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Chapter 14:
Efficient and Equitable
Econ 330: Public Finance
Dr. Reyadh Faras
Dr. Reyadh Faras
Optimal Commodity Taxation
 One economic policy question: At what rates should
various goods and services be taxed?
 The theory of optimal commodity taxation provides a
framework for answering this question.
 Knowing the right set of taxes depends on the
government’s goal.
 We assume that the only goal is to finance public
expenditures with minimum excess burden and without
using any lump sum taxes.
Dr. Reyadh Faras
 A person consumes 2 goods (X&Y) and leisure (l).
 The prices are PX, PY and w, respectively.
 The maximum number of hours for work is fixed at T,
per year.
 Assuming the person consumes all of its income, its
budget constraint is: w (T-l) = PX X + PY Y
 The LHS gives total earnings, and the RHS shows
how the earnings are spent.
 The equation may be rewritten as:
wT = PX X + PY Y+ wl
 The LHS is the value of time endowment.
 Assume that it is possible to tax X, Y and l by the same
Ad Valorem rate, t.
Dr. Reyadh Faras
 The after tax budget constraint becomes:
wT = (1+t) PX X + (1+t) PY Y+ (1+t) wl
 Dividing both sides by (1+t), we have:
(1/(1+t)) wT = PX X + PY Y+ wl
 Comparison between (14.3) and (14.4) shows that: A
tax on all commodities including leisure is equivalent to
reducing the value of time endowment from wT to
(1/(1+t)) wT
 Because w and T are fixed, wT is also fixed.
 Therefore, a proportional tax on time endowment is a
lump tax, which has no excess burden.
Dr. Reyadh Faras
 Conclusion: a tax at the same rate on all goods
including leisure is equivalent to a lump sum tax and
has no excess burden.
 It sounds good, but putting a tax on leisure time is
 The only available tax instruments are taxes on
commodities X and Y.
 Therefore, some excess burden is inevitable.
 The goal of the OCT is to select tax rates on X and Y
in a way that minimizes the excess burden of raising tax
 It might be appropriate to tax X and Y at the same
rate, so called neutral taxation.
 This will be shown laterDr. to
be inefficient.
Reyadh Faras
The Ramsey Rule
 To minimize overall excess burden, the marginal
excess burden of the last dollar of revenue raised from
each commodity must be the same.
 Otherwise, it would be possible to lower overall excess
burden by raising the rate on the commodity with the
smaller marginal excess burden, and vice versa.
 Assume X and Y are unrelated to each other, meaning
that a change in the price of either commodity affects
its own demand only.
Dr. Reyadh Faras
Figure 14.1
Dr. Reyadh Faras
 Suppose a unit tax (ux) is levied on X, followed by a
tax of 1, the total price becomes P0 + (ux + 1)
 Quantity demanded falls by Δx to X2 and the
associated excess burden is ½ Δx [ux + (ux +1)]
 With some algebra, the marginal excess burden is ΔX
and the marginal tax revenue as X1 - ΔX
 The marginal excess burden per last dollar of tax
revenue is ΔX/(X1 - ΔX)
 The condition for minimizing overall excess burden is
that the marginal excess burden per last dollar of
revenue be the same for each commodity, we must set
ΔX/(X1 - ΔX) = ΔY/(Y1 - ΔY)
Dr. Reyadh Faras
 This implies ΔX/X1 = ΔY/Y1
 Equation (14.7) says that to minimize total excess
burden, tax rates should be set so that the percentage
reduction in the quantity demanded of each commodity
is the same.
 This result is called the Ramsey Rule.
Dr. Reyadh Faras
A Reinterpretation of the Ramsey Rule
 It is useful to explore the relationship between the
Ramsey Rule and demand elasticities.
 Let ηx be the elasticity of demand for X, and tx be
the tax rate on X expressed as Ad Valorem tax.
 By definition of A.V. tax, tx is the percentage
increase in price induced by the tax.
 Hence, txηx is the percentage reduction in demand
for X induced by the tax (same applies for Y).
Dr. Reyadh Faras
 The Ramsey Rule says that to minimize excess
burden, the percentage of change for X and Y must
be equal: txηx = tyηy
 Now divide both sides by tyηx to obtain:
(tx/ty) = (ηy / ηx)
 Equation (14.9) is the inverse elasticity rule:
“As long as goods are unrelated in consumption,
tax rates should be inversely proportional to
 That is, the higher is ηy relative to ηx, the lower
should be ty relative totx .
 Efficiency doesn’t require all rates be set uniformly
Dr. Reyadh Faras
The Corlett-Hague Rule
 Corlett and Hague proved an interesting
implication of the Ramsey Rule: When there are
two commodities, efficient taxation requires taxing
the commodity that is complementary to leisure at
a relatively high rate.
 Since taxing leisure results in no excess burden,
but because it is impossible to tax leisure, taxing
goods used jointly with leisure, could lower the
demand for leisure.
Dr. Reyadh Faras
Equity Considerations
 The efficient tax theory may seem to have
unpleasant policy implication, as it relates tax
rates negatively with demand elasticities.
 Efficiency is one criterion to evaluate a tax system;
fairness is important as well.
 Tax system should have vertical equity:
 It should distribute burdens fairly across people
with different abilities to pay.
 The Ramsey Rule has been modified to account
for the distributional consequences of taxation.
Dr. Reyadh Faras
 If the poor spend high share of income on X than
the Rich, and vice versa for Y.
 Suppose the social welfare function puts more
weight on the utilities of the poor than the rich.
 Then even though demand for X is less elastic,
optimal taxation may require higher tax rate on Y.
 It is true this will create larger excess burden, but
it redistributes income toward the poor.
 Society may be willing to accept higher burden in
return for a more equal distribution of income.
Dr. Reyadh Faras
In general, the optimal departure from the Ramsey
Rule depends on two factors:
 How much society cares about equality.
 The extent to which consumption patterns differ
between the rich and the poor.
 If lump taxes were available, taxes could be raised
without any excess burden.
 Since lump sum taxes not available, minimizing the
excess burden requires setting taxes in a way that
reduces demand for all goods in same proportion.
 For unrelated goods, set tax rates inversely with
demand elasticities. An exception to this rule is if
equity is an issue.
Dr. Reyadh Faras
Optimal User Fees
 A user fee is the price paid by users of a good or
service provided by the government.
 Government production may be appropriate if there
are economies of scale; greater output reduces AC.
 In this case a single firm can supply the entire market.
 This phenomenon is called natural monopoly.
 A private firm may produce the commodity, while in
some cases produced by the public sector.
 Private monopolies often regulated by governments.
Dr. Reyadh Faras
Figure 14.3
Dr. Reyadh Faras
 Decreasing AC often leads to public sector
production, or regulated private sector production.
 Unregulated monopolist seeking profit produces
where MR=MC at quantity (Zm), price (Pm), and
making profit.
 Is this efficient?
 According to welfare economics, efficiency requires
 But at (Zm) price is greater than MC, hence (Zm) is
 Efficiency and existence of monopoly profits provide a
possible justification for government taking over the
production of (Z).
Dr. Reyadh Faras
 In this case the government should produce up to the
point where P=MC at output (Z*) and price (P*).
 Problem: in this case P<AC, the firm incurs a loss.
 How should the government confront this dilemma?
 Several solutions have been proposed.
1) Average Cost Pricing
 When P=AC, the firm breaks even.
 It produces (ZA) and charges price (PA).
 Note that (ZA) < (Z*), meaning that AC pricing fall
short of the efficient amount of output.
Dr. Reyadh Faras
2) Marginal Cost Pricing with Lump Sum Taxes
 Charge P=MC, and fill the gap with a lump sum tax.
 This price ensures efficiency in the market, while
lump sum taxes guarantees no new inefficiencies.
 However, there are two problems:
 1st: Lumps sum taxes are generally unavailable. Only
distorting taxes are available, which may outweigh
market efficiency from P=MC pricing.
 2nd: Fairness requires consumers of publicly provided
services to pay for it “benefits-received principle”.
Dr. Reyadh Faras
3) A Ramsey Solution
 Suppose the government runs several enterprises,
who as group cannot lose money, but any individual
one can.
 Suppose the government wants the financing to come
from users of each service enterprise.
 By how much should the user fee for each service
exceed its MC?
 This is similar to the optimal tax problem.
 The difference between the MC and the user fee is
just the tax that the government levies on the good.
Dr. Reyadh Faras
 The government has to raise certain amount of
revenue enough for the group to break even.
 The Ramsey Rule gives the answer: set the user fees
so that demands for each commodity are reduced
Dr. Reyadh Faras