Question

In: Economics

Consider a situation where the world consists of two countries, X, and Y. The total benefits...

Consider a situation where the world consists of two countries, X, and Y. The total benefits (B) and total costs (C) of greenhouse gas emissions abatement (A) for each country are given by the functions listed below. Utility is defined as the difference between the benefits and cost of greenhouse gas abatement. Notice that the benefits of greenhouse gas abatement for each country is determined by the combined total amount of greenhouse gas abatement by both countries. This is because of the nature of climate change as a transboundary pollution problem.

B_X = 8(A_X + A_Y)

B_Y = 5(A_X + A_Y)

C_X = 10 + 2A_X + 0.5A_X²

C_Y = 10 + 2A_Y + 0.5A_Y²

Obtain the privately optimizing level of abatement for country X, given that country Y decides to emit at the level of greenhouse gas emissions that country Y would emit in the cooperative equilibrium. (You should find that the answer as in Part (d), that country X does the same amount of abatement that it would have done in the non-cooperative case). What property or properties of the cost and benefit function used in this example cause this particular result?

Solutions

Expert Solution

B_X = 8(A_X + A_Y)

B_Y = 5(A_X + A_Y)

C_X = 10 + 2A_X + 0.5A_X²

C_Y = 10 + 2A_Y + 0.5A_Y²

Ux=8(A_X + A_Y)- 10 + 2A_X + 0.5A_X²

Uy=5(A_X + A_Y)- 10 + 2A_Y + 0.5A_Y²

Lets maximize Ux with respect to Uy

Ux=8(A_X + A_Y)- (10 + 2A_X + 0.5A_X² )

=8A_X +8A_Y-10 - 2A_X - 0.5A_X²

=6A_X+8A_Y-10 - 0.5A_X²

Uy=5(A_X + A_Y)- (10 + 2A_Y + 0.5A_Y²⁾

=5A_X + 3A_Y- 10 - 0.5A_Y²

6A_X+8A_Y-10 - 0.5A_X² +ƛ (5A_X + 3A_Y- 10 - 0.5A_Y²⁾

ux/∂y=8+3- A_Y

ux/∂x=6- A_X+5

ux/∂ ƛ=5A_X + 3A_Y- 10 - 0.5A_Y²

A_Y=11

A_X=1

5A_X + 3A_Y- 10 - 0.5A_Y² ₌₀

5+33-10-0.5A_Y² ₌₀

0.5A_Y² ₌₂₈

A_Y² ₌₅₆

Rahul Sunny answered 1 year ago

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