Chapter 10 worked example: Tax saliency

There are two goods bananas and money. If Ian buys bananas and has $ left his utility is

, 2 √

where is some parameter. Suppose that Ian has $ to start with and the price of a banana is

ignoring the tax and 1 taking into account the tax.

If Ian takes into account the tax he would maximize his utility subject to the budget

constraint 1 . The optimum is where the marginal utility from consuming

bananas equals the price. The marginal utility from consuming bananas is , √ . So at the optimum

√ 1 . Rearranging we get

1 . Note that we should round to the nearest integer. Plugging this into the budget constraint we

get 1 . ` If 10, 200 and 1 then with no sales tax ( 0) we get 101 1 0 100 200 1 1 0 100 100

From this we know that

2 √ 2 10 √100 100 300.

Tax revenue is clearly zero. With a five percent sales tax ( 0.05) we get 101 1 0.05 91 200 1 1 0.05 91 104.45. From this we know that

2 10 √91 104.45 295. Tax revenue is 0.05 91 $4.55. With a ten percent sales tax ( 0.1) we get 101 1 0.1 83 200 1 1 0.1 91 108.70. From this we know that

2 10 √83 108.70 291. Tax revenue is 0.1 83 $8.30. This is enough to fill in the numbers if Table 10.7

corresponding to the case of tax saliency.

Table 10.7

Tax rate

Is the tax salient?

no yes no yes no yes no yes

bananas bought money left utility tax revenue 0 100 100 $100.00 $100.00 300 300 $0.00 $0.00 0.05 100 91 $95.00 $104.45 295 295 $5.00 $4.55 1.1 100 83 $90.00 $108.70 290 291 $10.00 $8.30

If Ian does not take into account the sales tax he would maximize his utility subject to

the budget constraint . The optimum in this case is where

√ .

Rearranging we get

. In terms of money left we still take into account the sales tax and so 1 .

The story with no sales tax is identical to that where the tax is salient. With a five

percent sales tax ( 0.05) we get 101 100 200 1 1 0.05 100 $95. From this we know that

2 10 √100 95 295. Tax revenue is 0.05 100 $5.00. With a ten percent sales tax ( 0.1) we get 101 100 200 1 1 0.1 100 90. From this we know that

2 10 √100 90 290. Tax revenue is 0.1 100 $10. This fills in the rest of the numbers in Table 10.7.

The difference in utility if Ian takes account of the sales tax is ∆ , ,

which is given by

∆ 2 1 1 2 1 . This simplifies to

∆ 21 1 2 1 . We can simplify this further still to

∆ 1 1 1 11 1 . If 0.05 we get

∆ 10 0.051 1 0.05 0.23. If 0.1 we get

∆ 10 0.11 1 0.1 0.91.