Question

Efficient Allocations for Depletable Resources

n = 2 time periods.

Inverse Demand Curves:

P₁ = 10 - 0.4q₁ for period 1 and

P₂ = 10 - 0.4q₂ for period 2.

Marginal Costs for the two periods:

  MC₁ = $3.00

  MC₂ = $3.00

Discount rate = 15%

Resource Availability Constraint:

  Q = q₁ + q₂ = 25 billion units.

1. Calculate the dynamically efficient allocations q₁* and q₂* for periods 1 and 2.

Dynamic efficiency condition MNB₁ = λ = PV MNB₂

or P₁* – MC₁* = (P_2* - MC₂) / (1+r)

(10 - 0.4q₁*) - 3 = (10 - 0.4q₂* - 3) / (1.15)

7 - 0.4q₁* = (7 - 0.4q₂*)/1.15

Continue and we solve for q₂* as function of q₁*.

Use the above equation when you solve it for q₁* with the resource reserve constraint Q = q₁* + q₂* or  q₂* = 25 - q₁*  to solve for q₁* and q₂*.

2. Calculate PV(MUC) = MUC₁= MNB₁= P*₁ – MC* = l. You can also calculate it this way: PV(MUC₂) = PV(MNB2) = (P₂* - MC₂*)/[1+r] = l.

l. =PV(MUC1) =10 - 0.4q₁* – 3 by using period 1.

Plug q₂* =? from the solution of part 1 into this equation. This will be the PV of the cost of depletion (MUC₁) per barrel of oil in period 1.

3. Calculate MUC₂ for period 2 (without PV).  Is MUC rising over time? Hint: Look at the profile for MUC (marginal user cost).

(P₂* - MC₂*) = l₂

(10 - 0.4q₂*) – 3

Plug q₂* =? from the solution of #1 into this equation.

This will be the cost of depletion (MUC₂) per barrel of oil in period 1 without PV.

4. Calculate PV Producer’s net benefit (PNB_t) t= 1 and 2, and PV consumer’s net benefits (CNB_t) for periods 1 and 2.

Graph it

For period 1, Calculate P1* =10 - 0.4q₁*. Then

TNB1 = CNB1 + PNB1= area of triangle + area of rectangle for period 1.

Graph it

Calculate P2* =10 - 0.4q₂*. Then

PV (TNB2) = PV(CNB2) + PV (PNB2)

5. Calculate PV stream of TNB’s for periods 1 and 2. That is, PV(TNB₁, TNB₂) = TNB₁ + PV(TNB₂).

PV (stream of TNB’s in the two periods) = TNB1 + PV(TNB2)

Answer 1

P₁* – MC₁* = (P_2* - MC₂) / (1+r)

(10 - 0.4q₁*) - 3 = (10 - 0.4q₂* - 3) / (1.15)

7 - 0.4q₁* = (7 - 0.4q₂*)/1.15

q₁* = [ 7 - (7 - 0.4q₂*)/1.15 ] /  0.4

PUTTIN IT IN  Q = q₁* + q₂*

[ 7 - (7 - 0.4q₂*)/1.15 ] /  0.4 + q₂* = 25

q₂* = 12.15

q₁* + q₂* = 25

q₁* = 25 - 12.15

= 12.85

2) 10 - 0.4q₁* – 3

putting q₁* = 12.85

10 - 0.4q₁* – 3

= 10 - 0.4* 12.85 – 3

10 - 5.14 -3

= 1.86

3)  (10 - 0.4q₂*) – 3

plugging

q₂* = 12.15 in equation

10 - 0.4q₂*) – 3

10 - 0.4 * 12.15) – 3

10 - 4.86 - 3

= 2.14