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C ONJECTURES ON L EARNING IN
K RUSELL -S MITH -T YPE E CONOMIES
James Bullard, Federal Reserve Bank of St. Louis
2021 Bank of Canada Annual Economic Conference
Behavioral Macroeconomics and Finance: Implications for Central Bankers
Nov. 10, 2021
Any opinions expressed here are our own and do not necessarily reflect those of the FOMC.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
Introduction
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
N ATURE OF MONETARY POLICY
Bill Poole, the previous St. Louis Fed president, paraphrased:
Beginning of FOMC term: “Macroeconomics is about steady states punctuated by shocks.”
End of FOMC term: “Macroeconomics is about shocks punctuated by steady states.”
My argument illustrates the latter idea.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
L ARGE - SCALE HETEROGENEOUS - AGENT DSGE
Increasing interest in large-scale heterogeneous-agent DSGE models.
Realistic degrees of inequality—approaching observed Gini coefficients.
Ideally, “Krusell-Smith” stochastic structure: aggregate risk plus idiosyncratic risk.
Can the equilibrium of such an economy be learned by agents not initially possessing
rational expectations?
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
T HESE REMARKS
Sketch a heterogeneous-agent economy that has a known equilibrium that can be
calculated by “pencil and paper” methods.
Describe some of the features of the equilibrium.
Outline a few considerations in a possible implementation of social learning.
Conjecture that the learning process is likely to be especially slow in this environment.
Draw out some possible policy implications.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
The Classical Benchmarks Economy
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
B ULLARD -D I C ECIO (2021)
A heterogeneous-agent life-cycle 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.
Four policymaking authorities: (1) monetary, (2) fiscal, (3) labor and (4) education.
Bullard and DiCecio (2021) describe a competitive equilibrium in which the four
policymakers act in concert to attain the first-best allocation of resources defined in a
social welfare theorem.
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I NTRODUCTION
A
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
NOTE ON LIFE - CYCLE MODELS
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.
There is heterogeneity by age but also within cohorts.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
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.
Households also draw a scaling factor ξ from a lognormal distribution as they enter the
model: 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.
Rationale: Huggett, Ventura and Yaron (AER, 2011).
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
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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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
H OUSEHOLD CREDIT
The overlapping-generations structure creates a large private credit market essential
to good macroeconomic performance.
There is a key friction in the credit market: non-state contingent nominal contracting.
Under the proposed policy mix, the real rate of interest is equal to the stochastic
growth rate of the economy.
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I NTRODUCTION
A
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
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 to fund a government 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.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
Characterizing the Equilibrium
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
T HE MASS 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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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
H OURS 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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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
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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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
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 32%, calibrated to the
U.S. data value. The solid lines depict the limiting case with σmin = 0.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
N ET 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 72.4%, similar to values calculated in U.S.
data. The blue line depicts the limiting case with σmin = 0.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
Y OUNGER AND OLDER AGENTS HAVE HIGH MPC
poor hand-to-mouth
8 consumers
wealthy hand-to-mouth
consumers
6
4
2
1
0
0
60
120
180
240
quarters
F IGURE : Cross section: Marginal propensity to consume out of labor income by cohort. Notice that
the MPC does not depend on the endowment scaling factor, ξ.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
Social Learning
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
L EISURE CHOICES
Given the proposed policies, the household i first-order condition for leisure can be
written as
ē
`t,i (t + s) = (1 − η ) i
(1)
es,i
∑Ts=0 es,i .
A typical social learning implementation would allow leisure choices to initially
deviate from these values for all households.
for all i where ēi =
1
T +1
This would mean aggregate labor supply and output would initially deviate from
equilibrium values.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
C ONSUMPTION CHOICES
Given the proposed policies, the household i first-order condition for consumption is
linear in the contemporaneous value of the real wage and can be written as
(2)
ct,i (t + s) = η (1 − τ u ) 1 − τ f ēi w (t + s)
∑Ts=0 es,i .
A typical social learning implementation would allow consumption choices to
initially deviate from these values for all households.
for all i, where ēi =
1
T +1
This would mean the net asset holding distribution and the real interest rate would
initially deviate from equilibrium values.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
F ITNESS
We can think of an individual household in the social learning environment as
choosing a pair (c, `) at each date.
The consumption choice could be constrained to be a simple linear function of the
wage at that date.
The leisure choice could be constrained to be a simple linear function of the ratio of
total productivity to current productivity at that date.
Fitness is conceptually difficult—households would have to conjecture that if they
used their existing linear rules over their remaining life, they would attain higher
utility than using an alternative rule.
Alternative rules could be evaluated by households via crossover and mutation
operators.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
C ONVERGENCE
The life-cycle nature of this economy means that if any cohort or member of a cohort
makes choices inconsistent with the long-run equilibrium at any date, then, in
principle, one has to wait for that cohort or individual to exit the economy in order to
attain convergence.
In addition, under social learning, the households have to learn the correct decision
rules (in this formulation), which itself takes (perhaps considerable) time.
In principle, this means more members of cohorts making more choices inconsistent
with the long-run equilibrium for longer.
Conjecture: Learning in a Krusell-Smith environment would be “slow,” where slow is
appropriately defined.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
Policy
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
P OLICY CHOICES
The classical benchmarks economy has clearly delineated policies for the four
policymakers.
The monetary policymaker would likely be the most affected by the slow learning
process I have described.
In particular, real interest rates (which are supposed to equal the real output growth
rate) could be disturbed for long periods of time as the social learning process plays
out.
Other important quantities, such as Gini coefficients for consumption, income and
financial wealth, would also be disturbed.
One might want to develop policies that would speed up the social learning process.
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
C ONCLUSIONS
Conclusions
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I NTRODUCTION
C LASSICAL B ENCHMARKS
C HARACTERIZING THE E QUILIBRIUM
S OCIAL L EARNING
P OLICY
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.
These classic benchmarks may be useful in understanding the effects of
macroeconomic policy for models in this class going forward.
These remarks have emphasized, however, that with social learning an economy of
this type is more likely on a transition path at any point in time than actually at the
rational expectations equilibrium.
Whether the classic combination of policies could be altered to account for this is an
open question.
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