Classical Labor Supply: Kink and bunching ECON 34430: Topics in - - PowerPoint PPT Presentation

Classical Labor Supply: Kink and bunching

ECON 34430: Topics in Labor Markets

• T. Lamadon (U of Chicago)

Fall 2017

Agenda

1 Saez 2010 AEJ: Do Taxpayers Bunch at Kink Points?

• Analyze the kink among US tax-payers
• Estimate elasticity using kink
• Provide evidence on response in reporting

2 Blomquist, Liang and Newey (WP)

• Identification with non parametric preferences

Saez 2010 AEJ

Data

• Use IRS records: Individual Public Use Tax Files
• quasi-annually from 1960 to 2004
• 80k to 200k records per year
• focus on 2 sets of kinks:
• kinks in the EITC schedule (2 convex, 1 concave)
• kinks in Federal income tax

EITC structure

Earnings density distributions and the EITC

• bunching at the first kink
• not so much at kink 2 & 3

Use kinks to estimate elasticity

• Using simple model:

u(c, z) = sup

c,z c −

n 1 + 1/e z n

• (1)

s.t. c = (1 − t)z + R (2)

• Find relation between Bunching B at z ∗, density

h+(z ∗), h−(z ∗), taxes t1, t0 and elasticity e 2B = z ∗1 − t0 1 − t1 e − 1

• h−(z ∗) + h+(z ∗) ·

1 − t0 1 − t1 −e

Use kinks to estimate elasticity

2B = z ∗1 − t0 1 − t1 e − 1

• h−(z ∗) + h+(z ∗) ·

1 − t0 1 − t1 −e

• z ∗ and t1, t0 are known
• B , h+(z ∗), h−(z ∗) need to be estimated from data
• ignore convexity of h
• define δ regions and H−, H+, H
• use h−(z ∗) = H−/δ, h+(z ∗) = H+/δ
• and B = H − (H− + H+)
• get standard errors using Bootstrap or Delta method
• choice of δ?

• hump shaped because of additional frictions

Earnings density distributions and the EITC

• bunching at the first kink
• not so much at kink 2 & 3

Earnings density distributions and the EITC

• same for 2 children
• note that the kink is even stronger here

Wage earners Vs Self-Employed

• the bunching is mostly for self-employed workers
• this is also true for the 2 children workers

Evolution over time

• bunching becomes more and more pronounced at the first kink

Lessons from graphical evidence

1 bunching at the first kink 2 not so much at kink 2 & 3 3 bunching is mostly among self-employed workers 4 bunching becomes more pronounced over time

Table of elasticities

• results confirm graphical evidence

Table of elasticities

• estimates do grow over time
• estimates are very sensitive to choice of δ!

A model of tax reporting

1 wage earners do not display any evidence of response to tax

rate

• could be low elasticity
• could be lack of understanding of tax rules
• could be lack of ability to actually adjust hours
• or finally the inability to mis-report earnings (third party

reporting)

2 for self-employed bunching only happens at the first kink

• first kink is point of highest government transfer
• because EITC > pay-roll tax, create incentive to over-report!
• miss-reporting at other kinks?

A model of tax reporting

• assume linear preferences
• formal earnings w and informal earning y
• denote ˆ

y reported informal earnings

• taxes and transfers are based on w+y
• c = w + y − T(w + ˆ

y)

• administrative cost qa to report ˆ

y > 0

• moral cost qm to report ˆ

y = y

A model of tax reporting

• individual choose ˆ

y to maximize: w + y − T(w + ˆ y) − qa · 1[ˆ y > 0] − qm · 1[ˆ y = y]

• The authors shows that under some conditions ( T single

peaked at z ∗) the solution to this problem to do one of the following:

1 truthful reporting ˆ

y = y

2 complete evasion ˆ

y = 0

3 transfer maximization ˆ

y = z ∗ − w

• this:

1 creates bunching at z ∗ 2 does not generate bunching at other kinks since z ∗ maximizes

transfers

Kinks in the federal income tax

• a similar exercice can be conducted for the federal income tax
• focus on 2 periods: 1960-1972 and 1988-2004
• non-refundable tax credit:
• items that can reduce positive tax liability
• unlike EITC, can’t make taxes negative
• however can move a worker from one bracket to another
• this can create bunching beyond labor supply response
• child credit introduced in 1998
• shifts the kink for family with children

Taxable income density 1960-1969 - Married

• Bunching is present at the first kink

Taxable income density 1960-1969 - Single

• Bunching is less clear for singles

1960-1969 - Married - Itemized Vs Total

• Most of the response seems to come from changes in reported

differences between taxable and standard deduction

1960-1969 - Married - Evolution

• The figure reveals that workers might need time to adapt. In

1960, earnings and taxes were more stable.

1988-2020 - Married

• presence of 2 kinks, second one is much smaller

1988-2020 - Married - Itemized Vs Total

• similar conclusion to before, showing that part of the response

is due to itemizing

1988-2020 - Married - Shift due to children

• clear evidence that the bunch is due to the kink in tax

marginal rate

Saez 2010 AEJ conclusion

• clear bunching at first kink in all data sets considered
• for EITC, bunching is mostly due to self-employed
• for the federal tax, bunching is partly due to amount of

itemization

• overtime, optimal response might be subject to some friction
• this suggests twho margins of repsonse:
• adjustment in hours
• adjustment in reported income / reported items to deduct
• it seems to me that under convex cost of itemizing, we should

still see relatively strong bunching at the following kinks.

Individual Heterogeneity, Nonlinear Budget Sets and Taxable Income

Overview

• can we introduce preference heterogeneity?
• derive identification results using reveal preference argument
• apply method to tax reform in Sweden

The environment

• Individuals have static preferences U (c, y, η) :
• c is consumption
• y is taxable income
• η is unrestricted preference parameter

(can be multidimensional)

• U is increasing in c, decreasing in y, strictly quasi-concave in

(c, y)

• Under a linear budget set (ρ, R) the individual solves:

sup

c,y

U (c, y, η) s.t. c = y · ρ + R

Piece-wise linear budget sets

• A piece-wise linear budget set with J segments can be

described by a vector of parameters (ρ1...ρJ, R1...RJ) where

• ρj are the net of tax rates (slopes)
• Rj are the virtual incomes (intercept)
• the kink points are given by lj = (Rj+1 − Rj)/(ρj − ρj+1)
• B(y) is net income function
• B = {(c, y) : 0 ≤ c ≤ B(y), y ≥ 0} is the budget set
• We denote y(B, η) the choice of indiviudal η:

y(B, η) = arg max

y

U (B(y), y, η)

Additional definitions

• A convex budget B set corresponds to a concave B(y) and

will have ρj > ρj+1

• In the case where B(y) is linear with parameters (ρ, R) we

define:

• y(ρ, R, η) the response of individual η
• F(y|ρ, R) = Pr[y(ρ, R, ηi) ≤ y)] the distribution or resulting

taxable income

• For general budget sets with J components we can define the

response to individual segments: yj (η) = y(ρj , Rj , η)

• This paper will link F(y|ρ, R) to Pr[y(B, η) ≤ y|B]
• Pr[y(B, η) ≤ y|B] is our object of interest
• F(y|ρ, R) is a much smaller dimensional object
• E[y|B] can be used to get the average effect of changes in B

Additional definitions

• A convex budget B set corresponds to a concave B(y) and

will have ρj > ρj+1

• In the case where B(y) is linear with parameters (ρ, R) we

define:

• y(ρ, R, η) the response of individual η
• F(y|ρ, R) = Pr[y(ρ, R, ηi) ≤ y)] the distribution or resulting

taxable income

• For general budget sets with J components we can define the

response to individual segments: yj (η) = y(ρj , Rj , η)

• This paper will link F(y|ρ, R) to Pr[y(B, η) ≤ y|B]
• Pr[y(B, η) ≤ y|B] is our object of interest
• F(y|ρ, R) is a much smaller dimensional object
• E[y|B] can be used to get the average effect of changes in B

Hausman (1979)

• Hausman (1979) shows that when B(y) is concave, then:
• ∃!j s.t. yj(η) ≥ lj, yj+1(η) ≤ lj and then y(B, η) = lj
• or ∃!j s.t. lj−1 < yj(η) < lj and then y(B, η) = yj(η)
• The first point gives us an expression for masses at kink

points.

• The second part tells us tangency points will always be inside

segments, and there is only one.

• The intuition behind the proof:
• imagine that you have two tangency points on B(y)
• then linear combination is also in the budget set
• by concavity of the utility, this point has to be even better

Hausman (1979)

y B(y) 2 1 3

• If 1 and 2 are chosen then linear combination is feasible and

dominates

• Strict quasi-concave of U (c, y, η) in (c, y) gives that

U (B(y1) + B(y2) 2 , y1 + y2 2 , η)) > U (B(y1), y1, η)

Blumquist, Liang and Newey (2015-WP)

• Theorem 2 tells us that when
• B(y) is piece wise linear and concave
• yi = y(Bi, ηi) + ǫi with E(ǫi|Bi) = 0
• ηi is independent of Bi
• then

E(yi|Bi) = ¯ y(ρJii, RJii)+

Ji−1

• j=1
• µ(ρji, Rji, lji)−µ(ρj+1,i, Rj+1,i, ℓj+1,i)
• where
• ¯

y(ρ, R) =

• yF(dy|ρ, R)
• µ(ρ, R, ℓ) =
• 1[y < ℓ](y − ℓ)F(dy|ρ, R)
• the proof combines the Hausman results with some revealed

preference argument to rule out some terms

What have we gained?

• theorem 2 offers a huge dimension reduction of the state

space of E(yi|Bi)

• instead of a 2 · J-dimensional function:
• one 2-dimensional ¯

y(ρ, R)

• one 3-dimensional µ(ρ, R, ℓ)
• this breaks the curse of dimensionality for NP estimation

Blumquist, Liang and Newey (2015-WP) - Corollary 4

• Corollary 4 tells us that when B(y) is piece wise linear and

concave then,

• Pr[y(Bi, ηi) ≤ y|Bi] = F(y|ρi(y), Ri(y))
• so F(y|ρ, R) is identified for y values observed in a (ρ, R)

segment

• theorem 3 provides a more general results where this holds at

points with only local convexity, global convexity is not required

Blumquist, Liang and Newey (2015-WP) - Corollary 4

1 again, this provides a very important dimension reduction 2 F(y|ρ, R) is only identified for y, ρ(y), R(y) seen in the data 3 since the conditional mean expression involves the all

distribution, it is only identified for this ρ, R such that for every y, there is budget set ρ, R in the data at y.

4 an interesting insight is that varying over all convex budget

sets, does nothing more that tracing the linear budget set

• CDF. Convex budget sets provide no more information than

shifted linear budget sets.

Back to the Kink estimator 1/2

• we consider Saez isoelastic preferences:

U (C, A, ρ) = C − ρ 1 + 1/β A ρ 1+1/β

• for which we have shown that A = ρθβ for tax slope θ
• around the budget kink at K with taxes θ1, θ2 we get a

bunching: B = Kθ−β

2

Kθ−β

1

φ(ρ)dρ

• B is also equal to F2(K) − F1(K) where Fi(a) is the CDF

under linear tax θi

• The issue is that neither β nor φ is observed

Back to the Kink estimator 1/2

• If we are willing to make a parametric assumption then one

can learn from before and after the segment. F1(a) = Pr[ρθβ

1 ≤ a] = Φ(aθ−β 1 )

• otherwise we can’t deferentiate between two cases:
• B is large because β is large and people are not close to K in

counterfactual but can adjust at small utility cost

• B is large even-though β is small and income does not

respond, but everyone is extremely close to K.

Policy effects 1

• back to the Blomquist et Al paper
• the elasticity to a linear tax is given by

¯ y(ρ, R) =

• yF(dy|ρ, R)
• d log ¯

y d log ρ is the elasticity to change in net-of-tax rate

• d log ¯

y d log ρ is the elasticity to change in the intercept

• identifying it would require for (ρ, R) to be the marginal tax

rate for at least someone for each value of y

• plus, in the end, most individuals face a non-linear tax rate
• the paper focus on changes to non-linear budget set

Policy effects 2

• define the following g and G(a, b) functions:

g(ρ1...ρJ, R1...RJ) = E[yi|Bi = B] G(a, b) = g(ρ1 + a, ...ρJ + a, R1 + b...RJ + b)

• a tilts the budget constraints everywhere and b is like a

change in unearned income

• a policy relevant elasticity is

˜ ρ E[y|B] · ∂G ∂a (a = 0, b = 0)

• where ˜

ρ captures the aggregate slope and E[y|B] captures the income level

• identifying the effect of (a, b) require local shifting in slope

intercepts.

Policy effects 2

• identifying the effect of (a, b) require local shifting in slope
• intercepts. remember:

E(yi|Bi) = ¯ y(ρJii, RJii)+

Ji−1

• j=1
• µ(ρji, Rji, lji)−µ(ρj+1,i, Rj+1,i, ℓj+1,i)
• ¯

y(ρ, R) =

• yF(dy|ρ, R),

µ(ρ, R, ℓ) =

• 1[y < ℓ](y−ℓ)F(dy|ρ, R)
• the argument seems to be that the integral is on a path very

close to the one we actually observe. But still, the first term requires the change in F(y|ρ, R) with respect to ρJ at all y!

• the key problem seems to be with the kinks, but they appear

to cancel out in the region outside the current segment Pr(Y (B, η) = ℓj ) =

• 1[y ≤ ℓj ]F(dy|ρj+1, Rj+1)−
• 1[y < ℓj ]F(dy|ρj , Rj )
• they argue such variation will be present over time and

between individuals

Estimation

• some consideration to control for overall wage growth
• parametrize with series
• ¯

y(ρ, R) ≈

k ρm(k)Rq(k)

• µ(ρ, R, ℓ) ≈

k ρm(k)Rq(k)ℓr(k)

• alternatively, one can directly parametrize the F(y|ρ, R)
• one can even apply the Slutzky restrictions
• in practice:
• add covariates to F to alleviate independence concerns

between ηi and Bi

• use“control variates”to allow for some endogeneity in Bi

(refers to Blundell and Powell 2001)

• choose number of element in power series using Lasso (see

paper for more details, refere to Bernoulli and Chernozhukov 2010)

Application

• the paper applies the method to Swedish tax reform between

1993 and 2008

• married and co-habiting men
• information about income, tax, demographcis, housing

variables

• 17,000 people each year, total of 80,000 observations
• budget sets are constructed using FASIT, all tax code
• non-labor income is set to earnings if male receives no income

( they actually instrument non labor income to deal with endogeneity)

Preferred specification

• cross-validation gives productivity growth of 0.004
• standard errors are bootstrapped 50 reps

Effect of penalization

• at penalization less than 0.003 elasticities do not seem to

change much

Conclusion

• the paper provides a non-parametric approach to labor-supply

estimation

• extending Hausman result, estimation is made possible by

reducing the dimension of the state space under convex budget sets ( or locally convex)

• an important assumption is the independence of ηi and Bi
• covariates can be used to reduce concern
• Kline & Tartarri have an experiment that guarantees

independence!

• applied to the Swedish data, the preferred results are:
• net of tax elasticity of 0.60(0.13)
• income elasticity of −0.08(0.03).

Chetty and Al

Tax system in Danmark - overview

Kink and bunching in Danemark

Kink and bunching in Danemark

More...

• The paper develops a model with multiple margin of

adjustments

• Includes frictions and implications for the link between micro

and macro elasticities

• we will cover this in a later course

References