You have the following PMC, MD and SMB: PMC = 15 + 2Q MD = 1.2Q2 SMB = 70 - 3Q Find the Pigouvian tax t*

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Question restatement: You have the following PMC, MD and SMB: PMC = 15 + 2Q, MD = 1.2Q^2, SMB = 70 - 3Q. Find the Pigouvian tax t*. The provided answer option is: a. 30.

Step 1: Interpret the Pigouvian tax setup. In a standard Pigouvian framework, the social optimum is where SMB equals the social marginal cost (SMC). The social marginal cost equals the private marginal cost (PMC) plus the marginal external cost (MEC), which in this formulation is MD. Therefore, SMB = SMC = PMC + MD.

Step 2: Express the relationship with the given functions. We are given SMB = 70 - 3Q, PMC = 15 + 2Q, and MD = 1.2Q^2. The condition for the optimum is:
70 - 3Q = (15 + 2Q) + 1.2Q^2.

Step 3: Solve the equation for Q. Move all terms to one side to form a standard polynomial equation:
70 - 3Q - 15 - 2Q - 1.2Q^2 = 0
which simplifies to
55 - 5Q - 1.2Q^2 = 0.
Multiply by 10 to clear decimals:
550 - 50Q - 12Q^2 = 0
Rearrange to standard quadratic form:
12Q^2 + 50Q - 550 = 0.
Use the quadratic formula with a = 12, b = 50, c = -550:
Q = [-50 ± sqrt(50^2 - 4*12*(-550))] / (2*12) = [-50 ± sqrt(2500 + 26400)] / 24 = [-50 ± sqrt(28900)] / 24 = [-50 ± 170] / 24.
Positive root is Q* = (-50 + 170)/24 = 120/24 = 5.

Step 4: Compute the Pigouvian tax t*. By definition the tax equals the marginal external cost at the optimum, i.e., t* = MD(Q*).
MD(Q*) = 1.2*(Q*)^2 = 1.2*(5)^2 = 1.2*25 = 30.

Conclusion: The Pigouvian tax t* equals 30, which matches option a. If you examined other choices, they would fail to satisfy the condition SMB = PMC + MD at the socially optimal quantity or would yield a different Q* when solving the same equations.

IFTE0021_24-25 Mock Exam

You have the following PMC, MD and SMB: PMC = 15 + 2Q MD = 1.2Q2 SMB = 70 - 3Q Find the Pigouvian tax t*

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IFTE0021_24-25 Mock Exam

You have the following PMC, MD and SMB: PMC = 15 + 2Q MD = 1.2Q2 SMB = 70 - 3Q Find the Pigouvian tax t*

查看解析

登录即可查看完整答案

类似问题

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考试浏览器助手

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论文查重助手

文献引用助手

课堂转译助手

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A pollution charge gives the trucking industry an incentive to reduce its emissions, as long as the of reducing the emissions is .