price now.

Rahul Sunny answered 1 year ago

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