Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Who wins, who loses?
Tools for distributional policy evaluation
Maximilian Kasy
Department of Economics, Harvard University
Maximilian Kasy
Who wins, who loses?
Harvard
1 of 40
Introduction
Setup
I
I
Identification
Aggregation
Estimation
Application
Conclusion
Few policy changes result in Pareto improvements
Most generate WINNERS and LOSERS
EXAMPLES:
1. Trade liberalization
net producers vs. net consumers of goods with rising / declining
prices
2. Progressive income tax reform
high vs. low income earners
3. Price change of publicly provided good (health, education,...)
Inframarginal, marginal, and non-consumers of the good;
tax-payers
4. Migration
Migrants themselves; suppliers of substitutes vs. complements to
migrant labor
5. Skill biased technical change
suppliers of substitutes vs. complements to technology;
consumers
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
This implies...
If we evaluate social welfare based on individuals’ welfare:
1. To evaluate a policy effect, we need to
1.1 define how we measure individual gains and losses,
1.2 estimate them, and
1.3 take a stance on how to aggregate them.
2. To understand political economy, we need to characterize the
sets of winners and losers of a policy change.
My objective:
1. tools for distributional evaluation
2. utilitarian framework, arbitrary heterogeneity, endogenous prices
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Proposed procedure
1. impute money-metric welfare effect to each individual
2. then:
2.1 report average effects given income / other covariates
2.2 construct sets of winners and losers
2.3 aggregate using welfare weights
contrast with program evaluation approach:
1. effect on average
2. of observed outcome
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Contributions
1. Assumptions
1.1 endogenous prices / wages (vs. public finance)
1.2 utilitarian welfare (vs. labor, distributional decompositions)
1.3 arbitrary heterogeneity (vs. labor)
2. Objects of interest
2.1 disaggregated welfare ⇒
I
I
political economy
allow reader to have own welfare weights
2.2 aggregated ⇒ policy evaluation as in optimal taxation
3. Formal results
next slide
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Formal results
1. Identification
1.1 Main challenge: E [ẇ · l |w · l ]
1.2 More generally: E [ẋ |x ] causal effect of policy
conditional on endogenous outcomes,
1.3 solution: tools from vector analysis, fluid dynamics
2. Aggregation
social welfare & distributional decompositions
2.1 welfare weights ≈ derivative of influence function
2.2 welfare impact = impact on income - behavioral correction
3. Inference
3.1 local linear quantile regressions
3.2 combined with control functions
3.3 suitable weighted averages
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Literature
1. public - optimal taxation
Samuelson (1947), Mirrlees (1971), Saez (2001), Chetty (2009),
Hendren (2013), Saez and Stantcheva (2013)
2. labor - determinants of wage distribution
Autor et al. (2008), Card (2009)
3. distributional decompositions
Oaxaca (1973), DiNardo et al. (1996), Firpo et al. (2009), Rothe
(2010), Chernozhukov et al. (2013)
4. sociology - class analysis
Wright (2005)
5. mathematical physics - theory of differential forms
Rudin (1991)
6. econometrics - various
Koenker (2005), Hoderlein and Mammen (2007), Matzkin (2003),
Altonji and Matzkin (2005)
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Notation
I
policy α ∈ R
individuals i
I
potential outcome w α
realized outcome w
I
partial derivatives ∂w := ∂ /∂ w
with respect to policy ẇ := ∂α w α
I
density f
cdf F
quantile Q
I
wage w
labor supply l
consumption vector c
taxes t
covariates W
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Setup
Assumption (Individual utility maximization)
individuals choose c and l to solve
max u (c , l )
c ,l
s.t . c · p ≤ l · w − t (l · w ) + y0 .
(1)
v := max u
I
u, c, l, w vary arbitrarily across i
I
p, w , y0 , t depend on α
⇒ so do c, l, and v
I
u differentiable, increasing in c, decreasing in l, quasiconcave,
does not depend on α
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Objects of interest
Definition
1. Money metric utility impact of policy:
ė := v̇ ∂y0 v
2. Average conditional policy effect on welfare:
γ(y , W ) := E [ė|y , W , α]
3. Sets of winners and losers:
W := {(y , W ) : γ(y , W ) ≥ 0}
L := {(y , W ) : γ(y , W ) ≤ 0}
4. Policy effect on utilitarian welfare: SWF : v (.) → R
˙ = E [ω · γ]
SWF
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Marginal policy effect on individuals
Lemma
ẏ = (l̇ · w + l · ẇ ) · (1 − ∂z t ) − ṫ + y˙0 ,
ė =
l · ẇ · (1 − ∂z t ) − ṫ + y˙0 − c · ṗ.
(2)
Proof: Envelope theorem.
1. wage effect l · ẇ · (1 − ∂z t ),
2. effect on unearned income y˙0 ,
3. mechanical effect of changing taxes −ṫ.
4. behavioral effect b := l̇ · w · (1 − ∂z t ) = l̇ · n,
5. price effect −c · ṗ.
Income vs utility:
ẏ − ė = l̇ · n + c · ṗ.
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Example: Introduction of EITC (cf. Rothstein, 2010)
I
Transfer income to poor mothers made contingent on labor
income
1. mechanical effect > 0 if employed
< 0 if unemployed
2. labor supply effect > 0
3. wage effect
<0
for mothers and non-mothers
I
Evaluation based on
1. income (“labor”)
2. utility, assuming fixed wages (“public”)
3. utility, general model
I
I
1. mechanical + wage + labor supply
2. mechanical
3. mechanical + wage
Case 3 looks worse than “labor” / “public” evaluations
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Identification of disaggregated welfare effects
I
Goal: identify γ(y, W) = E[ė|y, W, α]
I
Simplified case:
no change in prices, taxes, unearned income
no covariates
I
Then
γ(y ) = E [l · ṅ|l · n, α]
I
Denote x = (l , w ).
Need to identify
g (x , α) = E [ẋ |x , α]
(3)
from
f (x |α).
I
Made necessary by combination of
1. utilitarian welfare
2. heterogeneous wage response.
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Assume :
1. x = x (α, ε), x ∈ Rk
2. α ⊥ ε
3. x (., ε) differentiable
Physics analogy:
I
x (α, ε): position of particle ε at time α
I
f (x |α): density of gas / fluid at time α , position x
I
ḟ change of density
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h(x , α) = E [ẋ |x , α] · f (x |α): “flow density”
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Stirring your coffee
I
I
If we know densities f (x |α),
what do we know about flow
g (x , α) = E [ẋ |x , α]?
Problem: Stirring your coffee
I does not change its density,
I yet moves it around.
I ⇒ different flows g (x , α)
consistent with a constant density f (x |α)
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Will show:
I
Knowledge of f (x |α)
I
I
I
I
Add to h
I
I
I
I
identifies ∇ · h = ∑kj=1 ∂x j hj
where h = E [ẋ |x , α] · f (x |α),
identifies nothing else.
h̃ such that ∇ · h̃ ≡ 0
⇒ f (x |α) does not change
“stirring your coffee”
Additional conditions
I
I
I
e.g.: “wage response unrelated to initial labor supply”
⇒ just-identification of g (x , α) = E [ẋ |x , α]
g j (x , α) = ∂α Q (v j |v 1 , . . . , v j −1 , .α)
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Density and flow
Recall
h(x , α) := E [ẋ |x , α] · f (x |α)
k
∇ · h := ∑ ∂x j hj
j =1
ḟ := ∂α f (x |α)
Theorem
ḟ = −∇ · h
(4)
h2+dx2h2
h1+dx1h1
h1
h2
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Proof:
1. For some a(x ), let
Z
A(α) := E [a(x (α, ε))|α] =
a(x (α, ε))dP (ε)
Z
=
a(x )f (x |α)dx .
2. By partial integration:
k
Ȧ(α) = E [∂x a · ẋ |α] =
Z
∑
∂x j a · hj dx
j =1
=−
Z
k
a·
∑ ∂x j hj dx = −
Z
a · (∇ · h)dx .
j =1
3. Alternatively:
Z
Ȧ(α) =
a(x )ḟ (x |α)dx .
4. 2 and 3 hold for any a ⇒ ḟ = −∇ · h. Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
The identified set
Theorem
The identified set for h is given by
h0 + H
(5)
where
H = {h̃ : ∇ · h̃ ≡ 0}
h (x , α) = f (x |α) · ∂α Q (v j |v 1 , . . . , v j −1 , α)
0j
v j = F (x j |x 1 , . . . , x j −1 , α)
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Proof of sharpness of identified set:
1. For any h s.t. ḟ = −∇ · h construct DGP as follows
2. Let ε = x (0, ε), f (ε) = f (x |α = 0)
3. Let x (., ε) be the solution of the ODE
ẋ = g (x , α), x (0, ε) = ε.
(existence: Peano’s theorem)
4. ⇒ consistent with h and with f
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Theorem
1. Suppose k = 1. Then
H = {h̃ ≡ 0}.
(6)
H = {h̃ : h̃ = A · ∇H for some H : X → R}.
(7)
2. Suppose k = 2. Then
where
A=
0
−1
1
0
.
3. Suppose k = 3. Then
H = {h̃ : h̃ = ∇ × G}.
(8)
where G : X → R3 .
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Proof: Poincaré’s Lemma.
Figure : Incompressible flow and rotated gradient of potential
h̃ = ∇ · (x 1 · exp(−x 21 − x 22 ))
2
H̃ = x 1 · exp(−x 21 − x 22 )
1.5
1
0.5
x2
0.5
0
0
−0.5
−1
−0.5
2
1
−1.5
2
1
0
−2
−2
0
−1
−1.5
−1
−0.5
0
x
1
0.5
1
1.5
2
x2
−1
−2
−2
x1
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Point identification
Theorem
Assume
∂
E [ẋ i |x , α] = 0 for j > i .
∂ xj
(9)
Then h is point identified, and equal to h0 as defined before.
In particular
g j (x , α) = E [ẋ j |x , α]
= ∂α Q (v j |v 1 , . . . , v j −1 , α).
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Controls; back to γ
Proposition
I
Suppose α ⊥ ε|W, and ∂∂x j E [ẋ i |x , W , α] = 0 for j > i. Then
E [ẋ j |x , W , α] = ∂α Q (v j |v 1 , . . . , v j −1 , W , α),
where v j = F (x j |x 1 , . . . , x j −1 , W , α).
I
If x j = n,
γ(y, W) =E [l · ṅ|y , W , α] =
E[l · ∂α Q(v j |v 1 , . . . , v j −1 , W , α)|y, W, α].
(10)
Instrumental variables: similar (see paper)
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Problems in practice
1. Equilibrium effects:
identification of causal effects using individual-level variation
requires “stable unit treatment values” (SUTVA)
I
I
Possible solution: city level variation (cf. Card, 2009)
or state level variation
⇒ appropriate definition of “treatment”
no issue for evaluation of historical changes
2. Unemployment:
our expression for ė requires:
only monetary constraints are affected by the policy change
policies shifting degree of involuntary unemployment ⇒
underestimate welfare impact
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Aggregation
I
Relationship
social welfare ⇔ distributional decompositions?
I
public finance welfare weights
≈ derivative of dist decomp influence functions
I
˙
Alternative representations of SWF
˙ :
⇒ alternative ways to estimate SWF
1. weighted average of individual welfare effects ė, γ
2. distributional decomposition for counterfactual income ỹ
(holding labor supply constant)
3. distributional decomposition of realized income
minus behavioral correction
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Welfare weights and influence functions
Consider θ : P y → R
Theorem
1. Welfare weights:
˙ = E [ω SWF · ė]
SWF
(11)
θ̇ = E [ω θ · ẏ ].
α
α
2. Influence function: Let s(y ) = ∂α f y (y )/f y (y ).
θ̇ = E [IF (y ) · s(y )] .
3. Relating the two:
ω θ = ∂y IF (y ).
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Introduction
Setup
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Estimation
Application
Conclusion
Alternative representations
Theorem
I
Assume ω SWF = ω θ = ω and ṗ = 0.
I
Let
ỹ α = l 0 · w α − t α (l 0 · w α ) + y0α ,
b = l̇ · n
Then ė = ỹ˙ = ẏ − b and
1. Counterfactual income distribution:
˙ = E [ω · ỹ˙ ]= E[ω · γ]
SWF
(12)
= E [IF (y ) · s̃(y )]
α
= ∂α θ P ỹ .
2. Behavioral correction of distributional decomposition:
˙ = E [ω · b].
θ̇ − SWF
(13)
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Estimation
1. First estimate the disaggregated welfare impact
γ(y , W ) = E [ė|y , W , α]
= E [l · ẇ · (1 − ∂z t ) − ṫ + y˙0 − c · ṗ|y , W , α]
(14)
2. Then estimate other objects by plugging in γb:
c= {(y , W ) : γb(y , W ) ≥ 0}
W
c= {(y , W ) : γb(y , W ) ≤ 0}
L
[
˙ = EN [ωi · γb(yi , Wi )].
SWF
(15)
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Estimation of g and γ
1) b
v j : estimate of F (x j |x 1 , . . . , x j −1 , W , α)
1. estimated conditional quantile of x j given (W , α, b
v 1, . . . , b
v j −1 )
2. estimate by local average
j
3. local weights: Ki for observation i around (W , α, b
v 1, . . . , b
v j −1 )
4.
j
b
vj =
j
EN [Ki · 1(xi ≤ x j )]
j
EN [Ki ]
(16)
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
2) b
g j : estimate of E [ẋ j |x , W α]
1. identified by slope of quantile regression
2. estimate by local linear Qreg
j
j
3. regression residual: Ui = xi − x j − g · αi
j
j
j
4. loss function: Li = Ui · (b
v j − 1(Ui ≤ 0))
5.
h
i
j
j
b
g j = argmin EN Ki · Li ,
g
3) γb(y , W ): estimate of E [l · ṅ|y , W , α]
1. n = x j ⇒ ė = l · ṅ = l · ẋ j
2. estimate γ by weighted average
γb(y , W ) =
EN Ki · l · b
gj
EN [Ki ]
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Introduction
Setup
Identification
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Estimation
Application
Conclusion
Inference
I
b
g j depends on data in 3 ways:
1. through x j , α ,
2. quantile b
vj,
v j −1 ).
3. controls (b
v 1, . . . , b
I
1 standard, 2 negligible, 3 nasty
I
to avoid dealing with 3: non-analytic methods of inference
I
I
I
I
bootstrap
Bayesian bootstrap
subsampling
validity?
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Application: distributional impact of EITC
I
Work in progress
I
Meyer and Rosenbaum (2001), Rothstein (2010), Leigh (2010)
I
March CPS sample from IPUMS
I
following Leigh (2010): Variation in state top-ups of EITC
across time and states
I
α = maximum EITC benefit available
(weighted average across family sizes)
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Mapping the model to the data
I
Need: l , w , n, y0 , t , p and covariates
I
tentative mapping:
1. l: weeks worked × usual hours worked per week
2. w: income from wages + self employment + farm work /l
3. n: w × (1− marginal federal tax rate)
4. p: CPI (just normalize everything)
5. y0 : all other income
6. t: ?
7. Covariates: age, sex, race, ...?
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Conclusion and Outlook
I
Most policies generate winners and losers
I
Motivates interest in
1. disaggregated welfare effects
2. sets of winners and losers (political economy!)
3. weighted average welfare effects (optimal policy!)
I
Consider framework which allows for
1. endogenous prices / wages (vs. public finance)
2. utilitarian welfare (vs. labor, distributional decompositions)
3. arbitrary heterogeneity (vs. labor)
Maximilian Kasy
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Main results
1. Identification
1.1 Main challenge: E [ẇ · l |w · l ]
1.2 More generally: E [ẋ |x ] causal effect of policy
conditional on endogenous outcomes,
1.3 solution: tools from vector analysis, fluid dynamics
2. Aggregation
social welfare & distributional decompositions
2.1 welfare weights ≈ derivative of influence function
2.2 welfare impact = impact on income - behavioral correction
3. Inference
3.1 local linear quantile regressions
3.2 combined with control functions
3.3 suitable weighted averages
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
Generalization of identification to dim(α) > 1
I
g (x , α) = E [∂α x |x , α] ∈ Rl ,
h(x , α) = g (x , α) · f (x |α)
I
∇ · h := (∇ · h1 , . . . , ∇ · hl )
I
Most results immediately generalize
I
In particular
Theorem
∂α f = −∇ · h
(17)
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Introduction
Setup
Identification
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Estimation
Application
Conclusion
Theorem
The identified set for h is contained in
h0 + H
(18)
where
H = {h̃ : ∇ · h̃ ≡ 0}
h (x , α) = f (x |α) · ∂α Q (v j |v 1 , . . . , v j −1 , α)
0j
v j = F (x j |x 1 , . . . , x j −1 , α)
I
open question: is this sharp?
I
does the model restrict the set of admissible g?
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Introduction
Setup
Identification
Aggregation
Estimation
Application
Conclusion
A partial answer
Lemma
The system of PDEs
∂α x (α) = g (α, x )
x (0) = x 0
has a local solution iff
∂α g j + ∂x g j · g
(19)
is symmetric ∀ j.
This solution is furthermore unique.
Proof: if: differentiation. only if: Frobenius’ theorem. I
cf. proof of sharpness in 1-d case
I
Q: what is the convex hull of all such g?
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Introduction
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Estimation
Application
Conclusion
Thanks for your time!
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