Mr. Inflexible's utility function is U(x₁,x₂)= min(x₁,x₂). The price of x₁is $2 including a $1 excise tax, the price of x₂is $2, and Mr. Inflexible's budget is $120. Imagine substituting a lump- sum tax for the excise tax. If both taxes raise the same amount of tax revenue, the lump- sum tax is: A)$20. B)$30. C)$10. D)$50.

Question

Quizplus · Accepted Answer

To determine the optimal consumption bundle, we need to set up the budget constraint: $2(x₁ + x₂) + T = $120 Where T is the tax revenue raised from either the excise tax or lump-sum tax. First, we'll solve for the optimal bundle under the excise tax: U(x₁, x₂) = min(x₁, x₂) Subject to $2(x₁ + x₂) + T = $120 We can solve for the optimal bundle by considering two cases: Case 1: x₁ < x₂ In this case, the utility function reduces to U(x₁, x₂) = x₁. Plugging this into the budget constraint and solving for x₁, we get: $x₁ = (60 - T/2)/2 Case 2: x₂ < x₁ In this case, the utility function reduces to U(x₁, x₂) = x₂. Plugging this into the budget constraint and solving for x₂, we get: $x₂ = (60 - T/2)/2 To find the optimal bundle, we need to compare the utilities in each case and choose the highest. If $2x₁ < $2x₂, then we know that x₁ is the limiting factor in the utility function, and thus we should set x₁ equal to (60 - T/2)/4, and x₂ equal to (60 - T/2)/2. If $2x₁ > $2x₂, then we know that x₂ is the limiting factor in the utility function, and thus we should set x₂ equal to (60 - T/2)/4 and x₁ equal to (60 - T/2)/2. Now, let's solve for the optimal bundle under the lump-sum tax. Since the lump-sum tax raises the same amount of revenue as the excise tax, we have: T = $2(x₁ + x₂) Plugging this into the budget constraint and rearranging, we get: $x₁ + x₂ = $59 We can then use the utility function to determine the optimal allocation between x₁ and x₂: U(x₁, x₂) = min(x₁, x₂) Subject to $x₁ + x₂ = $59 Since the utility function is symmetric, the optimal bundle is x₁ = x₂ = $29.5. Therefore, the lump-sum tax is $120 - $59 = $61, which is the same amount of revenue as the excise tax. The lump-sum tax is $61 - $30 = $31 higher than the excise tax.

Mr Inflexible's Utility Function Is U(x1,x2)= Min(x1,x2)

Mr Inflexible's Utility Function Is U(x₁,x₂)= Min(x₁,x₂)