EAE-1234: PUBLIC FINANCE

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.title[
# EAE-1234: PUBLIC FINANCE
]
.author[
### Pedro Forquesato <a href="http://www.pedroforquesato.com" class="uri">http://www.pedroforquesato.com</a> Office 217/FEA2 - <a href="mailto:pforquesato@usp.br" class="email">pforquesato@usp.br</a>
]
.institute[
### School of Economics, Business and Accounting University of São Paulo
]
.date[
### Topic 6: Tax incidence and deadweight loss 2026/2
]

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# Tax incidence

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## Equity implications of taxation

To understand whether a tax increases or reduces inequality, we have to understand who *ultimately* pays for it &mdash; this is the study of **tax incidence**

Of course, the law determines who formally pays the tax, but the market adjusts pre-tax and post-tax prices according to the *behavioral reaction* of consumers and producers

Therefore, even if a payroll tax is paid by the company, if it reduces the demand wage in response, then who is actually paying for the tax is the worker

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## Incidence

The first rule of incidence is that *under the micro assumptions*, **statutory incidence** does not matter for the **economic incidence** of taxes

If who disburses money is the consumer, the firm, or the worker, it should not matter how the market price adjusts to the tax, since it does not affect economic incentives &mdash; and these depend only on net and gross prices

But [SMT12] show that in Greece, changes in the payroll tax affect only those who actually pay for it (that is, the economic incidence equals the statutory incidence) &mdash; there is another example further on in which this rule fails

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## Incidence

Important, but sometimes overlooked, is that **tax shifting takes place through the market** &mdash; it is not (in competitive markets) an intentional decision by agents to shift the tax elsewhere, but the effects of market changes in the supply and demand curves

If the price (net of the tax) does not change with the tax introduction, then the real incidence is equal to the formal incidence

When we analyze not only consumers and producers of a specific good, but also other *substitute and complementary goods* and *factors of production*, we do a **general equilibrium incidence analysis**

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## Partial equilibrium incidence

Even so, we will focus on the *partial equilibrium incidence*, because it is more intuitive and it generally provides us a good approximation

Consider an initial tax situation `$t = 0$`, and add a fixed (infinitesimal) tax to be paid by the consumer `$dt$` &mdash; if before the equilibrium condition was `$S(p) = D(p)$`, now we add a **tax wedge** to the market

The market is still in equilibrium! But in the equilibrium with taxes, supply `$S$` equals demand `$D$` *with different prices*: the condition now becomes: `$$S(p^p) = D(p^c) = D(p^p + dt)$$`

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A tax that burdens the consumer (in the labor market, the firm, panel b) shifts the demand curve down, as now the good is 1$ "less desirable" &mdash; if it burdens the supplier (in labor market, the worker, panel a), the tax shifts the supply curve up, since now the marginal cost is $1 higher

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Partial equilibrium incidence &mdash; without tax, `$p^p = p^c = p$` (Saez)

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On the other hand, the tax creates a **tax wedge**: now `$p^c = p^p + dt > p^p$`  (Saez)

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The reduction in consumption makes firms operate in regions with lower MC, and eventually the price of the producer drops in response (Saez)

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## Partial equilibrium incidence

The market equilibrium condition is `$S(p^p) = D(p^p + dt)$`. Differentiating both sides in relation to `$t$`:

`$$S^{\prime}(p^p)\frac{dp^p}{dt} = D^{\prime}(p^p + dt)\left(1 + \frac{dp^p}{dt}\right)$$`

`$$\Rightarrow \frac{dp^p}{dt} \left[ S^{\prime}(p^p) - D^{\prime}(p^p + dt)  \right] = D^{\prime}(p^p + dt)$$`
`$$\therefore \ \frac{dp^p}{dt} = \frac{D^{\prime}(p^p + dt)}{S^{\prime}(p^p) - D^{\prime}(p^p + dt)}$$`
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## Partial equilibrium incidence

Elasticities are very useful in public sector economics because they are usually *sufficient statistics* for the behavior of producers and consumers (in competitive markets)

Defining `$p^c = p^p + dt$`, remember that `$\epsilon_S = \frac{p^p}{S(p^p)} S^{\prime}(p^p)$` and: `$$\epsilon_D = \frac{p^c}{D(p^c)} D^{\prime}(p^c) = \frac{p^p + dt}{D(p^p + dt)} D^{\prime}(p^p + dt)$$`

As `$dt$` is small, we can multiply the top and bottom numbers by `$p/Q$`: 
`$$\frac{dp^p}{dt} = \frac{D^{\prime}(p^p + dt) p/D(p)}{S^{\prime}(p^p)p/S(p) - D^{\prime}(p^p + dt)p/D(p)} = \frac{\epsilon_D}{\epsilon_S - \epsilon_D}$$`

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## Partial equilibrium incidence

Note that given `$\frac{dp^p}{dt} = \frac{\epsilon_D}{\epsilon_S - \epsilon_D}$` and `$p^c = p^p + dt$`, we have: `$$-1 \leq \frac{dp^p}{dt} \leq 0 \text{ and  } \frac{dp^c}{dt} = \frac{d(p^p + dt)}{dt} = 1 + \frac{dp^p}{dt} \in [0, 1]$$`

Consumers bear the full incidence of the tax when `$dp^p/dt = 0$`, and therefore `$dp^c/dt = 1$`, which happens if `$\epsilon_D = 0$` or `$\epsilon_S = \infty$`

Producers bear all the incidence of the tax when `$dp^p/dt = -1$`,  that is, if `$\epsilon_D = - \infty$` or `$\epsilon_S = 0$`

General rule: **the more inelastic side of the market bears most of the taxes**

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## Pass-through

The change in the price to the consumer caused by the tax is called tax **pass-through** rate: `$$\rho \equiv dp^c/dt = 1 + dp^p/dt$$`

It is believed that in the long run most industries operate at constant marginal costs (due to constant returns to scale): an (almost) infinitely elastic supply curve

It implies a **pass-through** rate of `$\approx 1$`: consumption taxes would be borne almost entirely by consumers &mdash; empirical evidence is consistent with pass-throughs between 0.8-1.0 for most taxes on consumer goods

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When demand is infinitely inelastic, the pass-through rate `$dp^c/dt$` is 1 (**full shifting**). When it is infinitely elastic, the incidence is all on the producer [Gru16]

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And the opposite occurs with the supply curve: the more elastic, the greater the pass-through [Gru16]

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## Incidence on competitive markets

*In competitive markets* we can approximate incidence by ignoring the **excess burden** (it is in *second order*) when calculating the welfare cost for consumers and producers

Although important (and the topic of next week), the excess burden is relatively small: Harberger (1959) estimated the deadweight loss of corporate taxation at ~8% of revenue; estimates for indirect taxes are similar (3-5%)

Thus, the loss of consumer surplus becomes `$\rho dt \cdot Q$` (which `$\rho$` is the **pass-through**) and the loss from the producer `$(1 - \rho)dt \cdot Q$` &mdash; both add up to (approximately) the government revenue `$R(dt) = Qdt$`

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## General equilibrium incidence

If the partial equilibrium analysis gives us a good intuition for tax incidence, it can be misleading if we do not consider the *general equilibrium effects* of taxation

A tax on beer sales affects the prices not only of beer, but also of substitute alcoholic drinks and input materials, as well as also burdens the capital and labor employed in the sector
 
Imagine a municipal tax on restaurants: if the demand is infinitely elastic, restaurants pay all the tax &mdash; but restaurants cannot pay taxes, *only people pay taxes!*

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## General equilibrium incidence

When we say that "restaurants bear the tax", in reality, we mean that *the factors of production* employed in production by restaurants bear the taxes

If the labor supply for restaurants is infinitely elastic and short-term capital infinitely inelastic (since invested capital is not convertible), then the incidence is all on restaurant owners

If we have several goods, the incidence will also depend on how *intensive* each factor is used in the production of the taxed goods (in relation to others)

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[BC19] analyzes the effect of a VAT cut in France of 14 p.p., finding a **pass-through** rate of 9,7%, which is the opposite of what we would expect given the argument that the incidence of consumer taxes is almost entirely on the consumer &mdash; there is evidence of *asymmetric incidence* of tax increases and decreases, at least in the short/medium term

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They find an incidence of 19% on wages, 60% on profit, 13% on the cost of raw materials, and only 8% was reverted to a reduction in consumer prices [BC19]

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In the real world, economic incidence often *depends* on statutory incidence, due to **bounded rationality**: in the US, taxes generally "appear" only in the cash register &mdash; in an experiment, [CLK09] found that compared to appearing on the price tag, the tax "in the cash register" only has 35% **salience**

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In the same way, at state level, increases in the *excise tax*, which appears on the price tag, are related to drops in beer consumption, but increases in *sales tax*, which does not appear (only in the cash register) much less [CLK09]

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# Efficiency costs of taxation

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## Efficiency costs taxation

So far we have discussed the redistributive effect of taxation: the **incidence** of indirect taxation, which is caused by changes *in the price* of products

The *efficiency cost* of taxation, on the other hand, is measured by the **excess burden** (aka deadweight loss), and it is related to changes *in the quantity* traded

Excess burden comes from the fact that taxes change *relative prices* faced by the economic agents, changing their behavior &mdash; for example, by making consumption more expensive in relation to leisure (non-taxable), it discourages work

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## Equivalent variation

We measure excess burden using the **equivalent variation**, a *money-metric* representation of welfare: how large a loss of income would have to be to have the same effect on the agent's utility as the introduction of the tax

An EV of `$-100$` means that introducing the consumption tax on a given good has the same effect for that consumer as taking R$100 from his income (in a lump-sum manner)

It is natural that if someone gives R$1 to the government, they will have a welfare loss of (at least) R$1 &mdash; but it will actually be larger than that, because transferring income between economic agents is costly (*leaky bucket*)

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## Lump-sum taxation and deadweight loss

This difference between the EV and government revenue is the excess burden: the loss of income equivalent to the loss of welfare (in monetary measure) generated by the tax that is *in excess of the revenue collected*

In this sense, *lump-sum taxation* does not generate deadweight loss, since the EV of removing R$100 reais in a person's income is exactly R$100, by the definition of equivalent variation

While consumption taxes change **relative prices**, and agents respond by "running away" from the tax &mdash; but this has a cost in terms of welfare that does not generate revenue (unlike with a lump-sum transfer)

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## Excess burden and equivalent variation

The deadweight loss arises from a **decrease in quantity**: the loss of trade surplus for trades that still happen equals the government revenue, but some trades that stop happening lower surplus without generating any revenue

The definition of excess burden as equivalent variation is useful because it has a clear interpretation: it is how much *excess* loss of income (beyond the government revenue) that would be *equivalent* to the implemented tax

Therefore, a excess burden of 10% means that for each R$ 1 collected, consumers and producers would be willing to pay R$ 1.10 instead if the prices were not distorted (a lump-sum transfer)

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## Substitution effect and income effect

It is important to always bear in mind that **what generates welfare loss is the substitution effect**:  income effect does not generate deadweight loss

In fact, lump-sum transfers also generate income effect, but with no change in relative prices. The equivalent variation is just the transfer of resources (private sector welfare loss is equal to the revenue)

The deadweight loss comes from **the change of relative prices**: universal transfers do not change the relative price (e.g. between consumption and leisure), and do not generate a **tax wedge**

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## Behavioral effects

In the end, economic inefficiency comes from the fact that there are relevant economic objects that the government cannot observe and tax: it is an **informational failure**

If the government could observe all the individuals' idiosyncrasies, it could condition taxation on immutable variables ("ability") and redistribute perfectly without generating behavioral changes

In the consumer market, when consumer and producer prices are not equal (**tax wedge**), there are *mutually beneficial exchanges* (and in the competitive market, therefore, socially beneficial) that are not carried out

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A market in undistorted equilibrium &mdash; all the *mutually beneficial exchanges* are carried out (Saez)

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With the tax, there are consumers willing to pay more for the good than its current marginal cost of production, and yet the exchange is not carried out: the quantity traded is reduced by `$dQ$` (Saez)

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The reduction in quantity traded creates the **Harberger triangle**: a surplus loss area that exceeds government revenue (Saez)

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## Harberger triangle

The **Harberger triangle** is (literally) a triangle, and under the conditions of the supply and demand graph (partial equilibrium, no income effect) we can calculate its area by elementary geometry

`$$\text{EB} = \frac{1}{2} dQ \times dt = \frac{1}{2} S^{\prime}(p) dp \times dt$$`
Which we use that `$Q = S(p)$`, and therefore, `$dQ = S^{\prime}(p)dp$`. Multiplying and dividing by `$pQdt$` and recalling last class that `$\frac{dp}{dt} = \frac{\epsilon_D}{\epsilon_S - \epsilon_D}$`:

`$$\text{EB} = \frac{1}{2} \frac{S^{\prime}(p) p}{Q} \frac{Q}{p} \frac{dp}{dt} \times (dt)^2 = \frac{1}{2}\frac{\epsilon_S \epsilon_D}{\epsilon_S - \epsilon_D} \frac{Q}{p} \times (dt)^2$$`

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Harberger triangle in this example has an area of `$\frac{1}{2} 10 \times 0,50 = 2,5$` billion [Gru16]

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## The deadweight loss formula

`$$\text{EB} = \frac{1}{2}\frac{\epsilon_S \epsilon_D}{\epsilon_S - \epsilon_D} \frac{Q}{p} \times (dt)^2$$`
Deadweight loss increases with the absolute value of elasticities `$\epsilon_S > 0$` and `$- \epsilon_D > 0$` : *it is more efficient to tax inelastic goods*

Deadweight loss increases *quadratically* in tax (*of second-order*) &mdash; the deadweight loss per BRL collected increases with taxation:

1. It is better to tax many goods little than fewer goods more;
2. It is better to finance extraordinary expenses (wars, pandemics) with debt, paid by taxes over a long period of time

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The more elastic the demand and supply, the greater is the deadweight loss  [Gru16]

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## Deadweight loss

As previously mentioned, deadweight loss is of second-order *when there are no distortions in economy* (triangle)

When there are distortions, such as market power, or other taxes (such as initial tax `$t \neq 0$`), the deadweight loss becomes a trapezoid &mdash; it is no longer negligible in calculating incidence and other effects

Calculation of welfare effect of deadweight loss taxation dates back to Dupuit in 1844, but the first to empirically estimate its size was Harberger (1954): he estimated the deadweight loss as `$~3\%$` of revenue, a very reasonable value given what we know today (see [Hin99])

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When we increase the rate of an existing tax, the deadweight loss is of *first-order* (trapezoid), and no longer of *second-order* (triangle) &mdash; this is due to pre-existing distortions, and it is also valid for any other market inefficiency, such as monopoly power [Gru16]

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## Window tax

[OS15] studies the case of the window tax instituted by King William III in the UK (1696)

As it is difficult to assess the value of properties (and it was much more in the past), King William decided to use as *proxy* of wealth the number of windows in the properties (an indicator of *ability to pay*): this would make sense (**tagging**) *if* the number of windows were not changeable &mdash; but it is

It was also unfair from the point of view of **horizontal equity**: a house in the country much cheaper than one in the city would certainly have way more windows

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## Window tax

> "In order to reduce the window tax, every window... was built up, and all source of ventilation was thus removed. The smell in this house was overpowering, and offensive to an unbearable extent. There is no evidence that the fever was imported into this house, but it was propagated from it to other parts of town, and 52 inhabitants were killed" Carlisle, 1781, apud [OS15]

Even worse, taxation had **notches**: points at which the average tax rate rises discontinuously, generating even greater distortions: there was no tax up to 9 windows, 6 pounds per window *in total windows* up to 14 windows, etc

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*Notches* in the tax rate generate anomalous masses just below the change &mdash; here in 9, 14 and 19 windows [Gru16]

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## Window tax

[OS15]  find that the deadweight loss generated by taxation on *households located "in the notch"*  was a terrible 62% of revenue: for every £1 collected by the government, they paid £1.62 lump-sum equivalent in lost utility

That is, they would be willing to pay £1,62 lump-sum transfer per £1 collected to end this tax &mdash; in general, the deadweight loss was 13,4% of collections: still 3-4x higher than other types of tax

Even with all this, the tax was only withdrawn in 1851, almost 160 years after its establishment, which demonstrates that very inefficient taxes can last for a long time for political reasons

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# References

[BC19] Y. Benzarti and D. Carloni. "Who really benefits from
consumption tax cuts? Evidence from a large VAT reform in France". In:
_American Economic Journal: Economic Policy_ 11.1 (2019), pp. 38-63.

[CLK09] R. Chetty, A. Looney, and K. Kroft. "Salience and taxation:
Theory and evidence". In: _American economic review_ 99.4 (2009), pp.
1145-77.

[Gru16] J. Gruber. _Public finance and public policy_. 5th ed.
Macmillan, 2016.

[Hin99] J. R. Hines. "Three sides of Harberger triangles". In: _Journal
of Economic Perspectives_ 13.2 (1999), pp. 167-188.

[OS15] W. E. Oates and R. M. Schwab. "The window tax: A case study in
excess burden". In: _Journal of Economic Perspectives_ 29.1 (2015), pp.
163-80.

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# References

[SMT12] E. Saez, M. Matsaganis, and P. Tsakloglou. "Earnings
determination and taxes: Evidence from a cohort-based payroll tax
reform in Greece". In: _The Quarterly Journal of Economics_ 127.1
(2012), pp. 493-533.