Econ 551 Government Finance: Revenues Fall,...

ECON 551: Lecture 4b 1 of 34

Econ 551

Government Finance: Revenues

Fall, 2019

Given by Kevin Milligan

Vancouver School of Economics

University of British Columbia

Lecture 4b: Optimal Commodity Taxation, Part II


Agenda:

1. Commodity taxation with many households

2. Commodity taxation and production efficiency

3. Application: Sugar-sweetened beverage taxation.


Diamond and Mirrlees 1971 AER:

This paper heralded the resurgence of interest in optimal taxation throughout the 1970s. See Myles

(p. 108-111); Atkinson-Stiglitz (p. 386-390).

Take a planner’s perspective: intro is about using taxes to control the

economy.

o This emphasizes a key point of this literature: we are simply

choosing a set of prices that is optimal. This is an important way to think about

taxes for reasons that will become clearer in a few minutes.

Study both the production and consumption sides of the economy. In

general equilibrium.

Used contemporary micro tools. Duality, etc.

They start with production, then move on to the consumer side. We’ll do it

in backwards order, though.


The setup:

H households. ℎ ∈ {1, … , 𝐻}

Linear commodity taxation available; no lump sum taxes, no income taxes.

No direct revelation problems; everyone reports the truth. (This will be different for income

taxation.)

Labour is the untaxed numeraire good.

Individual utility evaluated using indirect utility function (prices, wages, exogenous income).

𝑉ℎ = 𝑉ℎ(𝑞, 𝑤ℎ𝐼ℎ)

Welfare is aggregated using a Bergson-Samuelson SWF. Ψ(𝑉)


Today’s lemmata and notation:

𝜆ℎ =𝜕𝑉ℎ

𝜕𝐼ℎ Marginal utility of income.

Ψℎ =𝜕Ψ

𝜕𝑉ℎ Weight put on household h in the SWF.

𝛽ℎ = Ψℎ𝜆ℎ Social weight on household h’s marginal income.

𝜕𝑥𝑖

ℎ

𝜕𝑞𝑘= 𝑒𝑖𝑘

ℎ − 𝑥𝑘ℎ 𝜕𝑥𝑖

ℎ

𝜕𝐼ℎ Slutsky equation, where 𝑒𝑖𝑘ℎ is derivative of expenditure function with

respect to i and k.

𝜕𝑋𝑖

ℎ

𝜕𝑞𝑘= 𝐸𝑖𝑘

ℎ − ∑ 𝑥𝑘ℎ 𝜕𝑥𝑖

ℎ

𝜕𝐼ℎℎ Slutsky aggregated over households. (Capital letters for

aggregate quantities; lower-case for household level.)

𝑥𝑘ℎ = −

𝜕𝑉ℎ

𝜕𝑞𝑘

𝜕𝑉ℎ

𝜕𝐼ℎ

Roy’s identity


Start with the optimization problem:

max𝑞𝑘

Ψ(𝑉) 𝑠. 𝑡. ∑ 𝑡𝑖𝑋𝑖 ≥ 𝑅

𝑖

Set up a Lagrangian:

ℒ = Ψ(𝑉) + 𝜇 (∑ 𝑡𝑖𝑋𝑖

𝑖

− 𝑅)

First order conditions:

𝜕ℒ

𝜕𝑡𝑘= 0 = ∑

𝜕Ψ

𝜕𝑉ℎ

𝜕𝑉ℎ

𝜕𝑡𝑘ℎ

+ 𝜇 (𝑋𝑘 + ∑ 𝑡𝑖

𝜕𝑋𝑖

𝜕𝑞𝑘𝑖

)


Manipulate the results using Roy’s identity and Slutsky equation:

Plug in Roy’s identity to get:

∑𝜕Ψ

𝜕𝑉ℎ

𝜕𝑉ℎ

𝜕𝐼ℎ

ℎ

𝑥𝑘ℎ = ∑ 𝛽ℎ𝑥𝑘

ℎ

ℎ

= 𝜇 (𝑋𝑘 + ∑ 𝑡𝑖

𝜕𝑋𝑖

𝜕𝑞𝑘𝑖

)

Now plug in Slutsky equation to derivative on the far right, and rearrange:

∑ 𝑡𝑖

𝑖

𝐸𝑘𝑖 = ∑𝛽ℎ

𝜇𝑥𝑘

ℎ

ℎ

+ ∑ 𝑡𝑖

𝑖

∑ 𝑥𝑘ℎ

𝜕𝑥𝑖ℎ

𝜕𝐼ℎ− 𝑋𝑘

ℎ

Now just divide both sides by 𝑋𝑘 and we’ll have something to interpret.

∑ 𝑡𝑖𝑖 𝐸𝑘𝑖

𝑋𝑘= ∑

𝛽ℎ

𝜇

𝑥𝑘ℎ

𝑋𝑘ℎ

+ ∑ 𝑡𝑖

𝑖

∑𝑥𝑘

ℎ

𝑋𝑘

𝜕𝑥𝑖ℎ

𝜕𝐼ℎ− 1

ℎ


Interpretation of the result:


𝑋𝑘= ∑

𝛽ℎ

𝜇

𝑥𝑘ℎ

𝑋𝑘ℎ

+ ∑ 𝑡𝑖

𝑖

∑𝑥𝑘

ℎ

𝑋𝑘

𝜕𝑥𝑖ℎ

𝜕𝐼ℎ− 1

ℎ

1) The LHS is similar to what we saw for the Mirrlees index of discouragement.

2) On RHS, last part (-1) is a constant. Middle part represents extra tax revenue received when

h gets one more dollar. First part is what we want to interpret.

3) 𝛽ℎ

𝜇 is the marginal benefit to society of giving a dollar to h, divided by the cost of giving a

dollar to h. Say we get 15 utils for this h for an extra dollar, but the extra dollar costs us 1.5.

Then this ratio comes out as 10 dollars-worth of social benefit.

4) 𝑥𝑘

ℎ

𝑋𝑘 is the proportion of good k consumed by this household.


…continued


𝑋𝑘= ∑

𝛽ℎ

𝜇

𝑥𝑘ℎ

𝑋𝑘ℎ

+ ∑ 𝑡𝑖

𝑖

∑𝑥𝑘

ℎ

𝑋𝑘

𝜕𝑥𝑖ℎ

𝜕𝐼ℎ− 1

ℎ

5) ∑𝛽ℎ

𝜇

𝑥𝑘ℎ

𝑋𝑘ℎ is therefore like a correlation of welfare weight with proportion consumed.

6) The LHS is negative, so a larger value for the first term on the RHS means that the taxes

must be smaller so that proportionate demand change is smaller.

7) The RHS becomes more positive when the first term is larger. So, lower tax rates.

Summarized, this means that good consumed more by those who have high social weight should be

taxed less.


Comparing this result to one-consumer case:

1) If social marginal utility of income is the same for everyone (𝛽ℎ = 𝛽 ∀ℎ), then the

correlation will be zero and there is no gain from attempts to redistribute. Things reduce to

the one-person case.

2) If all individuals demand goods in the same proportions, then there will be no variation in

the proportion across households leading to a zero correlation and again it reduced to the

one-household case. Think of this as a signaling problem. The state is trying to infer whom

to tax by looking at who disproportionately consumes which goods. If there is no signal, it

can’t do anything about equity.


Agenda:



3. Application: Sugar-sweetened beverage taxation.


Breaktime!

Hungarian Mathematician Paul Erdős (1913-1996) traveled the world couch-

surfing and coauthoring.

Wrote 1525 published papers.

Had 511 coauthors.

People like to calculate their Erdős number, which counts the ‘collaboration

distance’ to Erdős.

Peter Diamond has Erdős number of 3.

o Peter Diamond-Eric Maskin-Peter Fishburn-Paul Erdős.

Mine is 5: Milligan---Courtney Coile---Peter Diamond---Maskin---

Fishburn--- Erdős.

IIRC, some other ‘entry points’ for economists to Erdős come from econometricians.

http://en.wikipedia.org/wiki/Paul_Erd%C5%91s

http://en.wikipedia.org/wiki/Erd%C5%91s_number

http://dx.doi.org/10.1111/j.1475-4991.2009.00320.x

http://www.sciencedirect.com/science/article/pii/S0047272701001293

http://econpapers.repec.org/article/rjebellje/v_3a10_3ay_3a1979_3ai_3aspring_3ap_3a282-316.htm

http://www.ams.org/mathscinet-getitem?mr=546968




Now we allow for a production side of the economy

Recall, this is the first part of Diamond Mirrlees (1971).

The question being posed here is whether we should tax intermediate inputs or not.

Diamond and Mirrlees show that, in general, it is not optimal to tax inputs. Instead, only final

goods should be taxed. This is referred to as the Production Efficiency Lemma.

We’re going to look at a diagram rather than go through all of the math…


Production efficiency and taxation

Taken from Myles (p. 128)

Input

Production

Possibility

Frontier

E0

E1

U0

Offer

Curve

Lumpsum tax budget

line, not through

origin.

Distortionary tax

through origin

U1


Walking through the result

Two goods; one input one output. Households supply input and consume output.

Production possibility set defined for some convex technology.

Ignoring taxes, we would be at a place like E0 where the MRTS =MRS.

However, tax revenue requirements shifts the set out to the left from the origin.

With lumpsum taxes, we can still get to E0. How? Adjust endowment along the input axis until

we are at the tangent point to E0. This is shown with dark black lumpsum tax budget line.

With distortionary taxes, price line must go through the origin.

Offer curve represents the locus of tangencies for ICs and price lines through the origin.

Optimal choice is at E1. Why?

o Want to get the most output possible for the least input.

o At interior point below E1, could have increased output without more input.

o At interior point to the left of E1, could have had same output with less input.


The lemma and its implications

Stating it formally:

If an optimum exists, then the optimum has production on the frontier of the production possibilities

set.

Implications:

1) Input taxes should not be differentiated among firms; all should face the same post-tax input

prices. (Why is this the case? If you are taxing inputs, you’re not going to be on the efficient

production frontier. Don’t tax inputs.)

2) We can mimic any set of input taxes using a tax on final goods, so why don’t we just tax final

goods instead? If we do that, we’re not distorting production decisions. That is, do not tax

intermediate goods. This is a major motivation for value-added taxation. Tax final

consumption; not intermediate goods.

3) Since capital is an input, we shouldn’t tax the return to capital. This is a conjecture; we’ll

see how it holds up….


Some caveats

1) In general, only holds with CRTS. However, later authors showed that with DRTS you could

still get the result if you taxed away 100% of the economic profit. In reality, this is hard to do

however, since distinguishing between economic profit and return to capital is difficult.

2) In imperfect competition settings, intermediate good taxation can be optimal.


Summary of optimal commodity taxation:

1. Can’t tax all goods uniformly or else no revenue raised.

2. When one good not taxed, Ramsey showed that uniform taxation is not generally desirable for

the taxed goods.

3. Ramsey-Mirrlees index of discouragement: taxation should affect goods’ demand equi-

proportionately.

4. With redistribution concerns, goods heavily consumed by poorer people should be taxed less.

5. If CRTS (or more precisely zero profits), then tax only final goods not intermediate goods.


Agenda:



3. Application: Sugar-sweetened beverage taxes


Start with Musgrave—‘Merit goods’

One of the concepts introduced in Musgrave’s 1959 book was the ‘merit good’. Here is Musgrave’s

definition.

“A different type of intervention occurs where public policy aims at an allocation of resources

which deviates from that reflected by consumer sovereignty. In other words, wants are satisfied that

could be serviced through the market but are not, since consumers choose to spend their money on

other things. … The reason, then, for budgetary action is to correct individual choice. Wants

satisfied under these conditions constitute a second type of public wants, and will be referred to as

merit wants.”

Q: Can you think of any examples?

Q: What is the difference between this and a public good?


Now, what about the opposite--demerit goods

Musgrave (1959) also talked about ‘demerit’ goods. The concept is the same, except that instead of

underconsuming them, we tend to overconsume them.

Musgrave specifically mentioned liquor.

Q: What other wants might be ‘demerit wants’.

Q: Does the idea of having your wants labeled as ‘good’ and ‘bad’ bother you? Is paternalism OK?


Modern take: libertarian paternalism

Sunstein and Thaler have pushed the idea of ‘libertarian paternalism’. Key features

of this approach:

Allow people to make choices, but restrict in some way the range or

presentation of those choices.

Embraces recent research on behavioural economics; how people actually

make decisions.

Critic: Glaeser (2006 UofC Law Review)

Quis custodiet ipsos custodies

Who gets to be the one who sets the default? Are we flawed voters

going to pick good default-pickers?

https://lawreview.sites.uchicago.edu/sites/lawreview.uchicago.edu/files/uploads/73.1/73_1_Glaeser.pdf


Considerations for a ‘sin’ tax:

See Alcott, Lockwood, and Taubinsky (2019 JEP; QJE)

Price elasticity of demand.

Externalities

Internalities

Distribution

Incidence


Soda facts:

Consumption grown 500% over 50 years.

7% of total calories consumed.

One 12-ounce can is 35-40g of sugar140 calories.

One sugary drink per day more 60% increase in obesity odds

$70.1 B in sales in US/year (2010), getting close to tobacco.

Price elasticity around -0.8 to -1.4.

Delicious on a hot day; goes well with salty snacks.

Question: What happens when tax on soda goes up?


What kind of responses might consumers have to soda tax?

Evidence from cigarette literature

o Does that only apply to addictive goods? (Is soda addictive?)

Substitutes?

o Diet soda

o Other sugary foods (to get your ‘kick’)

o Sugary drinks bought outside taxed jurisdiction.

Complements?

o Maybe with less soda you consumer less pizza or sugary foods….


Evidence: Fletcher et al. (2010) JPubE

Study state-varying tax rates using NHANES data, 1989-2006.

What is their empirical approach?

𝑌𝑖𝑠𝑡𝑞 = 𝛽1′𝑋𝑖𝑠𝑡𝑞 + 𝛽2 𝑇𝑠𝑡𝑞 + 𝜇𝑠 + 𝛿𝑡 + 𝛾𝑞 + 휀𝑖𝑠𝑡𝑞

ECON 551: Lecture 4b


Note that there is other evidence!

See Allcott et al. (2019 JEP) for summary of evidence….

From Grummon et al. (Science 2019)

An optimal tax of 1-2 cents per ounce (or 0.5 cents per gram of sugar).


Next time:

Optimal Income taxation

“Mirrlees, J.A. (1971), “An Exploration in the Theory of Optimum Income Taxation,” Review of

Economic Studies, Vol. 38, No. 2, pp. 175-208”

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