Question:
The total dollar value of bison killed from Huntington Forest is [tex] f(b) = 53b - 1.3b^2 [/tex] , where b is the number of bison killed. The marginal cost of killing bison is 0. What is the optimal bison-killing tax (per bison) to avoid the tragedy of the commons in this forest
Answer:
26.5
Explanation:
Given:
[tex] f(b) = 53b - 1.3 b^2 [/tex], where b is the number of bison killed.
Marginal cost, MC = 0
The socially efficient level: π=f(b)-MC(b)
π = 53b - 1.3b²
[tex] \frac{d\pi}{db} = 53 - 2.6b = 0 [/tex]
[tex] 2.6b = 53 [/tex]
[tex] b = \frac{53}{2.6} = 20.38 [/tex]
For tragedy of commons:
AR = AC = 0
[tex] AR = \frac{f(b)}{b} = 53 - 1.3b [/tex]
[tex] b' = \frac{53}{1.3} = 40.77 [/tex]
For imposed tax, t, we have:
AR = AC = 0 + t, where b = 20.38
53 - 1.3b = t
At [tex] b = b' = 20.38[/tex]
Let[tex] \frac{53}{2.6}[/tex] represent b
[tex] 53 - 1.3b = t [/tex]
[tex] 53 - 1.3(\frac{53}{2.6}) = t [/tex]
[tex] 53 - 0.5(53) = t [/tex]
[tex] 53 - 26.5 = t [/tex]
t = 26.5
Therefore, optimal bison-killing tax = 26.5
