Asset Price Shocks as Aggregate Demand
Shocks
Seth Leonard
November 18, 2015
Abstract
This paper makes two contributions to the macroeconomic literature that seeks to explain the fall in employment following the 2006
home price crash. First, I show that home prices have a significant impact on both consumption and the saving rate, and that binding debt
constraints cannot alone explain this result. Instead, I propose precautionary saving as a key link between nominal and real variables.
Second, to better asses the implications of this channel I construct
a county level dynamic stochastic and, to a limited extent, general
equilibrium model in which heterogeneous agents face idiosyncratic
unemployment risk and aggregate asset price risk. Under the parameterization that I use, a -10% shock to asset prices causes consumption
to fall by 1% and employment to fall by around 0.5%. This direct link
between asset prices and demand, and thus employment, depends on
(1) incomplete insurance markets (in this case a debt constraint) and
(2) nominal rigidities (in this case wage rigidity).
1
1
Introduction
In May 2006 U.S. housing prices began to fall from a peak of nearly double
their January 2000 values. They remained flat for almost a year before sliding into a free fall in April 2007. A few months later, in September 2007,
consumption began to slip. Two months after that, in November 2007, employment started a fall that would be the biggest since the Great Depression.
This paper makes two contributions to the macroeconomic literature that
seeks to explain the drop in employment following the 2006 home price crash.
First, I look at the role of precautionary saving in an empirical framework. In
a four variable non-linear VAR home price shocks have a significant impact
on either consumption or the saving rate implying, in line with Mian and Sufi
(2014), that home prices are an important determinant of demand. Moreover,
in line with Gan (2010), the significance of home price shocks remains even
when home prices are increasing; using data from 1963 to 2015 in a two
state smooth threshold VAR (STVAR) that switches on changes in the home
price I find no significant difference in either the saving rate response or the
consumption response to home price shocks between states. Moreover, an
STVAR that switches on the estimated1 delinquency rate, a measure of the
fraction of debt constrained households, again finds no significant difference
in the saving rate response between states. Finally, in a simple five variable
linear VAR the impact of stock price shocks on the saving rate is similar to
the impact of home price shocks on those variables. These findings suggest
that binding debt constraints cannot alone explain the demand response to
asset prices, though the fact that debt constraints mean insurance markets
are incomplete remains important. I propose that precautionary saving may
be a better explanation of the link between demand and asset prices.
Second, to theoretically asses the role of precautionary saving in transmitting asset price shocks to demand and thus employment I construct a
model of a county level economy in which heterogeneous agents can trade
a single infinitely lived bond up to a debt constraint. Whereas in the absence of financial frictions asset prices have no impact on real variables, the
introduction of a limited menu of assets with a fixed debt constraint gives
rise to the precautionary motive. Agents will self-insure by accumulating
buffer wealth yielding a unique optimal quantity of assets for each aggregate
state of the economy.2 When the asset price falls the value of this buffer
1
Following Gan (2010) I use delinquencies as a measure of the proportion of debt
constrained households. The mortgage delinquency rate is available on Datastream at
a quarterly frequency from 1979. Because the rest of my data is monthly from 1963 I
estimate missing observations based on 11 series available over the whole sample period.
2
This result in an extension of Carroll and Kimball (2006) and Carroll (2004); I offer
2
wealth is eroded so that the optimal quantity of assets increases; to achieve
this new higher quantity of assets agents increase their saving rate thereby
reducing consumption. Production in the county consists of a traded and
non-traded sector; while demand for traded sector output is exogenous, nontraded sector output must clear locally. Thus when demand for non-traded
sector output falls in the county, so does employment in that sector. Under
my chosen parametrization a 10% fall in asset prices from 1 to 0.9 causes
consumption to fall by 1% and employment to fall by 0.6%.
This paper has its roots in the financial accelerator literature. Kiyotaki
and Moore (1997), for example, construct a model in which the output of
the more productive sector is constrained by a limited ability to borrow; a
positive productivity shock increases the collateral value of capital and thus
the amount this sector can borrow and therefore produce. In Bernanke et al.
(1999) imperfect information reduces the number of entrepreneurs in the
economy; a positive productivity shock increases entrepreneurs’ wealth via
asset prices and thus ameliorates this information friction (as entrepreneurs
can put more of their own assets towards a project). In contrast, Mian
and Sufi (2014) and their book House of Debt show that falling U.S. home
prices in the 2007-2009 crisis led to higher unemployment through a drop
in demand, a phenomenon they term the housing net worth channel. Gan
(2010) also finds a significant effect of home prices on demand using credit
card data from Hong Kong prior to the crisis. Moreover, he finds that, for
a majority of households, this consumption response to home prices appears
to be due to changes in precautionary wealth, not binding debt constraints
or a direct wealth effect.
The broader empirical literature on precautionary saving and the impact
of wealth on consumption tends to focus on the concavity of the consumption
function and thus risk aversion. Gourinchas and Parker (2001), for example, use a non-parametric estimation of household consumption functions
from the U.S. Consumer Expenditure Survey and find that the estimated
consumption function is highly concave, a finding that matches theoretical
predictions based on precautionary saving. Gourinchas and Parker (2002),
Cagetti (2003), and Low et al. (2010) use data on the magnitude of income
shocks to match theoretical moments of consumption or wealth distributions
to observed moments. As a distinct exercise, I look at the state dependency
of both the consumption response and the saving response to home price
shocks in a VAR framework. I begin my analysis using a Markov switching
VAR to look at the consumption response to home prices conditional on the
business cycle. Like the above papers I find that the consumption response
a more detailed discussion in appendix A.
3
to home price shocks (my measure of household wealth) is non-linear with
a stronger short run response in business cycle contractions. However, in
an STVAR that switches on changes in the home price I find no significant
difference in either the saving rate response or the consumption response to
home price shocks between states. Moreover, in an STVAR that switches on
the estimated mortgage delinquency rate, a measure of the fraction of debt
constrained households in the economy, I again find no significant difference
in the saving rate response to a home price shock between states.
While the theoretical model in section three builds on the empirical framework of Mian and Sufi (2014), the channel linking asset prices to demand is
more closely related to the literature on risk shocks including, for example,
Carroll and Dunn (1997), Guerrieri and Lorenzoni (2009), Bloom (2009),
Challe and Ragot (2011), Guerrieri and Lorenzoni (2011), Christiano et al.
(2014), and Leduc and Liu (2014). Leduc and Liu (2014) is particularly
similar to this work in that higher risk (they use the term uncertainty), an
increase in the variance of a productivity or preference shock in their case,
depresses demand via the precautionary motive. However, instead of looking
at second or higher moments of variables, my model looks at the level of
asset prices. Asset price levels act as a risk shock by reducing the value of
buffer wealth and thus the ability of agents to self-insure against employment
risk. This in turn depresses demand as agents cut consumption in order to
re-establish a sufficient level of precautionary wealth.
The rest of the paper is organized as follows. Section two offers a more
substantial discussion of the related literature. Section three then introduces
my empirical models and results. In section four I construct a dynamic
stochastic and, to the extent that the non-traded goods sector must clear,
general equilibrium county level model in which heterogeneous agents face
endogenous employment risk and exogenous (state or national level) asset
price risk. Section five brings these results back to the existing literature and
concludes.
2
Related Literature
The causes of business cycle fluctuations, and the appropriate policy response, have been perhaps the most central and most contentious questions
in macroeconomics since Keynes published The General Theory of Employment, Interest and Money in 1936. While the RBC literature moved away
from the role of demand in the 1980s, New Keynesian models brought it back
to micro-founded theoretical models. Shocks, in this framework, could stem
from exogenous productivity fluctuations as in the RBC literature or from
4
nominal variables (including, potentially, wages). However, these models still
lack an explicit role for (non-monetary) demand shocks.
The empirical literature has found a much more central role for demand
shocks beginning with Blanchard and Quah (1989). Braun et al. (2009) offer
a more recent estimation of supply versus demand shocks, estimating that
the two contribute equally to business cycle volatility. As noted above, Mian
and Sufi (2014) explore what they call the housing net worth channel of
demand in the U.S. using county level panel data from 2007 to 2009. The
authors find that a deterioration in housing net worth is an important determinant of unemployment in the non-tradable sector while housing net worth
is uncorrelated with tradable sector employment. This suggests that home
price shocks impact the real economy through demand as non-traded goods
producers depend on local market conditions while traded goods producers
are more geographically diversified; if home price shocks propagated through
supply, one should observe a response to home prices in the traded goods
sector as well. Demand driven unemployment aligns well with business survey data: Mian and Sufi (2014) note that from 2006 to 2010 the percentage
of surveyed businesses reporting poor sales as the “most important problem
facing your business today” rose from 10 percent to over 30 percent. Financing and interest rate concerns over that period remained flat representing less
than 5 percent of responses. Gan (2010) looks at a similar question using
household credit card data from Hong Kong from 1992 to 2004. While his
work looks at the housing boom as opposed to the ensuing bust, his results
support the findings in Mian and Sufi (2014). Housing net worth (mostly
increases in home prices over the sample period) has a significant impact on
consumption. This impact remains regardless of whether households are debt
constrained or whether they refinanced a home mortgage at more favourable
rates and are thus able to realize some of the wealth effect from home price
increases. Gan (2010) proposes that precautionary saving can explain the
significant impact of home prices on all households.
A separate branch of this literature looks at the impact of uncertainty
on consumption directly. This channel is articulated well in Leduc and
Liu (2014)’s aptly named paper “Uncertainty Shocks are Aggregate Demand
Shocks”. The authors present both a theoretical model of uncertainty shocks
in a DSGE setting and empirical evidence of uncertainty on unemployment
in a simple VAR model using University of Michigan survey data on consumer sentiment, the CBI Industrial Trends Survey in the United Kingdom,
and the VIX. With a simple Choleski decomposition for identification the
authors show that uncertainty has a significant, positive, and persistent impact on unemployment and a negative impact on inflation — exactly the
response one would expect from a demand shock. Carroll and Dunn (1997)
5
also look at uncertainty shocks using survey data from the University of
Michigan and find a significant correlation between uncertainty, measured
as unemployment expectations, and consumption even after controlling for
future income. The authors go on to model explicitly durable (housing) and
non-durable consumption with debt constrained heterogeneous agents where
the aggregate state of the economy determines idiosyncratic unemployment
risk. Given an increase in unemployment risk durable purchases collapse
as potential home buyers put off acquisitions. The authors point out how
well the results of their model corresponded to industry interpretations of
the 1990 recession in the U.S.: “... a May 11, 1992 editorial (p. 12) in Automotive News read, in part: ‘[F]olks still aren’t buying cars ... and I am
convinced that most Americans are still concerned about their jobs. As long
as that insecurity exists, we are going to see a sluggish auto industry.’”
Another notable theoretical paper that models demand shocks is Lorenzoni (2009) in which agents’ observations of aggregate productivity are imperfect. This noisy public signal can give rise to “noise shocks” which look
like aggregate demand shocks: a positive noise shock increases output, employment, and prices in the short run. Guerrieri and Lorenzoni (2009) and
Challe and Ragot (2011) also explore the role of (non-monetary) demand in
a DSGE setting with productivity shocks. Both papers construct models in
which precautionary saving amplifies business cycles. Guerrieri and Lorenzoni (2011) construct a heterogeneous agents model in the style of Aiyagari
(1994) and Krusell and Smith (1998) with debt constrained agents and look
at the impact of an exogenous tightening of the borrowing constraint. The
result is a sharp fall in output followed by a slow recovery with a similar dynamic in interest rates; the interest rate is in fact negative for several periods
after the contraction in credit. Significantly, agents need not in fact be credit
constrained to generate this result; the tighter borrowing constraint increases
the (stochastic) steady state level of saving so that agents will reduce consumption even if they are not in fact debt constrained. Introducing pricing
rigidities and including a floor on the nominal interest rate at zero results
in a much larger initial drop in output than in the flex price case. However,
the authors abstract from the labor risk channel highlighted in their earlier
work.
Unlike previous empirical work on the relationship between asset prices
and demand the models in section two of this paper allow state specific impulse responses to home price shocks using monthly data over 53 years. My
approach goes beyond establishing the significance of home prices for consumption by suggesting that binding debt constraints alone cannot explain
this channel and that precautionary saving likely plays an important role.
While a large literature has established the importance of risk shocks in
6
business cycle volatility — Christiano et al. (2014) find that “fluctuations in
risk are the most important shock driving the business cycle” — the model
in section three of this paper is the first (to my knowledge) to illustrate that
shocks to asset price levels, as opposed to higher moments of variables, act
like a risk shock with important consequences for demand.
3
Evidence of Precautionary Saving
The 2007-2009 financial crisis, a period over which home prices fell by around
25%, saw the saving rate rise from around 3% in 2006 to a high of 8.1% in
2009 while real consumption fell by 5% over the second half of that period.
Several studies, notably Mian and Sufi (2014) and Gan (2010), have documented a direct effect of home prices on consumption. There are at least
three possible explanations what Mian and Sufi (2014) call the housing net
worth channel: a direct wealth effect whereby the reduction in consumption
is directly proportional to the fall in lifetime wealth; a tightening of debt
constraints due to lower collateral values which forces households off their
optimal consumption path; or a precautionary motive which induces households to increase their savings against possible future income shocks. These
three explanations are of course not mutually exclusive. Yet while both a
direct wealth effect and binding debt constraints contributed to the impact
of asset prices on saving and consumption in the financial crisis, this section
shows that these two explanations alone are insufficient to account for observed relationships between variables. In line with what Gan (2010) finds for
the 1988 - 2004 home price boom in Hong Kong, I find that macroeconomic
trends in consumption, employment, and home prices for the U.S. since 1963
are consistent with a precautionary saving motive. My econometric analysis
is organized as follows: section 3.1 describes the data I use and how I process it, section 3.2 summarizes my results, section 3.3 introduces my Markov
switching VAR model, section 3.4 introduces several STVAR models, and
section 3.5 concludes.
3.1
Data
I use monthly U.S. data from FRED over the periods January 1963 to August
2015 for the effective federal funds rate, unemployment, consumption, the
personal saving rate, and the median sale price of new homes. While the Case
Shiller home price index might be more desirable than simply the median
home price, the series for national data only goes back to 1975; using the
median sale price gives me an additional 144 observations. However using
7
Figure 1: Home Price Series
Ideally my home price series should capture changes in the entire
distribution of home prices. However, plotting the processed series
for the median sale price of new homes against the Case Shiller 10
city composite index since its inception in 1987 and the Case Shiller
national index shows that the median sale price of new homes does
quite well in this regard.
either median sale price or the Case Shiller index for a simple linear VAR
yields similar results; figure 1 illustrates that my processed series for home
prices is noisier than either the Case Shiller 10 city composite or the Case
Shiller national index, but that it is otherwise very similar.
I deflate home prices and consumption using the core consumer price
index, detrend both using a Hodrick Prescott filter, detrend the saving rate
as well as it tends downwards over the sample period, and standardize the
variance of each series. Figure 2 plots the processed data over the sample
Figure 2: Processed Data
8
period. I order the vector of observations in period t as
 
ft
ut 

yt = 
 it 
pt
where ft is the effective federal funds rate, ut is unemployment, it is either
consumption or the saving rate depending on the model, and pt home prices.
I identify all of the impulse response functions using this ordering assuming that there is a lag in the response of the federal funds rate to all the
other series, that unemployment does not react contemporaneously to consumption, the saving rate, or home price shocks and that consumption and
the saving rate do not react contemporaneously to home price shocks. As
home prices are an asset price I allow them to respond contemporaneously
to the other three variables. This ordering assumes that firms wait at least
one month to make hiring and firing decisions based on observed demand
and that consumption and saving are persistent at least within a month, due
perhaps to habit formation, in response to home price shocks but that they
can immediately adjust to unemployment shocks.
3.2
Summary of Results
I begin my analysis by confirming the results of previous empirical work on
precautionary saving and the relationship between home prices and demand.
Because this work looks at the consumption response to changes in wealth, I
also use consumption in my comparisons. However, because lower consumption may simply be due to lower income (which I do not observe), I prefer
using the saving rate to analyse the impact of home prices on real outcomes
and in trying to determine the importance of binding debt constraints for
this channel. Thus while my first two models are in the federal funds rate,
unemployment, consumption, and home prices I substitute the saving rate
for home prices in later models.
For households with a reasonably long planning horizon, short term asset
price volatility should not have a large impact on consumption via the wealth
effect; expectations of lifetime wealth should depend on asset price trends.
While the 2007-2009 crisis was a very large home price shock, as illustrated
by figure 3 the change in home price trends as measured by the median
price was more subdued. Because the data used in the following models is
in deviations from trend, the role of the wealth effect in impulse responses
should be small or absent. The fact that I continue to observe a significant
consumption or saving response to home price deviations from trend implies
9
Figure 3: Trend and Actual Real Home Prices
While home price trends did adjust downward in 2007, this change
was small relative to the short run home price boom and ensuing
crash.
that the wealth effect alone is not sufficient to explain the observed link
between nominal and real variables. Put differently, detrended home prices
represent transient changes in household wealth and, in line with for example
Pistaferri (2001), can be accounted for by precautionary saving or binding
debt constraints while changes in home price trends represent changes in
households’ permanent income.
Following the empirical literature on precautionary saving such as Gourinchas and Parker (2001) I begin my analysis with a Markov switching VAR
(or Hamilton filter) to look at the state contingent response of consumption
to home price shocks. In line with this literature I find that the consumption
response to this wealth shock is non-linear with a more pronounced response
in the state associated with rising unemployment. However, the consumption
response to home price shocks remains significant in both states.
Binding debt constraints could explain the significance of home prices for
consumption; as figure 4 illustrates mortgage debt comprises about 70% of
consumer credit in the U.S. Home prices are thus an important determinant
of access to credit. To test whether the significance of the demand response
to home prices depends on binding debt constraints I introduce two STVAR
specifications. Gan (2010) finds for Hong Kong that increases in home prices
resulted in increased consumption in the build up to the 2007-2009 financial
crisis. To examine this result using U.S. data the first STVAR model in
section 3.4 switches on changes in home prices. I also find that increases in
10
home prices have a significant impact on consumption. Moreover, I do not
find any significant difference in the impact of home prices on consumption
between the two states in the model. If debt constraints are more likely to
bind when prices are falling than this result weakens the ability of constrained
households to explain the link between home prices and demand. Because
a fall in consumption may be due simply to a fall in (unobserved) income
I run the same exercise replacing consumption with the saving rate for the
U.S. since 1963. The results are unchanged; the impact of home prices on
the saving rate is not significantly different between states. To take a closer
look at the explanatory power of binding debt constraints I then estimate
the loan delinquency rate for the U.S. since 1963 to provide a measure of the
fraction of households that are debt constrained. Using this estimate as my
switching variable I again find no significant difference in the response of the
saving rate to home price shocks over four years.
Figure 4: Mortgage Debt in the U.S.
Mortgage debt constitutes the majority of consumer credit in the
U.S., peaking at 73.7% in 2008 Q1. Source: Source: FRBNY Consumer Credit Panel/Equifax
To asses whether the significance of home price shocks in these models is
due to the fact that I omit other asset prices, I include in section 3.5 a broad
stock market index in a linear VAR. Inclusion of stock prices does not reduce
the significance of the saving rate response to home price shocks. In fact,
the saving rate responses to stock and home price shocks are very similar.
This again strengthens the precautionary saving story; unlike home values
stocks are not a common form of collateral against debt and thus binding
debt constraints cannot explain their significance for consumption.
11
Finally, it might be the case that the observed consumption response
to asset prices is driven by behavioral factors. Households may construct
expectations based only on very recent trends, and in line with Akerlof and
Shiller (2009) a collapse in asset prices may be associated with a collapse in
animal spirits. However, section 4 illustrates that behavioral considerations
are not necessary to explain observed trends at the county level;3 a fairly
simple rational expectations framework shows that demand should respond to
asset price shocks given incomplete insurance markets, and nominal rigidities
mean that the demand response will impact employment.
3.3
Markov Switching VAR
Gourinchas and Parker (2001) show that the estimated household consumption policy function is concave, a finding in line with, for example, Carroll and Kimball (2006) and the results in appendix A. To ensure that my
empirical findings are consistent I begin by looking at the consumption response to home prices in a two state Markov switching VAR following Hamilton (1989) and Hamilton (2005). Because actual states are unobserved this
regime switching model assigns a probability of being in a given state, ξjt ,
based on the likelihood of the observed data at time t and the previous state
probabilities ξi,t−1 . The estimated state of the economy is thus endogenously
determined (instead of exogenously imposed, such as a seasonally switching
model as in Lutkepohl (2006)) by all the variables though unlike, for example, Kim et al. (2008), I do not allow for time varying transition probabilities.
That is, I assume my transition matrix Π is fixed.
Suppose that the unobserved state of the economy
is determined
by a
π11 π12
Markov process with transition probabilities Π =
. Defining zt =
π21 π22
1 yt−1 . . . yt−l where l is the number of lags my model is
B1 zt + 1t if s = 1
(1)
yt =
B2 zt + 2t if s = 2
Following Hamilton (2005) denote the probability of being in state j in
period t given observations 1 : t and the parameters of the model Θ =
{B1 , B2 , Σ1 , Σ2 , Π} as
ξjt = P r(st = j|y1:t ; Θ)
3
The behavioral arguments in Akerlof and Shiller (2009) may, however, be necessary
to explain the asset price volatility that I take as exogenous.
12
Figure 5: State Probabilities against Unemployment
Probability of state 1 (blue) plotted against unemployment (green).
The model appears to switch on changes in the business cycle with
state 1 denoting either no change or expansions and state 2 denoting
contractions. As indicated by the estimated transition matrix, state
1 is far more persistent.
so that the maximum likelihood estimate for Bj assuming I know the state
probabilities ξjt is
" T
#" T
#−1
X
X
(2)
B̂j =
ξjt (yt zt0 )
ξjt (zt zt0 )
t=1
t=1
Similarly, the maximum likelihood estimate for Σj is
#−1 T
" T
X
X
(3)
Σ̂j =
ξjt
ξjt (yt − Bj zt )(yt − Bj zt )0
t=1
t=1
Finally, I can estimate π11 and π22 as
T −1 −1 T −1
P
P
π̂11 =
ξ1t
ξ1,t ξ1,t+1
t=1
(4)
Tt=1
−1 T −1
−1
P
P
ξ2t
ξ2,t ξ2,t+1
π̂22 =
t=1
t=1
Appendix C provides the complete derivation of these results; I derive the
actual estimates for Bj used to construct impulse response functions via the
EM algorithm.
My estimated transition matrix for this framework using 6 lags is
0.93 0.07
Π̂ =
0.41 0.59
13
Figure 6: State Contingent Consumption IRFs
One standard deviation confidence bands for the consumption response to a home price shock. The short run response of consumption to home price shocks is larger in contractions (red, no outline)
than otherwise (blue, solid outline) though it quickly falls (within
one year) so that the cumulative impact is zero in the long run. In
normal times and expansions the short run consumption response
is more subdued but the long run cumulative response is positive.
indicating that the model spends more time in state 1 than state 2 and
that state 1 is far more persistent. Figure 5 plots the state probabilities
against unemployment. Instead of switching on the level of unemployment,
the model tends to switch on changes in unemployment. Thus state 1 denotes
both “normal times” and expansions while state 2 denotes contractions. As
illustrated by figure 6 the short run the consumption response is more pronounced in state 2 when unemployment is increasing and home prices tend
to be low, thus this result is consistent with previous work on the concavity of the consumption policy function. The cumulative long run response,
however, is zero in contractions. In state 1, on the other hand, the short run
response is more subdued but the cumulative long run response is positive.
Notably, the consumption response to home price shocks is significant in both
states.
3.4
Smooth Transition VAR Models
Gan (2010) shows that for Hong Kong at the household level increases in
home prices correspond to increases in consumption prior to the 2007-2009
financial crisis and attributes this result to the precautionary motive. To
see whether this result holds in aggregate for the U.S. over a longer period I
construct a smooth threshold VAR that switches on changes in home prices
14
using data from 1963 to the present. For a closer analysis of whether binding
debt constraints can explain this result, I then estimate the loan delinquency
rate over this period and use it as the switching variable in a STVAR to look
at the state contingent saving rate response to home price shocks.
The STVAR model has the form
(5)
yt = Φ(α + βst ) B1 zt + 1t + 1 − Φ(α + βst ) B2 zt + 2t
where Φ(·) is the standard normal cumulative distribution function and st is
the continuous variable or variables that determines the state of the model in
period t. Defining ξ1t = Φ(α + βst ) and ξ2t = 1 − Φ(α + βst ) my maximum
likelihood estimates4 for Bj and Σj given ξjt are
"
(6)
B̂j =
T
X
ξjt (yt zt0 )
#" T
X
t=1
#−1
ξjt (zt zt0 )
t=1
and
"
(7)
Σ̂j =
T
X
t=1
#−1
ξjt
T
X
ξjt (yt − Bj zt )(yt − Bj zt )0
t=1
Of course I do not know the parameters α and β in advance. One option
would be to numerically search for the values that maximize the likelihood
function using equations 6 and 7 for a given {α, β}. However, Matlab’s
fminsearch is much better behaved over the loss function
loss =
T X
2
X
ξjt 0jt jt
t=1 j=1
thus in practice I numerically search for the values of {α, β} that minimize
this loss as opposed to the values that maximize the likelihood (or log likelihood).
Beginning with the four variable model in the federal funds rate, unemployment, consumption, and home prices using 6 lags, the STVAR selects a
hard threshold (the state probabilities are all zero or one) at around the 50th
percentile so that half of all observations are assigned to each state. In the
falling price state, state 2 in this model, home price shocks have a significant
impact on consumption. Like Gan (2010) I also find that in the rising price
state, state 1 in this model, home price shocks have a significant impact on
4
Appendix C again gives the full derivation.
15
Figure 7: State Contingent IRFs (left) and Difference Between States (right)
The response of consumption to a price shock when home prices
are high (blue, solid outline) versus low (red, no outline) are similar
(left panel shows one standard deviation confidence bands). The
response of consumption to a home price shock is not significantly
different between states (right panel shows the one s.d. confidence
band for the statistic Φ1p→c,t − Φ2p→c,t ).
consumption. The left hand panel of figure 7 plots the one standard deviation confidence intervals for the consumption response to a home price shock
in each state. As is evident from the figure, I find no significant difference
in the impulse responses of consumption to a home price shock between the
two states; the right hand panel of figure 7 plots a one standard deviation
confidence band for the statistic Φ1p→c,t − Φ2p→c,t where Φip→c,t is the IRF for
a price shock to consumption in period t and state i. If debt constraints are
more likely to bind when home prices are falling then this result casts doubt
on their ability to explain the link between home prices and consumption.
Replacing consumption with the saving rate yields similar results; again there
is no significant difference in the saving rate response to a home price shock
between states.
To better examine whether binding debt constraints can explain the link
between home prices and demand I estimate the delinquency rate for home
mortgages in the U.S. The actual delinquency rate is available on Datastream
from 1979 at a quarterly frequency. Because my model is monthly from 1963
I estimate the missing observations from 11 housing related series available
over the whole sample period5 . Using this as my switching variable I estimate
5
These 11 variables are the employment population ratio, the unemployment rate,
detrended consumption, detrended durable consumption, the median sale price of new
homes, housing starts, the saving rate, detrended industrial production, the federal funds
rate, the 3 year T-bill rate, and the 10 year T-bill rate.
16
Figure 8: State Contingent IRFs (left) and Difference Between States (right)
The response of the saving rate to a price shock when the mortgage
delinquency rate is high (blue, solid outline) versus low (red, no
outline) are similar (left panel shows one s.d. confidence bands). At
six months out from the shock the response when the delinquency
rate is high is slightly larger (in magnitude); otherwise they are not
significantly different (right panel shows one s.d. confidence band
for the statistic Φ1p→s,t − Φ2p→s,t ).
an STVAR in the federal funds rate, unemployment, the saving rate, and
home prices to look at the state contingent response of the saving rate to
home price shocks6 . Again, I find that home prices have a significant impact
on the saving rate both when mortgage delinquencies are high, implying a
high fraction of debt constrained households, and when delinquencies are low,
implying a low fraction of debt constrained households. And again, with a
slight exception at 6 months after the shock, I find no statistically significant
difference in the response of the saving rate to home price shocks across
states. This result makes a strong case against binding debt constraints as
the primary explanation of the link between home prices and demand; if this
were the case I should observe a significantly stronger response to home price
shocks when more households are debt constrained.
3.5
Conclusions
As a final look at the impact of asset prices on the saving rate, figure 9
adds the Wilshire 5000 stock price index as a broad measure of asset prices
6
A potential problem with this approach is that even if B estimated for the full sample
is stationary, the state contingent parameters Bj may not be. The problem arises in this
specification. One can either ignore the problem or bias the estimated Bj towards zero;
for the results in this section I do the latter. Specifically, I bias the estimates for the low
delinquency state towards zero; if anything this works against the result I find.
17
to the federal funds rate, unemployment, saving rate, and home price data.
Unlike homes, stocks do not play a large role as collateral against consumer
debt in the U.S. That the response of the saving rate to stock price and
home price shocks in the U.S. is similar indicates that perhaps the housing
net worth channel applies to a broader range of assets. Put differently, it
may be that what is important for demand is not home values per say, but
households’ total wealth. Of course, homes comprise a large portion of this
wealth in the U.S. Quantitatively, a (positive) one standard deviation home
price shock corresponds to a decrease in the saving rate of 0.0787 (the saving
rate is a percent of income) and a one standard deviation stock price shock
corresponds to a decrease in the saving rate of 0.14. The larger response to
stock price shocks is due to the fact that a one standard deviation stock price
shock is nearly three times larger than a one standard deviation home price
shock; a stock price shock that causes a 10% deviation from trend corresponds
to a 0.33 change in the saving rate while a home price shock that causes a
10% deviation from trend corresponds to a 0.53 change in the saving rate.
Thus for an exogenous change of equal magnitude to home prices and stock
prices, the home price has a slightly larger impact. However, these IRFs are
not significantly different at most horizons.
The message from the three econometric models in this section is the
same: asset prices have a significant impact on consumption regardless of
the state of the economy, and neither the wealth effect nor binding debt
constraints can fully explain this result. That binding debt constraints are
not a sufficient explanation of the housing net worth channel should not be
interpreted as a claim that debt constraints do not matter. Debt constraints
imply that income risk cannot be insured away; if a household is hit by a series
of negative shocks it may be pushed up against this constraint. It is exactly
risks such as these that give rise to precautionary saving; to mitigate this risk
households accumulate precautionary wealth. If asset prices fall reducing the
value of households’ buffer wealth they will increase their saving rate until
they re-establish a reasonable buffer. Thus while binding debt constraints
cannot fully explain the link between asset prices and demand, the fact that
debt constraints mean markets are incomplete remains important. To explore
whether precautionary saving can in fact explain the link between asset prices
and demand, section three presents a model of precautionary saving with
exogenous price shocks.
7
The values presented in this section are the maximum responses of the median IRFs.
18
Figure 9: IRFs Including Stock Prices
Adding the Wilshire 5000 stock price index to unemployment, consumption, and home price data yields a saving rate response to
stock price shocks similar to the saving rate response to home price
shocks. This suggest that there is not anything particularly special
about home prices, but rather that all asset prices have an impact
on demand.
4
A Model of Precautionary Saving with Exogenous Price Shocks
In a classical monetary model, for example that in Gali (2015) chapter 2,
nominal variables have no impact on real variables. The same will be true in
a frictionless version of this model; in the frictionless case for the individual
agent there is no consumption or saving response to asset prices. This section
illustrates that restricting asset trading to a single asset with a limit on debt
creates a channel through which the asset price, a nominal variable, impacts
real variables via the precautionary motive in combination with labor market
frictions.
While comparing the results of a county level model with national level
empirical findings is imperfect, the similarities between the two are worth
noting. For the U.S. since 1963 I find that a stock price shock causing a 10%
fall in stock prices corresponds to a 0.6% fall in consumption and a 0.5%
increase in unemployment.8 The model in this section predicts that a 10%
fall in asset prices from 1 to 0.9 corresponds, at the county level, to a 1% fall
in consumption and a 0.6% fall in employment.
8
Again I use the maximum response of the median IRF.
19
Following the empirical framework of Mian and Sufi (2014), the economy
I model is a county populated by heterogeneous (in terms of employment and
wealth but not preferences) agents subject to exogenous asset price volatility.
Asset prices are in fact the only aggregate shock to the economy; including
aggregate productivity fluctuations is not necessary to the precautionary
channel of asset price shocks and entails a large computational cost due to the
increased dimensionality of the problem. Production occurs in two sectors,
a traded sector in which demand is determined outside the county and a
non-traded sector for which markets must clear locally. Employment in the
non-traded sector thus depends on local demand. Despite the strong response
of non-traded sector employment to home price shocks, Mian and Sufi (2014)
find no statistically significant response of wages to home prices from 2007 to
2009 at the county level. In my model I therefore take the case of perfectly
rigid wages; I fix wages so that in simulation employment approaches 1 as the
asset price rises. Section 4.1 introduces the dynamic programming problem
faced by individual agents. Section 4.2 then illustrates the precautionary
channel in partial equilibrium by solving the problem of an individual agent
when aggregate variables are exogenous and contrasts this result with the
frictionless case. Section 4.3 introduces the profit maximization problem of
firms and sections 4.4 and 4.5 state the conditions which define a steady state
and an equilibrium respectively. Section 4.6 describes my solution technique,
an extension of Krusell and Smith (1998) and Krusell and Smith (2006), and
section 4.7 presents my results.
4.1
Agents
Agents solve the dynamic programming problem
(8)
n
o
V (qt , bt , nt , Φt ) = max u c(qt , bt , nt , Φt ) + βEt V (qt+1 , bt+1 , nt+1 , Φt+1 )
ct ,bt+1
subject to
(9)
y(nt ) + (qt + d)bt = P (qt , Φt )c(qt , bt , nt , Φt ) + qt bt+1
and
(10)
bt > b
where u(·) is the period utility function, c(·) household consumption, P (·)
the aggregate price index, and y(·) income. Assets bt pay a constant dividend
d but are subject to exogenous fluctuations in price qt . Assuming that asset
20
prices are exogenous reflects the assumption that prices are determined at
a more aggregated level than the county. As employment volatility is only
in the extensive margin, an agent’s income depends only on his employment
state: income, y(nt ) = wnt + ζ(1 − nt ), is either w if the agent is employed
(nt = 1) or an exogenously determined level of unemployment benefits ζ if
the agent is unemployed. Profits accrue outside the county, thus the asset
price qt and the distribution of agents Φt are only important in determining
demand and thus, since non-traded goods markets have to clear within the
county, non-traded sector employment.
Period consumption is a constant elasticity of substitution (CES) aggregate of a traded and non-traded good
i γ
h
γ−1
γ−1 γ−1
1
1
T
N γ
γ
γ
γ
+ δ (ct )
(11)
ct = (1 − δ) (ct )
where δ measures the consumer’s preference for the non-traded good. Total
consumption expenditures for each household (the price of the traded good
is fixed at 1) are
cTt + pt cN
t = ct P t
thus a household’s demand for each good is given by
−γ
−γ
1
pt
T
N
ct ct = (1 − δ)
ct
(12)
ct = δ
Pt
Pt
and the aggregate price index is
1
1−γ
Pt = δp1−γ
+
1
−
δ
t
Non-traded consumption is in turn a CES aggregate of a continuum of nontraded good firms
ϕ
 ϕ−1
 1
Z
N,i ϕ−1

ϕ di
cN
=
(c
t )
t
0
so that demand for each brand of non-traded good is
i −ϕ
pt
N,i
ct =
cN
t
N
pt
Since firms are homogeneous it follows that cN,i
= cN
t
t ; the motivation for
this nested CES form is to reduce the monopoly power of non-traded goods
firms and allow the elasticity of substitution between traded and non-traded
goods γ to be less than one.
21
Before moving on to the structure of firms in section 4.3 and the characteristics of an equilibrium in sections 4.4 and 4.5, it is useful to look at
the individual’s problem in isolation. The key to the model is the reduction
in consumption that arises from an exogenous price shock, a feature most
clearly illustrated by solving the problem for an individual for whom income
and employment probabilities are exogenous.
4.2
The Individual’s Problem
The individual’s problem in this section modifies equations 8, 9, and 10 so
that there are only two state variables, employment nt and the asset price
qt , and the (homogeneous) goods price is 1. Income is
y(nt ) = wnt + ζ(1 − nt ) = {1, 0.5}
Assuming constant relative risk aversion (CRRA) utility u(ct ) =
Euler equation arising from this problem is
(13)
c1−σ
t
1−σ
the
−1 −σ
Pt−1 c−σ
t qt ≥ βEt Pt+1 ct+1 (qt+1 + d)
where the inequality is due to the debt constraint bt ≥ b = −3. This first
order condition is identical to that for the general equilibrium model, though
in this case since there is only one homogeneous good Pt = Pt+1 = 1.
In order to eliminate the possibility of expected capital gains assets in the
model must follow a (nearly) random walk; the nature of the grid space used
to solve the problem requires that unlike a true random walk asset prices
must be bounded above and below. I use 0.1 as a lower bound on asset
prices and 2 as an upper bound yielding, with twenty grid points, intervals
of 0.1 between asset price states. The asset price transition matrix in this
and subsequent problems is


.9 .1 0 0 0 · · ·
.1 .8 .1 0 0 · · ·


 0 .1 .8 .1 0 · · ·
(14)
Πq = 

 0 0 .1 .8 .1 · · ·


.. .. .. .. .. . .
.
. . . . .
While a random walk is not variance stationary, the shock driving this process
does have a well defined variance. For the above transition matrix a change in
the asset price, when realized, corresponds to approximately a two standard
deviation shock.
22
Another potential issue is that, since dividends are constant, the interest
rate paid on an asset changes with its price. For example, if the dividend
is d = 0.04 and the asset price is 1.5 (and remains 1.5) then the asset pays
an interest rate of 2.67 %. If instead the asset price was 0.7 (and remained
0.7) with the same dividend the interest rate is 5.7%. A lower asset price
thus corresponds to a higher interest rate which will induce the agent to tilt
consumption away from the present by saving more. To avoid this effect I
therefore set the dividend to zero for the individual’s problem in this section
(though I use a more normal monthly rate of d = 0.0036 in the full model
to match real returns on the Wilshire 5000). Finally, it could be the case
that the wealth effect influences these results: if the price goes from .7 to 1.1
and asset prices follow a random walk then an agent with bt = 1 experiences
an increase in expected lifetime wealth of 0.4. However, not only would the
impact of this wealth effect be small in a given period, it would not exist if
the agent had zero assets and would be negative if the agent had negative
assets.
Figure 10: Consumption Policy Functions for Different Asset Prices
In the frictionless case consumption does not respond to changes in
the asset price. With a debt constraint, however, the consumption
policy function is increasing in the asset price so that a fall in the
price corresponds to a fall in consumption.
Because the dividend (interest rate) is zero, removing the debt constraint
corresponds to an economy without financial frictions.9 Equilibrium in this
case requires that β = 1 and that assets will follow a random walk (unlimited
9
As Aiyagari (1994) points out a positive interest rate with a single asset implies a
debt constraint of yt = rt bt where yt is income and rt the interest rate. That is, a natural
23
borrowing is ruled out by the transversality condition). Under the calibration
I use for this example when the wage is 3, unemployment benefits are 1.5, and
the probability of unemployment is 0.04, the agent will save an additional
0.06
assets every period when employed and borrow an additional 1.44
assets
qt
qt
every period when unemployed. In this way average asset holdings will be
− .04 × 1.44
= 0) and consumption will be constant at 2.94,
zero (.96 × 0.06
qt
qt
illustrated by the line labelled frictionless policy in figure 10. If, however, I
introduce a debt constraint (in this case at -3) the agent can no longer perfectly smooth consumption (this requires β < 1; see appendix A for further
discussion). Instead, he will accumulate precautionary wealth against the
risk of being pushed up against the debt constraint when unemployed. The
agent’s target quantity of assets is illustrated by the intersection of the debt
constrained policy functions and the line labelled constant assets in figure
10; this is the state contingent stochastic steady state (stochastic because
the agent forms expectations based on the transition matrices for employment and the asset price, steady state because when employed and in the
absence of shocks asset holdings will be constant). The target quantity or
Figure 11: Saving Rate for Different Asset Prices
The agent’s saving rate (conditional on the employment and bond
states) is decreasing in the asset price.
steady state level of assets is increasing in the asset price as a fall in the asset
price erodes the value of this buffer wealth. Thus when the asset price falls,
the agent increases his saving rate thereby cutting consumption in order to
debt limit exists when all of an agent’s income is used to service debt. For a single noncontingent asset a frictionless economy thus requires that the interest rate be zero.
24
reach the new target level of assets. For the fixed level of assets illustrated
in figure 11, in this case zero, the saving rate is decreasing in the asset price.
Again, this is due to the fact that for a low asset price the quantity zero
(the fixed quantity of bonds illustrated) is below the stochastic steady state
quantity of assets. When the difference between zero and the steady state
level of bonds increases as the price falls further still, the saving rate increases
to make up this difference faster. When asset prices are high, the opposite
dynamic is true.
4.3
Firms
In the county level model firms produce in a traded and a non-traded sector.
The traded good price is set by the world market and fixed at 1. Traded
goods firms have decreasing returns to scale and maximize
n
o
πtT = max aTt (lT )α − wltT
ltT
so that labor in the traded sector is constant at
1
T 1−α
αa
(15)
ltT =
w
Non-traded goods firms have constant returns to scale, are monopolistically
competitive, and choose the non-traded good price to maximize
πtN,i
n
o
cN,i
t
i N,i
= max
p t ct − w N
a
pit
RR N
where the market clearing condition for non-traded goods is ytN,i =
ct Φt dbt dnt .
N,i
N N,i
Because the non-traded sector has constant returns to scale, ct = a lt , the
price of the non-traded good is the standard constant mark-up over marginal
cost
ϕ w
(16)
pt =
ϕ − 1 aN
t
and labor in the non-traded sector is, using the fact that cN,i
= cN
t
t ,
(17)
ltN =
cN
t
aN
t
where cN
t is given using prices and households’ maximization conditions,
equations 11, 13, and the constraints, across all the idiosyncratic households.
25
Total aggregate employment in the economy is then10
Lt = ltT + ltN
(18)
For simplicity I assume all agents have an equal probability of employment,
thus the state dependent idiosyncratic transition matrix (the probability of
transitioning between employment and unemployment) is
Lt 1 − Lt
ΠL =
Lt 1 − Lt
This transition matrix assumes that both the employed and unemployed face
the same probability of finding a job next period. This assumption is not
necessary; the unemployed could face a lower probability of finding work next
period than the employed. However, since the duration of unemployment in
the appendix B models does not seem to have a large effect on the consumption policy functions of employed agents I do not consider more complicated
transition matrices (which require additional transition states) to be worth
the computational cost.
4.4
Steady State
Heterogeneity in the model is in asset holdings bt and employment state nt ,
thus the distribution of agents, Φ(bt , nt ), is defined over those two dimensions. The only aggregate state is the asset price, qt , thus I define a steady
state as the stable distribution Φqj arising when an asset price qj is realized
indefinitely. While qj and Φqj are constant in a steady state (there is one for
every asset price) agents still form expectations as if a shock to qj could be
realized using the appropriate transition matrix. Define Π:,qi as the column
of Πq corresponding to the next period asset price qi and ΠL (qj , qi ) as the
matrix of employment probabilities given the current steady state distribution Φqj and the next period price realization qi . Then the transition matrix
faced by an individual agent is
Πqj = [Π:,q1 ⊗ ΠL (qj , q1 ) Π:,q2 ⊗ ΠL (qj , q2 ) Π:,q3 ⊗ ΠL (qj , q3 ) · · · ]
where rows and columns index the aggregate-idiosyncratic states
{qt , nt } = {1, 1}, {1, 0}, {2, 1}, {2, 0}, · · ·
10
To clarify the notation for labor: l(nt ) is the labor of a specific household, ltT aggregate
labor in the tradedR sector,
ltN aggregate labor in the non-traded sector, and Lt aggregate
R
employment Lt =
l(nt )Φt dbt dnt .
26
While the distribution of individuals Φqj is constant in the steady state for
the asset price qj individual states are not; within a steady state agents
transition in and out of unemployment and in doing so change their asset
holdings to smooth consumption or accumulate precautionary wealth.
Formally, let bt+1 = g(st , Φt ) be the policy function that solves the household’s problem, equation 8 subject to 9 and 10, where consumption of each
type of good (tradable and non-tradable) is given by equation 12. For notational convenience let st = {bt , nt , qt }. Let B be the set of all possible bond
holdings and N be the set of all possible employment state realizations (in
practice there are only two: employed and unemployed) so that B × N is the
set of all possible individual types. The distribution over all types is then
Φ(bt , nt ) with the marginal distributions
Z
Φ(nt ) =
Φ(bt , nt )dbt
bt ∈B
and
Z
Φ(bt ) =
Φ(bt , nt )dnt
nt ∈N
Let Φt+1 = Γ(qt , Φt , qt+1 ) be the aggregate law of motion for the distribution
of individual types (it depends on future aggregate states because the probability of being employed in period t + 1 depends on the aggregate state that
period). Then Φt+1 = Γ(qt , Φt , qt+1 ) should satisfy
Z Z
(19)
Bt+1 =
g(st , Φt )Φt (bt , nt )dnt dbt
b∈B n∈N
where Bt =
R
bt Φt (bt )dbt is the aggregate level of wealth in the economy
b∈B
and
Z
(20)
Z
Lt =
nt Φ(bt , nt )dbt dnt
b∈B n∈N
which is the aggregate level of employment.
Definition 1 The steady state for qj is a wealth distribution Φqj , a policy
function g(st , Φqj ), a level of non-traded goods consumption yqNj , and a level
of labor inputs lqTj and lqNj , such that
1. the policy function bt+1 = g(st , Φqj ) solves the agents’ optimization
problem, equation 8 subject to equations 9 and 10,
27
2. firms maximize profits so that ljT and ljN satisfy equations 15 and 17,
3. the distribution of agents Φqj is a fixed point in Φt+1 = Γ(qj , Φt , qj ),
4. prices satisfy equation 16 and the non-traded goods market clears
Z Z
N
cN
∀t
t (st , Φqj )Φqj dbt dnt = Cj
b∈B n∈N
4.5
Equilibrium
The steady state is an equilibrium in which aggregate shocks to the economy
are not realized. The concept of an equilibrium away from the steady state
is similar to the previous equilibrium with the key difference that the current
distribution of individuals, Φt , is taken as given and subsequent distributions
are determined by the aggregate law of motion for wealth and employment,
Φt+1 = Γ(qt , Φt , qt+1 ).
Definition 2 Given a current distribution of types Φ0 an equilibrium is a
sequence of policy functions {g(st , Φt )}, a sequence of distributions {Φt },
a sequence of non-traded goods consumption {ytN } and a sequence of labor
inputs {ltT } and {ltN } such that
1. the policy functions bt+1 = g(st , Φt ) solve the agents’ optimization problem, equation 8 subject to equations 9 and 10,
2. firms maximize profits so that ljT and ljN satisfy equations 15 and 17,
3. the distribution of households Φt+1 = Γ(Φt ) satisfies equation 19 and
equation 20,
4. prices satisfy equation 16 and the non-traded good market clears
Z Z
N
cN
∀t
t (st , Φt )Φt dbt dnt = Ct
b∈B n∈N
4.6
Solution Technique
As this model incorporates the precautionary motive and heterogeneous
agents, I use policy function iteration to find a numerical solution for my
chosen parametrization.11 While this solution approach is global in theory,
11
The Matlab programs used to solve the model are available on my website
www.setonleonard.wix.com/economics.
28
in practice it enables me to solve the model over my chosen state space. As
is often the case with such an approach, the dimensionality of the problem
limits the size of this space, though using a sparse grid for the asset state
and interpolating intermediate points using a cubic spline greatly alleviates
the burden of dimensionality. While the agent need choose only one variable
(next period assets in this case; consumption is given by the next period
asset choice through the budget constraint), the states he must consider are
the asset price state in Q, his individual bond state in B, and his individual
employment state in N . In practice it is simple to limit N to employed or
unemployed, {0, 1}. However, I cannot reduce in this manner the size of
the state spaces for Q and B since they determine the accuracy of the solution. Q can be much coarser, though at either end of the grid space the
solution will be distorted by opportunities for capital gains (if, for example,
the price cannot go any lower), while B should imitate as closely as possible a continuous variable. I therefore define Q over [.1, 2] in increments
of .1 for 20 grid points and B over [−3, 10] in increments of 0.01 for 1301
grid points. The debt constraint is then the lower bound of the grid space,
-3, which matches the (fixed) monthly wage in the model. The full set of
states in S = Q × B × N contains 52,040 points. A sparse grid over the
bond state allows me to reduce the state space to 6000 points (the choice
of bonds remains in the full grid space); the sparse grid is constructed so
that the density of points is higher at lower values of bt as the curvature
of the consumption policy function will be greater at lower asset quantities.
However, this does not yet include the distribution of agents Φ(bt , nt ), which
also enters an individual’s maximization problem. In a steady state for asset
price qj this distribution, Φqj , is time invariant and agents form expectations
based on this constant distribution of types.
4.6.1
Steady State
The key steady state variables for my solution technique in section 4.6 are
aggregate asset holdings in the steady state corresponding to qj , denoted
B̄j , and aggregate non-traded consumption (which in turn determines nontraded sector employment), denoted C̄jN . To find these variables I begin my
iterations with an initial guess of gj,0 (st ) = bt ∀st (that is, my initial guess is
constant savings and the policy function is specific to the steady state associated with qj ). Given this policy function, the program then searches for the
asset choice bt+1 ∈ B that satisfies the Euler equation 13 using the budget
constraint, equation 9, to substitute out consumption for every state in the
sparse grid. This asset choice for each state becomes the new policy function gj,1 (st ) (where intermediate points are interpolated), and the program
29
continues until convergence, defined as gj,i (st ) = gj,i+1 (st ) ∀st .
The individual’s problem, outlined above, is nested within an aggregate
problem of finding a stationary distribution that satisfies definition 1. This
process is also iterative; I begin with an initial guess for a distribution Φ0qj
(uniform) and solve the individual’s problem. Following Aiyagari (1994) I
then calculate the new long term distribution for the aggregate state qj based
on the policy functions calculated under Φ0qj , which yields Φ1qj . I then repeat
this process to get Φ2qj and repeat until convergence, defined as Φiqj = Φi+1
qj .
N
I then save B̄j and C̄j for each of the twenty aggregate states and use
these variables to form a linear approximation of the information in the
law of motion for the distribution of types, Φt+1 = Γ(qt , Φt , qt+1 ), relevant
to individuals’ policy functions; in this case the only information agents
need in order to form rational expectations is state dependent next period
employment probabilities.
4.6.2
An Exogenous Asset Price Shock
If the precautionary motive can explain at least part of the consumption and
thus employment response to asset price shocks observed in practice, a fall
in the exogenous asset price in the model should correspond to a reduction
in non-traded goods consumption and thus non-traded sector employment as
agents accumulate assets against the risk of unemployment.
As usual in heterogeneous agent models with aggregate uncertainty the
main difficulty lies in calculating the law of motion for the distribution of
agents. In this model the only information agents need from the distribution
is next period aggregate non-tradable consumption from which they can calculate next period employment probabilities. Krusell and Smith (1998) and
Krusell and Smith (2006) propose positing an aggregate law of motion for
each aggregate state. However, the number of aggregate states in my model
(there are 20) makes this approach computationally burdensome. Instead I
posit the laws of motion
(21)
B̂j,t+1 = γb B̂j,t
(22)
N
Ĉj,t
= γc,j B̂j,t
where Bt is the aggregate level of bond holdings in the economy in period t,
B̄j is the steady state level of aggregate bond holdings for the asset price qj ,
B̂j,t+1 = Bt − B̄j , and similarly CtN is the aggregate level of non-traded goods
consumption in period t, C̄jN is the steady state level of aggregate non-traded
N
goods consumption for the asset price qj , and Ĉj,t
= CtN − C̄jN . I calculate
30
state specific steady state values for bonds and non-traded consumption as
outlined in section 4.6.1. Equipped with these values, I then guess values for
γb and γc,j . Next period aggregate bonds are then
Bt+1 = γb (Bt − B̄j ) + B̄j
(23)
where j is the state in period t, and state dependent next period aggregate
consumption is
N
Ci,t+1
= γc,i (Bt+1 − B̄i ) + C̄iN
(24)
N
With Ci,t+1
I can calculate state dependent next period employment probabilities from equations 17 and 18 and thus solve the model for a given
realization of asset states Q. Having solved the model, I then regress B̂j,t+1
N
on B̂j,t to update γb and Ĉj,t
on B̂j,t to update γc,j . As in Krusell and Smith
(1998) and Krusell and Smith (2006) I repeat this updating process until the
model converges on values for γb and γc,j . While the coefficient for B̂t+1 is
N
not state dependent, the coefficient for Ci,t+1
depends on the state in period
t + 1, thus for each simulation I will be estimating 21 parameters.
4.7
Results
The nature of the results is evident from the individual’s problem in section
4.2: the steady state quantity of bonds for employed agents is decreasing in
the asset price. In terms of transition dynamics, this means that when asset
prices fall agents will increase their saving rate in order to reach the new,
higher steady state level of assets. In aggregate, this is a demand shock.
Consumption falls over both types of good (traded and non-traded); the fall
in non-traded goods consumption causes employment in that sector to fall.
4.7.1
Calibration
While the qualitative results of the model are not sensitive to the calibration, the magnitude of the consumption response to an asset price shock is
sensitive to the level of risk faced by agents in the model, agents’ level of
risk aversion σ, and their time preference β. Unemployment benefits are one
of the biggest determinants of risk in the model; higher benefits reduce the
fall in income due to unemployment and thus reduce the consumption risk
between employment states. Higher benefits therefore reduce the incentive
to accumulate precautionary assets, and when unemployment benefits are
high the fall in consumption due to an asset price shock is low. I set unemployment benefits to half the wage. This is on the high end of unemployment
31
benefits in the U.S., which are typically between 40% and 50% of wages and
last for only 90 days. Choosing the high end of the range gives me a conservative result and avoids overstating the level of risk in the economy. A
second important determinant of risk is the debt constraint, with a higher
debt constraint corresponding to a higher risk environment. Because debt in
this model is uncollateralized, I set the debt limit equal to the wage. That
is, agents can borrow up to what they make in one month when employed.
Following Backus et al. (1992), among many others, I let the coefficient of
relative risk aversion σ = 2. As the model is monthly I set the exogenous
dividend to d = 0.0036 to match real returns on the Wilshire 5000 when
the asset price is 1. I then choose the time preference so that optimal asset
holdings fall within the lower half of the asset grid space for most asset price
realizations. While the lower bound on asset holdings should impact the
problem, the upper bound from the construction of the asset space should
not. To this end I let β = 0.995. The left hand panel of figure 12 illustrates
the policy functions for three steady states under this calibration: those for
q = 0.9, q = 1, and q = 1.1; the right hand panel illustrates the wealth
distributions over the 15 highest asset price states.
Figure 12: Steady State Policy Functions and Wealth Distributions
The model is calibrated so that steady state savings are at the
lower end of the asset space for most asset prices (left). Due to
the precautionary motive the steady state wealth distributions are
decreasing in the asset price (right).
While risk aversion, the time preference, the interest rate, and the level
of risk faced by agents determine the consumption response to an asset price
shock, the size of the non-traded sector and productivity in that sector determine the aggregate employment response. Because the traded sector is
perfectly competitive and the non-traded sector monopolistic the non-traded
goods price is always high relative to the traded goods price. This in turn
makes employment in the non traded sector low — under most calibrations
32
slightly less than employment in the traded sector. Mian and Sufi (2014),
on the other hand, estimate non-traded sector employment to be two times
traded sector employment. While the construction of the model prevents it
from matching this observation, the bias towards higher traded sector employment is working against the overall employment response to an asset
price shock.
Parameter
σ
γ
φ
δ
pt
d
β
α
an
at
w
η
Table 1: Parameter Values
Value
Motivation
2
To match macro literature such as Backus
et al. (1992)
0.74
From Corsetti et al. (2012)
4
High end of Corsetti et al. (2012)
0.5
Equal preference for traded and non-traded
good
1
0.0036
Real returns on Wilshire 5000 at an asset price
of 1
.995
Low to keep steady state savings away from
the upper bound of the asset space
0.6
Labor share in traded sector
0.8
Traded sector slightly more efficient
0.9
See below
3
at above combined with a wage of 3 means
that in simulation labor approaches 1 in expansions.
w/2
Unemployment benefits on the high end of
those in the U.S. as a conservative estimate
This value for β is low compared with most of the rest of the macro literature; it implies an annualized
time preference of 0.94. The reason for this choice is to keep steady state asset holdings away from the
upper bound of the asset grid; while the lower bound should impact consumer choices, since it is
intended to model a debt constraint in the real economy, the upper bound should not. However, the
standard assumption of β = 0.96 at an annual rate based on real returns does not apply to models of
precautionary saving as it is calculated assuming a deterministic steady state. In models of
precautionary saving, denoting real steady state returns R, equilibrium requires Rβ < 1.
Values for the elasticities between traded and non-traded goods, the elasticities between non-traded goods, and the preference for the non-traded
good (which I keep at 0.5, implying no bias towards non-traded goods) come
33
from Corsetti et al. (2012). Finally, I set the (fixed) wage to 3 so that unemployment in the initial steady state (for q = 1) is 4% to match the top of the
business cycle in the U.S. over the last 50 years, and so that labor markets
clear in a simulated expansion (due to increasing asset prices). Table 1 gives
the value for each free parameter used to calculate the results in this section.
4.7.2
An Exogenous Asset Price Shock
The key variables for plotting the impulse responses of aggregate assets, consumption, and employment to a fall in the asset price are the state specific
steady states for aggregate bonds, B̄j , aggregate non-traded good consumption, C̄j , and the coefficients that determine the estimated law of motion
for these variables, γb and the γc,j ’s. With these values I can calculate the
impulse response for aggregate assets from equation 21, for aggregate nontraded consumption from equation 22, for non-traded sector employment
from 17, and for total employment from 18. Simulating the model over 500
periods12 I estimate γb = 0.97 from a regression of B̂j,t+1 on B̂j,t with an R2
greater than 0.99. In the simulation over 500 periods some of the asset states
at the extreme end of the distribution are not realized; for these states I use
the coefficient γc,j from the closest realized state. For those states that are
realized, my estimated parameter values range from 0.004 to 0.1. Using the
N
appropriate state contingent parameter in the regression Cj,t
= γc,j B̂j,t my
2
R is 0.95 indicating that the state contingent linear approximation fits the
true model well, though not as well as in the case of next period bond holdings. Figure 13 plots the state contingent expected realizations for Bt+1,j ,
N
, and employment against the actual realizations.
Ct+1,j
Using the estimated coefficients γb and γc,j as well as the state contingent
steady state values of aggregate bonds and aggregate non-traded goods consumption figure 14 plots the impulse response functions for the saving rate,
consumption, and employment given a fall in the asset price from 1 to 0.9.
The size of this demand and thus employment response to a change in the asset price depends both on the difference between steady state bond holdings
in the initial and new asset price state and on the state contingent consumption response γc,j . When the difference in steady state bonds between price
states is small, the demand and thus employment response will be small. As
illustrated by figure 12, this difference will be smaller when asset prices are
higher. However, the difference in steady state bonds between asset price
states is offset by the fact that the consumption response to a deviation in
bonds from the steady state, given by the coefficients γc,j , is increasing in the
12
This number of periods to simulate is low due to the computational intensity of the
problem.
34
Figure 13: Forecasted versus Realized Values
Forecasts for aggregate bonds are nearly perfect with R2 > 0.99
while (state dependent) forecasts for aggregate non-traded goods
consumption are slightly less accurate with an R2 of 0.95. Forecasted employment is just a linear function of forecasted non-traded
goods consumption and thus reflects its imperfections.
asset price. Table 2 illustrates this fact for several different scenarios. Thus
Table 2: Contemporaneous Employment Drop for Asset Price Scenarios
Asset Price
SS Bonds
γc,j
Fall in Total
Employment
1.4 to 1.3 (7%) -2.33 to -2.29 0.04
0.4
1 to 0.9 (10%) -2.01 to -1.78 0.01
0.6
0.6 to 0.5 (17%) -0.45 to 1.66 0.004
0.5
The employment response to an asset price shock is remarkably constant for a fixed
change in the asset price (0.1 in this case).
while the change in steady state saving becomes large as the asset price falls,
the contemporaneous employment response is remarkably stable: for a 0.1
fall in the asset price (a two standard deviation shock) employment falls by
around 0.5% in each case.
While a comparison between a stylized county level theoretical model and
national level empirical models is not ideal, the results from this section and
section 3 are strikingly similar. In the county level model a 10% fall in the
asset price from 1 to 0.9 corresponds to a 1% reduction in consumption and
a 0.6% fall in employment; in a simple linear VAR in the federal funds rate,
unemployment, consumption, home prices, and stock prices a 10% fall in
the stock price corresponds to a 0.6% fall in consumption and a 0.4% fall
in employment (home price shocks have a slightly larger impact). Moreover,
in my county level model I find that the consumption and employment response to an asset price shock is remarkably stable over a range of initial
35
Figure 14: Model Impulse Responses
When the asset price falls from 1 to 0.9 in period one individuals
begin to increase their asset holdings, corresponding to an increase
in aggregate assets (left hand panel) and the aggregate saving rate
(right hand panel).
asset prices. This finding parallels my empirical result that the state contingent responses of consumption or the saving rate to an asset price shock are
not significantly different across states. There are, of course, many reasons
why this comparison may be imperfect. First and foremost, the model is
designed to represent county, not national, level trends. Thus in the model
both the interest rate and the wage rate are fixed and exogenous, while at
the national level both may adjust. Additionally, the model does not allow for habit formation in consumption, a feature that would moderate the
contemporaneous consumption response to an asset price shock. For these
reasons the model results may be overstated. On the other hand, there are
also reasons why the model results may understate the consumption and thus
employment response to a financial crisis. One potential problem, as Carroll
and Dunn (1997) point out, is that the majority of the action in consumption
volatility comes from durable consumption; the model in this paper includes
only non-durable consumption which tends to be much more stable. Because
households find adjustments to durable consumption less painful (it’s easier
to put off buying a new car for a month than to put off buying groceries
over the same period), the durable consumption response to an asset price
shock is liable to be much larger than the non-durable consumption response
illustrated here. Another issue is that the model may underestimate the
risks faced by households in the U.S. First, the shock modelled here only
concerns asset prices, not volatility. In practice a financial crisis will likely
36
Figure 15: Model Impulse Responses
The increase in the saving rate corresponds to a fall in non-traded
good consumption (left). Since the non-traded goods market must
clear locally and wages are rigid this causes employment to fall
(right).
be associated with higher moments of variables; higher volatility corresponds
to a higher risk environment and thus a higher desired level of buffer assets.
Second, the asset price shock in this model is not associated with a tighter
borrowing limit (though the interest rate is increasing as the asset price falls
due to the fixed dividend). Third, unemployment benefits in the U.S. expire
after 90 days, thus the model likely underestimates income risk. Nevertheless
the message of the model is clear: given imperfect income insurance asset
prices have a direct impact on demand due to the precautionary motive; if
goods or labor prices cannot immediately adjust then this demand shock will
impact employment.
4.7.3
Summary
Section 4.2 establishes that for the individual it is optimal to increase savings
and cut consumption in the face of a negative asset price shock. The new
(lower) asset price represents a new state of the economy with a new steady
state quantity of assets. This new optimal quantity of assets is larger than
that in the former (high asset price) state. Section 4.7 illustrates the resulting
dynamic in a county level model. When asset prices fall employed agents
increase their saving rate which in turn reduces employment in the nontraded sector (which must clear locally). Non-traded sector employment
then recovers as the saving rate falls back towards zero in each progressive
period. Quantitatively, under the calibration used in section 4.7, a fall in
37
the asset price from 1 to 0.9 corresponds to a contemporaneous 1% fall in
non-traded sector employment or a 0.6% fall in total employment.
Figure 16: Simulated Results
In a simulation over 500 periods changes in the saving rate, and
thus consumption, drive unemployment; the correlation between
the two variables is 0.98. While in the beginning of the sample
the fraction of debt constrained agents moves with the saving rate
and unemployment, in the second half of the simulation households
accumulate more assets and fewer are constrained
While these results depend on incomplete markets and thus the existence
of a debt constraint, the contribution of binding debt constrains to employment volatility is negligible. Figure 16 plots the saving rate, the unemployment rate, and the fraction of debt constrained households for a simulation
over 500 periods. Changes in the saving rate, and thus changes in consumption, drive unemployment. Unemployment in turn effects the fraction of debt
constrained households; agents do not become debt constrained unless they
are unemployed and borrowing to smooth consumption. Thus causality runs
primarily from employment to the fraction of debt constrained households;
the variance of unemployment is in fact higher in the second half of the simulation when the fraction of debt constrained households is fairly constant
at around zero.
5
Discussion and Conclusion
In contrast to the existing financial accelerator literature, I propose that asset prices may have a direct impact on employment via demand due to the
precautionary motive. This mechanism accords with the findings in Mian
and Sufi (2014), Gan (2010) and firm surveys that the build up and subsequent collapse of home prices from 2004 to 2009 influenced output primarily
through demand. Moreover, my own findings in section 3 indicate that the
38
role of precautionary saving in transmitting nominal shocks to real variables
in the U.S. is not unique to the recent crisis. Using data from 1963 to the
present I find that home prices have a significant impact on the saving rate
regardless of the state of the economy or percentage of debt constrained
households, and that home prices and stock prices have a similar impact on
the saving rate. This rules out binding debt constraints as a sufficient explanation of the link between asset prices and demand. Nor can my results
be explained by the wealth effect alone; the data I use is in deviations from
trend while lifetime wealth for households with a reasonably long planning
horizon should be determined by changes in trends. These results suggest
that the precautionary motive plays an important role. Quantitatively, I find
that a shock causing a 10% fall in stock prices implies a 0.33 increase in the
saving rate or a -0.6% consumption deviation from trend.
The county level dynamic stochastic and, to the extent that non-traded
goods markets must clear, general equilibrium model in section 4 shows that
the level of asset prices should have a direct impact on demand and thus
employment via the precautionary motive. This result depends on two frictions: incomplete insurance markets in the form of a debt constraint and
nominal rigidities in the form of fixed wages. Incomplete insurance markets
in the form of a debt constraint imply that if an agent’s savings are too low
he risks being unable to borrow to smooth consumption against a negative
employment shock. This risk gives rise to the precautionary motive; agents
seek to maintain a sufficient level of buffer assets to avoid the sharp drop
in consumption that would result from hitting the debt constraint. A fall in
asset prices will reduce the value of this buffer wealth thereby increasing consumption risk, thus if asset prices fall agents will save in order to re-establish
the value of their precautionary wealth. The precautionary motive will arise
whenever insurance markets are incomplete. However, with flexible prices
labor markets will still clear when demand falls; wage rigidity implies that
this fall in demand causes a fall in employment as well. Quantitatively, under
the parametrization used in section 4.7, a fall in the asset price from 1 to 0.9
— approximately a two standard deviation shock — corresponds to a 1% fall
in consumption and thus a 0.6% fall in total employment. The magnitude
of the response to a two standard deviation shock is remarkably stable over
the asset price, in line with my empirical findings.
The mechanism at work in the section 4 model is the same as that in the
risk shock literature. Of the existing work on risk and uncertainty shocks this
paper is most similar to Leduc and Liu (2014); in their work the interaction
between employment search frictions and sticky prices gives rise to the precautionary motive so that changes in the variance of exogenous shocks cause
demand to fall. The authors do not, however, incorporate a debt constraint
39
(agents can buy and sell one period bonds in any amount). Nevertheless, precautionary saving is a feature in any model with consumption risk: Jensen’s
inequality implies that for any u(·) where u0 (·) > 0, u00 (·) < 0, u(3) (·) > 0
(thus u0 (·) is convex) the Euler equation
u0 (ct ) = β(1 + r)Et u0 (ct+1 )
ensures that for a given Et (ct+1 ), ct is decreasing in the volatility of ct+1 .
Because it depends on second and higher moments of ct+1 , the precautionary
motive is not captured in the popular solution approach that uses a deterministic steady state (which assumes no volatility) and is thus absent from
most log linear solution techniques.
This work highlights the need to consider heterogeneity, asset prices, and
the precautionary motive in theoretical models of business cycles. A next
step in this line of research is to find ways to incorporate these features
without undue computational complexity. This paper relies on several simplifying assumptions, mainly that the economy is a county (or small open
economy) in which wages, dividends, and the asset price are exogenous. Closing the economy and endogenizing the asset price is an important direction of
study. The solution technique used here is a potentially useful tool towards
that end. While I use numerical techniques to solve individual policy functions, allowing for both the precautionary motive and heterogeneity (since
some agents are employed and others are not), I use a linear approximation13 for key variables from the law of motion for the distribution of agents
Φt+1 = Γ(qt , Φt , qt+1 ). Unlike Krusell and Smith (1998), I posit these laws
of motion in terms of deviations from a steady state in which agents expect
shocks to occur (and shocks to individual variables such as employment are
in fact realized). This allows me to solve the model for a large number of
steady states. Finally, this paper hints at policy responses that could help
to mitigate demand shocks arising from a riskier environment, whether that
risk is due to lower precautionary wealth on account of lower asset prices
or due to higher moments of variables. The most important determinant of
risk in this model is unemployment insurance. If this insurance were state
contingent with higher benefits in the lower asset price (or higher risk) state
it would be possible, at least in a theoretical setting, to reduce and even
eliminate the demand shock arising from asset price volatility. While the
total elimination of this demand shock isn’t practical in the real economy,
counter-cyclical benefits may offer a higher return in terms of mitigating
unemployment than traditional expansionary fiscal policy.
13
Using a higher order approximation in this approach is as simple as including squared
N
or higher order terms in the regressions for B̂j,t+1 and Ĉj,t
.
40
The recent financial crisis has once again highlighted the role of asset
prices in determining real outcomes. This paper shows that precautionary
saving provides a direct link between the two via demand, and that this link
is clearly supported by empirical observations.
41
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44
A
The Role of Precautionary Saving
The first friction necessary to my results is imperfect income insurance which
gives rise to the precautionary motive. Carroll and Kimball (2006) provide
a definitive exposition of precautionary saving and precautionary wealth.
Central to my analysis is the fact that these models give rise to a unique
state contingent optimal level of wealth (see Carroll (2004) for a proof). As
an illustration, consider the dynamic programming problem
n
o
v(st , bt ) = max u(bt + y(st ) − qbt+1 ) + βEt v(st+1 , bt+1 )
bt+1
c1−σ
t
is a CRRA utility function, and q the discount on one
where u(ct ) = 1−σ
period bonds bt . Figure 17 reproduces figure 2 in Carroll and Kimball (2006)
Figure 17: Consumption Policy Functions
Whereas with complete markets and no aggregate risk β < q implies
consumption will asymptotically approach zero (perfect foresight
consumption is always greater than constant savings consumption),
incomplete insurance markets mean there is a unique target level of
savings at b̃.
using the above optimization problem with β = .9 and q = .92.14 The figure
depicts three consumption functions. The first is the solution to the model
.9
Other parameter values are σ = 2, st = [H, L] with transition matrix π =
.5
y(st ) = [1 .5], and a debt limit at bt = 0.
14
45
.1
,
.5
if there is no uncertainty (yt = ȳH ), labelled “Perfect Foresight”. The second, labelled “Constant Savings” gives the level of consumption necessary
to maintain a constant level of bond holdings (bt = bt+1 ) in the high income
state. Finally, the debt constrained consumption policy gives the high income state consumption policy function of a consumer with uncertain income
under incomplete markets, in this case a debt constraint at bt = 0. Because
β < q under perfect foresight consumption will asymptotically approach zero
as the agent tilts consumption towards the present period; this consumption
function is always greater than the constant saving function. However, given
financial frictions the precautionary motive induces the agent to save if bond
holdings get too low. The concavity of the consumption function implies that
the marginal incentive to save due to the precautionary motive is decreasing
in wealth: as bond holdings increase the difference between β and q overpowers the precautionary motive so that above b̃ the debt constrained agent
consumes more than the constant savings level. Below b̃ the opposite is true
so that b̃ gives the unique target level of savings.
If the state of the economy in this model changes, so will the optimal
level of asset holdings b̃. One possible change of state could be an increase
in uncertainty (such as an increase in the variance of yt ). This would in
turn increase the precautionary motive pushing b̃ to the right; Leduc and
Liu (2014) analyse this dynamic in a general equilibrium setting. Another
possible change in state is a fall in the asset price as illustrated by this paper.
B
The Structure of the Labor Market
While the debt constraint alone generates some employment volatility, I also
assume, in contrast to homogeneous agent models, that labor adjusts in the
extensive margin. This reflects the fact that for most people a working day
is a fixed number of hours and a full time employee works a fixed number
of days in a year. This structure could arise from union demands, labor
regulations, fixed costs of employment (such as healthcare in the U.S.), or
cultural expectations. What it means is that if the firm wants to reduce its
labor input it must do so by firing workers, not by reducing the number of
hours every worker spends on the job. If markets are complete and workers
can fully insure against temporary spells of unemployment, this feature will
not impact individual consumption functions. However, if markets do not
provide full insurance against unemployment the structure of marginal adjustments to labor can have a large impact on workers’ consumption policy
function.
As an example I consider an agent’s optimization problem described by
46
the value function
n
o
v(st , bt ) = max u(bt + n(st )α − qbt+1 ) + βEt v(st+1 , bt+1 )
bt+1
c1−σ
t
where u(ct ) = 1−σ
, n, labor, is a function of the exogenous state (st ), and
q is the discount on one period bonds bt . The aggregate economy produces
either a high or low value of output depending on the aggregate state, but
in one scenario there is no unemployment and all adjustments to labor are
in the intensive margin; this scenario corresponds to a representative agent
economy. In a second scenario labor only adjusts in the extensive margin so
that an individual is either working full time (n = 1) or unemployed, in which
case he receives an exogenously determined level of unemployment benefits
(for this example unemployment benefits are half of labor income in the low
representative agent state).
Figure 18: Consumption Policy Functions
In the heterogeneous agents case with unemployment risk (extensive
margin) the consumption policy function is always less than the
policy function in the representative agent case (intensive margin).
Figure 18 contrasts these two examples where the dashed red line plots
the consumption policy function when all the adjustment to labor happens
through the number of hours worked (which corresponds to the representative agent case) and the solid blue line plots the consumption policy function
when all the adjustment to labor happens through unemployment (which requires heterogeneous agents). The financial friction from appendix A remains
in both cases. As the figure illustrates, for any given level of savings (bond
47
Table 3: State Dependent Income
High Output
Low Output
Intensive Margin
1
0.66
Employed Unemployed
Extensive Margin
1
.83
.33
The income distribution in the heterogeneous agents model that allows for unemployment
(extensive margin above) is a mean preserving spread of the representative agent income
distribution (intensive margin above). Income when employed is lower in the extensive
margin low output case because the employed must pay the benefits of the unemployed.
holdings) agents consume less in the model that allows unemployment, although the two policy functions converge at the upper end of the asset space.
The expected value of lifetime income is the same in both scenarios but the
possibility of unemployment makes the extensive margin case higher risk;
as table 3 illustrates the heterogeneous agent income distribution is a mean
preserving spread of the representative agent income distribution.
When labor adjustment happens in the extensive margin the scale of
unemployment benefits determines the strength of the precautionary saving
motive: higher benefits reduce the cost of unemployment. The duration of
unemployment can also alter an individual’s consumption policy function. In
this case a longer duration of unemployment with the same level of aggregate
output means that the probability of becoming unemployed is less but once
out of work the chance of becoming employed again is also less. However,
the impact in this example appears to be small and non-monotonic.
In summary, the inability of individuals to insure against unemployment via
state contingent assets means that they choose to self insure by accumulating precautionary wealth. Rigid wages mean that labor markets do not
necessarily clear, and labor adjustment in the extensive margin generates a
mean preserving spread of the intensive margin income distribution thereby
increasing the precautionary motive. When agents face unemployment risk
the strength of the precautionary motive depends on the size of unemployment benefits.
C
C.1
Derivations for Regime Switching Models
STVAR Models
Denote st the observed variable
or set ofvariables that determine the state of
0
the model. Defining yt = ft ut ct pt where ft is the federal funds rate, ut
48
is unemployment, ct consumption, and pt prices, and zt = 1 yt−1 . . . yt−l
where l is the number of lags the smooth threshold VAR model has the form
(25)
yt = Φ(α + βst ) B1 zt + 1t + 1 − Φ(α + βst ) B2 zt + 2t
where Φ(·) is the standard normal cumulative
distribution function. Defining
ξ1t = Φ(α+βst ) and ξ2t = 1−Φ(α+βst ) I can write the likelihood function
for this model as
T X
2
Y
1
− 21
0 −1
− k2
(26)
L=
ξjt (2π) |Σj | exp − (yt − Bj zt ) Σj (yt − Bj zt )
2
t=1 j=1
so that, given ξjt my maximum likelihood estimates for Bj and Σj are
"
(27)
B̂j =
T
X
ξjt (yt zt0 )
#" T
X
t=1
#−1
ξjt (zt zt0 )
t=1
and
"
(28)
Σ̂j =
T
X
t=1
#−1
ξjt
T
X
ξjt (yt − Bj zt )(yt − Bj zt )0
t=1
Of course I do not know the parameters α and β in advance. Because Matlab’s fminsearch is much better behaved over the loss function15
loss =
T X
2
X
ξjt 0jt jt
t=1 j=1
than the likelihood function, in practice I numerically search for the values
of {α, β} that minimize this loss as opposed to the values that maximize the
likelihood (or log likelihood).
C.2
Markov Switching VAR
Suppose that the unobserved state of the economy is determined by a Markov
0
π
π
process with transition probabilities Π = 11 12 . Defining yt = ft ut ct pt
π21 π22
where ft is the federal funds rate, ut is unemployment, ct consumption, and
15
I choose this form for the loss function because minimizing it with respect to the
parameters Bj conditional on ξj yields the conditional maximum likelihood estimates I
use above.
49
pt prices, and zt = 1 yt−1 . . . yt−l where l is the number of lags my
model is
B1 zt + 1t if s = 1
(29)
yt =
B2 zt + 2t if s = 2
Following Hamilton (2005) denote the probability of being in state j in
period t given observations 1 : t and the parameters of the model Θ =
B1 , B2 , Σ1 , Σ2 , Π as
ξjt = P r(st = j|y1:t ; Θ)
Denote the p.d.f. of yt given state j, observations 1 : t−1, and the parameters
Θ as
1
− k2
0 −1
− 12
ηjt = f (yt |st = j, y1:t−1 ; Θ) = (2π) |Σj | exp − (yt − Bj zt ) Σj (yt − Bj zt )
2
Then, as in Hamilton (2005), the density of observation t given observations
1 : t − 1 and the parameters is
(30)
f (yt |y1:t−1 ; Θ) =
2 X
2
X
πij ξi,t−1 ηjt
i=1 j=1
the joint density f (yt , st = j|y1:t−1 ; Θ) is
(31)
f (yt , st = j|y1:t−1 ; Θ) =
2
X
πij ξi,t−1 ηjt
i=1
so that, following Bayes’ Rule, the probability of state j given observations
1 : t − 1 and the parameters conditional on observation t is
2
P
(32)
ξjt =
πij ξi,t−1 ηjt
i=1
f (yt |y1:t−1 ; Θ)
Suppose for now that I know the state probabilities ξjt for periods 1 : T . In
that case I could write the likelihood function for this model as
T X
2
Y
1
− k2
− 21
0 −1
(33)
L=
ξjt (2π) |Σj | exp − (yt − Bj zt ) Σj (yt − Bj zt )
2
t=1 j=1
50
or L =
T
Q
f (yt |y1:t−1 ; Θ). To find the maximum likelihood estimates of the
t=1
parameters I can differentiate the log likelihood function with respect to the
parameters yielding
T
∂ log(L) X
1
∂f (yt |y1:t−1 ; Θ)
=
∂Θ
f (yt |y1:t−1 ; Θ)
∂Θ
t=1
For the matrix of coefficients Bj I have
∂f (yt |y1:t−1 ; Θ)
−1
0
0
= f (yt , st = j|y1:t−1 ; Θ) Σ−1
j yt zt + Σj Bj (zt zt )
∂Bj
so that
T
∂ log(L) X f (yt , st = j|y1:t−1 ; Θ) −1 0
0
Σj yt zt + Σ−1
=
j Bj (zt zt )
∂Bj
f (yt |y1:t−1 ; Θ)
t=1
or
T
∂ log(L) X
−1
0
0
ξjt Σ−1
y
z
+
Σ
B
(z
z
)
=
t
j
t
t
t
j
j
∂Bj
t=1
so that the maximum likelihood estimate for Bj , assuming I know the state
probabilities ξjt , is
"
(34)
B̂j =
T
X
ξjt (yt zt0 )
#" T
X
t=1
#−1
ξjt (zt zt0 )
t=1
Similarly, the maximum likelihood estimate for Σj is
"
(35)
Σ̂j =
T
X
#−1
ξjt
t=1
T
X
ξjt (yt − Bj zt )(yt − Bj zt )0
t=1
Finally, I can estimate π11 and π22 as
π̂11 =
T −1
P
π̂22 =
Tt=1
−1
P
(36)
ξ1t
−1 T −1
P
ξ2t
−1 Tt=1
−1
P
t=1
ξ1,t ξ1,t+1
ξ2,t ξ2,t+1
t=1
Of course, these derivations assume I in fact know the sequences of state
probabilities ξ1,1:T and ξ2,1:T . However, I can solve this problem iteratively
51
using the expectation maximization (EM) algorithm in the above framework.
To do so, I begin with an initial guess of the parameters Θ0 and a guess of
the initial state probabilities ξi1 . Thus equipped, I can use equations 30
through 32 to solve iteratively for the sequences of state probabilities; this
constitutes the E step of the EM algorithm. Once I have ξ1,1:T and ξ2,1:T
I can solve for new parameter estimates Θ1 using equations 34 through 36;
this is the M step of the algorithm. I then go back to the E step to find a
new sequence of state probabilities, and iterate until state probabilities and
parameter estimates converge.
52