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C LASSIC P OLICY B ENCHMARKS FOR E CONOMIES
WITH S UBSTANTIAL I NEQUALITY
James Bullard, Federal Reserve Bank of St. Louis
Riccardo DiCecio, Federal Reserve Bank of St. Louis
Heterogeneous Agents or Heterogeneous Information:
Which Route for Monetary Policy?
Banque de France and CEPR
Dec. 6, 2019
Paris, France
Any opinions expressed here are our own and do not necessarily reflect those of the FOMC.
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Introduction
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A N ACADEMIC TALK
This is an academic talk.
This material has been presented before at conferences in Hong Kong, Barcelona, St.
Louis and Pretoria during 2019. Those presentations are available on my web page.
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L ARGE - SCALE HETEROGENEOUS - AGENT DSGE
Increasing interest in large-scale heterogeneous-agent DSGE models.
Realistic degrees of inequality—approaching observed Gini coefficients.
What is the role of monetary policy in such a model?
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O UTLINE OF THE ARGUMENT
We construct a heterogeneous-agent economy featuring:
Three aggregate shocks: (1) total factor productivity, (2) labor supply and (3) aggregate
demand.
Both permanent and temporary idiosyncratic risk at the household level.
A simple and symmetric structure.
Income, wealth and consumption inequality on the same scale as in observed economies.
We include four policymaking authorities: (1) monetary, (2) fiscal, (3) labor market
and (4) education.
We describe a competitive equilibrium in which the four policymakers act in concert
to attain a first-best allocation of resources.
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CLASSIC VIEW
The policymaker roles are “classic.”
The monetary authority reacts to shocks each period in order to achieve the Wicksellian
natural real rate of interest for the economy.
The fiscal authority raises revenue via a non-state contingent linear labor income tax on all
households.
The labor market authority runs an unemployment insurance program.
The education authority minimizes the variance of beginning-of-life human capital
endowments.
Hence, the main result is that classic policy prescriptions can achieve the first-best allocation
of resources in this benchmark heterogeneous-agent economy.
This result may be helpful in understanding more complicated economies that deviate
from this benchmark.
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S OME SURPRISING FINDINGS
The proposed classic policies appear broadly similar to actual policies in place in
many economies.
The monetary authority meets often and reacts to current developments.
Simple linear labor income taxes set for the long run can be used without distorting the
labor supply; age-dependent taxation is unnecessary.
The best policy combination drives the consumption Gini toward zero but leaves
income and financial wealth Ginis substantially positive—suggesting that some
observed income and financial wealth inequality is due to life-cycle effects alone.
The model has a paper-and-pencil solution despite the three aggregate shocks and the
idiosyncratic risk.
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ECONOMY ASSUMPTION
The benchmark economy analyzed in this talk is a closed economy.
I will make some comments on possible extensions to open economy versions at the
end of the talk.
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S OME RECENT LITERATURE
Kaplan, Moll and Violante (AER, 2018). “HANK.”
Bhandari, Evans, Golosov and Sargent (Working paper, NBER, 2018). Ramsey meets
HANK.
Bullard and DiCecio (Working paper, St. Louis Fed, 2019). Substantial heterogeneity,
even under optimal monetary policy.
Bullard and Singh (JMCB, 2019 forthcoming). NGDP targeting in a simpler version.
Azariadis, Bullard, Singh and Suda (JEDC, 2019). Effective lower bound.
Heathcote, Storesletten and Violante (JPubE, 2019 forthcoming). Age-dependent
taxation.
Koenig (IJCB, 2013). NGDP targeting.
Sheedy (BPEA, 2014). NGDP targeting.
Huggett, Ventura and Yaron (AER, 2011). Idiosyncratic risk at the beginning of the life
cycle.
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Environment
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L IFE - CYCLE MODELS
We construct a general-equilibrium life-cycle economy with “symmetry
assumptions” that could be relaxed in a computational exercise.
Each period, a new continuum of households enters the economy, makes economic
decisions over the next T + 1 = 241 periods (“quarterly”), then exits the economy.
The symmetry assumptions:
All households have log preferences defined over consumption and leisure.
All households have a discount factor of unity.
All households are endowed with a personal productivity profile at the beginning of their
life cycle that begins low, rises to a peak exactly in the middle of the life cycle, and then
declines in a symmetric fashion.
The aggregate production technology is linear in the aggregate effective labor input. We
allow variable utilization of the labor input.
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P REFERENCES
Households are uncertain about how much they may wish to consume in the future.
For the agent entering the economy at date t, preferences are given by
T
Vt =
∑ η ln c̃t (t + s) + (1 − η ) ln `t (t + s) ,
(1)
s=0
where c̃t (t + s) ≡ ct (t + s) D (t + s) and the state of demand D is given by
D (t + 1) = δ (t, t + 1) D (t) ,
(2)
where δ follows an AR (1) process with mean unity.
The preferences for other households are analogous, and the demand shock is
experienced by all households at each date.
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P RODUCTION TECHNOLOGY
The linear production technology is given by
Y (t) = Q (t) D (t) N (t) L (t) ,
where L (t) is the aggregate effective labor input; Q (t) is the state of technology,
which follows
Q (t + 1) = λ (t, t + 1) Q (t) ,
(3)
(4)
where λ follows an AR (1) process with mean λ̄ > 1; N (t) is the population index,
which follows
N (t + 1) = ν (t, t + 1) N (t) ,
(5)
where ν follows an AR (1) process with mean ν̄ > 1.
The population index is understood to measure the simultaneous increase or decrease
in the size of all cohorts so as to maintain symmetry. It can be viewed as
“immigration.”
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L IFE - CYCLE PRODUCTIVITY
Each household is randomly assigned a personal productivity profile
e = {e0 , e1 , ..., e240 } when entering the model.
Profiles are symmetric—they begin low, rise and peak exactly in the middle of life, then
decline symmetrically back to the low level.
Profiles are restricted to be consistent with interior solutions to all household
problems—households will choose to work low hours but not zero.
Households also draw a scaling
factor ξ from a lognormal distribution as they enter the
model, i.e., ln ξ ∼ N µ, σ2 .
The product of their scaling factor and their assigned productivity profile permanently
determines their life-cycle productivity, that is, ξe.
Accordingly, there will be arbitrarily rich and arbitrarily poor households in the economy.
For illustrative purposes, ξ is drawn from a uniform distribution U [a, b] in the figures
shown later.
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A PERMANENT LIFE - CYCLE PRODUCTIVITY ASSIGNMENT
The assignment of productivity profiles is a stand-in for the human capital
development that takes place before age 20 in actual economies, including schooling,
parenting and any job experience.
Huggett, Ventura and Yaron (AER, 2011) argue that differences in initial conditions
(human capital at age 23) are more important than subsequent shocks in explaining
lifetime income.
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B ASELINE LIFE - CYCLE PRODUCTIVITY
1.5
1
0.5
0
0
60
120
180
240
quarters
F IGURE : A baseline personal productivity endowment profile. The profile is symmetric and peaks in
the middle period of the life cycle at a level about 50% greater than at the beginning or end. A full
model would include a set of symmetric profiles with differing shapes.
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OF LIFE - CYCLE PRODUCTIVITY
F IGURE : The mass of endowment profiles with the scaling factor drawn from a uniform distribution
U [0.05, 1.95]. Drawing from a lognormal distribution is harder to visualize, but such a distribution
would include arbitrarily rich and arbitrarily poor households. The endowment Gini is about 35%.
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H AND - TO - MOUTH HOUSEHOLDS
It is possible to add “hand-to-mouth” households to this economy without changing
qualitative results.
The baseline endowment profile for these households is flat, with es = 1 for
s = 0, . . . , 240.
This baseline can also be scaled up and down by ξ.
These households will not borrow or lend, and instead will consume labor income in
each period.
The theorem below would go through with hand-to-mouth agents, but we do not
pursue this issue further in these slides.
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T HE MASS OF LIFE - CYCLE PRODUCTIVITY WITH HAND - TO - MOUTH AGENTS
F IGURE : The mass of endowment profiles for life-cycle households (blue shaded area) and
hand-to-mouth households (red shaded area) with the scaling factor drawn from a uniform
distribution U [0.05, 1.95].
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A DDITIONAL IDIOSYNCRATIC RISK
Households can earn income in a competitive economywide labor market by
supplying hours along with the productivity they have available at that date.
At the beginning of each period, each household may be randomly unemployed.
The household earns no income from work on dates of unemployment.
The unemployment probability is i.i.d. and uncorrelated with the aggregate shock.
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H OUSEHOLD CREDIT
The overlapping-generations structure creates a large private credit market essential
to good macroeconomic performance.
The key asset is therefore privately issued household debt.
As practical motivation, think of privately issued debt = “mortgage-backed
securities.”
U.S. household debt in the second quarter of 2019 was about $13.9 trillion, which was
equal to about two-thirds of GDP.
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NOMINAL CONTRACTING
There is a key friction in the credit market: non-state contingent nominal contracting.
There are two aspects to this assumption.
The non-state contingent aspect means that real resources are misallocated via this friction.
The nominal aspect means that the monetary authority may be able to fix the distortion to
the equilibrium through appropriate monetary policy.
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I NFORMATION ASSUMPTIONS
All households have private information concerning their scaling factor ξ and their
productivity profile e.
Households can choose a level of effort, or intensity of work, which is linear in the
life-cycle productivity profile.
Accordingly, households can potentially pretend to possess lower productivity profiles
than they actually have.
This does not occur in the equilibrium we study.
The unemployment draw is public information.
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A DDITIONAL ASSUMPTIONS
There are no borrowing constraints.
See Azariadis et al. (JEDC, 2019) for a version with some households excluded from credit
markets.
There is stochastic labor force (= population) growth in this version.
There is no default.
Prices are flexible.
Capital is fixed.
We ignore the effective lower bound in this version; see Azariadis et al. (JEDC, 2019).
All policies are set credibly for all time t ∈ (−∞, +∞) .
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Four Policymakers
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ENTITIES
There are four policymaking entities.
The monetary authority can observe the three aggregate shocks at the beginning of date t
and then set the price level P (t) .
The fiscal authority can set taxes on labor or capital income to raise an exogenously
specified fraction of available real output.
The labor market authority observes household-specific unemployment shocks, sets taxes
and provides household-specific transfers.
The education authority can control the initial dispersion of life-cycle productivity profiles
by controlling the standard deviation of the scaling factor ξ up to some limit σmin ≥ 0.
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T HE PROPOSED POLICY MIX
The proposed policy mix is as follows:
The monetary policymaker follows an NGDP targeting rule similar to Koenig (IJCB, 2013)
and Sheedy (BPEA, 2014).
The fiscal authority sets a linear tax on all labor income earned that is sufficient to meet its
revenue requirement.
The labor market authority sets a linear tax on all labor income earned that is sufficient to
provide appropriate transfers to unemployed households.
The education authority minimizes the dispersion of life-cycle productivity profiles by
setting σ = σmin .
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T HE W ICKSELLIAN NATURAL REAL RATE OF INTEREST
T HEOREM
Under the proposed policy mix, the real interest rate is exactly equal to the stochastic aggregate
output growth rate at every date, and an equal-treatment social planner that discounts at this rate
will conclude that this is a social optimum.
C OROLLARY (E QUITY SHARE CONTRACTING )
Any two households that share the same productivity profile consume the same amount at each
date, and consumption growth is equalized for all households.
C OROLLARY
Equilibrium labor supply over the life cycle depends on the shape of the productivity profile alone.
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Characterizing the Policies
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THE POLICIES
We first describe how the monetary policy works.
We then characterize the equilibrium in a series of schematic graphs representing the
cross-sectional distribution of households at each date.
In these illustrative graphs, we have set the mass of hand-to-mouth agents to zero.
In the graphs, we will show both the case where σmin = 0 and the case where
σmin > 0.
We can interpret this latter case as one where the education policymaker cannot drive the
variance of the scaling factor all the way to zero.
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M ONETARY POLICY
The monetary policymaker controls the price level P (t) directly and follows an
NGDP targeting rule given by
P (t) =
Rn (t − 1, t)
P (t − 1) ,
λ (t − 1, t) ν (t − 1, t) δ (t − 1, t)
(6)
where Rn (t − 1, t), the gross nominal interest rate, is equal to the expected gross rate
of NGDP growth between dates t − 1 and t.
The NGDP targeting rule works because it provides a form of insurance for all
households against current and future aggregate shocks.
This policy rule is not unique. See Bullard and Singh (JMCB, 2019 forthcoming).
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OF AN AGGREGATE SHOCK
1.06
1.02
1.04
1
1.02
0
5
0
10
quarters
5
10
quarters
1.4
1.06
1.2
1.04
1.02
1
0
5
quarters
10
0
5
10
quarters
F IGURE : Monetary policy responds to a decrease in the natural rate, i.e., a decrease in λ, ν or δ, by
increasing the inflation rate in the period of the shock. Subsequently, inflation converges to its
long-run equilibrium value from below. The nominal interest rate drops in the period after the
shock. The Phillips curve correlation is high in this model.
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TAX POLICY
Given the monetary policy, the labor market authority sets a tax τ u that is linear in
labor earnings.
The fiscal authority sets a tax τ f that is also linear in labor earnings.
The household i first-order condition for leisure can then be written as
`t,i (t + s) = (1 − η )
where ē =
1
T +1
∑Ts=0 es and ēi =
1
T +1
ēi
ē
= (1 − η ) , ∀i,
es,i
es
(7)
∑Ts=0 es,i .
The scaling factor ξ and the two taxes τ u and τ f are all linear in e and therefore cancel
out in this expression—so taxing in this manner is nondistortionary.
This result requires that all labor income is taxed at these rates for all time.
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L EISURE CHOICES AND M IRRLEESIAN CONSIDERATIONS
f
Because τ ui = τ u ∀i and τ i = τ f ∀i, households will not be incentivized to withhold
effort by misrepresenting their type.
Under these policy choices, all households choose the same leisure and hours-worked
profile over the life cycle, and aggregate output is as large as it would be in an
economy without taxation.
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WORKED OVER THE LIFE CYCLE
1
0.5
0
0
60
120
180
240
quarters
F IGURE : Cross section: Leisure decisions (green), labor supply decisions (blue) and fraction of work
time in U.S. data, 19% (red). The labor/leisure choice depends on age only. High-income
households plan to work the same hours as low-income households at each age. A certain
percentage of the continuum of households in each cohort is unemployed but insured.
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E DUCATION POLICY
“Education policy” influences the productivity profile dispersion parameter σ.
One could interpret this as an idealized insurance market that operates before
households enter the economy at age 20.
Limiting case: σmin = 0, all households receive the same profile.
This would be a “perfectly equal” economy in that the talent distribution would
collapse to just one life-cycle pattern.
This would drive the consumption Gini to zero.
However, the income and wealth Gini coefficients would remain close to observed
values—these are driven mostly by the life-cycle structure.
We will show both the idealized case σmin = 0 and cases where σmin > 0 in a later
slide.
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Characterizing the Equilibrium
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L ABOR INCOME
Households want to work more when they are in their peak earning years in the
middle of the life cycle.
This creates substantial labor income inequality.
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L ABOR INCOME MASS
F IGURE : Cross section: Labor income profiles with unemployment insurance. Personal productivity
peaks at the middle of the life cycle, and households work more at that time as well, making income
even more concentrated in the peak earning years. The blue line depicts the limiting case with
σmin = 0.
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C ONSUMPTION MASS
F IGURE : Cross section: Schematic consumption mass (red) and labor income mass (blue). Under
optimal monetary policy, the private credit market reallocates uneven labor income into perfectly
equal consumption along each productivity profile. The consumption Gini is 31.7%, similar to
values calculated from U.S. data. The solid lines depict the limiting case with σmin = 0.
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C ONSUMPTION EVOLUTION
F IGURE : Time series: Evolution of the distribution of log consumption (shaded area) and examples
of individual log consumption profiles (colored lines). Under optimal monetary policy, individual
consumption profiles share the same stochastic trend as aggregate consumption.
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ASSET HOLDING MASS
F IGURE : Cross section: Schematic net asset holding mass relative to GDP by cohort. Borrowing, the
negative values to the left, peaks at stage 60 of the life cycle (age ˜35), while positive assets peak at
stage 180 of life (age ˜65). The financial wealth Gini is 72.7%, similar to values calculated in U.S.
data. The blue line depicts the limiting case with σmin = 0.
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T HREE NOTIONS
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OF INCOME
Three notions of income:
Pretax labor income, Y1 .
Pretax labor income plus non-negative capital income, Y2 .
The non-negative component of total income, Y3 .
Gini coefficients of income distributions: GY1 = 56.1%, GY2 = 51.5%, GY3 = 59.5%.
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L ABOR INCOME + NON - NEGATIVE CAPITAL INCOME
F IGURE : Cross section: Profiles of labor income and non-negative capital income. The blue line
depicts the limiting case with σmin = 0.
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N ON - NEGATIVE TOTAL INCOME
F IGURE : Cross section: Profiles of non-negative total income. The blue line depicts the limiting case
with σmin = 0.
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Inequality
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D ATA ON INEQUALITY IN THE U.S.
Consumption (Heathcote, Perri and Violante, RED, 2010): GC,U.S. = 32%.
Income (CBO, 2016): pre-taxes/transfers GY,U.S. = 51%; post-taxes/transfers
GY,U.S. = 43%.
Financial wealth (Davies, Sandström, Shorrocks and Wolff, EJ, 2011): GW,U.S. = 80%.
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G INI COEFFICIENTS
Wealth
W
Y1
Income
Y2
Y3
51%
Consumption
C
U.S. data
80%
32%
Uniform
72.7%
56.1%
51.5%
59.5%
31.7%
Lognormal
72.4%
55.7%
51.1%
59.0%
32%
TABLE : Gini coefficients in the U.S. data and in the model with uniform and lognormal productivity.
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P RODUCTIVITY DISPERSION AND G INI COEFFICIENTS
1
0.653
0.5
0.443
0
0
0.5
1
1.5
2
2.5
3
3.5
F IGURE : As the dispersion of productivity profiles, σ, increases, the Gini coefficients increase. The
ordering GW > GY > GC is preserved. The case where σmin = 0 has GC = 0, but GW = 65.3% and
GY = 44.3%. The model can match any single Gini with a sufficiently large choice of σ.
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EXTENSIONS
The model could be extended to an open economy setting.
Two countries could have very different levels of income, wealth and consumption
inequality.
Two countries could be growing at substantially different rates, based in part on
growth in human capital.
International investors would seek higher returns in faster growing economies,
eventually leading to equalized real returns in the two economies.
Slower growing economies would have trade deficits.
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I NEQUALITY
C ONCLUSIONS
Conclusions
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I NTRODUCTION
E NVIRONMENT
F OUR POLICYMAKERS
P OLICY
E QUILIBRIUM
I NEQUALITY
C ONCLUSIONS
C ONCLUSIONS
A classic combination of policies can deliver a first-best allocation of resources in this
environment even with substantial inequality in income, wealth and consumption.
A monetary policymaker provides period-by-period insurance against aggregate shocks.
This enables nondistortionary linear labor income taxes to fund government expenditures
as well as an unemployment insurance program.
Education policy can drive the consumption Gini toward zero but would leave income
and wealth Ginis at positive levels.
These classic benchmarks may be useful in understanding the effects of
macroeconomic policy for models in this class going forward.
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@FedFRASER