Chapter 14:

Efficient and Equitable Taxation

Econ 330: Public Finance

Dr. Reyadh Faras

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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.

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Example 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.

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The after tax budget constraint becomes:wT = (1+t) PX X + (1+t) PY Y+ (1+t) wl (14.3)

Dividing both sides by (1+t), we have:

(1/(1+t)) wT = PX X + PY Y+ wl (14.4)

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.

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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 impossible. 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 revenues. It might be appropriate to tax X and Y at the same rate, so called neutral taxation. This will be shown later to be inefficient.

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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.

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Figure 14.1

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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)

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This implies ΔX/X1 = ΔY/Y1 (14.7) 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.

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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).

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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) (14.9)

Equation (14.9) is the inverse elasticity rule:

“As long as goods are unrelated in consumption, tax rates should be inversely proportional to elasticities”.

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

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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.

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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.

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Example 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.

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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.

Summary 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.

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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.

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Figure 14.3

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Monopolist 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 MC=P. But at (Zm) price is greater than MC, hence (Zm) is inefficient.

Efficiency and existence of monopoly profits provide a possible justification for government taking over the production of (Z).

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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.

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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”.

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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.

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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 proportionately.

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