1
Chapter 14 – Efficient and
Equitable Taxation
Public Economics
2
Optimal Commodity Taxation
? Assume that the goal is to finance
expenditures with a minimum of excess
burden.
? Assume lump sum taxes are infeasible.
? 3 commodities:
– Good X,Y,and leisure
– Prices PX,PY,and w.
3
Optimal Commodity Taxation
? Time endowment is fixed at,T
? The full budget constraint can be written
as:
w T P X P Y wlX Y? ? ?
4
Optimal Commodity Taxation:
Case 1 – All goods can be taxed
? If all commodities can be taxed,imposing
equal ad-valorem tax rates yields:
? ? ? ? ? ?w T t P X t P Y t wlX Y? ? ? ? ? ?1 1 1
? ?
w T
t
P X P Y wlX Y
1 ?
? ? ?
5
Optimal Commodity Taxation:
Case 1 – All goods can be taxed
? In this case,the inability to impose a lump
sum tax is irrelevant.
? The government can effectively take away
a lump sum amount through equal taxes
on all commodities (including leisure).
? No excess burden.
6
Optimal Commodity Taxation:
Case 2 – Not all goods can be
taxed
? May be impossible to tax non-market work.
? Assume only taxes can be applied to
goods X and Y.
? In general,some excess burden is
inevitable,Key question is how to select
rates on X and Y to minimize excess
burden subject to the revenue constraint.
7
Optimal Commodity Taxation:
Ramsey Rule
? Consider the idea of marginal excess
burden
– The additional inefficiency from
incrementally raising a tax by a small
amount.
– Figure 14.1 shows the initial excess burden
as a triangle (abc),and the marginal excess
burden as a trapezoid (fbae).
Figure 14.1
9
Optimal Commodity Taxation:
Ramsey Rule
? The marginal excess burden of taxing
good X is approximately,ΔX.
? The marginal tax revenue raised is
approximately,X1.
? Therefore the marginal excess burden
per dollar of tax revenue is:
?X
X1
10
Optimal Commodity Taxation:
Ramsey Rule
? Similar reasoning is used for good Y.
? Optimization therefore leads to:
? ?X
X
Y
Y1 1
?
? Ramsey rule says that to minimize total
excess burden,tax rates should be set so
the percentage reduction in the quantity
of each good demanded is the same.
11
Optimal Commodity Taxation:
Ramsey Rule Reinterpreted
? Recall the formula for excess burden for
good X:
EB P XtX X X? 12 2?
? Planner’s optimization problem is to
minimize total excess burden by choose
taxes on goods X and Y,subject to a
revenue constraint.
12
Optimal Commodity Taxation:
Ramsey Rule Reinterpreted
? Setting up the LaGrangian:
? ?m i n,,L P Xt P Yt R P Xt P Ytt t X X X Y Y Y X X Y Y
X Y ?
? ? ?? ? ? ? ?12 122 2
13
Optimal Commodity Taxation:
Ramsey Rule Reinterpreted
? Solving leads to a relationship between
tax rates and elasticities,
t tX X Y Y? ??
? Or rearranging we have the inverse
elasticity rule:
t
t
X
Y
Y
X
?
?
?
14
Optimal Commodity Taxation:
Ramsey Rule Reinterpreted
? Implication of the inverse elasticity rule:
– As long as goods are unrelated in
consumption (neither complements nor
substitutes),tax rates should be inversely
proportional to elasticities.
– When good Y is relatively inelastic,tax it
more.
15
Optimal Commodity Taxation:
Equity Considerations
? Is it,fair” to tax inelastic goods like food
and medicine?
– Clearly it is not.
? Another criteria for a tax system is
vertical equity,it should distribute
burdens fairly across people with
different abilities to pay.
16
Optimal Commodity Taxation:
Equity Considerations
? Ramsey rule has been modified to
account for the distributional issues.
? Degree of departure from original rule
depends on:
– How much society cares about equity
– Extent to which consumption patterns of
rich and poor differ
17
Optimal User Fees
? If government produces a good or
service,must directly choose a user fee.
– A user fee is price paid by users of the
good or service to the government.
– For example,natural monopoly.
? What is the,best” fee?
18
Optimal User Fees
? Consider the natural monopoly in Figure
14.2.
– Continually decreasing average costs
– Marginal cost lies everywhere below
average cost
Figure 14.2
20
Optimal User Fees
? A private firm would set MR=MC,and
choose Zm,This output level leads to
inefficiency.
? See Figure 14.3
Figure 14.3
22
Optimal User Fees
? Efficiency would require P=MC,or output
at Z*.
? Key problem is that at this quantity,price
is less than average cost,so the
operation suffers losses.
23
Optimal User Fees
? Policy solutions:
– Average cost pricing,Zero profits,but
ZA<Z*.
– Marginal cost pricing with Lump Sum
Taxes,Set P=MC,provide Z* at a loss,and
finance it with a lump sum tax.
? Assumes such a tax is available
? Equity considerations – who uses the good?
24
Optimal User Fees
? Second principle is called the benefits-
received principle – consumers of a
publicly provided service pay for it.
? A Ramsey Solution
– If government is running several
enterprises,choose markup over marginal
costs subject to a breakeven constraint.
25
Optimal Income Taxation
? Edgeworth’s model implies a radically
progressive tax structure,marginal tax
rates on high income individuals are
100%.
? Key problem is work incentives are not
accounted for.
26
Optimal Income Taxation:
Modern studies
? Account for work disincentives.
? Tax schedule is characterized by:
r e v e n u e t I n c o m e? ? ? ??
? Figure 14.4 shows this equation
Figure 14.4
28
Optimal Income Taxation:
Modern studies
? This schedule is referred to as a linear
income tax schedule (or a flat income
tax).
? Higher values of t mean more progressive
tax but larger excess burdens.
? Optimal income tax finds right
combination of α and t.
29
Optimal Income Taxation:
Modern studies
? Typical findings of optimal income tax
problems:
– Allowing for modest amount of substitution
between leisure and income leads to
income tax rates considerably less than
100%.
30
Other Criteria for Tax Design
? Horizontal equity,People in equal
positions should be treated equally
– Measures represent outcomes of people’s
decisions so it is difficult to figure out
whether they were initially in equal position.
? Costs of running a tax system
– Tax evasion
– Tax avoidance
31
Tax Evasion
? Tax evasion is failing to pay legally due
taxes.
? Tax cheating difficult to measure,and
probably manifests itself in a number of
ways:
– Keeping two sets of books
– Moonlighting for cash
– Barter
– Deal in cash
32
Tax Evasion
? Suppose person cares only about maximizing
expected income
– Goal is to choose R,the amount that is hidden from
authorities
– Marginal benefit of hiding income is the tax rate
? Assume authorities randomly audit with
probability ρ,and increasing penalty for greater
amounts hidden.
33
Tax Evasion
? Figure 14.5 shows that optimal
underreporting occurs when the
expected marginal benefit from doing
so exceeds the marginal cost.
– Implications,Cheating increases with
tax rates and decreases with
enforcement.
Figure 14.5
35
Tax Evasion
? Ignores a number of real-world
aspects:
– Psychic costs of cheating
– Risk aversion
– Work choices
– Probabilities of audit
36
Recap of Efficient and Equitable
Taxation
? Optimal Commodity Taxation
– All goods taxed
– Only some taxed
? User fees
? Optimal Income Taxation
? Tax Evasion
Chapter 14 – Efficient and
Equitable Taxation
Public Economics
2
Optimal Commodity Taxation
? Assume that the goal is to finance
expenditures with a minimum of excess
burden.
? Assume lump sum taxes are infeasible.
? 3 commodities:
– Good X,Y,and leisure
– Prices PX,PY,and w.
3
Optimal Commodity Taxation
? Time endowment is fixed at,T
? The full budget constraint can be written
as:
w T P X P Y wlX Y? ? ?
4
Optimal Commodity Taxation:
Case 1 – All goods can be taxed
? If all commodities can be taxed,imposing
equal ad-valorem tax rates yields:
? ? ? ? ? ?w T t P X t P Y t wlX Y? ? ? ? ? ?1 1 1
? ?
w T
t
P X P Y wlX Y
1 ?
? ? ?
5
Optimal Commodity Taxation:
Case 1 – All goods can be taxed
? In this case,the inability to impose a lump
sum tax is irrelevant.
? The government can effectively take away
a lump sum amount through equal taxes
on all commodities (including leisure).
? No excess burden.
6
Optimal Commodity Taxation:
Case 2 – Not all goods can be
taxed
? May be impossible to tax non-market work.
? Assume only taxes can be applied to
goods X and Y.
? In general,some excess burden is
inevitable,Key question is how to select
rates on X and Y to minimize excess
burden subject to the revenue constraint.
7
Optimal Commodity Taxation:
Ramsey Rule
? Consider the idea of marginal excess
burden
– The additional inefficiency from
incrementally raising a tax by a small
amount.
– Figure 14.1 shows the initial excess burden
as a triangle (abc),and the marginal excess
burden as a trapezoid (fbae).
Figure 14.1
9
Optimal Commodity Taxation:
Ramsey Rule
? The marginal excess burden of taxing
good X is approximately,ΔX.
? The marginal tax revenue raised is
approximately,X1.
? Therefore the marginal excess burden
per dollar of tax revenue is:
?X
X1
10
Optimal Commodity Taxation:
Ramsey Rule
? Similar reasoning is used for good Y.
? Optimization therefore leads to:
? ?X
X
Y
Y1 1
?
? Ramsey rule says that to minimize total
excess burden,tax rates should be set so
the percentage reduction in the quantity
of each good demanded is the same.
11
Optimal Commodity Taxation:
Ramsey Rule Reinterpreted
? Recall the formula for excess burden for
good X:
EB P XtX X X? 12 2?
? Planner’s optimization problem is to
minimize total excess burden by choose
taxes on goods X and Y,subject to a
revenue constraint.
12
Optimal Commodity Taxation:
Ramsey Rule Reinterpreted
? Setting up the LaGrangian:
? ?m i n,,L P Xt P Yt R P Xt P Ytt t X X X Y Y Y X X Y Y
X Y ?
? ? ?? ? ? ? ?12 122 2
13
Optimal Commodity Taxation:
Ramsey Rule Reinterpreted
? Solving leads to a relationship between
tax rates and elasticities,
t tX X Y Y? ??
? Or rearranging we have the inverse
elasticity rule:
t
t
X
Y
Y
X
?
?
?
14
Optimal Commodity Taxation:
Ramsey Rule Reinterpreted
? Implication of the inverse elasticity rule:
– As long as goods are unrelated in
consumption (neither complements nor
substitutes),tax rates should be inversely
proportional to elasticities.
– When good Y is relatively inelastic,tax it
more.
15
Optimal Commodity Taxation:
Equity Considerations
? Is it,fair” to tax inelastic goods like food
and medicine?
– Clearly it is not.
? Another criteria for a tax system is
vertical equity,it should distribute
burdens fairly across people with
different abilities to pay.
16
Optimal Commodity Taxation:
Equity Considerations
? Ramsey rule has been modified to
account for the distributional issues.
? Degree of departure from original rule
depends on:
– How much society cares about equity
– Extent to which consumption patterns of
rich and poor differ
17
Optimal User Fees
? If government produces a good or
service,must directly choose a user fee.
– A user fee is price paid by users of the
good or service to the government.
– For example,natural monopoly.
? What is the,best” fee?
18
Optimal User Fees
? Consider the natural monopoly in Figure
14.2.
– Continually decreasing average costs
– Marginal cost lies everywhere below
average cost
Figure 14.2
20
Optimal User Fees
? A private firm would set MR=MC,and
choose Zm,This output level leads to
inefficiency.
? See Figure 14.3
Figure 14.3
22
Optimal User Fees
? Efficiency would require P=MC,or output
at Z*.
? Key problem is that at this quantity,price
is less than average cost,so the
operation suffers losses.
23
Optimal User Fees
? Policy solutions:
– Average cost pricing,Zero profits,but
ZA<Z*.
– Marginal cost pricing with Lump Sum
Taxes,Set P=MC,provide Z* at a loss,and
finance it with a lump sum tax.
? Assumes such a tax is available
? Equity considerations – who uses the good?
24
Optimal User Fees
? Second principle is called the benefits-
received principle – consumers of a
publicly provided service pay for it.
? A Ramsey Solution
– If government is running several
enterprises,choose markup over marginal
costs subject to a breakeven constraint.
25
Optimal Income Taxation
? Edgeworth’s model implies a radically
progressive tax structure,marginal tax
rates on high income individuals are
100%.
? Key problem is work incentives are not
accounted for.
26
Optimal Income Taxation:
Modern studies
? Account for work disincentives.
? Tax schedule is characterized by:
r e v e n u e t I n c o m e? ? ? ??
? Figure 14.4 shows this equation
Figure 14.4
28
Optimal Income Taxation:
Modern studies
? This schedule is referred to as a linear
income tax schedule (or a flat income
tax).
? Higher values of t mean more progressive
tax but larger excess burdens.
? Optimal income tax finds right
combination of α and t.
29
Optimal Income Taxation:
Modern studies
? Typical findings of optimal income tax
problems:
– Allowing for modest amount of substitution
between leisure and income leads to
income tax rates considerably less than
100%.
30
Other Criteria for Tax Design
? Horizontal equity,People in equal
positions should be treated equally
– Measures represent outcomes of people’s
decisions so it is difficult to figure out
whether they were initially in equal position.
? Costs of running a tax system
– Tax evasion
– Tax avoidance
31
Tax Evasion
? Tax evasion is failing to pay legally due
taxes.
? Tax cheating difficult to measure,and
probably manifests itself in a number of
ways:
– Keeping two sets of books
– Moonlighting for cash
– Barter
– Deal in cash
32
Tax Evasion
? Suppose person cares only about maximizing
expected income
– Goal is to choose R,the amount that is hidden from
authorities
– Marginal benefit of hiding income is the tax rate
? Assume authorities randomly audit with
probability ρ,and increasing penalty for greater
amounts hidden.
33
Tax Evasion
? Figure 14.5 shows that optimal
underreporting occurs when the
expected marginal benefit from doing
so exceeds the marginal cost.
– Implications,Cheating increases with
tax rates and decreases with
enforcement.
Figure 14.5
35
Tax Evasion
? Ignores a number of real-world
aspects:
– Psychic costs of cheating
– Risk aversion
– Work choices
– Probabilities of audit
36
Recap of Efficient and Equitable
Taxation
? Optimal Commodity Taxation
– All goods taxed
– Only some taxed
? User fees
? Optimal Income Taxation
? Tax Evasion
