Why Piketty Says r − g Matters for Inequality
Supplementary Lecture Notes “Income and Wealth Distribution”
Benjamin Moll
Princeton
June 1, 2014
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These Notes
My version of a hybrid of
1
Section 5.4 of Piketty and Zucman (2014)
http://gabriel-zucman.eu/files/PikettyZucman2014HID.pdf
2
Benhabib, Bisin and Zhu (2013)
http://www.econ.nyu.edu/user/benhabib/lineartail31.pdf
Warning: high probability of algebra mistakes. If you find one, please email me
[email protected]
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These Notes
Summary:
• Standard explanation of high observed wealth concentration
(e.g. top 1% own 30%): idiosyncratic capital income risk
• In theories with capital income risk, r¯ − g is one main
determinant of top wealth inequality
• Theory suggests slight modification: r¯ − g − c̄ where c̄ is the
marginal propensity to consume out of wealth for rich people
What these notes are not about:
• the aggregate capital-output ratio K /Y : different story
(inequality across groups)
• see Piketty and Zucman (2014)
• and critical reviews by Ray and Krusell-Smith:
http://www.econ.nyu.edu/user/debraj/Papers/Piketty.pdf
http://aida.wss.yale.edu/smith/piketty1.pdf
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Outline
1
Simplest possible case
• Brownian capital income risk
• exogenous MPC
2
Generalizations
• endogenous savings/MPC
• labor income risk
• more general capital income processes
3
Transition dynamics
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Simplest Possible Case
• Continuum of individuals, heterogeneous in
• wealth b
• labor income w
• Wealth evolves as
dbt = [wt + rt bt − ct ]dt
• Labor income wt grows deterministically wt = we gt
(e.g. GDP grows and constant labor share: wt = (1 − α)Yt )
• Capital income rt is stochastic
rt = r¯ + σdWt
where Wt is a standard
Brownian motion, that is
√
dWt ≡ lim∆t→0 εt ∆t, with εt ∼ N (0, 1)
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Simplest Possible Case
• Combining
dbt = [wt + r¯bt − ct ]dt + σbt dWt
• bt is non-stationary because wt is growing
• ⇒ define detrended wealth: at = bt e −gt
• Using dat /at = dbt /bt − gdt:
dat = [w + (¯
r − g )at − ct ]dt + σat dWt
• For now: assume exogenous MPC out of wealth ct = c̄a.
Assume c̄ > r − g
• Endogenize saving behavior later
• reinterpret c̄ = lima→∞ c(a)/a where c=consumption policy fn
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Stationary Wealth Distribution
• De-trended wealth follows stationary stochastic process
dat = [w + (¯
r − g − c̄)at ]dt + σat dWt
(∗)
• Characterize stationary distribution?
• Definition: a has a Pareto tail if there exists C > 0 and
ζ > 0 such that
lim aζ Pr(ã > a) = C .
a→∞
• Note: ζ = “tail parameter.” Top wealth inequality = 1/ζ
• Result: The stationary wealth distribution has a Pareto tail
with tail parameter (recall c̄ > r − g )
r¯ − g − c̄
> 1,
σ 2 /2
• Observations:
ζ =1−
1
2
1
σ 2 /2
= 2
ζ
σ /2 − (¯
r − g − c̄)
inequality 1/ζ increasing in r¯ − g
but also depends on c̄, σ (decreasing in c̄, increasing in σ)
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Proof of Result
• Wealth distribution f (a, t) satisfies Kolmogorov
Forward/Fokker-Planck equation
r − g − c̄)a)f (a, t)) +
∂t f (a, t) = −∂a ((w + (¯
σ2
∂aa (a2 f (a, t))
2
• Stationary wealth distribution f (a) satisfies:
0 = −∂a ((w + (¯
r − g − ρ)a)f (a)) +
σ2
∂aa (a2 f (a))
2
• Guess and verify f (a) ∝ a−ζ−1
0 = w (ζ + 1)a−ζ−2 + ζ(¯
r − g − c̄)a−ζ−1 + (ζ − 1)ζ
σ 2 −ζ−1
a
2
• We are interested in f as a → ∞: first term drops!
σ2
2
• Collecting terms yields formula on previous slide.
0 = ζ(¯
r − g − c̄) + (ζ − 1)ζ
•
Note: swept some technical issues under the rug e.g. existence of stationary distribution. Should follow
from fact that (∗) is Kesten process (random growth process with intercept). See Benhabib-Bisin-Zhu.
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The Effect of Taxes on Wealth Inequality
• Introduce taxes
• labor income tax τw
• capital income tax τr
dbt = [(1 − τw )wt + (1 − τr )rt bt − ct ]dt
dat = [(1 − τw )w + ((1 − τr )¯
r − g − c̄)at ]dt + σ(1 − τr )at dWt
• Result: Formula for tail parameter generalizes to
ζ =1−
(1 − τr )¯
r − g − c̄
σ 2 (1 − τr )2 /2
(1 − τr )2 σ 2 /2
1
=
ζ
(1 − τr )2 σ 2 /2 − (¯
r − g − c̄) + τr r¯
• Observations:
1
2
inequality decreasing in τr for two reasons: capital income pays
lower return r¯, and is less volatile
inequality does not depend on labor income tax
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Discussion
• Other sources of randomness in wealth growth
• Piketty-Zucman (Section 5.4) have stochastic
savings/bequests rather than stochastic capital income
• this is mathematically isomorphic: everything identical if set
rt = r¯,
ct (a) = c̄a + σadWt
• what matters is that at follows random growth process like (∗)
• randomness in bequests would work similarly (e.g. in more
general model with OLG structure and Poisson death)
• while mathematically isomorphic, economics obviously different
• Partial vs. General Equilibrium
• obviously both r¯ and g are endogenous, and so the above
analysis is potentially misleading
• GE extension interesting/desirable, especially for
counterfactuals/policy
• but PE with exogenous r¯, g =useful starting point
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Generalizations
1
Optimally chosen savings
2
stochastic labor income wt
3
more general process for capital income rt
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Optimal Savings + Stochastic wt
• Individuals solve
V (b, w̃ ) = max E0
{ct }
Z
∞
e −ρt u(ct )dt
s.t.
0
dbt = [w̃t + rt bt − ct ]dt
d w̃t = (g + µw (w̃t ))dt + σw (w̃t )dWt
rt = r¯ + σdWt
bt ≥ 0,
(b0 , w̃0 ) = (b, w̃ )
• Assume CRRA utility
u(c) =
c 1−γ
,
1−γ
γ>0
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Generalizations
• Detrended problem: wt = w̃t e −gt , at = bt e −gt
v (a, w ) = max E0
{ct }
Z
∞
e −ρt u(ct )dt
s.t.
0
dat = [wt + (¯
r − g )at − ct ]dt + σat dWt
dwt = µw (wt )dt + σw (wt )dWt
• HJB equation:
at ≥ 0,
(a0 , w0 ) = (a, w )
ρv (a, w ) = max u(c) + ∂a v (a, w )(w + (¯
r − g )a − c) + ∂aa v (a, w )
c
σ 2 a2
2
σw2 (w )
2
with a state constraint boundary condition to enforce the
borrowing constraint.
+ ∂w v (a, w )µw (w ) + ∂ww v (a, w )
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Tail Saving Behavior & Implied Inequality
Proposition (Asymptotic Linearity)
Consumption policy functions are asymptotically linear, i.e. MPCs
out of wealth are asymptotically constant:
ρ − (1 − γ)(¯
r − g)
σ2
c(a, w )
= c̄ =
+ (1 − γ)
a→∞
a
γ
2
lim
Corollary
Formula for tail parameter becomes
2
(¯
r − g − ρ)/γ − (1 − γ) σ2
ζ = 1−
,
σ 2 /2
1
σ 2 /2
=
2
ζ
r − g − ρ)/γ
(2 − γ) σ2 − (¯
Observations:
1
2
inequality still depends on r¯ − g
but quantitative mapping different, e.g. depends on γ
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Proof of Linearity Prop.: Homogeneity
auxiliary result from Achdou, Lasry, Lions and Moll (2014)
Proposition (Homogeneity)
For any ξ > 0,
v (ξa, w ) = ξ 1−γ vξ (a, w )
where vξ solves
ρvξ (a, w ) = max u(c) + ∂a vξ (a, w )(w /ξ + (¯
r − g )a − c) + ∂aa vξ (a, w )
c
+ ∂w vξ (a, w )µw (w ) + ∂ww vξ (a, w )
Corollary
σ2
2
σw2 (w )
2
For large a, individuals behave as if they had no labor income:
lim
a→∞
v (a, w )
=1
ṽ (a)
where ṽ (a) solves
ρṽ (a) = max u(c) + ṽ ′ (a)((¯
r − g )a − c) + ṽ ′′ (a)
c
σ 2 a2
2
(∗∗)
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Proof of Linearity Proposition
• Next step: find explicit solution for policy function of (∗∗)
ρṽ(a) =H(ṽ ′ (a)) + ṽ ′ (a)(¯
r − g )a + ṽ ′′ (a)
H(p) = max u(c) − pc =
c
σ 2 a2
2
γ−1
γ
p γ
1−γ
• Guess and verify ṽ (a) = Ba1−γ , ṽ ′ (a) = (1 − γ)Ba−γ ,
ṽ ′′ (a) = −γ(1−γ)Ba−γ−1 , H(ṽ ′ (a)) =
γ−1
γ
((1−γ)B) γ a1−γ
1−γ
1
ρ = γ((1 − γ)B)− γ + (1 − γ)(¯
r − g ) − γ(1 − γ)
σ2
2
• from FOC, c̃(a) = c̄a, c̄ = ((1 − γ)B)−1/γ and hence
ρ − (1 − γ)(¯
r − g)
σ2
+ (1 − γ)
γ
2
• Asymptotic Linearity Proposition follows directly from
Homogeneity Proposition and above.
c̄ =
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Further Generalization: General r Process
• Individuals solve
V (b, w̃, r ) = max E0
{ct }
Z
∞
e −ρt u(ct )dt
s.t.
0
dbt = [w̃t + rt bt − ct ]dt
d w̃t = (g + µw (w̃t ))dt + σw (w̃t )dWt
drt = µr (rt ))dt + σr (rt )dBt
(b0 , w̃0 , r0 ) = (b, w̃ , r )
• Assume CRRA utility
u(c) =
c 1−γ
,
1−γ
γ>0
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Further Generalization: General r Process
• Detrended problem: wt = w̃t e −gt , at = bt e −gt
v (a, w , r ) = max E0
{ct }
Z
∞
e −ρt u(ct )dt
s.t.
0
dat = [wt + (rt − g )at − ct ]dt
dwt = µw (wt )dt + σw (wt )dWt
drt = µr (rt )dt + σr (rt )dBt
(a0 , w0 , r0 ) = (a, w , r )
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Tail Saving Behavior
Following similar steps as above, one can show:
Corollary
Consumption policy functions are asymptotically linear, i.e. MPCs
out of wealth are asymptotically constant:
lim
a→∞
c(a, w , r )
= c̄(r )
a
The task is therefore to characterize the stationary distribution
f (a, w , r ) of the following Kesten-type process:
dat = [wt + (rt − g − c̄(rt ))at ]dt
dwt = µw (wt )dt + σw (wt )dWt
drt = µr (rt )dt + σr (rt )dBt
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Stationary Wealth Distribution
• Here’s how to do it, based on Gabaix (2010) “On Random
Growth Processes with Autocorrelated Shocks”
Proposition (Gabaix)
Assume w and r are stationary processes. Then the process for a
has a stationary distribution with a Pareto tail
f (a, w , r ) ∼ φ(w , r )a−ζ−1 where the tail parameter ζ satisfies an
eigenvalue problem
1
0 = ζ(r − g − c̄(r ))e(w , r ) + µw (w )∂w e(w , r )] + σw2 (w )∂ww e(w , r )
2
1 2
(E)
+ µr (r )∂r e(w , r ) + σr (r )∂rr e(w , r )
2
for some eigenfunction e ≥ 0.
• Need to solve (E) numerically
• But can handle very general class of r -processes
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Proof
• Stationary distribution satisfies
0 = − ∂a [(w + (r − g − c̄(r ))a)f (a, w , r )]
1
− ∂w [µw (w )f (a, w , r )] + ∂ww [σw2 (w )f (a, w , r )]
2
1
− ∂r [µr (r )f (a, w , r )] + ∂rr [σr2 (r )f (a, w , r )]
2
−ζ−1
• Guess f (a, w , r ) = φ(w , r )a
and substitute in.
• Divide by a−ζ−1 and use that we’re interested in the tail as
a → ∞ and hence w /a drops:
0 =ζ(r − g − c̄(r ))φ(w , r )
1
− ∂y [µw (w )φ(w , r )] + ∂ww [σw2 (w )φ(w , r )]
2
1
− ∂r [µr (r )φ(w , r )] + ∂rr [σr2 (r )φ(w , r )]
2
• Using KF equation for (w , r ), obtain (E).
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Transition Dynamics
Figure 10.6. Wealth inequality: Europe and the U.S., 1810-2010
100%
Share of top decile or percentile in total wealth
90%
80%
70%
60%
50%
40%
30%
Top 10% wealth share: Europe
20%
Top 10% wealth share: U.S.
Top 1% wealth share: Europe
10%
Top 1% wealth share: U.S.
0%
1810
1830
1850
1870
1890
1910
1930
1950
1970
1990
2010
!"#$%&'(%)*+%,-&'%.("&/012%3(4$&'%*"(5/4$*&1%346%'*7'(0%*"%8/09:(%&'4"%*"%&'(%!"*&(+%;&4&(6<%
Sources and series: see piketty.pse.ens.fr/capital21c.
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Transition Dynamics
• So far: only focussed on stationary distributions
• But Piketty’s whole point: world is not stationary (see
Figure on previous slide)
• wants to argue: that’s because r¯t − gt varies over time
• Most interesting questions require extension to transition
dynamics
• simplest case: characterize f (a, t) satisfying
∂t f (a, t) = −∂a ((w + (¯
rt − gt − c̄t )a)f (a, t)) +
σ2
∂aa (a2 f (a, t))
2
• probably need to go numerical
• Economics should be similar to comparing steady states
• inequality depends on r¯t − gt − c̄t :
• how much does c̄t vary over time relative to r¯t − gt ?
• Open question: how fast (or slow) are transitions?
• e.g. if r¯t − gt − c̄t ↑, how long until inequality ↑?
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Richer Models
• Why would capital income be stochastic?
• One answer: entrepreneurship
• Quadrini (1999, 2000)
• Cagetti and DeNardi (2006)
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Summary
• Standard explanation of high observed wealth concentration
(e.g. top 1% own 30%): idiosyncratic capital income risk
• In theories with capital income risk, r¯ − g is one main
determinant of top wealth inequality
• does not rely on weird assumptions about saving behavior
(instead optimization w/ CRRA utility)
• but theory suggests slight modification: r¯ − g − c̄
• other factors also potentially important, e.g. σ
• Changes in inequality over time? Open questions:
• speed of transitions?
• relative (quantitative) importance of different factors
• e.g. how much does c̄t vary over time relative to r¯t − gt ?
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