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ECON 551: Lecture 5b 2 of 41
Agenda:
- 1. Indirect vs direct taxation
- 2. Atkinson-Stiglitz Theorem
- 3. Joint income taxation
Econ 551 Government Finance: Revenues Fall, 2019 Given by Kevin - - PowerPoint PPT Presentation
Econ 551 Government Finance: Revenues Fall, 2019 Given by Kevin - - PowerPoint PPT Presentation
Econ 551 Government Finance: Revenues Fall, 2019 Given by Kevin Milligan Vancouver School of Economics University of British Columbia Lecture 5b: Optimal Income Taxation, Part II ECON 551: Lecture 5b 1 of 41 Agenda: 1. Indirect vs direct
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ECON 551: Lecture 5b 3 of 41
Indirect and Direct Taxation
John Stuart Mill (1806-1873)
‘direct taxes’ those that were levied on the people who actually must pay them. ‘Indirect taxes’ on the other hand, could be passed onto other people through higher prices. This definition is far from satisfactory, because for many taxes some part of the burden can be shared. This is true for both income and commodity taxes. A better definition is given by Atkinson and Stiglitz (p. 427): “Direct taxes may be adjusted to the individual characteristics of the taxpayer, whereas indirect taxes are levied on transactions irrespective of the circumstances of the buyer and seller.” Even this falls apart:
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ECON 551: Lecture 5b 4 of 41
Point of sale reductions in sales tax for ‘children’s clothes’ in BC’s PST.
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ECON 551: Lecture 5b 5 of 41
Indirect and Direct Taxation
Which is better? The debate about which to use – or both – has persisted. Here is an excerpt from the British House of Commons debate, spoken by William Gladstone, four-time Prime Minister in the 19th Century: “I never can think of direct and indirect taxation except as I should think of two attractive sisters . .. differing only as sisters may differ. I cannot conceive any reason why there should be unfriendly rivalry between the admirers of these two damsels. I have always thought it not only allowable, but even an act of duty, to pay my address to them both.”
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ECON 551: Lecture 5b 6 of 41
Indirect and Direct Taxation
Arguments one way or the other include: Two instruments; two targets. Use direct taxation for equity and indirect for efficiency. Direct taxation would be swell, but it requires a high level of administrative
- sophistication. So, indirect taxation is desirable because it is easier to administer.
Indirect taxes are undesirable because they distort relative prices whereas direct taxes don’t. This is only true if labour is inelastically supplied, of course!
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ECON 551: Lecture 5b 7 of 41
Agenda:
- 1. Indirect vs direct taxation
- 2. Atkinson-Stiglitz Theorem
- 3. Joint income taxation
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ECON 551: Lecture 5b 8 of 41
The Atkinson-Stiglitz Theorem
We will approach the question systematically, building on the work of Atkinson and Stiglitz (1976). These authors developed an eponymously named theorem to describe the circumstances in which direct and indirect taxation should be chosen. Good background reading on this topic includes the A&S paper itself, as well as the Boadway and Pestieau article on the reading list.
- B&P provide a nice summary of advances in the area since A&S published
their original work.
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ECON 551: Lecture 5b 9 of 41
“The result is arguably the most relevant result for policy purposes to emerge from the
- ptimal income tax literature initiated by Mirrlees (1971)”
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ECON 551: Lecture 5b 10 of 41
Notation:
Two types: 𝑗 ∈ {1,2} Indexes ability. So, type 2 doesn’t have to work as hard… 𝑜𝑗 size of population of type i. 𝑑𝑗 disposable income. x and z goods. 𝑚𝑗 labour income 𝑥𝑗 wage: 𝑥1 < 𝑥2 𝑧𝑗 earned income, 𝑧 = 𝑥𝑚. 𝑣((𝑦, 𝑨), 𝑚) direct utility—note separability in labour/leisure. ℎ(𝑟, 𝑑𝑗) indirect utility of household i over goods t tax on good z; no tax on good x. q tax inclusive price of z.
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ECON 551: Lecture 5b 11 of 41
if 𝑢 = 0, then we should use income taxation since that implies tax on z and x should be the same ⟷ uniform commodity tax.
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ECON 551: Lecture 5b 12 of 41
Planner’s objective:
Maximize social utility (using utilitarian SWF) subject to revenue constraint and self- selection constraint. max
𝑑1,𝑑2,𝑟 ∑ 𝑜𝑗𝑤𝑗 𝑗=1,2
(ℎ(𝑟, 𝑑𝑗), 𝑧𝑗 𝑥𝑗) subject to revenue constraint: ∑ 𝑜𝑗[(𝑧𝑗 − 𝑑𝑗) + 𝑢𝑨𝑗(𝑟, 𝑑𝑗)] ≥ 𝑆
𝑗=1,2
(multiplier λ) And selection constraint: 𝑤2 (ℎ(𝑟, 𝑑2), 𝑧2 𝑥2) ≥ 𝑤2 (ℎ(𝑟, 𝑑1), 𝑧1 𝑥1) (multiplier γ) For the selection constraint, the utility to type 2 must be higher when he puts in more effort than when he puts in less and mimics the type 1.
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ECON 551: Lecture 5b 13 of 41
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ECON 551: Lecture 5b 14 of 41
Set up the Lagrangian and solve
ℒ = ∑ 𝑜𝑗𝑤𝑗
𝑗=1,2
(ℎ(𝑟, 𝑑𝑗), 𝑧𝑗 𝑥𝑗) + 𝜇 [ ∑ 𝑜𝑗[(𝑧𝑗 − 𝑑𝑗) + 𝑢𝑨𝑗(𝑟, 𝑑𝑗)]
𝑗=1,2
] + 𝛿 [𝑤2 (ℎ(𝑟, 𝑑2), 𝑧2 𝑥2) − 𝑤2 (ℎ(𝑟, 𝑑1), 𝑧1 𝑥1)] FOCs wrt 𝑑1, 𝑑2, 𝑟: 𝑜1𝑤ℎ
1ℎ𝑑 1 − 𝜇𝑜1 (1 − 𝑢 𝜖𝑨1
𝜖𝑑1) − 𝛿𝑤 ̂ℎ
2ℎ
̂𝑑
2 = 0
(We use the ‘hats’ for a type 2 mimicking a type 1. Which is true with 𝛿 > 0. 𝑜2𝑤ℎ
2ℎ𝑑 2 − 𝜇𝑜2 (1 − 𝑢 𝜖𝑨2
𝜖𝑑2) + 𝛿𝑤ℎ
2ℎ𝑑 2 = 0
∑ 𝑜𝑗𝑤ℎ
𝑗 ℎ𝑟 𝑗 𝑗=1,2
+ 𝜇 ∑ 𝑜𝑗 (𝑨𝑗 + 𝑢 𝜖𝑨𝑗 𝜖𝑟 )
𝑗=1,2
+ 𝛿(𝑤ℎ
2ℎ𝑟 2 − 𝑤
̂ℎ
2ℎ
̂𝑟
2) = 0
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ECON 551: Lecture 5b 15 of 41
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ECON 551: Lecture 5b 16 of 41
Combine the FOCs and solve
Start by multiplying the first two FOCs by 𝑨1 and 𝑨2. Also remember how to use Roy’s identity here:
𝜖𝑤𝑗 𝜖𝑟 𝜖𝑤𝑗 𝜖𝑑𝑗
=
ℎ𝑟
𝑗
ℎ𝑑
𝑗 = −𝑨𝑗. So, ℎ𝑟
𝑗 = −𝑨𝑗ℎ𝑑 𝑗 .
Using all this, we can plug the first two FOCs into the first term of the third: −𝜇𝑜1𝑨1 (1 − 𝑢 𝜖𝑨1 𝜖𝑑1) − 𝛿𝑤 ̂ℎ
2ℎ
̂𝑑
2𝑨1 − 𝜇𝑜2𝑨2 (1 − 𝑢 𝜖𝑨2
𝜖𝑑2) + 𝛿𝑤ℎ
2ℎ𝑑 2𝑨2 +
𝜇 ∑ 𝑜𝑗 (𝑨𝑗 + 𝑢 𝜖𝑨𝑗 𝜖𝑟 )
𝑗=1,2
+ 𝛿(𝑤ℎ
2ℎ𝑟 2 − 𝑤
̂ℎ
2ℎ
̂𝑟
2) = 0
Let’s move lamdas to the left and gammas to the right….
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ECON 551: Lecture 5b 17 of 41
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ECON 551: Lecture 5b 18 of 41
−𝜇𝑜1𝑨1 (1 − 𝑢 𝜖𝑨1 𝜖𝑑1) − 𝜇𝑜2𝑨2 (1 − 𝑢 𝜖𝑨2 𝜖𝑑2) + 𝜇 ∑ 𝑜𝑗 (𝑨𝑗 + 𝑢 𝜖𝑨𝑗 𝜖𝑟 )
𝑗=1,2
= 𝛿𝑤 ̂ℎ
2ℎ
̂𝑑
2𝑨1 − 𝛿𝑤ℎ 2ℎ𝑑 2𝑨2 − 𝛿(𝑤ℎ 2ℎ𝑟 2 − 𝑤
̂ℎ
2ℎ
̂𝑟
2)
Collect terms; use Roy’s identity again on RHS… 𝜇𝑢 ∑ 𝑜𝑗 [(𝜖𝑨𝑗 𝜖𝑟 ) + 𝑨𝑗 𝜖𝑨𝑗 𝜖𝑑𝑗]
𝑗=1,2
= 𝛿[𝑤 ̂ℎ
2ℎ
̂𝑑
2(𝑨1 − 𝑨̂2) − (𝑤ℎ 2ℎ𝑑 2𝑨2 + 𝑤ℎ 2ℎ𝑟 2)]
One more application of Roy’s identity on RHS pushes last term to zero… 𝜇𝑢 ∑ 𝑜𝑗 [(𝜖𝑨𝑗 𝜖𝑟 ) + 𝑨𝑗 𝜖𝑨𝑗 𝜖𝑑𝑗]
𝑗=1,2
= 𝛿[𝑤 ̂ℎ
2ℎ
̂𝑑
2(𝑨1 − 𝑨̂2)]
Note that on the LHS, the 2nd term inside the square bracket is compensation for a price
- change. (If there were only an income effect from a tax change, 𝑨𝑗
𝜖𝑨𝑗 𝜖𝑑𝑗 would put you right
back where you started. Use the Slutsky equation….)
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ECON 551: Lecture 5b 19 of 41
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ECON 551: Lecture 5b 20 of 41
Atkinson and Stiglitz result:
Let’s call 𝑨̃𝑗 the compensated demand for good i ….and we’re done. 𝑢 ∑ 𝑜𝑗 [(𝜖𝑨̃𝑗 𝜖𝑟 )]
𝑗=1,2
= 𝛿 𝜇 [𝑤 ̂ℎ
2ℎ
̂𝑑
2(𝑨1 − 𝑨̂2)]
Since type 2 is mimicking type 1 at 𝑨̂2, incomes are the same. Type 2 gets more leisure, but they report same income. If incomes are the same, and if leisure is separable from consumption, then both types will choose the exact same consumption; 𝑨1 = 𝑨̂2. The RHS collapses to zero, so therefore t must be zero. We don’t impose a commodity tax; just income tax. Recall that the RHS is negative (own price elasticities…) so that means if 𝑨1 < 𝑨̂2 then 𝑢 > 0.
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ECON 551: Lecture 5b 21 of 41
If leisure and z are complements, then mimicker will likely choose more z than type
- 1. Why? Because he has more leisure when they both have the same income since he
is higher ability.
- So 𝑨1 < 𝑨̂2 and 𝑢 > 0.
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ECON 551: Lecture 5b 22 of 41
Implications
If leisure and goods are separable, then we should tax income only. Commodity taxation is useless.
- Mimickers are earning and consuming the same as low types, so planner just
can’t see any way to distinguish them. Saez (2002) and others have re-examined the case. For example, heteroegenous preferences will generate 𝑢 > 0 if high types prefer the good. Big implications for capital income taxation. If z is future consumption, we will only want to tax z different than x if z is complementary to leisure. Is retirement consumption complementary to leisure? In Saez’s model, do high types have higher preference for saving or future consumption? But why do we see both direct and indirect coexisting in the real world? Administration, ignorance, factors outside the model, not separable, income taxes aren’t optimal so A-S doesn’t apply. All possible explanations. (This is from Boadway and Pestieau.)
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ECON 551: Lecture 5b 23 of 41
Agenda:
- 1. Indirect vs direct taxation
- 2. Atkinson-Stiglitz Theorem
- 3. Joint income taxation
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ECON 551: Lecture 5b 24 of 41
Application: taxing couples
Longstanding question in taxation: what is the right ‘tax unit’? Individuals? Families? Something else? This was a hot policy issue in Canada in 2013-15 as the government allowed some measure of income splitting between members of a couple—but that is now gone. Many countries tax couples jointly—Canada’s individual-based system is a bit of an
- utlier. The issue of the tax unit remains a contentious policy issue.
We’ll stick mostly to the theory here, but first let’s look at a fairly surprising graphs…
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ECON 551: Lecture 5b 25 of 41
Gender: Who is the ‘primary earner’?
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ECON 551: Lecture 5b 26 of 41
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ECON 551: Lecture 5b 27 of 41
Not driven by age; some pattern by education
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ECON 551: Lecture 5b 28 of 41
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ECON 551: Lecture 5b 29 of 41
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ECON 551: Lecture 5b 30 of 41
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ECON 551: Lecture 5b 31 of 41
Optimal Tax Research on Couples
First optimal tax paper on this topic was Boskin and Sheshinksi (1983 JPubE)
- ‘Second earners’ face same marginal tax rate as ‘primary earners’ under family
taxation.
- But second earners have higher labour supply elasticities.
- This paper emphasized the efficiency consequences of this.
Kleven Kreiner and Saez (2009) took a new look at the question.
- Introduced idea of ‘negative jointness’ vs ‘positive jointness’
- If tax rate of second earner is decreasing in the earnings of the primary earner,
then tax schedule exhibits negative jointness. They show this is optimal in case
- f uncorrelated spousal ability.
- This happens because the efficiency effects cancel, but the redistribution to
‘both low’ is valued highly. Frankel (2014 AEJ) extends the KKS work to a case with positive assortative matching in the marriage market.
- We will take a look at that one.
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ECON 551: Lecture 5b 32 of 41
The Couple’s Problem
max
𝑍𝑛,𝑍𝑥 ∑ (𝑍𝑗 − 𝐷𝑗(𝑍𝑗|𝜄𝑗)) − 𝜐(𝑍𝑛, 𝑍𝑥) 𝑗=𝑛,𝑥
Where: Individuals 𝑗 ∈ {𝑛, 𝑥} in a couple have earnings type 𝜄𝑗. Face cost of effort 𝐷𝑗(𝑍𝑗|𝜄𝑗). Pay tax (or receive transfer) of 𝜐(𝑍𝑛, 𝑍𝑥). Utility quasi-linear in money, so no income effects in labour decisions.
- Do spouses of high income people not work because of taxes or income?
Individual types are binary: 𝜄𝑗 ∈ {𝐼, 𝑀}. Joint type 𝜾 = 𝜄𝑛𝜄𝑥. Joint type space 𝜾 ∈ Θ = {𝐼𝐼, 𝐼𝑀, 𝑀𝐼, 𝑀𝑀}, with population shares 𝛽𝜄 satisfying ∑ 𝛽𝜄
𝜄
= 1.
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ECON 551: Lecture 5b 33 of 41
Types may be correlated within couple: 𝜍 ≡ 𝛽𝑀𝑀𝛽𝐼𝐼 − 𝛽𝑀𝐼𝛽𝐼𝑀.
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ECON 551: Lecture 5b 34 of 41
The Government’s Problem
max
𝜐(𝑍𝑛,𝑍𝑥)𝑇𝑋 = 𝛽𝑀𝑀𝑉𝑀𝑀 + 𝜇 ∑
𝛽𝜄𝑉𝜄
𝜄≠𝑀𝑀
𝛽𝑀𝑀 + 𝜇(1 − 𝛽𝑀𝑀) Where: SW is the social welfare function All non-LL households receive welfare weight 𝜇 ∈ [0,1].
- Can do equal-weight utilitarian or Rawlsian here.
LL household gets welfare weight=1. Subject to incentive compatibility against all other types 𝜄′: ∑ (𝑍
𝜄 𝑗 − 𝐷𝑗(𝑍 𝜄 𝑗|𝜄𝑗)) − 𝜐𝜄 𝑗=𝑛,𝑥
≥ ∑ (𝑍
𝜄′ 𝑗 − 𝐷𝑗(𝑍 𝜄′ 𝑗 |𝜄𝑗)) − 𝜐𝜄 𝑗=𝑛,𝑥
…and budget balance: ∑ 𝛽𝜄𝜐𝜄
𝜄
= 0.
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ECON 551: Lecture 5b 35 of 41
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ECON 551: Lecture 5b 36 of 41
Two definitions:
Jointness: Say that a tax system has zero jointness if 𝑢𝜄
𝑗 = 𝑢𝜄′ 𝑗 whenever 𝜄𝑗 = 𝜄′𝑗 . It has negative
jointness if 𝜄𝑗 = 𝜄′𝑗, 𝜄−𝑗 = 𝑀, 𝑏𝑜𝑒 𝜄−′𝑗 = 𝐼imply that 𝑢𝜄
𝑗 ≥ 𝑢𝜄′ 𝑗 . It has positive jointness
if this last inequality is reversed. Separability: A tax schedule is separable if it has zero jointness and if there exists 𝜐𝜄𝑗
𝑗 for each 𝑗 =
𝑛, 𝑥 and each 𝜄𝑗 = 𝑀, 𝐼 such that 𝜐𝜄𝑛𝜄𝑥 = 𝜐𝜄𝑛
𝑛 + 𝜐𝜄𝑥 𝑥 .
Zero jointness does not imply separability, because zero jointness is about marginal tax rates; separability is about tax levels.
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ECON 551: Lecture 5b 37 of 41
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ECON 551: Lecture 5b 38 of 41
Main Result:
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ECON 551: Lecture 5b 39 of 41
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ECON 551: Lecture 5b 40 of 41
Main Result:
Negative jointness can be seen in Panel A at low correlation range: LL pays higher tax than LH. This difference shrinks until hitting some threshold level or correlation. After that threshold, zero jointness. It doesn’t slide into positive jointness because an incentive compatibility constraint binds: the HL would try to look like the LL and take the lower tax rate. For output in panel B, it is the opposite: Lower output for LL because they face higher tax rates Above the correlation threshold, no output difference because no MTR difference.
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