Transcript Document

Chapter 13 –
Taxation and Efficiency
Public Finance
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McGraw-Hill/Irwin
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
Introduction
• Are people unaffected by a tax increase if
they pay zero in taxes afterwards?
– No, consumption may have changed in
response to the tax increase
– Bundle consumed is less desirable
• Excess burden is a loss of welfare
above and beyond the tax revenues
collected.
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Excess Burden Defined
• Two commodities
– Barley and corn
• Fixed income
• Pb and Pc are prices of goods
• No distortions such as externalities,
imperfect competition, public goods, etc.
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Excess Burden Defined
• Figure 13.1 shows the budget constraint
(AD), with utility maximized at bundle E1.
• Ad-valorem tax levied on barley at rate tb
raises the price to (1+tb)Pb and rotates
the budget constraint along the x-axis.
The new budget constraint is AF.
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Figure 13.1
Excess Burden Defined
• At each consumption level of barley, the
vertical distance between AD and AF
shows tax payments in terms of forgone
corn.
• Normalize Pc=$1, so that vertical distance
can be measured in either quantity of
corn or dollars.
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Excess Burden Defined
• Figure 13.2 shows new optimizing choice
with the higher prices along budget
constraint AF.
• Utility maximized at bundle E2.
• Vertical distance between old and new
budget constraints is GE2, the “tax bill.”
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Excess Burden Defined
• Any tax will lower utility, but is there an
alternative tax that raises the same
revenue, GE2, but entails a smaller utility
loss? Or greater revenue with the same
utility loss?
• If so, the tax on barley leads to excess
burden.
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Figure 13.2
Excess Burden Defined
• Equivalent variation is the amount of income
we would have to take away (before any tax
was imposed) to induce a move to the lower
indifference curve (Increasing price is
equivalent to a income reduction).
• Taking away income is equivalent to a parallel
movement inward on the budget constraint.
• Budget constraint HI in Figure 13.3 shows this.
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Figure 13.3
Excess Burden Defined
• Note that ME3=GN>GE2, but both give
the consumer the same utility.
• Thus, the difference E2N is the excess
burden of the barley tax. The barley tax
makes the person worse off by an
amount that exceeds the revenue it
generates.
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Excess Burden Defined
• Lump sum tax is a tax that must be paid
regardless of the taxpayer’s behavior.
• Budget constraint HI satisfies this.
Revenue yield exactly equals the
equivalent variation.
– Conclusion: Lump sum tax has no excess
burden.
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Questions and Answers
• Why aren’t lump sum taxes widely used?
– Construed as unfair because people’s abilities to pay
vary
• How do lump sum taxes relate to welfare
economics?
– The equilibrium conditions become:
MRS bc
1  tb  Pb


Pc
Pb
 MRTbc 
Pc
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Questions and Answers
• Intuitively, when MRS>MRT the marginal
utility of substituting barley consumption
for corn consumption exceeds the
change in production costs from doing so.
• In the presence of the tax, there is no
financial incentive to do so.
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Questions and Answers
• Does an income tax entail excess burden?
– It usually does entail excess burden, if a third
commodity, leisure, exists.
• If demand for a commodity is perfectly
inelastic, is there excess burden?
– Yes, see Figure 13.4.
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Figure 13.4
Questions and Answers
• In Figure 13.4, the ordinary (uncompensated)
demand curve is inelastic – B1=B2 when the
price increases.
• But this is because the income effect offsets the
substitution effect.
• The substitution effect is the part necessary to
compute excess burden. The compensated
demand curve (which holds utility constant as
prices change) is the relevant one, and the
elasticity for it is non-zero.
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Excess Burden Measurement with
Demand Curves
• Consider a compensated demand curve,
such as the one in Figure 13.5.
• Impose an ad-valorem tax on barley, so
that its price increases to (1+tb)Pb.
• Equivalent to the supply curve shifting
upward.
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Figure 13.5
The measurement of excess burden
•The concept of excess burden can be reinterpreted using (compensated)
demand curves. Assume that the compensated demand curve for barley
can be represented by the straight line Db For convenience, we continue to
assume that the social marginal cost of barley is constant at Pb. Suppose
that a tax at percentage rate tb is levied on barley, so the new price, (1 +
tb)Pb, is associated with supply curve Sb.
•The excess burden of the tax, fid, is one-half the product of its base (the
tax-induced change in the quantity of barley) and height (the tax per pound),
that is
•-(1/2)( P) (q)…………………..(*)
•Since q=(q1/pb) (P)e, with P=tbPb , some simple algebra leads to the
conclusion that the excess burden (*) can be written
•-(1/2)ePbql(tb)2
(10.3)
•where e is the value of the compensated price elasticity of demand (in
general e <0) for barley.
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•A high absolute value of e indicates that the compensated
quantity demanded is quite sensitive to changes in price.
Thus, the presence of e in Equation (10.3) makes intuitive
sense--the more the tax distorts the (compensated)
consumption decision, the higher the excess burden.
•Finally, the presence of (tb)2 suggests that as the tax rate
increases, excess burden goes up with its square. Doubling a
tax quadruples its excess burden, other things being the
same. Because excess burden in- creases with the square of
the tax rate, the marginal excess burden from raising one
more dollar of revenue exceeds the average excess burden.
That is, the incremental excess burden of raising one more
dollar of revenue exceeds the ratio of total excess burden to
total revenues. This fact has important implications for costbenefit analysis.
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 Similarly, the concept of excess burden can be
reinterpreted using (compensated) supply curves.
Assume that the social marginal cost of barley is
upward sloping. The excess burden of the tax can
be approximated by
-(1/2)( ps) (q)…………………..(**)
 Since q=(q1/ps) (ps)h, with ps=-tsp , some simple
algebra leads to the conclusion that the excess
burden (**) can be written
 (1/2)hpq(ts)2
(10.**)
 where h is the value of the compensated price
elasticity of supply (in general h >0) for barley.
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Excess Burden Measurement with
Demand Curves
• Excess burden equal to triangle fid.
• Through some mathematical manipulation,
this can be expressed as:
1
2
EB  h Pb qt t b
2
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Excess Burden Measurement with
Demand Curves
• Implications of formula:
– Higher (compensated) elasticities lead to
larger excess burden.
– Excess burden increases with the square of
the tax rate.
– The greater the initial expenditure on the taxed
commodity, the larger the excess burden.
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Differential Taxation of Inputs
• Some inputs are taxed differently
depending on where they are used:
– Capital used in the corporate sector is subject
to a higher tax rate than capital used in the
noncorporate sector.
– Labor used in the household is untaxed
• Figure 13.8 measures the efficiency cost
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Figure 13.8
Differential Taxation of Inputs
• In this figure, total amount of labor is fixed at
OO’. Moving along the x-axis simply shifts labor
from the labor market to the household sector.
• VMP is the value of marginal product, or the
dollar value of the additional input produced
from an hour of work.
• VMP declines with hours worked in a sector.
Optimal allocation of hours equates margins,
such that OH* is spent in household production,
and O’H* is spent in the market.
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Differential Taxation of Inputs
• If a tax is levied on market work, but not
household production, then the “effective”
VMP curve for market work rotates
downward.
• Figure 13.9 shows the effects.
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Figure 13.9
Differential Taxation of Inputs
• People shift hours into nonmarket work.
– Household production increases from OH* to
OHt, while market work decreases from O’H*
to O’Ht.
• Excess burden equal to abe.
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Overall excess burden
•
To summarize: Given that the tax on rum was already in place, the tax on gin
creates an excess burden of efd in the gin market and simultaneously
decreases excess burden by cbhg in the rum market. If cbhg is sufficiently
large, the tax can actually reduce overall excess burden. This is an example of
the theory of the second best: In the presence of existing distortions, policies
that in isolation would increase efficiency can decrease it and vice versa.
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This discussion is a special case of the result that the excess burden of a set
of taxes generally depends on the whole set of tax rates, as well as on the
degree of substitutability and complementarity among the various
commodities. Specifically, suppose that n commodities are subject to taxation.
• Let Pi be the before-tax price of the ith commodity; ti the ad valorem tax
• on the ith commodity; and Sij, the compensated response in the demand of the
ith good with respect to a change in the price of the jth good. Then the overall
excess burden is
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Recap of Taxation and Efficiency
• Excess Burden Defined
• Questions and Answers
• Excess Burden Measurement with
Demand Curves
• Differential Taxation of Inputs
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