Post on 06-Jul-2020
Lecture 3: Power Law Models
Fatih GuvenenFebruary 20, 2019
University of Minnesota
What is a Power Law?
▶ General definition: A power law (PL) is defined as a relationshipbetween two variables, x and y, where:
y = a× x−α, (1)
for some scaling constant k.
α : is the power law exponent, and is a key parameter.
▶ Eq. (1) implies:log y = −α× log x,
so a log-log plot of y and x should be a straight line with slopeα, which allows us to see a power law visually (without fittingequation (1)).
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Pareto Distribution
▶ Let w be a random variable whose distribution obeys therelationship:
P(w > x) = a× x−α
where P(w > x) is the counter-CDF of w, for some a and apositive α.
→ w follows a PL or, alternatively, w has a Pareto distribution.
▶ Asymptotic Power Law:Sometimes the power law holds only above a threshold, x. In thatcase, w is said to be asymptotically PL.
▶ PLs are pervasive in nature, including in many distributionssocial scientists are interested in.
▶ Important property: a PL has finite moments only up to the αth
moment.If α = 1 (called Zipf’s law): mean does not exist.if α = 1.5, variance does not exist.
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Size Distribution for Firms, α = 1.0495
1 2 3 4 5 6 7 8 9
S ≡ ln(employees)
7
8
9
10
11
12
13
14
15y≡
ln(No.firm
swithS
ormrore
employees)
US Data, 2011
Linear regression line
y = 16.364− 1.0495× S, R2 = 0.9999
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US Wealth Distribution, α = 1.51
1M 10M 100M 1B 10B 100BWealth (Log scale)
-16
-14
-12
-10
-8
-6
-4
-2
0L
og
Co
un
ter-
-CD
F:
F(
w >
Wea
lth
)US Data, Forbes 400
US Data, Survey of Consumer Finances
Regression Line: 21.67 - 1.51 x log(wealth)
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US Cities Size Distribution, α = 1.0315
250K 500K 1M 5M 10MMSA Population
1
10
50
100
200M
SA
Ran
kUS Data, 2010
ln(Rank) = 18.147− 1.0315×ln(Pop.), R2 = 0.980
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Annual Income Growth Distribution, US Males
-6 -4 -2 0 2 4 6yt+1 − yt
-12
-10
-8
-6
-4
-2
0
2LogDen
sity
US Data, 1997–98
N (0, 0.512)
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A few more examples
▶ The average frequency of space debris (meteors and such)hitting the earth’s atmosphere, call y, is inversely proportionalto the surface area of the debris, or its squared diameter, call itx2.
The law holds remarkably well across 10 orders of magnitude:▶ micro particles (1 micron diameter) hitting the space shuttle at rate
1012 per year▶ meteors with 100+ meter diameter hitting earth every 10,000 years
(10−4 per year).▶ meteors of 10,000+ meters hitting every 100 million years.
▶ Per Krusell’s “discovery”:
Fraction of population with a Masters degree, 1Phd, 2Phds, 3Phds,4Phds, are all 1/10th of each other.
▶ Schroeder (1991)’s book “Fractals, Chaos, Power Laws,” is afascinating read on this.
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Features of Power Laws
▶ Combining different distributions in a model can quicklycomplicate the analysis.
So, distributions that have convenient properties are very usefulin research.An important reason why Normal, Poisson, Frechet, etc. are verypopular.
▶ PL also has very convenient properties that makes it veryuseful.
e.g., the sum, product, min, max of PL distributions are all PL.In most combinations, the thickest PL (lowest α) dominates.
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PL Models
▶ What kinds of mechanisms can generate a power law?
▶ Which of these mechanisms are plausible or relevant for agiven PL distribution?
▶ For example:“In all places and all times, the distribution of income remainsthe same. Neither institutional change nor egalitarian taxationcan alter this fundamental constant of social sciences.”
▶ What forces are behind this law? and for wealth, firm sizes,income shocks, etc?
▶ Note: there are other thick-tailed distributions that don’t havethe PL structure, such as Cauchy, Levy, etc. We won’t talk aboutthem today but they are also used.
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Simple(st) Framework
▶ Proportional random growth model:
st+1 = αtst (2)
where αt is a positive i.i.d random variable.
▶ Which yields:log st = log s0 +Σt−1
n=1αn
▶ Two results:1 Assuming αt has a well-behaved distribution so that the centrallimit theorem applies, 1
t st converges to a log-normal distribution.2 The distribution of st spreads without bound, so it has nostationary distribution (over time).
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Power Law Models: Simple Mechanics
▶ Wit :wealth of individual i and Wt : average (per-capita) wealth.
▶ Since wealth grows over time, define: wit ≡Wit
Wt.
▶ Proportional random growth: wit+1 = Ait × wti , where Ait is i.i.dover t and i, with density f(A).
▶ Define: Gt(wt) := P(wt > x) as the counter-CDF, which evolves:
Gt+1(x) = P(wit+1 > x) = P(Aitwit > x) > P(wit >xAit)
=
∫ ∞
0
Gt(xA )f(A)dA.
▶ If there is a steady state: G(x) =∫∞0G( xA )f(A)dA.
▶ Guess Pareto distr.: G(x) = cxα and plugging in yields:
1 =
∫ ∞
0
Aαf(A)dA = E(Aα).
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A Full-Blown GE Model: Benhabib-Bisin-Zhu (2011, Ecma):
▶ OLG model. Each individual lives for T years. Replaced by singleoffspring.
▶ wn: initial wealth of generation n. After estate tax b :
wn+1(0) = (1− b)wn(T).
▶ “Warm glow” utility from bequests: ϕ (wn+1) , with ϕ′ > 0, wheren denotes generation number.
▶ Labor income (yn) and return on wealth (rn) are both stochasticacross generations.
▶ Incomplete markets. No aggregate uncertainty.
▶ Wealth accumulation: wn(τ) = rnwn(τ) + yn − cn(τ),for τ = 0 toT.
▶ Key point: rn has a multiplicative effect whereas yn only entersadditively.
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Key Assumptions
▶ Assumption 1: Preferences satisfy u(c) = c1−σ
1−σ andϕ(w) = χw1−σ
1−σ �. Furthermore, σ ≥ 1, rn ≥ ρ and χ > 0.
▶ Assumption 2: The stochastic process(rn, yn) is a real,irreducible, aperiodic, stationary Markov chain with finite statespace and P(rn, yn|rn−1, yn−1) = P(rn, yn|rn−1). (Markovmodulated chain).
▶ Assumption 2 allows a certain degree of persistence butimplies that rn is independent of past yn−k’s.
▶ In other words: it assumes that one factor that drives rn to alsodrive yn but not vice versa.
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Results
▶ One can derive a wealth evolution equation of this form:
wn+1 = αn (rn)wn + βn (rn, yn) ,
where αn (rn) and βn (rn, yn) are stochastic processes inducedby (rn, yn) *and* the solution of the household’s problem (sothey depend of the deep parameters of the model).
Theorem 1Consider wn+1 = αn (rn)wn + βn (rn, yn) ,and let (rn, yn) satisfyassumption 2 (and a regularity condition). Then, the tail of thestationary distribution of wn, P(wn > w) is asymptotic to a Paretolaw:
Pr (wn > w) ∼ kw−µ.
where µ > 1 satisfies
limN→∞
(EΠN−1
n=0 (α−n)µ)1/N
= 1.
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Two More Results
▶ When αn is i.i.d. the previous condition in Theorem 1 reduces toE (α)
µ= 1 (Kesten 1973).
▶ Remark: Theorem 1 implies that the properties of income riskhas no effect on the tail of wealth distribution.
▶ Propositions 1-4: The tail index µ :
1 decreases with the size of rate of return risk.
2 decreases with the bequest motive χ�.
3 increases with the estate tax b and with the capital income tax ζ.
4 decreases with the persistence of αif it follows an AR(1) or MA(1).
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A Quantitative Model
Households
▶ OLG demographic structure.
▶ Mortality risk: ϕh is uncond. probability of survival to age h.
▶ Accidental bequests are inherited by (newborn) offspring.
▶ Preferences: E0
(∑Hh=1 β
h−1ϕhu(ch, ℓh))
▶ Individuals:
Two exogenous characteristics: yih (labor mkt productivity) andzih (entrepreneurial productivity)
Make three decisions:
1 consumption-savings
2 leisure vs. labor supply
3 whether to engage in entrepreneurial activity
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1. Labor Market Productivity
▶ Labor market efficiency of household i at age h is
log yih = κh︸︷︷︸life cycle
+ θi︸︷︷︸permanent
+ ηih︸︷︷︸AR(1)
▶ Individual-specific labor market efficiency θi is imperfectlyinherited from parents:
θchildi = ρθ × θparenti + εθ
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2. Entrepreneurial Productivity
▶ Key source of heterogeneity: entrepreneurial ability zih.
▶ Household i produces xih units of intermediate good i:
xih = zihkih,
where kih is capital.
▶ zih has a permanent and a stochastic component:
zih = f( zpi︸︷︷︸perm. comp.
, Iih︸︷︷︸stoch. comp.
)
▶ zpi is constant over the lifecycle and inherited imperfectly:
log(zpchild) = ρz log(zpparent) + εz.
▶ Iih is governed by transition matrix Πz, specified in a moment.Fatih Guvenen Lecture 3: Power Law Models 19 / 54
3. Competitive Final Good Producer
▶ Efficiency-adjusted aggregate capital:
Q =
(∫(xih)µdidh
)1/µ
, µ < 1
▶ Efficiency-adjusted aggregate labor:
L =∫(yih(1− ℓih))didh
(individuals supply labor to the aggregate firm, not to produce xih)
▶ Final good production: Y = QαL1−α
▶ Price of intermediate good xi: pi (xi) = αxµ−1i × Qα−µL1−α
▶ Wage rate (per efficiency unit of labor): w = (1− α)QαL−α
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Back to Entrepreneurial Productivity: Specifics
▶ The lifecycle pattern of wealth accumulation for the very richmatters greatly:
1 steady accumulation of wealth: the rich today have highexpected returns tomorrow.
▶ Larger % of wealthy prefer wealth taxation (rel. to capitalincome).
2 extremely fast growth that tapers off: rich today have lowexpected returns tomorrow.
▶ Smaller % of wealthy prefer wealth taxation.
▶ With µ < 1, returns fall as wealth increases, but not fast enoughif zih = zi for all h.
▶ So, we consider a process that allows for both scenarios.
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Back to Entrepreneurial Productivity: Specifics
▶ Over the life cycle, entrepreneurial ability evolves as follows:Iih ∈ {H, L, 0}
zih = f(zpi , Iih) =
(zpi)λ if Iih = H where λ > 1
zpi if Iih = Lzmin if Iih = 0
with transition matrix:
Πzs =
1− p1 − p2 p1 p20 1− p2 p20 0 1
▶ λ : degree of superstar productivity.▶ p1: annual probability of losing superstar productivity▶ p2 : annual probability of losing entrepreneurial productivitycompletely→ become a passive saver.
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4. Bond Market & Entrepreneur’s Problem
▶ a : assets owned by individual. k : capital used in entrepr. prod.
▶ Individuals can borrow for production up to: k ≤ ϑ (z)× aIf ϑ(z) = 1 ⇒ cannot borrow or lend
Borrowing capacity is nondecreasing in ability: dϑ(z)/dz ≥ 0
▶ Individuals can lend, at interest rate r determined inequilibrium (zero net supply).
▶ Entrepreneur’s (static!) capital choice:
maxk≤ϑ(z)a
{[R× (zk)µ + (1− δ)k]− (1 + r)(k− a)}
= (1 + r)a+ π∗(a, z)
▶ After-production wealth:
Π(a, z) =a+ [ra+ π∗(a, z)]
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Individual’s Budget Constraint
▶ During working life:
c+ a′ = Π(a, z) + wyhn
and a′ ≥ 0.
▶ During retirement:
c+ a′ = Π(a, z; τ) + yR(θ, η)
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Functional Forms and Parameters
▶ Preferences:
u(c, ℓ) =(cγℓ1−γ
)1−σ
1− σ
▶ Pension system:yR(θ, η) = Φ(θ, η)× Y where Y is the average labor income ineconomy, and
Φ(θ, η) is a concave replacement rate function taken from SocialSecurity’s OASDI system.
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Calibration Target
▶ Goal: Match the fraction of Forbes 400 wealthiest that areself-made (54%, we get 50%)
▶ Permanent z alone does not create enough self-made Forbes400 rich.
It takes too long (many generations) to get into Forbes 400.
▶ We set: λ = 5, p1 = 0.05, and p2 = 0.03.
Πzs =
0.92 0.05 0.03
0 0.97 0.03
0 0 1
▶ We also have robustness analysis with constant productivity:λ = 1, p1 = 0, and p2 = 0.
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Parameters Set Outside the Model
Table 1: Benchmark Parameters
Parameter ValueCurvature of utility σ 4.0Curvature of CES aggregator of varieties µ 0.90Capital share in production α 0.40Depreciation rate of capital δ 0.05Interg. persistence of invest. ability ρzP 0.10Interg. persistence of labor efficiency ρθ 0.50Persistence of labor efficiency shock ρη 0.90Std. dev. of labor efficiency shock σεη 0.20
τk = 25%, τℓ = 22.4%, and τc = 7.5% (McDaniel, 2007)
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Calibration Targets and Outcomes
▶ ρz = 0.1 is set based on Fagereng et al (2016) for Norway. (Wehave also experimented with values up to 0.5)
▶ We calibrate 4 remaining parameters (β, γ, σεzp , σεθ ) to match 4data moments:
Table 2: Benchmark Parameters Calibrated Jointly in EquilibriumParameter Value MomentDiscount factor β 00.948 Capital/Output 3.00∗Cons. share in U γ 0.46 Avg. Hours 0.40∗σ of entrepr. ability σεzp 0.072 Top 1% share 0.36∗σ of labor fix. eff. σεθ 0.305 σ(log(Earn)) 0.80∗
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Moments
Table 3: Benchmark vs. Wealth Tax Economy
US Data Benchmark Wealth TaxTop 1% 0.36∗ 0.36Capital/Output 3.00∗ 3.00Bequest/Wealth 1–2%00 0 1.0%σ(log(Earnings)) 0.80∗ 0.80Avg. Hours 0.40∗ 0.40
▶ Calibrated model generates:total tax revenues
GDP : 25% vs. 24.8% in the datacapital tax revenuetotal tax revenue : 25% vs. 28% in the datacorporate debt
GDP : 1.29 vs. 1.26 in the data
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µ = 0.9 and Pareto Tail
1e+06 1e+07 1e+08 1e+09 1e+10 5e+10
Wealth (log scale)
-16
-14
-12
-10
-8
-6
-4
-2
0
Lo
g C
ou
nte
r-C
DF
Pareto Tail Above $1000000
US DataRegression LineModelRegression Line
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Wealth measured in Present Value
1e+06 1e+07 1e+08 1e+09 1e+10 5e+10
Wealth (log scale)
-16
-14
-12
-10
-8
-6
-4
-2
0Log C
ounte
r-C
DF
Pareto Tail Above $1000000
US DataRegression LineModelRegression Line
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µ = 0.8 and Pareto Tail
1e+06 1e+07 1e+08 1e+09 1e+10 5e+10
Wealth (log scale)
-16
-14
-12
-10
-8
-6
-4
-2
0Log C
ounte
r-C
DF
Pareto Tail Above $1000000
US DataRegression LineModel (µ=0.8)
Regression Line
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Return Heterogeneity: Model vs. Data
Table 4: Deviation of percentiles of the distribution of lifetime returnsrelative to the median
p10 p25 p75 p90 p99 p99.9Norwegian Data -2.4% -1.3% 2.1% 4.1% 9.7% 19.9%Ages 25-65 -3.4% -2.9% 4.7% 8.0% 13.6% 19.9%Ages 20-24 -10.0% -5.6% 4.9% 13.3% 32.7% 55.4%
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Two Tax Experiments
1 Tax reform: Calibrate to current US economy with capitalincome taxes.
Baseline: Replace capital income taxes with wealth taxes so as tokeep government revenue constant.
2 Optimal taxation: Government maximizes utilitarian socialwelfare choosing:
1 linear labor income and capital income taxes, or2 linear labor income and wealth taxes,
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Tax Reform: Aggregate Variables
Table 5: Benchmark vs. Wealth Tax Economy
Benchmark Wealth Tax % Changeτk 0025.0% 000.00τa 0.00 1.13%k 19.4Q 24.8w 8.7Y 10.1L 1.3C 10.0
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Reallocation of Wealth Across Agents
Table 6: Tax Reform from τk to τa: Change in Wealth Composition
Productivity group (Percentile)Top x% 0-40 40-80 80-90 90-99 99-99.9 99.9-99.99 99.99+
1 -12.0 -13.0 -10.8 10.5 11.2 9.8 6.95 -8.2 -3.3 1.6 8.3 8.9 8.1 6.210 -6.4 -1.3 2.9 6.4 6.9 6.3 5.050 -2.5 0.9 1.8 1.6 1.2 1.1 1.1
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Welfare Analysis: Two Measures
Let s0 ≡ (θ, z,a0), and V0 and V0 be lifetime value functions inbenchmark (US) and counterfactual economies, respectively.
▶ Measure 1: Compute individual-specific consumptionequivalent welfare and integrate:
V0((1 + CE1(s0))c∗US(s0), ℓ∗US(s0)) = V0(c(s0), ℓ(s0))
CE1 ≡∑s0
ΓUS(s0)× CE(s0)
▶ Measure 2: Fixed proportional consumption transfer to allindividuals in the benchmark economy:∑s0
ΓUS(s0)×V0((1+CE2)c∗US(s0), ℓ∗US(s0)) =∑s0
Γ(s0)×V0(c(s0), ℓ(s0)).
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Tax Reform: Average Welfare Change
Baseline Baseline + SSCE1 CE2 CE1 CE2
Average CE for newborns 7.40% 7.86% 5.58% 4.71%
Average CE 3.14% 5.14% 4.95 4.10%
% in favor of reform 67.8% 94.8%
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Tax Reform: Who Gains? Who Loses?
Productivity group (Percentile)Age 0-40 40-80 80-90 90-99 99-99.9 99.9-99.99 99.99+20 7.0 7.3 7.9 8.9 10.6 11.6 12.4
21–34 6.5 6.3 6.3 6.6 7.0 6.9 5.735–49 5.1 4.4 3.9 3.3 1.7 0.4 -2.250–64 2.3 1.8 1.4 0.8 -0.6 -1.7 -3.565+ -0.2 -0.3 -0.4 -0.6 -1.2 -1.7 -2.7Note: Each cell reports the average of CE1(θ, z, a, h)× 100 within each ageand productivity group
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Optimal Taxation
2. Optimal Taxation
Two Optimal Tax Problems
Compare:
1 (linear) labor taxes and capital income taxes
2 (linear) labor taxes and wealth taxes.
The government maximizes ex ante (expected) lifetime utility ofnewborns.
Then analyze:
▶ Benchmark vs. Optimal tax (either capital income or wealth)
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Optimal Tax Structure and Outcomes
τk τℓ τa k/Y Top 1%Benchmark 025% 22.4% – 3.0 0.36Tax reform – 22.4% 1.13% 3.25 0.46Opt. τk –34.4% 36.0% – 4.04 0.56Opt. τa – 14.1% 3.06% 2.90 0.47Opt. τa w/ threshold – 14.2% 3.30% 2.86 0.47
ThresholdE = 25% 63% of pop. taxed
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Wealth Taxes – Distortions and Misallocation
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Tax Revenue from K / Total Tax Revenue
-40
-30
-20
-10
0
10
20
30
40
PercentChange
k, τkk, τa
▶ Wealth tax reduces k less than capital income tax.Fatih Guvenen Lecture 3: Power Law Models 43 / 54
Wealth Taxes – Distortions and Misallocation
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Tax Revenue from K / Total Tax Revenue
-40
-30
-20
-10
0
10
20
30
40
PercentChange
k, τkk, τaQ, τkQ, τa
▶ Q, declines less than k under wealth taxes. Opposite undercapital income taxes.
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Optimal Taxes: Aggregate Variables
∆K ∆Q ∆L ∆Y ∆w ∆w ∆r ∆r% change (net) (net)Tax reform 19.37 24.79 1.28 10.10 8.70 8.70 –0.25 –0.90Opt. τk 68.97 79.57 –1.16 25.51 26.97 4.72 –1.51 –0.87Opt. τa 2.76 10.26 3.90 6.40 2.41 13.42 0.68 –1.92Opt. τa 0.41 8.12 3.67 5.42 1.70 12.48 0.78 –2.07Threshold
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Optimal Taxes: Welfare
τk τℓ τa CE2 Vote
(%) (%)Benchmark 025% 22.4% – – –Tax reform – 22.4% 1.13% 7.86Opt. τk –34.4% 36.0% – 6.28Opt. τa – 14.1% 3.06% 9.61Opt. τa – 14.2% 3.30% 9.83Threshold Threshold
E = 25%
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Welfare: Levels vs. Redistribution
Tax Reform Opt. τk.Opt. τaCE2 (NB) 7.86 6.28 9.61
ConsumptionTotal 8.27 5.90 11.02Level 10.01 21.04 8.28Dist. –1.58 –12.51 2.53
LeisureTotal –0.38 0.36 –1.27Level –0.66 0.73 –2.21Dist. 0.27 –0.38 0.76
Formula
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Optimal Capital Income Tax: Welfare
Optimal Capital Income Tax - Welfare Gain by Age/ProductivityProductivity group (Percentile)
Age 0-40 40-80 80-90 90-99 99-99.9 99.9-99.99 99.99+20 4.0 5.6 7.2 9.5 13.0 14.8 16.1
21–34 3.7 5.0 6.2 7.9 10.4 11.4 11.235–49 2.7 3.3 3.8 4.0 3.5 2.7 0.750–64 1.1 1.4 1.6 1.5 0.6 -0.2 -1.965+ -0.1 0.1 0.2 0.2 -0.2 -0.7 -1.6
Political Support
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Optimal Wealth Tax: Welfare
Optimal Wealth Tax - Welfare Gain by Age/ProductivityProductivity group (Percentile)
Age 0-40 40-80 80-90 90-99 99-99.9 99.9-99.99 99.99+20 10.0 9.7 10.1 11.1 13.1 14.3 15.3
21–34 9.2 7.9 7.3 7.1 6.6 5.9 3.135–49 6.8 4.9 3.7 2.1 -1.3 -3.9 -8.850–64 2.7 1.4 0.6 -0.8 -3.7 -5.8 -9.365+ -0.6 -0.9 -1.2 -1.8 -3.2 -4.3 -6.3
Political Support Political Support + Threshold
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Optimal Taxes: Welfare
Baseline
τk τℓ τa CE2 Vote
(%) (%)Benchmark 025% 22.4% – – –Tax reform – 22.4% 1.13% 7.86 67.8Opt. τk –34.4% 36.0% – 6.28 69.7Opt. τa – 14.1% 3.06% 9.61 60.7Opt. τa – 14.2% 3.30% 9.83 78.9Threshold
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Power Law Models: Wrapping Up
Measuring the Tail
▶ Researchers do not always have access to micro data to plotthe log density vs log x graph and see linear relationship.
▶ A Pareto distribution can be verified and tail index estimated ina simpler way.
▶ First: If P(y > x) = k× x−α, the conditional mean of y above anyy is E(y|y > y) = y× α
1−α .
Two Implications:
1 E(y|y>y)y = α
1−α . LHS can be measured by IRS tabulations.
2 E(y|y>y1)E(y|y>y2
=y1y2. Ratio of top income (or wealth) share must be
constant.
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Jones and Kim (2018, JPE) extending Saez (2001)
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Jones and Kim (2018, JPE), very top end
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