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201Winter2010Classes6to8

# 201Winter2010Classes6to8 - Class 5 An Optimal Taxation...

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Class 5: An Optimal Taxation Application Q . Suppose that the government must collect exactly revenue R from a large number of consumers by value taxes. Should the government impose a value tax on only one good or on all goods? Remark . We assume that all consumers have identical preferences. Thus we can examine the question with one representative consumer . De°nition . A value tax ° on good 1 increases the price of good 1 from p 1 to p 1 (1 + ° ) per unit. Remark . The seller of good 1 collects revenue p 1 x 1 and the government collects the tax revenue °p 1 x 1 : Remark . If the budget constraint is p 1 x 1 + p 2 x 2 ° m without any taxes, with a value tax ° on good 1 the budget constraint is p 1 (1 + ° ) x 1 + p 2 x 2 ° m: The budget line therefore changes from p 1 x 1 + p 2 x 2 = m to p 1 (1 + ° ) x 1 + p 2 x 2 = m: FIGURE 5.1 HERE Remark . In what follows we use the terms budget constraint and budget line interchange- ably. 1

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Remark . If the budget constraint is p 1 x 1 + p 2 x 2 ° m without any taxes, with a value tax ± both on good 1 and on good 2 the budget constraint is p 1 (1 + ± ) x 1 + p 2 (1 + ± ) x 2 ° m: The budget line therefore changes from p 1 x 1 + p 2 x 2 = m to p 1 (1 + ± ) x 1 + p 2 (1 + ± ) x 2 = m: FIGURE 5.2 HERE Let ( x ° 1 ; x ° 2 ) denote the optimal choice with a value tax, denoted °; on only one good. FIGURE 5.3 HERE Remark . This assumption implies that MRS at ( x ° 1 ; x ° 2 ) of the indi/erence curve that goes through ( x ° 1 ; x ° 2 ) equals the slope of the budget constraint with a value tax ° on only good 1. Remark . The slope of the budget constraint with a value tax ° on only good 1 is ± p 1 (1+ ° ) p 2 . (Step 1: Divide both sides of the budget constraint by p 2 : Step 2: Rearrange x 2 on the left-hand side and all other terms on the right-hand side). 2
Let (^ x 1 ; ^ x 2 ) denote the optimal choice with value tax, denoted ±; on both goods. FIGURE 5.4 HERE Remark . The MRS at (^ x 1 ; ^ x 2 ) of the indi/erence curve that goes through (^ x 1 ; ^ x 2 ) equals the slope of the budget constraint with a value tax ± on both goods. Remark . The slope of the budget constraint with a value tax ± on both goods is ± p 1 p 2 : (Step 1: Divide both sides of the budget constraint by (1 + ± ) p 2 : Step 2: Rearrange x 2 on the left-hand side and all other terms on the right-hand side). Remark . The key question is: where are the two budget constraints (the budget constraint with tax ° on one good vs. the budget constraint with tax ± on both goods) \ FIGURE 5.5 HERE 3

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Result : With value tax ± on both goods, the government°s revenue is the same regardless of the consumer°s choice. Proof : The government°s revenue is ±p 1 x 1 + ±p 2 x 2 : The consumer°s optimal choice satis±es (1 + ± ) p 1 x 1 + (1 + ± ) p 2 x 2 = m; which can be written as x 2 = m p 2 (1 + ± ) ± p 1 p 2 x 1 : Substituting x 2 = m p 2 (1+ ± ) ± p 1 p 2 x 1 for x 2 in the expression for the government°s revenue, ±p 1 x 1 + ±p 2 x 2 , gives ±p 1 x 1 + ±p 2 ° m p 2 (1 + ± ) ± p 1 p 2 x 1 ± = ±p 1 x 1 + ±p 2 m p 2 (1 + ± ) ± ±p 2 p 1 p 2 x 1 = ±p 1 x 1 + ± m 1 + ± ± ±p 1 x 1 = ± m 1 + ±
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201Winter2010Classes6to8 - Class 5 An Optimal Taxation...

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