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Unformatted text preview: Economics 3010
Fall 2010 Professor Daniel Benjamin
Cornell University Problem Set 5 Solutions Optional Problem. (Taxation: The lump‐sum principle) [ You can find solutions to some parts of this question in ch. 5.6 (except that ch. 5.6 examines the case of a quantity tax, while this question is about an ad valorem tax). Feel free to use what you learn from that part of the textbook in writing up your answer, but try the problem first with the textbook closed, and make sure you write up solutions in your own words. ] (a) Write down a standard 2‐good budget constraint, and draw a diagram. Now write down the budget constraint after an ad valorem tax is imposed on Good 1, and draw it on the same diagram. Draw indifference curves to show the consumer’s choice before and after the imposition of the tax. Throughout this problem, we are assuming (to keep the analysis as simple as possible) that the price of the good remains constant, despite the tax. Of course, we know from the supply/demand model that, in general, a tax will change the equilibrium price. Assuming that the tax does not affect the price amounts to assuming implicitly that the supply curve is perfectly elastic. Show on a supply/demand diagram that if supply is perfectly elastic, then a tax will be fully borne by demanders (regardless of whether it is imposed on demanders or suppliers) and will not affect the equilibrium price. (B1) Original Budget Line p1 x1 + p2 x2 = m (B2) With Ad Valorem Tax on Good 1 (rate ) p1(1+) x1 + p2 x2 = m x2 (I1) (I2) *
x1 (B1) x1 * Taxation with perfectly elastic supply (for reading, please refer to Chapter 16.6 ‐ 7) 1 Demand
Price p*+t S’
D Q In the case of a perfectly elastic supply curve the tax is completely passed along to the consumers. (b) Now consider a lump‐sum tax that raises the same amount of revenue. Write down the budget constraint, and draw it on the same budget constraint/indifference curve diagram as you drew the ad valorem tax. Draw an indifference curve to show the consumer’s choice after the imposition of the lump‐sum tax. Show that the consumer must be better off with the lump‐sum tax than with the revenue‐equivalent ad valorem tax. (The fact that lump sum taxes are the welfare‐maximizing way of raising a given amount of revenue is sometimes called “the lump‐sum principle.”) From (B2) the tax revenue of the government (R) = p1 x1* Hence, if the lump‐sum tax is imposed, the budget line is: p1 x1 + p2 x2 = m – p1 x1* (B3) x2 (I2) (I3) (I1) *
x2 (B2) (B3) (B1) x1 (c) We’ve shown graphically that the consumer must be better off with the lump‐sum tax; now we’ll show it analytically. With the ad valorem tax, what is the consumer’s marginal rate of substitution? (Hint: Remember that at an optimum for the consumer, it must equal the slope of the budget line.) *
x1 2 What is the slope of the budget line with the lump‐sum tax? Show that the consumer will be better off with the lump‐sum tax because he will consume more of Good 1 (and less of Good 2). p (1 )
MRS with ad valorem tax = 1 MRS with lump‐sum tax = 1 p2 p2 At (x1*, x2*), the MRS is p1 (1 )
. But the income tax allows the consumer to trade at a rate of p2 exchange of – p1/p2. Hence, the consumer will increase theconsumption of Good 1 and reduce the consumption of good 2, achieving a higher point of utility. (d) In lecture, we’ve discussed how lump‐sum taxes are extremely unpopular in practice. Why do you think people generally consider lump‐sum taxes to be “unfair”? Lump‐sum taxes are regarded as unfair because the same tax is charged regardless of people’s abilities to pay. (e) There are sneaky ways of effectively imposing a lump‐sum tax that people do not generally consider to be unfair. Show that imposing an ad valorem tax with the same tax rate on both goods is economically equivalent to imposing a lump‐sum tax. (Hint: Write down the budget constraint in this case.) p1(1+) x1 + p2 (1+) x2 = m (B1) ad valorem tax with the same tax rate on both goods p1x1 + p2x2 = m – m( / (1+)) (B2) Impose an equivalent amount of lump‐sum tax (R) If we rearrange the terms in (B2), it is exactly same as (B1) (f) Of course, we have only been considering the individual’s choice between consumption goods. In fact, imposing a “consumption tax” (a tax on all consumption goods) may distort the tradeoff between consumption and leisure. Write down the consumption‐leisure budget constraint. Argue that having a tax system that fully replicates a lump‐sum tax would also require a labor income subsidy with the subsidy rate on labor income equal to the tax rate on consumption. (Note that because the subsidy will cost the government money, the tax rate on consumption will now have to be higher to raise the same amount of revenue.) As we derived in class, the budget line for consumption and leisure can be written as: pc wR M wL where c: total amount of consumption p: price of unit of consumption R: amount of leisure (hours) w: wage rate M: non‐labor income 3 L : total time in the day Re‐arrange terms and we have pc w (R L ) M (B1) As we’ve seen in (e), imposing a consumption tax with the same tax rate on both goods is economically equivalent to imposing a lump‐sum tax, i.e. p(1 )c w (1 )(R L ) M (B2) When we write (B2) in terms of labor (ℓ, where ℓ = L ‐ R): p (1+) c = M + w (1+) ℓ Hence, it is equivalent that the consumer gains subsidy of (1+) on the wage (w). (g) In reality, many countries do have a consumption tax, such as a sales tax or a Value Added Tax (VAT), that is imposed on all consumption goods. (In some cases, the consumption tax rates are quite high. For example, the U.K. has a 17.5% VAT.) Of course, governments generally tax labor income, rather than subsidizing it. Economists often analyze the extent to which income taxes distort individuals’ consumption‐leisure tradeoff. Explain why analyses that ignore consumption taxes will underestimate the extent to which income taxes distort individuals’ consumption‐leisure choice. When the income tax is imposed and wage decreases, the supply of labor is expected to decrease. In addition, consumption tax also decreases the labor supply as it makes the consumption of other goods more expensive and at the same time consumption of leisure relatively cheaper (substitution effect). Hence, considering income tax effect alone underestimates the extent to which an individual’s labor supply decision will deviate from what he would choose in markets without taxes. 4 ...
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