Question

Efficient Allocations for Depletable Resources n = 2 time periods. Inverse Demand Curves: P1 = 10...

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* = (P2* - 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 (PNBt) t= 1 and 2, and PV consumer’s net benefits (CNBt) 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)

Homework Answers

Answer #1

P1* – MC1* = (P2* - 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

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