1. society
2. work
3. unemployment

Endogenous Growth, Household Leverage, and
Learning
Emily C. Marshall∗
Bates College and Dickinson College
Hoang Nguyen†
Bates College
Paul Shea‡
Bates College
March 31, 2015
Abstract
We add heterogeneous agents and collateral constraints to an endogenous growth model.
We find two major results. First, changing households’ maximum leverage ratio has potentially large effects on the steady state growth rate. Second, if agents form expectations
through adaptive learning, then faster learning increases the average growth rate. The direction of the former effect depends mainly on the substitutability between labor for the
production sector and labor for research and development, and the magnitude of the effect
is larger when labor supply is relatively elastic. When labor supply is highly elastic and
households do not try to smooth their labor supply between the two types of labor, the
growth rate decreases from 11.6% to approximately zero as the debt to capital ratio rises
from 0 to 1.38. However, if households instead have a strong preference for smoothing
their labor supply between production and R&D, then growth increases from 2.91% to
3.83% as the debt-to-capital ratio rises from 0 to 1.55. Relaxing the assumption of rational
expectations in favor of adaptive learning, we find that as agents learn faster, growth tends
to be higher than when agents learn at a slower rate.
JEL Classification: E21, E22, E44, O40
Keywords: Credit Constraints, Learning, Financial Markets, Endogenous Growth
∗
ecmars2@gmail.com
hnguyen6@bates.edu
‡
pshea@bates.edu
†
1
1
Introduction
Over the past decade, a considerable literature has examined the impact of assuming that
households can only borrow up to a fraction of their capital stock. The analysis of these credit
constraints has mostly focused on how their existence allows demand shocks to be amplified
and propagated through financial accelerator effects.1 In this paper, we add credit constraints
with heterogeneous households into a research and development based endogenous growth
model similar to Jones (1995).2 Households must choose how much labor to supply to the
productive sector and how much to supply to research and development (R&D), the latter of
which determines the economy’s steady state growth rate.
We identify two important drivers of economic growth in this setup. First, the maximum
household leverage ratio may have enormous effects on the balanced growth path. In one example, we assume both that households have little desire to smooth their labor supply between
R&D and productive labor, and that aggregate labor supply is highly elastic. Here, patient
households (who always act as creditors) are wealthier than impatient households (who act as
creditors). When borrowing is not allowed, the balance growth rate is 11.6% per year. As
impatient households are allowed to borrow so that their debt to capital ratio equals 1, growth
falls to 3.03%. By the time the leverage ratio reaches 1.4, growth is almost zero. This result
occurs because more debt increases the wealth of creditors. They respond by substituting toward leisure and away from labor supply, including R&D. When we allow labor supply to be
less elastic, the effect is smaller, but still large for even very inelastic labor supply.
We also find cases where more debt increases growth. When we assume that households
prefer to smooth their labor between production and R&D, and that labor supply is relatively
elastic, growth is now driven more by the R&D choices of borrowers. As they become more
1
2
See Iacoviello (2005) and Kiyotaki and Moore (1997).
For a more complete discussion of models with growth driven by endogenous technological change, see Romer
(1990), Grossman and Helpman (1991a,b,c), and Aghion and Howitt (1992).
2
indebted, they supply more of all types of labor, including R&D. The steady state growth rate
rises from 2.91% without debt, to 3.00% when the leverage ratio equals 1, and 3.83% when the
leverage ratio equals 1.55. The magnitude of these effects decreases as labor supply becomes
less elastic.
When we simulate the dynamics of the model around its steady state, we find a second
determinate of the long run growth rate. We assume that agents form expectations using adaptive learning where they use least squares to forecast the model’s endogenous variables. We
find that as agents learn faster (place more weight on recent observations), the growth rate
of the economy is significantly higher. This occurs because faster learning adds volatility to
the model. Because we simulate the non-linear model, the volatility added by learning has
asymmetric effects. For the case where agents like to smooth their labor supply (where growth
increases along with leverage), deviations from the steady state to consumption and R&D are
larger in magnitude when they are positive than when they are negative. This causes average
growth to be higher as volatility is added.
In much of the credit constraints literature, the leverage ratio is treated as a constant determined by the exogenous ability of lenders to recover collateral from borrowers in the case of
default. We argue that it is important to examine how a variable leverage ratio affects growth
because there is ample evidence that the leverage ratio is neither constant nor immune from
policy. Figure 1 plots the ratio of household debt to physical capital for the United States
between 1980 and 2011:
3
Figure 1: U.S. Household Debt to Capital
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
1980
1984
1988
1992
1996
2000
2004
2008
Source: St. Louis Fed.
Because these data are aggregate, they do not compare directly to the leverage ratio in our
model which applies only to specific types of households. They do, however, illustrate that
access to credit has both been trending upward and has been volatile around its trend. Figure
2 shows the leverage ratio (total debt-to-assets) for households in the lowest two percentiles of
net worth from 1989 to 2010:3
3
Source: Federal Reserve Board of Governors Survey of Consumer Finances Table 12
4
Figure 2: Leverage Ratio by Percentile of Net Worth
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
1989
1992
1995
1998
2001
Less than 25th Percentile
2004
2007
2010
25-49.9th Percentile
This chart again provides evidence that the leverage ratio is variable. In addition, Figure 2
shows that for the poorest households, the leverage ratio often exceeds 1.
There are also ample anecdotal accounts of policy makers attempting to influence leverage
ratios. The existence of the mortgage giants Fannie Mae and Freddie, government sponsored
enterprises in the United States, may fairly be viewed as a policy attempt to facilitate greater
access to mortgage debt. Provisions in the Dodd-Frank Wall Street Reform and Consumer
Protection Act of 2010 attempt to induce lower leverage among American firms, and the Basel
Accords may be viewed as an international effort to minimize leverage ratios in the banking
sector. We thus consider it important to understand how changing leverage ratios affect growth.
This is the first paper to combine an endogenous growth model with a R&D sector with
a model that includes heterogeneous households and credit constraints. There is, however, a
small literature that incorporates collateral constraints, using the Kiyotaki and Moore (1997)
framework in growth models. Several papers use liquidity constraints in an endogenous growth
5
model where growth is determined by capital accumulation instead of technological advancement through R&D. Jappelli and Pagano (1994) find that credit constraints increase the savings
rate of households which leads to a higher growth rate. Amable et al. (2004) use a model in
which net worth determines borrowing capacity. They examine the impact of a change in the
interest rate on growth and conclude that when the interest rate increases, growth decreases
due to the negative impact on retained earnings and the negative leverage effect. Finally, Bencivenga and Smith (1993) consider a model in which all investment activities are externally
financed and find that an increase in capital production technology magnifies the negative effects of credit rationing and reduces growth. These papers, however, either do not examine the
effects of changes in leverage on growth, or do not find that the leverage ratio has a significant
impact on growth rates. In contrast, we find that the effects of altering the leverage ratio are
often dramatic.
Our paper also contributes to the small literature on learning and growth. Evans et al. (1998)
develop an endogenous growth model with two stable steady state growth rates. They find
that learning allows the economy to endogenously switch between the neighborhoods of these
steady states. Arifovic et al. (1997) finds that allowing agents to form expectations through
learning provides a better explanation of the transition dynamics between a low-income steady
state and high-income steady state compared to perfect foresight.
Our paper also contributes to the mostly empirical literature on growth and financial development. There is considerable debate over whether growth causes financial development
or vice-versa. Levine (2005) provides an overview and argues that the bulk of the evidence
suggests that greater financial development does promote growth. Although it is not obvious
how to equate leverage to general financial development, our paper presents a novel channel
where changes in the financial sector affect growth.
The paper is organized as follows. Section 2 outlines the theoretical model. Section 3
shows the steady state results and the impact of varying access to credit. Section 4 describes
6
the adaptive learning algorithm and the effects of learning on economic growth. Section 5
concludes.
2
The Model
The contribution of this paper is to examine the effects of changes in household access
to credit on economic growth by combining the heterogeneous agent modeling devices of the
credit constraint literature and the setup of endogenous growth models. There are two types of
agents, impatient and patient households, defined by their relative discount factors. The agents
are infinitely lived and on a continuum on the unit interval. Both types of households produce,
consume, and supply labor to the research and development sector (R&D) and the production
sector. Technology used by the lenders and borrowers is the same and evolves according to a
recursive structure:
0
λ
At+1 − At = µ(Lλa,t + La,t
)
(2.1)
0
where At represents technology in time t, and La,t and La,t are labor supplied to the technology
sector by lenders and borrowers respectively. We select a value of µ that yields a growth rate
of approximately 3% for m = 1. Consider a case where La,t changes by one unit such that
dLa,t = 1. Solving for dLa,t+1 yields:
dLa,t+1 = −
La,t
La,t+1
λ−1 1 + gt+2
1 + gt+1
(2.2)
where
gt+1 =
At+1 − At
0λ
= µ(Lλa,t + La,t
)
At
7
(2.3)
In order for At+2 to remain unchanged, if labor is increased by one unit in time t, labor in time
t + 1 changes by dLa,t+1 .
Patient households are savers and lend to the impatient households. They maximize expected lifetime utility, which is a function of consumption (ct ), hours worked for R&D (La,t ),
and hours worked for production (Ly,t ), subject to a flow of funds constraint.
Max
ct ,La,t ,Ly,t ,kt+1
E0
∞
X
η
χ(La,t + Ly,t ) lnct −
η
βt
t=0
!
where the expectation operator is E0 , the discount factor is β (calibrated at a value of 0.99),
χ is the weight placed on the disutility associated with supplying labor, is a parameter that
dictates the substitutability between labor supplied to R&D and production, and η is the Frisch
inverse elasticity of labor supply. The real budget constraint is given by:
ct + kt+1 + bt ≤ Rt−1 bt−1 + Yt + (1 − δ)kt
where ct is patient household consumption, kt is patient household capital stock, δ is the depreciation rate, and Yt is patient household output. The borrower and lender relationship between
the two households is defined by their relative discount factors. The discount factor of the impatient households is less than that of patient households. As a result, the impatient households
are the borrowers and patient households are the lenders in the model. The loan structure is
such that in time t, a loan of amount bt is made from the patient households to the impatient
households. In the subsequent period, the impatient households pay off the debt at an interest
rate of rt , where Rt = 1 + rt . Therefore, in time t, the value of the loan plus interest from time
t − 1 is an additional source of income for the patient households.
In addition, the patient households produce according to the following constant returns to
scale (in labor and capital, given technology) production function:
8
(1−α)
Yt = (At Ly,t )α kt
The first order conditions for the patient households are:
β
1 + gt+1
−α
=
(1 − δ + (1 − α)Lαy,t+1 k˜t+1
)
c˜t
c˜t+1
(2.4)
1 + gt+1
βRt
=
c˜t
c˜t+1
(2.5)
˜
αLα−1
y,t kt
c˜t
η
χ(La,t +Ly,t ) −1 L−1
a,t
1−α
η
= χ(La,t + Ly,t ) −1 L−1
y,t
(2.6)
λ−1 α
˜1−α
βλµαLλ−1
η
La,t
1 + gt+2
a,t Ly,t+1 kt+1
−1
−1
=
+βχLa,t+1 (La,t+1 +Ly,t+1 ) c˜t+1 (1 + gt+1 )
La,t+1
1 + gt+1
(2.7)
where equation 2.4 is patient household demand for capital, equation 2.5 is the Euler equation,
equation 2.6 is patient household labor supply to the production sector, and equation 2.7 is
patient household labor supply to the research and development industry. Consumption and
capital have been detrended such that c˜t =
ct
,
At
k˜t =
kt
,
At
and b˜t =
bt
.
At
Impatient households, denoted by the prime symbol, work, consume, produce, and borrow
from the patient households. They maximize expected lifetime utility:
0
0
Max
0
E0
0
ct ,La,t ,Ly,t ,kt+1
∞
X
0
β
η
0
χ(La,t
+ Ly,t
)
lnct −
η
0t
0
t=0
subject to a budget constraint:
0
0
0
0
ct + kt+1 + Rt−1 bt−1 = bt + Yt + (1 − δ)kt
9
!
and a credit constraint:
0
Rt bt ≤ mt kt+1
where mt evolves according to an AR(1) process such that
m
em,t
mt = mρt−1
(2.8)
and em,t is a random shock. The detrended credit constraint is given by:
˜0
˜bt ≤ (1 + gt )mt kt+1
Rt
(2.9)
The collateral constraint on the impatient households requires that the value of their debt plus
interest cannot exceed the amount of recoverable future capital. In this setup, the threat of default matters but borrowers are not allowed to actually be insolvent.4 We interpret the variable
mt as representing access to credit. Consider the credit constraint evaluated at the steady state.
Provided that the impatient household discount rate is less than the patient household discount
0
rate (β < β), at the steady state, the collateral constraint will be binding.
˜
˜b = (1 + g)mk
R
At the steady state, R =
(1+g)
,
β˜
0
and β is calibrated to be 0.99. Using an approximation for
β = 1, plugging in the value for R, and rearranging yields:
m=
˜b
k˜0
(2.10)
which is the debt-to-capital ratio. The variable m can be interpreted as a leverage ratio. It is
4
Marshall and Shea (2014) relax this assumption allowing agents to explicitly default and show that this causes a
discrete drop in housing prices and output.
10
analogous to capital requirements on firms or a loan-to-value (LTV) ratio for households. The
data show that capital requirements and LTV ratios are highly variable. In addition, access to
credit can be influenced by policy as outlined in Section 1. Impatient households also produce
using their own labor and capital:
0 (1−α)
0
0
Yt = (At Ly,t )α kt
The first order conditions for the impatient households are:
0
0
0
−α
α
β (1 − δ + (1 − α)Ly,t+1
k˜t+1
)
(1 + gt+1 )
=
+ m˜
γt (1 + gt+1 )
0
0
c˜t
c˜t+1
(2.11)
0
β Rt
1
γ˜t Rt + 0
= 0
c˜t+1 (1 + gt+1 )
c˜t
0 (α−1)
αLy,t
c˜t
(2.12)
0
(1−α)
k˜t
0
0
0
0 (−1)
η
−1
= χ(La,t
+ Ly,t
) Ly,t
0
0 (λ−1)
0
(2.13)
0 (1−α)
α
k˜t+1
β λµαLa,t Ly,t+1
χ(La,t + Ly,t ) La,t =
0
c˜t+1 (1 + gt+1 )
!λ−1 0
L
0 (−1)
η
1 + gt+2
0
0
0
a,t
−1
+β χLa,t+1 (La,t+1 + Ly,t+1 )
0
1 + gt+1
La,t+1
0
0
η
−1
0 (−1)
(2.14)
where equation 2.11 is impatient household demand for capital, equation 2.12 is the Euler
equation, equation 2.13 is impatient household labor supply to the production sector, and equation 2.14 is impatient household labor supply to the R&D industry. The Lagrange multiplier
on the credit constraint is detrended by γ˜t = γt At .
Other equations in the system include:
11
Y˜t = Lαt k˜t1−α
(2.15)
0 (1−α)
0
0
Y˜t = Ltα k˜t
(2.16)
Rt−1˜bt−1
+ Y˜t + (1 − δ)k˜t
c˜t + k˜t+1 (1 + gt+1 ) + ˜bt ≤
(1 + gt )
(2.17)
0
0
Rt−1˜bt−1 ˜
0
0
c˜t + k˜t+1 (1 + gt+1 ) +
≤ bt + Y˜t + (1 − δ)k˜t
(1 + gt )
(2.18)
where 2.15 and 2.16 are the detrended production function for the patient and impatient households respectively and 2.17 and 2.18 are the detrended budget constraints for the patient and
impatient households respectively. There are two other equations that enter the model to introduce a technology shock:
A
At = Aρt−1
eA,t
(2.19)
ρu
Ut = Ut−1
eu,t
(2.20)
and a demand shock:
where eA,t and eu,t are mean zero and ρA and ρu are the AR(1) coefficients.
3
The Balanced Growth Path and Leverage
This section shows that the effects of varying m on g may be in either direction, depending
on our calibration, and that they are potentially large. Two parameters are critical. When 12
is close to one, suggesting that households have little desire to smooth their labor supply, the
higher leverage reduces growth. High values of , suggesting a desire to smooth labor supply,
allow for more leverage to increase growth. The magnitude of the effects are large when η is
low (suggesting elastic labor supply) and smaller when it is high. Because is novel to our
paper and the correct calibration of η is controversial, we consider alternate calibrations.
We also consider the optimal leverage ratio, calibrated using a social welfare function evaluated at the steady state. These results are sensitive to both our calibration and our choice of
social welfare functions, specifically whether we use a Utilitarian function that generally tracks
the utility of patient households or a Rawlsian function that generally tracks the utility of impatient households. We find cases where optimality occurs at m = 0 suggesting that allowing
any debt is welfare reducing, and cases where the optimal leverage ratio is greater than one,
suggesting that households should be allowed to go underwater on their debt.
As far as possible, we follow the related literature in calibrating our model. We set α = 2/3,
a standard value for the Cobb-Douglas production function. We set β = 0.99, implying a real
0
interest rate of 4% for quarterly data. We set β = 0.95, and, as in Jones (1995), we set λ = 1,
and we later consider the effects of lower values.5 We set δ = 0.025%, another standard value.
For each simulation, we fix µ so that the steady state growth rate for m = 1 is near 3%. We
then examine the effects of changing m. Table 1 summarizes the baseline calibrations.
5
As γ → β debt approaches zero and the effects changing m also go to zero.
13
α
β
0
β
χ
λ
δ
0
β
σa
σe
σm
ρa
ρm
3.1
Table 1: Calibration
labor’s share in production function
patient households’ discount factor
impatient households’ discount factor
weight on labor supply in utility function
returns to scale on R&D
capital depreciation rate
impatient households’ discount factor
st. dev. of innovations to productivity
st. dev. of innovations to demand
st. dev. of shocks to LTV
AR(1) coefficient for productivity shocks
AR(1) coefficient for LTV shocks
0.67
0.99
0.95
1
1
0.025
0.95
0.001
0.001
0.001
0.95
0.95
Simulation 1: = 1.1, η = 1.1, µ = 0.01222
In this simulation, is low suggesting that households have little taste for smoothing their
labor supply across types. In addition, labor supply is highly elastic (η = 1.1). The key
implication of having a low is that impatient households, having a lower discount factor,
supply very little R&D, instead allowing patient households to almost entirely drive growth.
For all of our simulations, we choose µ to yield about a 3% growth rate at m = 1. In this
simulation, growth equals 11.60% when m = 0. Here there is no interaction between the two
types of households, except that each of their R&D benefits the other. As shown in Figure 3,
the growth rate continues to fall as m increases, reaching 3.03% at m = 1, and basically zero
at m = 1.38
14
Figure 3: Effects of Varying m on Growth and Utility
0.14
80
70
0.12
60
50
0.10
40
0.08
30
20
0.06
10
0.04
0
-10
0.02
-20
0.00
-30
0.0
0.1
0.2
0.3
0.4
Growth
0.5
0.6
0.7
m
Patient Utility
0.8
0.9
1.0
1.1
1.2
1.3
Impatient Utility
The effect of a change in access to credit on growth in this simulation is dramatic. Growth
becomes virtually nonexistent when the impatient households are allowed to be highly leveraged. At the steady state, higher leverage transfers wealth from impatient households to patient
households. Impatient households respond by increasing their productive labor. But because
they supply almost no R&D, this has almost no impact on growth. Figure 4 summarizes the
impact of a change in access to credit on impatient households.
15
Figure 4: Effects of Varying m on Impatient Households
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
m
Prod. Labor
0.8
0.9
1.0
1.1
1.2
1.3
Consumption
As m increases, patient households become wealthier. They respond by substituting away
from both types of labor towards leisure. Because they drive growth, the reduction in their R&D
causes the aggregate growth rate to collapse. For very high leverage ratios, patient households
are content to live almost entirely on debt payments while supplying little labor and causing
very low growth. Figure 5 shows the effects of a change in m on the patient households. The
values for all variable are normalized so that they equal 100 when m = 0.
16
Figure 5: Effects of Varying m on Patient Households
180
160
140
120
100
80
60
40
20
0
0.0
0.1
0.2
0.3
0.4
Consumption
0.5
0.6
0.7
m
Capital
0.8
0.9
Prod. Labor
1.0
1.1
1.2
1.3
R&D Labor
We now evaluate the welfare implications of different values of m using the steady state
levels of utility for each type of household. Iterating the utility functions forward at the steady
state and taking infinite geometric series yields:
"
#
η
(La + Ly ) ln(g)
1
ln(c) − χ
+
U=
1−β
η
1−β
(3.1)
"
#
η
0
0
((La ) + (Ly ) ) 1
ln(g)
0
U =
ln(c ) − χ
+
1−β
η
1−β
(3.2)
0
These utility levels are included in Figure 3. Steady state utility is maximized for impatient households at m = 0 because here their wealth and growth are highest, and for patient
households at m = 1.38 (the largest value of m in this calibration). A utilitarian social welfare
function based on steady state utilities is maximized at m = 1.38, while a Rawlsian is welfare
function is maximized at m = 0 because it perfectly tracks impatient utility.
17
This simulation illustrates how higher leverage can cause a dramatic decline in growth.
Increasing access to credit causes both lower growth and much higher levels of income inequality.
3.2
Simulation 2: = 10, η = 1.1, µ = 0.0042
In this case, we continue to assume that labor supply is highly elastic. But now, we set
= 10 so that households try to smooth their labor supply between the two types of labor. We
also lower µ in order to keep growth rates close to 3%. Figure 6 shows the effects of varying
m on the growth rate and utility.
Figure 6: Effects of Varying m on Growth and Utility
0.040
0.038
0.036
0.034
0.032
0.030
0.028
0.0
0.1
0.2
0.3
0.4
0.5
Growth
0.6
0.7
0.8
m
Patient Utility
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Impatient Utility
When m = 0, the annualized growth rate is 2.91%. As m rises to one, the growth rate exhibits
a small but important increase to 3.00%. As m increases to 1.55, however, the increase in the
growth rate accelerates, rising to 3.83%.
By assuming that households wish to smooth their consumption across types, we now in18
duce impatient households to supply significant levels of R&D. As m increases and wealth is
transferred from impatient households to patient households, the former now respond by increasing their supply of both types of labor while the latter respond by decreasing both of their
labor supplies. In this simulation, the effect on impatient households is dominant and growth
increases along with m.
Figure 7 shows how impatient households change their steady state levels of labor and
consumption for different values of m while Figure 8 shows the effects on patient households.6
Figure 7: Effects of Varying m on Impatient Households
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.0
0.1
0.2
0.3
0.4
0.5
R&D Labor
6
0.6
0.7
0.8
m
Prod. Labor
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Consumption
For Figure 8 the values for all variables are normalized so that they equal 100 when m = 0.
19
Figure 8: Effects of Varying m on Patient Households
120
100
80
60
40
20
0
0.0
0.1
0.2
0.3
0.4
0.5
Consumption
0.6
0.7
0.8
m
Capital
0.9
1.0
Prod. Labor
1.1
1.2
1.3
1.4
1.5
R&D Labor
A striking result is how the effect on the growth rate accelerates when households are
allowed to be significantly underwater on their debt. To understand this result, consider the
model when m is below one. When impatient households are allowed to borrow more, they do
so (at the steady state), and their steady state consumption falls accordingly. They respond to
this both by supplying more productive labor and acquiring more capital. With a LTV ratio of
less than one, any additional capital increases debt less than one-to-one, and increased capital
thus has a positive effect on the households’ wealth. This secondary effect dampens the overall
reduction in wealth and, as a result, all variables exhibit relatively small changes, including the
growth rate.
Now suppose that m is above 1. Once again increased access to credit reduces impatient
wealth and increases impatient capital. Now however, an extra unit of capital results in more
than a one-to-one increase in debt. If m is far enough above 1, the effect of extra debt payments will overcome that of a positive marginal product of capital so that more capital actually
20
reduces impatient households’ wealth. The initial effect on wealth is now amplified instead
of dampened so that all variables, including the growth rate, exhibit much larger changes in
response to varying m.
Impatient household individual welfare and a Rawlsian social welfare function are maximized when m = 0.88. Because growth is now increasing in m, impatient households no
longer are better off without any debt even though no debt eliminates the negative wealth effect.
Patient household welfare and a utilitarian social welfare function are highest when m = 1.55
(the largest value in this calibration).
3.3
Simulation 3: = 1.1, η = 3, µ = 0.0148
Our first two simulations assume that labor supply is highly elastic. The correct calibration
of η remains controversial.7 We now consider a variation of 3.1 where we continue to assume
that = 1.1 but where labor supply is now inelastic where η = 3. Recall, a low implies
that households do not have a strong preference for smoothing their labor supply across types.
Figure 9 reports the results:
7
See Chetty et al. (2011) for further discussion of the controversy over the correct calibration of the Frisch elasticity of labor supply.
21
Figure 9: Effects of Varying m on Growth and Utility
0.04
140
0.035
120
100
0.03
80
0.025
60
0.02
40
0.015
20
0.01
0
0.005
-20
0
-40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
m
Growth
Patient Utility
Impatient Utility
The growth rate is 3.45% at m = 0 and is 3.00% at m = 1. Growth decreases to around
0.75% as access to credit increases up to the point where m = 1.50.
The mechanisms for this simulation are qualitatively the same as for 3.1. Impatient households again supply almost no R&D. As patient households become wealthier, they substitute
away from R&D (and productive labor) towards leisure, and growth declines. But because
labor supply is less elastic, the changes in R&D and growth are less dramatic, but still large, as
compared to 3.1.
Figure 10 shows the effects on impatient households. As in 3.1, impatient household labor
supply to the R&D sector is basically zero.
22
Figure 10: Effects of Varying m on Impatient Households
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
m
Prod. Labor
Consumption
As m increases, impatient households once again acquire more capital and supply more
R&D labor. However, the change in the growth rate is driven by patient households as shown
in Figure 11:8
8
Again, the values of the variables are normalized to equal 100 when m = 0.
23
Figure 11: Effects of Varying m on Patient Households
240
220
200
180
160
140
120
100
80
60
40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
m
Consumption
Capital
Prod. Labor
R&D Labor
Patient households again supply less R&D and productive labor, but their consumption
increases due to the increase in wealth from debt payments. Now, the decline in patient households’ R&D causes the decline in growth seen in Figure 9.
Patient utility is increasing throughout this range of m. Impatient utility peaks at m =
0.86. A Rawlsian social welfare function again tracks impatient utility in this simulation. A
utilitarian social welfare function more closely tracks patient household utility. Once again,
the optimal value of m depends on the social welfare function specification, but lies between
0.86 and 1.50.
3.4
Simulation 4: = 10, η = 3, µ = 0.0044
In the last simulation, labor supply is less elastic and households prefer to smooth their
labor between production and R&D, as in 3.2. The results here are similar to that of 3.2 except
that the effects are dampened due to the reduction in labor supply elasticity. Figure 12 shows
24
the impact of a change in m on growth and utility:
Figure 12: Effects of Varying m on Growth and Utility
0.06
300
250
0.05
200
0.04
150
100
0.03
50
0.02
0
-50
0.01
-100
0.00
-150
0.0
0.1
0.2
0.3
0.4
0.5
Growth
0.6
0.7
0.8 0.9
m
Patient Utility
1.0
1.1
1.2
1.3
Impatient Utility
Growth increases from 2.94% when m = 0 to 4.77% when m = 1.69. The growth is driven
by an increase in impatient household labor supply to the production sector, and as a result
of the preference for smoothing labor across types, R&D labor supply also increases. Patient
household utility is increasing as access to credit increases. Impatient household utility begins
to decline significantly once the LTV ratio exceeds one.
Impatient household utility and Rawlsian social welfare are maximized when m = 0.87.
The LTV ratio is much higher at m = 1.69 (the largest possible value) when maximizing
patient household utility. Utilitarian welfare is maximized when m = 1.61.
25
4
Learning
We now consider the dynamics of our non-linear model around its steady state. We assume
that agents form expectations using constant gain adaptive learning.9 Agents run a set of OLS
regressions and use the resulting coefficients to form expectations, which they then use to
choose consumption and labor supply from the model in Section 2. The gain in the learning
algorithm represents the weight that agents place on the most recent observation. A higher gain
is similar to using a shorter sample period. We show that, for the example of the steady state
from Section 3.3, a higher gain results in higher economic growth.
0
0
In period t, we assume that agents’ information set is zt = [k˜t , k˜t , ˜bt , At , mt , Ut ] . Agents
thus observe the current values of the model’s three shocks and the predetermined capital
stocks, but only the lagged value of debt, which is endogenous. We assume that agents then
use the following specification to form their expectations:
yt = ψ1 + ψ2 zt + ut
(4.1)
0
0
0
0
where yt = [˜
ct , c˜t , La,t , La,t , Ly,t , Ly,t , Rt , ˜bt ] , the set of variables to be forecasted. The vector
ψ1 and the matrix ψ2 are regression coefficients. For given values of these, the model works in
the following way. First, agents use 4.1 to form Et [yt ] their contemporaneous expectations of
the model’s endogenous variables. Second, they then use Et [yt ] and 2.18, 2.17, and 2.9, and
their knowledge of the model’s stochastic shocks to form Et [zt+1 ]. Third, they use Et [zt+1 ]
and 4.1 to obtain Et [yt+1 ], a vector of future expected endogenous variables. Finally, they use
Et [yt+1 ] in the model of Section 2 to choose their consumption and labor supplies, which then
determine the growth rate in the next period.
After each period, we assume that agents update their learning algorithm using recursive
least squares:
9
For a thorough discussion of adaptive learning, see Evans and Honkapohja (2001).
26
i
yt+1
= yti + γRt−1 zt (yti − ψˆi zt )
0
Rt−1 = (1 − γ)Rt + γt zt zt
(4.2)
(4.3)
0
where yti is the ith element of yt and ψˆ = [ψ1 , ψ2 ] . The gain, γ, is the weight placed on the
most recent observation and may be interpreted as the speed at which agents learn. If it is
set equal to t−1 , known as decreasing gain learning, then (4.2)-(4.3) are equivalent to OLS.
We, however, assume that the gain is constant which implies that our learning algorithm is
equivalent to weighted least squares. Whereas decreasing gain learning might converge to
fixed learning coefficients, constant gain learning instead might converge to an equilibrium
where the learning coefficients are a stochastic distribution with variance proportional to the
gain.10
We assume that agents include the true state space as regressors in their learning algorithms.
Their algorithms are thus misspecified only in the sense that they are using linear regression
techniques in a non-linear model. The model’s non-linear nature is crucial for our results.
A random positive change in a learning coefficient may, for example, lead to a increase in
R&D that is larger in magnitude than the decline in R&D that results if the leaning coefficient
decreases by the same amount. This type of non-linearity allows for learning to influence the
long term output level of the economy.
We analyze the simulation from Section 3.3 under learning where = 10 and η = 1.1.
We consider two cases, one with a low gain where γ = 0.0001 and a higher gain case where
γ = 0.0005. After allowing a 30,000 period burn so that the initial conditions no longer affect
the results, we plot the ratio of At in the high gain economy to that of the low gain economy,
10
Sargent (2001) offers constant gain learning as an appropriate framework in settings where agents must deal
with structural change within a model.
27
where At in each economy is initially set equal to one:
Figure 13: Ratio of At in High Gain (0.0005) to Low Gain (0.0001) Economies
1.6
1.5
1.4
1.3
1.2
1.1
1
0
1000
2000
Periods
3000
Figure 13 shows that the high gain economy diverges from the low gain economy through
higher growth. A higher gain adds volatility to the model. But because households’ utility
function are concave, the effects of increased volatility are asymmetric in the non-linear model
so that households choose larger increases in their consumption (when they need to reduce their
marginal utility of consumption) than decreases (when they need to decrease their marginal
utility of consumption. As shown in Section 3, impatient households, who drive growth for
this calibration, tend to increase their consumption and R&D simultaneously. By increasing
mean impatient consumption, more rapid learning thus causes higher growth as well.
Our preliminary results in this section thus suggest that learning affects the long term
growth rate of the economy. Only a few other papers find a similar result. Evans et al. (1998)
sets up a model with multiple equilibrium growth rates and shows that learning can cause the
economy to transition between them endogenously. Davies and Shea (2010) develop a model
28
of international trade where learning causes a unit root on debt under rational expectations to
behave explosively, eventually resulting in financial crises.
5
Conclusion
Our results are preliminary. We feel the example of how learning affects the growth of
the economy in Section 4 is intriguing and we intend to expand our analysis. We expect that
higher gains will result in even more dramatic divergence than that of Figure 13. In addition,
just as the effect of m on growth depends on our calibration, it is also possible that alternate
calibrations could allow for learning to have different effects. We thus intend to examine these
different cases under learning. Finally, we will consider different values of λ (returns to scale
on R&D labor) and the impact on the steady state results as well as learning.
29
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