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S OCIAL L EARNING FOR THE M ASSES
James Bullard
Federal Reserve Bank of St. Louis
Computational & Experimental Economics Workshop
Simon Fraser University
Feb. 4, 2023
Vancouver, British Columbia
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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T HE INTELLECTUAL LEGACY
OF J ASMINA
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A RIFOVIC
Jasmina Arifovic (JA) was a pioneer in the application of artificial intelligence to
macroeconomics, helping us to gain insight into the question, “How is equilibrium
achieved?”
JA ideas will be even more important in the decades ahead as macroeconomists work
with more and more granular models: more agents, more details, more shocks, more
frictions.
This paper: How is equilibrium achieved in these more complex environments?
An earlier and more preliminary version of this talk was given under the title
“Conjectures on Learning in Krusell-Smith-type Economies” at the 2021 Bank of Canada
Annual Economic Conference on Nov. 10, 2021.
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MORE COMPLEX ECONOMY
I study a stylized DSGE heterogeneous agent life cycle model with a known
competitive equilibrium featuring Gini coefficients close to those in the U.S. data.
The model features three aggregate shocks as well as idiosyncratic risk, but also
features policies that can mitigate both the aggregate risk and the idiosyncratic risk.
A welfare theorem states the sense in which these policies can achieve an optimal
allocation of resources.
The subtext in this talk is that models in this class represent, broadly, the current and
future direction of macroeconomics, and that the learning literature will have to continue
to refine methods to provide insight for these environments.
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L EARNING
I then turn to discuss how agents might learn in this relatively complex
macroeconomic setting if agent behavior is at some point disturbed.
I will conclude that social learning as promoted by Jasmina Arifovic is likely to
provide the best path forward.
Unmodified concepts of econometric learning promoted and studied extensively in the
existing literature are less likely to be appropriate in this environment.
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Core argument
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A N OLDER TRANSITION TIME RESULT
Suppose the economy is initially on a balanced growth path but is suddenly
disrupted by a “one-time, large, unanticipated shock.”
This shock is above and beyond the shocks envisioned within the ambient stochastic
environment of the model.
To fix ideas, think of an unanticipated “financial crisis” or an unanticipated “pandemic.”
In a related class of hetergeneous agent models, an earlier generation of quantitative
study emphasized perfect foresight transition times following a disturbance of this
type.
That literature found that transition times are long—measured in years or
decades—in this related class of models.
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W HY SLOW CONVERGENCE ?
The slow convergence was because the disrupted agents experiencing the shock
would have to complete their life cycle and exit the model before the long-run
balanced growth path can be achieved.
Taken literally, one might conclude that actual macroeconomies subject to occasional
“large, unanticipated shocks” would nearly always be in transition, even if
households had rational expectations following the large shock.
Examples: Auerbach and Kotlikoff (Dynamic Fiscal Policy, 1987); see also Cogley and
Sargent (JME, 2008) in which the large shock twists the priors of a Bayesian learner and
leads to slow learning over subsequent decades.
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R ELATIVELY
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FAST CONVERGENCE IN THE
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DATA
I will calibrate the DSGE model used in this paper to U.S. data assuming U.S.
macroeconomic policies are in fact the optimal ones the model requires.
I will then provide prima facie evidence that actual convergence times in U.S.
economic data following a “large, unanticipated shock” are an order of magnitude
shorter than in the earlier literature.
In particular, these transition times are measured in quarters rather than years or
decades.
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FOR THE MASSES
This suggests that in reality, the U.S. economy—despite its complexity—does not
seem to follow the types of slow adjustment paths emphasized in some of the earlier
literature.
I will suggest that the rapid convergence observed in the U.S. data could occur if
there is substantial communciation across the society—social learning as
implemented by JA.
In the economy I describe, this can occur because there are many millions of agents
that have already learned and retained the “DNA” of optimal decision rules for
consumption, assets and hours worked before the shock occurred.
Other agents that may not know these decision rules can learn relatively quickly from
those that do.
I call this phenomenon “social learning for the masses.”
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Environment
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E NVIRONMENT BASICS
At each date t, a new continuum of households enters the economy, makes economic
decisions over T + 1 = 241 dates, then exits the economy. (To fix ideas, think of ≈ 1m
agents per quarterly cohort.)
This corresponds to an agent entering the economy as a decision-maker at age 20 and
exiting as a decision-maker at age 80, inclusive of end points, and making economic
decisions at a quarterly frequency.
Results are perfectly general for the choice of T, with higher values corresponding to
decision-making at more frequent intervals.
This class of models has a “paper-and-pencil” equilibrium solution, and so it
provides a simple benchmark model for heterogeneous-agent macroeconomies with
aggregate shocks.
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R ISKS FACED
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BY HOUSEHOLDS
There are both aggregate risk and idiosyncratic risk.
Idiosyncratic risk is borne as a productivity-profile scaling shock as the agent enters
the economy, and also in the form of simple i.i.d. unemployment risk at each date.
Monetary and fiscal policymakers provide a form of insurance against the aggregate
risk, and a labor market authority provides unemployment insurance.
The idiosyncratic risk borne as the agent enters the economy via the
productivity-profile scaling shock is uninsurable.
A welfare theorem describes the sense in which the equilibrium studied here
represents a first-best allocation of resources.
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A SSETS
There are three nominally denominated assets: privately issued debt, publicly issued
debt and capital.
We think of these as representing U.S. data counterparts: (1) mortgage-backed
securities (MBS), (2) federally issued debt and (3) physical capital, respectively.
In the U.S. data, MBS net out, but federally issued debt and physical capital are in
positive net supply and we target a value of the assets-to-GDP ratio equal to
1.23 + 3.32 = 4.55.
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N OMINAL CONTRACTING
The credit market friction is non-state contingent nominal contracting (NSCNC): All
debt contracts are stated in nominal terms, with a stated nominal interest rate, and
repayment is not state-contingent.
The role of monetary policy is to adjust the price level each period in order to convert
these nominal, non-state contingent contracts into real, state-contingent contracts.
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H OUSEHOLD TYPES
Household types: “life cycle” (LC) and “hand-to-mouth” (HTM).
The life-cycle households are assigned a hump-shaped productivity profile at the
beginning of their life cycle. Accordingly, they need to use credit markets (hold
assets) to smooth life-cycle consumption.
The hand-to-mouth households are assigned a perfectly flat productivity profile as
they enter the economy. Accordingly, they never need to use credit markets and
instead consume their labor income each period.
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ATTAINING THE CORRECT ASSET LEVEL
The economy with only LC households wants to hold assets equal to A/4Y = 5.71, a
value which is considerably higher than the value observed in the U.S. data, which is
4.55.
The economy with only HTM households would be “Spartan,” and would hold no
assets at all.
We will adjust the fraction of HTM households in order to match the assets-to-GDP
ratio in the U.S. data.
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P REFERENCES
Each household i ∈ (0, 1) entering the economy at date t has preferences (the same for
both LC and HTM types)
T
Ut,i =
∑ [η ln c̃t,i (t + s) + (1 − η ) ln ℓt,i (t + s)] .
s=0
We define c̃t,i (t + s) = D (t + s) ct,i (t + s) , where D (t + s) is the state of aggregate
demand at date t + s. The state of demand evolves as
Dt = δ(t − 1, t)Dt−1 ,
where δ(t − 1, t) is the gross growth rate of demand, which follows an appropriate
stochastic process that keeps D(t) > 0 ∀t.
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P RODUCTIVITY PROFILES
Agents entering the economy draw a scaling factor x from a lognormal distribution
and receive a productivity profile that is a scaled version of a baseline profile, es :
where for LC agents eLC
s
es,i = x · es ,
s−p2 4
, and where p1 , p2 and p3 are
= 1 + p1 exp − p3
chosen to match calibration targets given below, and for HTM agents
eHTM
= h (1/T ) ∑Ts=0 eLC
s
s where h ∈ (0, 1) .
Huggett, Ventura and Yaron (AER, 2011) argue that differences in initial conditions are
more important than differences in shocks for lifetime earnings.
We think of all endowments at each date as containing linear labor tax factor (1 − τ u ) ,
with τ u set for all households in each cohort by the labor authority to fund unemployment
insurance. This type of tax will not distort labor supply in this model.
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T ECHNOLOGY
Aggregate real output Y (t) is given by
Y (t) = [D (t) Q (t) N (t)]1−α K (t)α [L (t)]1−α ,
(1)
where K (t) is the real value of the physical capital stock, L (t) is the aggregate
effective human capital supply (hours × productivity of various households), Q (t) is
a productivity index, N (t) indexes the size of the labor force, and D (t) is the state of
aggregate demand.
Q, N and D grow at stochastic gross rates λ, ν and δ respectively.
These assumptions mean that real output grows at the stochastic rate λνδ each period.
The aggregate demand assumption is a simple version of Bai, Rı́os-Rull and Storesletten
(unpublished, 2019).
The labor force growth assumption affects all cohorts proportionately and can be
interpreted as “immigration.”
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N OMINAL CONTRACTING AND
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TIMING PROTOCOL
Under the assumptions outlined, the contract nominal interest rate is given by
c̃t,i (t)
P (t)
.
Rn (t, t + 1)−1 = Et
c̃t,i (t + 1) P (t + 1)
(2)
The timing protocol is: (1) Nature assigns new entrant productivity profiles and also
draws aggregate shocks; (2) The fiscal authority issues nominal debt; (3) The
monetary authority sets the price level; (4) Households choose date t consumption,
hours worked and net asset holding.
Households will be able to make date t decisions without reference to future
uncertainty, as the monetary policymaker is providing a type of perfect insurance.
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T HE FISCAL AUTHORITY
The fully credible nominal debt issuance process is given by
B (t) = Rn (t − 1, t) B (t − 1) ,
(3)
where B (t) is the total level of nominal debt and B (0) > 0.
The fiscal authority is issuing enough new debt to maintain the level of assets in the
economy at the appropriate level.
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T HE MONETARY AUTHORITY
The monetary authority controls the price level directly and implements a price-level
path criterion
Rn (t − 1, t)
P (t) =
P (t − 1) .
(4)
δ (t − 1, t) λ (t − 1, t) ν (t − 1, t)
This criterion implements countercyclical price-level movements relative to the
expectation embodied in the contract rate Rn (t − 1, t) .
See Koenig (IJCB, 2013) and Sheedy (BPEA, 2014) on NGDP targeting.
See Andolfatto, et al. (unpublished, 2021, p. 14) for a discussion of how this criterion
relates to a similar New Keynesian “targeting criterion” developed by Giannoni and
Woodford ( 2004, pp. 101-2 ).
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C OMPETITIVE EQUILIBRIUM AND SOCIAL WELFARE
Solution: Guess and verify that there is a competitive equilibrium in which the real
rate of interest is always equal to the stochastic rate of real output growth.
The “Wicksellian natural real rate of interest” for this economy.
A social planner would conclude that the allocation of resources is a social optimum
provided (i) the planner places equal weight on all households for all time, (ii) the
planner discounts backward and forward in time at the stochastic real rate of interest,
(iii) the planner cannot alter the distribution of productivity profiles within the
cohort, which are decided by nature at the beginning of the life cycle, (iv) the planner
cannot alter the tax rate τ u .
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Calibration
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M APPING TO THE DATA
Adjust cohort size based on data from the U.S. Census Bureau.
Set the baseline hump-shaped life-cycle productivity profile such that households
endogenously choose to work the hours worked by age in the U.S. data.
Choose η to match average time devoted to market work across the economy.
Set the fraction of HTM households (who do not hold assets) such that the aggregate
level of assets to output, A/ (4Y) , matches the U.S. data (4.55), with net assets
defined as capital, K/ (4Y) = 3.32, plus government issued debt, B/ (4Y) = 1.23.
Choose the within-cohort standard deviations of productivity for life-cycle and
hand-to-mouth households, σlc and σhtm , respectively, to approach the
pre-taxes-and-transfers Gini coefficients for income and financial wealth in the U.S.
data.
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B ASELINE LIFE - CYCLE PRODUCTIVITY
1.5
1
0.5
0
60
120
180
240
quarters
F IGURE : Baseline endowment profile of life-cycle agents.
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OF LIFE - CYCLE PRODUCTIVITY
3
2
1
0
0
60
120
180
240
quarters
F IGURE : The mass of endowment profiles: life-cycle agents (blue) and hand-to-mouth agents for
h = 0.5 (red). The dashed lines denote the 25th and the 75th percentile of the endowment
distributions.
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WORKED BY AGE
0.3
0.2
0.1
0
Data
Model
0
60
120
180
240
quarters
F IGURE : Hours worked by age for life-cycle households: U.S. data (blue) and calibrated model (red).
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P OPULATION WEIGHTS
10-3
5
4
3
Data
Smoothed data
2
1
0
60
120
180
240
quarters
F IGURE : Population weights: U.S. data (blue) and 4th degree polynomial smoothed (red).
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G INI COEFFICIENTS
h
σlc
σhtm
A/ (4Y)
GW
GY
GC
Model
0.50
1.24
1.03
4.55
0.74
0.66
0.62†
U.S. data
−
−
−
4.55
0.78
0.63
0.32‡
TABLE : Parameter values and associated assets-to-output ratio and Gini coefficients in the model
equilibrium vs. the U.S. data.
† The consumption Gini in the model is based on a pre-taxes-and-transfers income concept.
‡ The consumption Gini in the data is based on a post-taxes-and-transfers income concept.
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T HE CONSUMPTION G INI
The consumption (out of pre-taxes-and-transfers income) Gini in the model
equilibrium is Gc = 0.62.
In the U.S. data, the consumption (out of post-taxes-and-transfers income) Gini is 0.32,
about half as large.
The model is saying that the net effect of taxes and transfers in the U.S. data is enough
to reduce consumption inequality by half.
Some evidence: Using German data, Haan, Kemptner, and Prowse (working paper, 2018)
use a life-cycle model to estimate that the tax-and-transfer system is sufficient to offset
54% of the inequality in lifetime earnings.
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M ARGINAL PROPENSITIES TO CONSUME SCHEMATIC
Life-cycle agents
Hand-to-mouth agents
8
6
4
2
1
0
0
60
120
180
240
quarters
F IGURE : Cross section: Schematic of the marginal propensity to consume out of labor income by
cohort for life-cycle agents. The MPC does not depend on the endowment scaling factor.
Hand-to-mouth agents have a MPC of one.
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OF EQUILIBRIUM FIT TO
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DATA
The model is calibrated to match hours worked by cohort for life-cycle households.
Heckman: Appropriately specified “wage regressions” will suggest hours changes
are independent of real wage changes.
The model can be calibrated to fit U.S. real output growth exactly, attributing the
growth in part to technological improvement, labor force growth uniform across
cohorts, and the state of aggregate demand.
The model predicts that consumption growth will be equalized across households at
different ages and different income levels: economic growth gets “shared out”
appropriately.
The income, wealth and consumption distributions are maintained by a smoothly
operating credit market with the correct interest rate.
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Learning
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N OMINAL RETURNS
The model equilibrium predicts equalized nominal and real returns for three assets
under optimal monetary policy: capital, MBS and Treasuries.
These assets are not further differentiated inside the model.
To compare with the data, we need an asset representing a return to capital in a
format with risk characteristics similar to MBS and Treasuries.
One candidate is a high-quality corporate bond.
I will use a seven-year nominal investment-grade corporate bond metric. In the
model and the data, this type of bond has a seven-year horizon but can be refinanced
each period.
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GROWTH
The model equilibrium predicts that the nominal return on the assets should be equal
to the nominal consumption growth rate, or, equivalently in the model, the nominal
output growth rate.
This prediction holds in periods of relative stability with optimal monetary, fiscal,
and labor market policy.
In these circumstances the private sector is able to set nominal debt contracts relying
on the monetary authority to set the price level that ratifies those debt contracts
ex-post.
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SHOCKS
I will argue that the U.S. economy has been disturbed by two large unanticipated
shocks since 2005: (1) the global financial crisis (GFC), and (2) the global pandemic.
For my purposes, these events are simply “large disturbances” outside the scope of
this model.
The interim period, 2011-2019, fits the model assumptions better and we may expect
the model to provide a better fit to the data during this time frame.
It does not take long for the equilibrium conditions to be met after the GFC.
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DATA
Percent
30
20
Nominal consumption growth
Lewis-Mertens-Stock index + core PCE inflation
7-year high-quality bond yield
The economy has returned to equilibrium
10
0
-10
Disturbances
-20
Jan-05
Jan-08
Jan-11
Jan-14
Jan-17
Jan-20
F IGURE : In ”normal times,” nominal consumption growth and nominal yields are close, as predicted
by the model.
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W HAT THE CHART SHOWS
Measures of U.S. nominal consumption growth and nominal GDP growth on a
12-month basis are approximately equal to the nominal return on a 7-year high
quality corporate bond between 2011 and 2019, as predicted by the model
equilibrium.
However, nominal growth rates and interest rates are considerably different during
large, unanticipated shocks like the GFC and the pandemic.
The chart suggests that the conditions of macroeconomic equilibrium were
re-established relatively quickly after the GFC, and also appear to be close to being
re-established following the pandemic.
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E CONOMETRIC LEARNING
The standard approach to learning—replace agents in the model with
econometricians as in Cogley and Sargent (JME, 2008)—might interpret the large
shocks as moments where rational expectations were badly disturbed across all
agents in the economy: young and old, rich and poor.
Forecasts that placed considerable weight on the chaotic observations from the crisis
could lead to important changes in economic behavior, which could then feed back
and continue to keep the economy away from its long-run equilibrium for some time.
This vision of learning seems to be at odds with the data in the figure.
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S OCIAL LEARNING
This model has decision rules for LC households i ∈ (0, 1):
= xlc ηēw (t) ,
ē
ℓt−s,i (t) = (1 − η ) ,
es
("
#
T )
s
at−s,i (t)
s+1
= xlc w (t)
∑ ej − T + 1 ∑ ej ,
P (t)
j=0
j=0
c̃t−s,i (t)
(5)
(6)
(7)
for s = 0, ..., T, where ē is the average baseline endowment for LC agents and xlc is the
scale factor for agent i within the cohort, and for HTM agents s = 0, ..., T :
c̃htm
t−s,i (t)
ℓhtm
t−s,i
ahtm
t−s,i
= xhtm ηhēw (t) ,
(t) = 1 − η,
(t) = 0.
(8)
(9)
(10)
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S EEDED WITH DNA
Households in the equilibrium of this model have very different incomes, levels of
consumption, and assets.
Nevertheless, they can learn from each other due to the fact that these optimal
decision rules are transferable across agents because they adjust for age and
productivity in the appropriate way.
Furthermore, most agents would have had to learn these decision rules before the
large, unanticipated shock occurred.
The economy is in effect seeded with a sort of “DNA”—known, previously learned
decision rules—even after the large shock occurs and begins to dissipate.
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S OCIAL LEARNING
After the GFC, for instance, there would be some cohorts whose only experience as
decision-makers in the economy was during the crisis.
However, there would be many more agents in the society, 95% or more, that would
have knowledge of optimal decision-making in normal times.
These known decision rules are relatively simple and can propagate exponentially
quickly through the population following the large shock, returning the economy to
equilibrium in short order.
This “social learning for the masses” is more likely to be the successful learning
concept in large heterogeneous agent economies.
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Conclusions
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J ASMINA A RIFOVIC ’ S
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CONTRIBUTIONS
JA was a pioneer in the application of methods from artificial intelligence to
macroeconomics to try to help answer the question, “How is equilibrium achieved?”
I have argued here that the combination of her insights and the likely future direction
of macroeconomic research suggests that this work will be even more important in
the decades ahead.
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