Question

the inverse demand function for the depletable resource is P = 8 - 0.4q and the marginal cost of supplying it is $4. (a) If 12 units are to be allocated between two periods, in a dynamic efficient allocation, determine how much would be allocated to the first period and how much to the second when the discount rate is .10. (b) Calculate the efficient price for the two periods. (c) Calculate the marginal user cost in each period.

Answer 1

a)

Given

P=8-0.4q

MC=4

Let us see calculate the output when P=MC

8-0.4q=4

q=10

We find that required resources (i.e. 2*10=20 units) are less than the available resources (i.e. 12 units). We go for dynamically efficient allocation.

Marginal Net benefit=P-MC=8-0.4q-4=4-0.4q

PV of net benefit in period 1=PV(MNB₁)=4-0.4q₁

PV of net benefit in period 2=PV(MNB2)=(4-0.4q2)/(1+i)=(4-0.4q₂)/(1+0.10)=(4-0.4q₂)/(1.1)

Set PV(MNB₁)=PV(MNB₂)

4-0.4q₁=(4-0.4q₂)/(1.1)

4.4-0.44q₁=4-0.4q₂

0.4-0.44q₁+0.4q₂=0

We know that combined availability is 12 units

i.e. q₁+q₂=12

q₂=12-q₁

So,

0.4-0.44q₁+0.4*(12-q₁)=0

0.4-0.44q₁+4.8-0.4q₁=0

5.20-0.84q₁=0

q₁=Allocation in first period=6.19 units

q₂=Allocation in second period=(12-6.19)=5.81 units

b)

We are given

P=8-0.4q

For period 1

P₁=8-0.4q₁=8-0.4*6.19=$5.52

For period 2

P₂=8-0.4q₂=8-0.4*5.81=$5.68

c)

MUC₁=P₁-MC=5.52-4=$1.52

MUC₂=P₂-MC=5.68-4=$1.68