Question

Assume that there are 50 units of an exhaustible resource that can be used in either period 1 or period 2. The demand curve is given by: P=100-Q. The interest rate is 50%. The industry is perfectly competitive. There are no costs of extraction. a) Solve for the efficient level of output and price in period 1 and period 2 b) Suppose that the government imposes a tax of $30 per unit. Solve for the level of output and price in both period 1 and period 2. c) Suppose that the government imposes a tax of $30 on period 1 sales and $45 on period 2 sales. Solve for the level of output and price in both period 1 and period 2 d) Is the tax described in part (b) or in part (c) more efficient (causes less deadweight loss)? Explain.

Answer 1

a) Suppose efficient quantity in period 1=Q then in priod 2=50-Q for efficient quantity Revenue=P₁Q+P₂(50-Q)

Profit₁=(100-Q)*Q;Profit₂=(100-(50-Q))(50-Q)-(50-Q)*(100-Q)*0.5(Interest) Total Profit=100Q-Q²+(2500-Q²)-0.5(5000-150Q+Q²⁾ for maximizing Profit differentiating dProfit/dQ=175-5Q=0 this gives Q=35 hence in Period 1 Q=35 P=65 and in 2 Q=15 P=(100-15)=85

b)If tax of $30 introduced then Profit₁=(100-Q-30)*Q;Profit₂=(100-(50-Q)-30)(50-Q)-(50-Q)*(100-Q)*0.5(Interest) Total Profit=70Q-Q²+(1000-Q²+30Q)-0.5(5000-150Q+Q²⁾ dProfit/dQ=175-5Q=0 this gives Q=35 hence price and quantity will remains as previous

c) with tax as 30 and 45 Total Profit= 70Q-Q²+(250-Q²+45Q)-0.5(5000-150Q+Q²) differentiating will give Q=38 hence hence in Period 1 Q=38 P=62 and in 2 Q=12 P=(100-12)=88

d) for Efficiency total gain in case b=35*65+15*85-0.5*15*65+30*50=4562.5 for case c=38*62+12*88-0.5*12*62+30*38+45*12=4720 hence part c is more efficient