Question

Consider the following supply and demand equations for a depletable energy resource, with subscripts representing the time period. D0=100-2Q0 S0=50 : D1=100-Q1 S1= 30

A) If the discount rate is 2%, and the total stock of the resource at the beginning of time period 0 (Q-bar) is 90 units, determine the allocation of Q in each period that maximizes the Net Present Value of Total Benefits (NPVTB) across both periods. This is known as the “optimal allocation” or “dynamically efficient allocation.”

B) Calculate the scarcity rent associated with your answer in part a.

Answer 1

a) At the dynamically efficient allocation, PV of net MB should be same across periods

This implies that PD - PS in period 1 should be equal to discounted PD - PS in period 2

100 - 2Q0 - 50 = (100 - Q1 - 30)/(1+2%) and Q0 + Q1 = 90

Solve the two equations

1.02*(50- 2Q0) = (70 - Q1)

51 - 2.04Q0 = 70 - (90 - Q0).............. (Using Q1 = 90 - Q0)

71 = 3.04Q0

This gives Q0 = 23.36 and Q1 = 66.64. This is the optimal allocation” or “dynamically efficient allocation.”

b) Scarcity rent is marginal user cost and is given by Pd - Ps

= (100 - 23.36*2) - 50

= $3.29 per unit.