econ 551 government finance: revenues fall,...
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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
ECON 551: Lecture 4b 2 of 34
Agenda:
1. Commodity taxation with many households
2. Commodity taxation and production efficiency
3. Application: Sugar-sweetened beverage taxation.
ECON 551: Lecture 4b 3 of 34
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.
ECON 551: Lecture 4b 4 of 34
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)
ECON 551: Lecture 4b 5 of 34
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
ECON 551: Lecture 4b 6 of 34
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
)
ECON 551: Lecture 4b 7 of 34
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
ECON 551: Lecture 4b 8 of 34
Interpretation of the result:
∑ 𝑡𝑖𝑖 𝐸𝑘𝑖
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.
ECON 551: Lecture 4b 9 of 34
…continued
∑ 𝑡𝑖𝑖 𝐸𝑘𝑖
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.
ECON 551: Lecture 4b 10 of 34
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.
ECON 551: Lecture 4b 11 of 34
Agenda:
1. Commodity taxation with many households
2. Commodity taxation and production efficiency
3. Application: Sugar-sweetened beverage taxation.
ECON 551: Lecture 4b 12 of 34
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.
ECON 551: Lecture 4b 13 of 34
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…
ECON 551: Lecture 4b 14 of 34
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
ECON 551: Lecture 4b 15 of 34
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.
ECON 551: Lecture 4b 16 of 34
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….
ECON 551: Lecture 4b 17 of 34
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.
ECON 551: Lecture 4b 18 of 34
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.
ECON 551: Lecture 4b 19 of 34
Agenda:
1. Commodity taxation with many households
2. Commodity taxation and production efficiency
3. Application: Sugar-sweetened beverage taxes
ECON 551: Lecture 4b 20 of 34
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?
ECON 551: Lecture 4b 21 of 34
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?
ECON 551: Lecture 4b 22 of 34
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?
ECON 551: Lecture 4b 23 of 34
Considerations for a ‘sin’ tax:
See Alcott, Lockwood, and Taubinsky (2019 JEP; QJE)
Price elasticity of demand.
Externalities
Internalities
Distribution
Incidence
ECON 551: Lecture 4b 24 of 34
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?
ECON 551: Lecture 4b 25 of 34
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….
ECON 551: Lecture 4b 26 of 34
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 27 of 34
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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).
ECON 551: Lecture 4b 34 of 34
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”