Download Quantitative Easing and the American Economy: How Saving is

Quantitative Easing and the American Economy:
How Saving is Saving Us From Inflation
Mark Nasca „12
Colgate University
April 30th, 2012
Abstract
Following the Great Recession (2007-09), the Federal Reserve (Fed) utilized monetary policy instruments that had
never been used in previous economic recoveries. With interest rates near zero, the Fed undertook rounds of
quantitative easing (QE), a non-standard policy, in an attempt to stimulate the economy and help bring the nation out
of the recession. In this study, the theoretical model presented by Nasca (2011) will be expanded to show that price
level can be stabilized when saving and the money supply increase in tandem, all else constant. Following the
theoretical discussion, this study will then utilize an intertemporal model with heterogeneous agents to describe the
U.S. economy in order to analyze factors affecting consumer saving decisions when QE policies are enacted
following an economic crisis. The goal of this model is to show how the saving decision is affected by the enactment
of QE in a crisis environment, given the lower prevailing interest rate scenario and elevated levels of economic
instability. This study finds that the increase in saving rate observed in the unfavorable rate environment can
potentially be attributed to the increased uncertainty in future income expectations and heightened levels of risk
aversion that are characteristic of a post-crisis economy. This serves as a theoretical justification for why saving has
risen during this stimulative period, thereby stabilizing price levels, as demonstrated empirically by Nasca (2011).
Keywords: Decision-Making under Risk and Uncertainty, Inflation, Saving, Money Supply, Monetary Policy
JEL Codes: D18, E21, E31, E51, E52
1
Introduction
Since the end of the Great Recession, generally assumed to be June of 2009, the United States
economy has experienced two rounds of quantitative easing (QE), and there is still discussion by
the Fed about the potential for a third round. According to basic economic theory, an increase in
the money supply, ceteris paribus, will lead to a decrease in the value of money, an increase in
the price level, thereby creating inflation. With some relatively loose assumptions, this
conclusion can be proved using simple theoretical models. It is interesting that with an increase
in the monetary base of 147% in two years, journalists have mentioned the risk of deflation. It is
clear that there are some factors that are preventing the onset of inflation.
The route to recovery from the Great Recession is fundamentally different than previous
economic recoveries because the specific economic shock experienced was an aggregate demand
shock, rather than an aggregate supply shock like the recessions seen in the 1970s. For the first
time in recent history, we are seeing a significant trend of deleveraging by households (Figure 1)
and de-risking in consumer portfolios.1,2 In addition to this uncommon consumer behavior, the
Federal Reserve (Fed) is currently enacting unprecedented policies in America; aggressive QE
policies were used for the first time in the United States during the current recovery. Recent
empirical research by Nasca (2011) suggests that the money supply and personal saving rate
affect the price level in opposite directions. The empirical model used in this study yielded
coefficients on these variables of similar magnitudes, with the personal saving rate having a
slightly larger effect. According to the data, the movement of these variables during the recovery
has been quite similar. The study concludes that it is possible that significant inflation has been
deferred, or perhaps even prevented, by the escalation of the personal saving rate during this
period of monetary expansion.
The goal of this paper is to expand the theoretical model introduced by Nasca (2011) and
attempt to uncover the levers affecting saving rate during a recovery. Specifically, this study
seeks to explore what is driving the deleveraging activity seen in the current recovery that was
not observed in previous recession recoveries. In a period of monetary expansion, interest rates
are depressed and thereby should lead to lower levels of saving relative to an environment in
1
2
Source: Hilsenrath and Simon, “Spenders Become Savers, Hurting Recovery,” WSJ, Oct. 22nd , 2011
Source: Light, Pilon and Silver-Greenberg, “The Young and the Riskless,” WSJ, Nov. 5th, 2011
1
which rates have not dropped. However, in actuality, we are seeing increased saving levels,
despite the sharp drops in interest rates. This is a puzzle that requires exploration because it is
contrary to economic theory. By highlighting the link between saving and price level, and
uncovering the levers which affect saving decisions, this research will help increase
understanding of which variables the Fed should pay the closest attention to when planning its
exit strategy from QE, in order to avoid massive inflation as the economy recovers.
Motivated by an expansion of the theoretical work by Champ and Freeman (2001) and
Nasca (2011), a theoretical intertemporal economic model with heterogeneous consumers will be
created in this piece. Since QE programs are enacted during time periods of economic stress, the
model of the consumer will be specified, through the utilization of an isoelastic utility function,
in such a way that incorporates risk aversion behavior. Uncertainty will be accounted for by the
inclusion of an expected value function for second period income that is affected by general
economic outlook. Inspired by Disyatat (2010) and applied to agents rather than financial
intermediaries, a consumer‟s expected value of future income will be affected by their perception
of the economy. By accounting for economic stress in the consumer, this model accurately
portrays an economy where agents are prone to de-risk and potentially save more, given
heightened uncertainty in the economy. The goal of this model is to show how the saving
decision is affected by ambiguity in future income, given the lower prevailing interest rates and
elevated levels of economic stress.
Given the characteristics of a post-crisis economic environment, it is plausible, from the
model specified in this study, that agents will elect to save more, despite the significantly
depressed yields on saving caused by the enactment of QE programs. As expected, the results
produced by the model underestimate the saving rate in the economy and overestimate the
percent change in the rate observed. These differences are present because, unlike the real
economy, the model only considers saving in the form of a risk free asset. Since the saving rate
includes holdings of risky assets, the omission of said assets from the model leads to rates being
underestimated. The overestimation of the percent change is the result of an increased preference
by agents to augment their holding of risk free assets, relative to risky assets, in a time of
economic uncertainty. Therefore, it can be said that although the rates produced are slightly
different than the true values, they differ in intuitive ways such that the results can be considered
2
reasonable approximations of the economy of interest in this study. Given these outcomes, we
find that the increase in saving in such an unfavorable rate climate that is currently being
observed in the U.S. can potentially be the result of elevated levels of risk aversion and
heightened uncertainty in future income expectations. Although QE programs are intended to be
stimulative, this study shows that increased uncertainty in future income expectations and
augmented risk aversion may potentially overwhelm the effect of lower interest rates and thereby
cause agents to save more in a rate environment designed to induce consumption.
2
Literature review
Given how recently the Fed‟s QE policies were enacted in the U.S., there is understandably a
lack of published research on this practice and its effects on the U.S. economy. The goal of this
research is to fill this void and attempt to rationalize specific consumer behaviors that we are
observing, post-crisis. In studying about the historical economy of the United States, there is
specifically little relevant research regarding QE. Therefore, the bulk of the literature which
motivates this piece comes from studies on QE in England and Japan. Both economies are
comparable to the U.S. in that they experienced a time period of monetary expansion, post-crisis,
in a very low interest rate environment. Taking into consideration these circumstances, research
on these countries can be transitively applied to studies on QE policies in America.
Bricker, Bucks, Kennickell, Mach and Moore (2011) present an analysis of the change in
household saving and consumption behavior in the U.S. since the recent recession. They utilize
panel data from the Survey of Consumer Finances (SCF) for the years 2007 and 2009. Through
their analysis, the authors find that respondents to the survey exhibit a universal desire to
increase precautionary saving, independent of the magnitude of the change in income
experienced from the crisis. Respondents attribute their desire to augment saving to an increase
in future income uncertainty and reduced levels of risk tolerance. Since the goal of our study is to
accurately replicate the U.S. economy, the inclusion of these behavioral elements is essential for
completeness of the model. Thus, functional forms for future income expectations and utility will
be utilized, allowing for the incorporation of different levels of uncertainty and risk aversion,
relative to the period being analyzed.
3
In Disyatat‟s 2010 piece, he introduces new factors that he argues should be considered
in a lending authority‟s optimization problem. Through a complex derivation, Disyatat makes a
sound argument that a bank‟s balance sheet strength, the quality of its assets and its economic
outlook must be considered when determining lending behavior. The end result is a model which
helps to intuitively explain bank lending (or lack thereof) post-recession. Disyatat is directly
contesting the conclusion of Bernanke and Blinder (1988), who argued that increased reserves
will always lead to increased lending. While providing banks with increased liquidity may make
lending more attractive, the economic climates in which such QE policies occur tend to be
uncertain and saving-inducing. The macroeconomic data collected from the current recovery in
the U.S. shows that there has been a significant increase in saving and household deleveraging,
despite the large amount of QE that the Fed has undertaken. Whereas traditional bank models fail
to explain this decrease in lending in this environment, Disyatat‟s model provides a sound
rationalization by accounting for the economic climate in the bank‟s lending decision.
Much as a negative economic outlook affects a bank‟s prospective lending behavior, this
same economic perception will also alter the saving and consumption decisions of the individual
agents in the economy. As with any type of economic crisis, it can be assumed that uncertainty in
future income for agents increases as the economic climate deteriorates (i.e., default and
unemployment risks increase). Inclusion of a factor that allows an agent‟s expectations of future
income to vary, depending on the economic climate, will help explain the increasing saving rates
during, and immediately following, a recession. This variable will also allow for experiments
with a theoretical model, showing the differences in outcomes had QE been undertaken in a time
of economic prosperity. It is possible that this research will show that the positive results from
QE will be attenuated in such a time of de-risking and deleveraging, a similar conclusion to the
one reached by Disyatat. Since there is no accurate measure for such a qualitative variable, this
study will use aggregate default risk in the economy as a proxy for economic outlook. This
variable captures the overall economic climate, in that uncertainty is shown to be associated with
higher rates of default. Also, default rates tend to be higher in times of economic stress.
Since QE was used over a decade ago in Japan, the body of research available on its
effects is quite broad. Initially, with interest rates already near zero, it was adopted in 2001 as a
policy to stave off deflation and stimulate their economy. While no inflation was produced,
4
according to a lengthy study conducted by Schenkelberg and Watzka (2011), the program led to
short run increases in industrial activity. Their conclusion states that the QE in Japan, relative to
their domestic product (GDP), was on a “small” scale. This monetary policy program was only
6% of Japan‟s GDP, and it took about four years to complete. In the United States, QE programs
undertaken have totaled over $1.7 trillion USD, or 10% of the U.S.‟s GDP, in just two years.
Relative to Japanese programs, the U.S. QE policies are twice the magnitude in GDP terms and
were conducted in half the time. With these stark differences in the programs highlighted, it is
quite possible that the conclusions of Schenkelberg and Watzka concerning Japan may not be
directly applicable to the United States. One of the goals of this study is to fill this gap in
research by suggesting that while changes in money supply should, theoretically, yield inflation,
other variables are influencing increases in saving, despite the rate environment, and thereby
stabilizing the price level.
Bedford, Berry, Nikolov, Young and Robson (2009) provide the reasoning behind QE
and explain the program‟s desired outcomes in England. Using the same rationale as the Bank of
Japan, the Bank of England enacted this policy when interest rates were low and additional
stimulus was required to reach an inflation target. The authors cite various pieces of research that
confirm, through empirical analyses, the link between price level and money supply. The
uncertainty is not in whether or not prices will rise, but rather when will they rise. The intent of
this study is to explain how consumer and lending behavior changes after a recession and during
a period of monetary expansion. By understanding which factors have contributed most to the
deleveraging and risk avoidant behavior being observed and, in turn, those that have most
significantly aided in stabilizing price level, we will have greater understanding of those
variables and data points that the Fed should pay closest attention to over time. One of the goals
of this paper is to uncover those variables so that the Fed will, when seeking to avoid explosive
inflation, know which factors to watch most diligently as they plan the timing of their exit
strategy from this QE.
3
Modeling inflation and saving – standard framework3
In order to model how increases in the money supply and saving decisions in the United States
economy can cause changes in the price level within an economy, we begin with the two period
3
Portions of this section are adapted from a similar derivation presented in Nasca (2011)
5
overlapping generation model developed by Samuelson (1958). We assume two groups of
consumers in the economy, differing in their initial endowments and incomes, and a single
monetary authority. Motivated by the work of Champ and Freeman (2001), growth in the money
supply is defined as
where
is the gross rate of monetary expansion and each time period is denoted by . Because
this proof is used to explore price level in a period of monetary expansion, we will assume
is
greater than 1. Sargent and Wallace (1983) show that consumers are subject to a budget
constraint each period defined as
where
is consumption,
is the value of money in terms of the consumption good,
quantity of money held by an agent,
is a consumer‟s income, and
is the
is initial endowment
in terms of the consumption good. All variables are shown in terms of time period, and in terms
of consumer type of , which takes a value of
, where one type is a borrower and the other
is a saver.
We aggregate the budget constraints of all consumers in the model economy and find the
equality of supply and demand in the money market to be
(
(
where
is the aggregate money supply,
)
(
))
is the proportion of the population of type i, and
is the population, all listed in terms of time period, . Next, we must isolate
with
(
(
)
We know that saving in each period is equal to
and thus we can simplify the former equation to
6
(
))
which leaves us
(
(
)
(
))
To find changes in the price level, we must first observe how the rate of return of money changes
by period. To explore this relationship, we define rate of return of money to be
(
(
)
(
(
(
)
(
))
))
Since QE policies involve large injections of cash in relatively short periods of time, we assume
that population and the proportion of the population belonging to each consumer type remain
constant between periods. We also know that average gross saving in the economy,
, is equal
to
(
)
(
)
Therefore, the above formula can be simplified to
(
)
( )
(
)
( )
( )
The price level is inversely related to the value of money, therefore
where
is the price level in time period, . The ratio of price levels between periods can be used
to inspect inflation. This ratio is defined as
( )
7
Through some simple algebra, the price level in the second period,
, can be defined in terms
of
( )
While the formula above appears quite simple, it has very powerful implications.
Presuming the assumptions of the model hold true, a value of
unchanged, will lead to
that is greater than
greater than , with gross saving
and therefore, inflation in the economy.
Economic data shows, however, that aggregate saving in the United States economy has
increased significantly during the current recovery. Since we assume the proportions in each
category do not change, when added together, the saving in each group yield the average gross
saving in the economy. Since the decisions of all agents collectively drive price level, the model
can be simplified to
( )
where
( )
( )
(1)
is the change in average gross saving in the second period. Shown by this simple proof,
and assuming the assumptions of the model hold, unless rate of growth in the money supply is
equal to rate of growth in gross saving, we will see price level increases. Since, according to
Keynesian theory (Hicks, 1980), the augmentation of the money supply depresses interest rates,
it would appear counterintuitive to witness an increase in saving during the same time period.
Contrary to accepted theory, given that we are, following a period of rapid monetary expansion,
currently seeing stable price level, combined with significant increases in personal saving rate,
one must conclude that there are other factors at work. In the intertemporal model to be
presented in this paper, we will seek to explore what is driving the increase in saving rate that is
currently being observed in the economy.
4
A modified intertemporal model which accounts for risk
In this paper, a two period intertemporal saving and consumption model will be created in which
there are three different types of consumers. The model contains selected aggregate variables that
capture the general economic climate, as well as variables which are specific to individuals and
8
capture the desired heterogeneity in the behavior of actors in the economy. All monetary policies
will be enacted exogenously, since the goal of this research does not include analyzing the
implication of such policies from the government‟s fiscal perspective. As a result, the banking
intermediary will not be explicitly modeled. Observations will be made about consumer behavior
to assess the efficacy of QE policies enacted during recession recoveries. The goal of this
theoretical model is to show how the consumer‟s saving decision is altered by increased
uncertainty in future income and lower prevailing interest rates as a result of QE policies.
4.1
Agent specification
Within the model economy, there are three different types of consumers. For presentation
simplicity, we will suppress the subscript which denotes agent type, , from the variables in the
model. If these subscripts were displayed, all the variables in the model except those that do not
vary by consumer type would possess them. Thus, all individualistic parameters would contain
subscripts to denote agent type. While not shown explicitly, these subscripts are still utilized
when parameterizing and running experiments with the model. The duration of each period is a
single year. Agents begin the period with a net worth of
. We make the assumption that an
agent‟s wealth is stored in mostly illiquid assets such as real estate (commercial and residential)
and private equity. Coupled with the fact that we consider each period to last only a single year,
the net worth of an agent is only considered as determinant for which quintiles of the population
we will use for the study. Beyond this determination, while we include
in the model
derivation and discussion for completeness, we do not consider it in the experiments. If we were
to include
, agents would then have the ability to essentially “consume” their house and all
their other endowed assets within a single year, which is something almost no rational agent
would ever consider to be reasonable. In the first period, consumers earn
with absolute
certainty. Since the labor decision is not discussed and the government is not an actor in the
economy, taxes have been omitted from the model. In the first period, agents can chose to either
save or consume. Saving yields an exogenously set rate of return of , while borrowing occurs at
a rate of
, such that
is greater than
. Since we allow for no default in this model, we
assume all borrowing is repaid in full in the second period.
In order to incorporate different income expectations, relative to the economic climate, an
expected value function is used for second period income. In the second period, consumers
9
expect to earn
, subject to (
), where
is a measure of economic uncertainty (Disyatat,
2010). Past research provides evidence that the magnitude of uncertainty in future income should
be different for each wealth level. In America, recent years have shown a significant increase in
the displacement of low-skill workers by technological advances. As a result, unemployment
rates are different between high-skill and low-skill groups of workers (Dreze & Sneessens,
1997). High-skill workers, in the aggregate, are more educated and earn more, as their jobs pay
higher wages given the limited labor supply. The converse argument can be made for the lowskill workers and thus, it is understandable that the gap between the two groups‟ unemployment
rates continues to widen as technology advances. Thus, it can be assumed that low-skill
employees (i.e., low-wage earners), have a higher uncertainty in their future income, as their jobs
may disappear because of technological innovations. Alternatively, high-skill workers are less
susceptible to this substitution effect and therefore have higher certainty in future income. While
it would be ideal to capture this effect by placing a different weight on
in each respective agent
group‟s expected value function, determining the actual magnitude of this weight proves to be
nearly impossible. Since this value cannot accurately be determined, we have elected to allow
agents of all classes to be subject to the same weight of . In doing so, we avoid including
parameter values that may be scrutinized for being deduced in a spurious manner and thereby
preserve the integrity of the model. As previously stated, consumers must repay all loans in this
period prior to consuming goods. We can distill all of this information into the agent‟s
intertemporal budget constraint
( )
To capture risk aversion behavior in the economy (Bicker et al., 2011), an isoelastic
utility function is used in the following form (Arrow, 1963; Pratt 1964)
(
where
)
(
)
(
)
represents an agent‟s coefficient of relative risk aversion (CRRA). The CRRA has been
shown to be independent of income and, therefore, will differ based only on the economic
climate (Kaplow, 2003). Agents with a high CRRA will have a low elasticity of intertemporal
substitution and, therefore, will seek to smooth consumption as much as possible. This
10
characteristic of the utility function captures a consumer‟s risk aversion behavior and is the
reason why it was selected for this model. The chosen form of the consumer model allows us to
explore the effects of a recessionary shock on consumer saving behavior. This shock will be
represented by increased economic uncertainty via increases in CRRA and downward revisions
of future income expectations. While the aforementioned implications of a recessionary shock,
ceteris paribus, can easily be shown to lead agents to save more, the lower rates yielded by QE
policies (which will influence agents to save less) make the saving decision somewhat
ambiguous.
The discount factor in the utility function,
, is essentially a measure of an agent‟s
patience. Fisher (1930) was one of the first to make the argument that low income agents, all else
constant, tend to exhibit the highest degree of impatience, “partly from the thought that provision
for the present is necessary both for the present itself and for the future as well, and partly from
lack of foresight and self-control” (p. 73). More recent empirical research has affirmed this
conclusion and shown that low income agents do, in fact, tend to be less patient than high
income ones and therefore will likely put a lower value on consumption in future periods, and, as
a result, consume more in the current period (Lawrance, 1991). Thus, the associated
for each
consumer type will be directly related to the agent‟s income. Higher income agents will possess
higher values for
and thus, will be “more patient”, whereas lower income agents will have
a lower value and thereby appear to be “less patient”. The use of differing values for
of
allows for
the model to more realistically represent the U.S. economy which, like all other systems,
contains heterogeneous agents with varying time-preferences and income levels. Since
is
specific to the individual consumers, and variables are already included in the model that account
for economic climate and relative risk aversion, the discount factor will not change when the
economy is shocked.
The major simplification of this model is the agent‟s decision to either save or consume.
Our rationalization behind considering saving (essentially, a safe allocation) as the only way an
agent can transfer wealth to the next period is actually quite close to what is being observed in
the U.S. economy. Previously, residential real estate and equity investments have been favorable
ways of allocating portions of income that are not consumed, in order to preserve and enhance
wealth. Since the recession, however, agents have expressed a unilateral desire to reduce the risk
11
of their asset holdings (Bicker et al., 2011). Additionally, the relatively lower rates of return
when compared to historical indexes, and further, the highly uncertain sentiments regarding the
probability of return on these investments (Figures 2 and 3) have led to a significant decrease in
utilization of these traditional investment choices. The high uncertainty in equity markets has
caused trading volumes to drop to their lowest levels since 2008. 4 Many are citing the lack of
participation of “mom and pop investors” as the reason we are seeing such depressed volumes. 5
Thus, it is clear that many households are holding their wealth in cash positions, rather than
diversifying through other investment products that may have riskier outcomes and thus, our
model is not far from the reality we are currently observing.
Another significant simplification of this model is that we consider all of the functions
and decisions of the financial intermediary to be exogenous. While an ideal model would contain
a dynamic, well-specified “bank” to facilitate saving and lending, as well as the determination of
market rates for both of these actions, the simplifications required to include a functioning bank
were far too extensive for the model to be considered realistic. Thus, since it is quite difficult to
adequately and convincingly model a bank, we are considering the decisions of the financial
intermediary to be exogenous. These decisions, however, will be motivated by economic data
and past research to ensure that they closely reflect the U.S. economy, thereby yielding a model
from which we can draw tangible, convincing conclusions.
4.2
General equilibrium and optimization problem
Based on the assumptions presented in the previous section, we can solve the optimal solutions
for the saving and consumptions decisions of this model through numerical optimization by
using Lagrange multipliers. The derivation for the solutions can be found in Appendix AI. For
convenience, the optimal solutions for
,
, and
are shown below
( )
(2)
4
Source: Wang, “Stock Trading Lowest in U.S. Since 2008 After Fund Withdrawals, Job Cuts,” Bloomberg, Jan.
23rd, 2012
5
Source: Cox, “Stock Market Rally Still Missing One Thing: A Crowd,” CNBC, Jan. 20th, 2012
12
( )
[
]
(3)
( )
(4)
The relationship between each optimal allocation and other variables will now be discussed. The
explanations in these sections will be general and, following the model‟s parameterization in
Section 6, we will discuss specific conclusions that can be drawn from the model with respect to
the U.S. during the current recovery.
Optimal consumption in period 1,
Although certain effects are somewhat ambiguous, the optimal solution for first period
consumption gives insight into which variables positively and negatively influence the decision
to consume. To begin, it is quite clear that all forms of wealth (
,
, and ( )) positively
affect consumption in the first period. Given an increase in any of these variables, ceteris
paribus, consumption should increase. Since the goods available for consumption in this
economy are normal goods, the amount consumed should increase as wealth increases.
Conversely, as wealth decreases, agents will consume less of the good, as would be expected.
The effect of
is also unambiguous; as this variable approaches 1 and agents discount future
period consumption less, their consumption in the current period declines. Given the form of the
equation, it is quite hard to see by eye which direction
with respect to
will influence
is shown in Appendix B, Part II. Since
derivative is negative. This shows that as
. A partial derivative of
is always less than 1, this partial
increases, first period consumption deceases. Lastly,
the association between interest rate, , and
is found also to be an inverse relationship. We
will talk in greater depth about the effect of interest rate on the model in a succeeding discussion
on optimal first period saving
13
Optimal consumption in period 2,
Similar to
, we again see the income effect at work in
, as increases in any form of wealth
will cause an increase in consumption in the second period. This, again, makes intuitive sense
because of the normal nature of the consumption good considered in this economy. As the
discount factor placed on second period consumption, , approaches 1,
will increase as an
increase in this variable signifies an increased preference for second period consumption. Again,
given the placement of
in an exponent, it is far easier to see its association with
of a partial derivative. A partial derivative of
Appendix B, Part II. Since
that as
with respect to
in the form
is, again, presented in
is always less than 1, this partial derivative is positive. This shows
increases, second period consumption also increases. The relationship between
and
is easy to see; clearly as the interest rate increases, second period consumption will also increase,
all else constant.
Optimal saving in period 1,
When looking at the optimal solution for first period saving, it is immediately clear that the
effects of the various forms of wealth on this decision are varying. We find that as
increases,
saving also increases. This positive association, however, is not mimicked by the other variables.
Both
and ( ) are inversely related to first period saving. Because saving is defined as
income less consumption, we see that optimal saving is simply first period income less
this fact in mind, we can easily deduce the relationships between
increase in any factor that is positively associated with
have a negative relationship with saving. Thus, , , and
will cause
. With
and the other variables. An
to decrease and thus, will
are all positively associated with first
period saving. In a situation where all else is held constant, a decrease in
is associated with a
decrease in saving. Dynamic economies like the U.S., however, rarely experience periods during
which only a single variable in the economy undergoes change. Although we have seen interest
rates plunge to their lowest levels in recent history, agents are still choosing to increase their
saving (Figure 4). Clearly, there must be other variables at work that are influencing this
decision, because it runs counter to economic intuition. With this model, we will later simulate
the U.S. economy and offer possible rationalizations for why agents are increasing saving in an
environment that offers historically low returns.
14
4.3
Effects of uncertainty on saving and consumption decisions
The model created in this study differs from traditional intertemporal models in a few respects,
most notably the inclusion of a variable to account for economic climate and an expected value
function for future income that utilizes an agent‟s perception of the economy. While the
relationship of ( ) with respect to each saving and consumption variable was discussed in
Section 4.2 in passing, here we will discuss the effects in terms of their partial derivatives, as
well as how specific aspects of the expected value function alter the value of ( ) and thereby
how they affect
,
, and
with respect to (
.
)
Solving for the partial derivative of
with respect to ( ), we find
( )
(
)
(5)
It must be noted that this partial derivative is greater than zero, meaning that an increase
(decrease) in the expected value of second period income will lead to an increased (decreased)
amount of consumption in the first period. Since we have simplified this economy such that
agents can chose to either save or consume in each period, it is intuitive that greater expectations
for future income will lead agents to consume more in the current period, thereby saving less.
with respect to (
)
When we solve for the partial derivative of
with respect to ( ), we find
(
( )
)
(
)
(6)
Again, we have a result that matches economic intuition, further confirming the validity of the
model‟s specification. The partial derivative shows that an increase (decrease) in expected future
period income will yield an increase (decrease) in future period consumption. It is quite logical
15
that as income increases in a period, an agent will consume more. Thus, this partial derivative
shows a positive relationship between the two variables.
with respect to (
)
With the partial derivative of
with respect to ( ), we can explicitly show the negative
relationship between optimal saving level and expected future income
( )
(
)
(7)
In words, the above partial derivative shows that a decrease (increase) in expected future income
yields an increase (decrease) in saving in the current period. Since we have established that
expected future income is a function of the economic climate in the current period, this model
predicts that rational agents will save more as the economy deteriorates (i.e., as the value of their
expected future income decreases). By including a variable that accounts for current economic
outlook in the function for expected future income, we have created forward looking agents that
respect risk in the economy and prepare accordingly, based on information available to them in
the current period. This point is noted here because, given the relationship between expected
future income and saving, we can now see a clear link between the saving decision and economic
outlook. This is an idea we will expand upon later in this piece.
5
Utilization of model to explain current recovery
It has been made clear in this research that consumer saving behavior during the current
economic recovery has been paradoxical. QE increases available liquidity and drives down
interest rates in an economy, in an attempt to lure agents away from saving and toward
consumption. Traditional models that fail to include variables that create a fully economically
“conscious” consumer (i.e., one who not only looks at rates, but who also perceives risk and
uncertainty and adjusts accordingly) will predict this result of increased consumption in a
depressed rate environment. This is the link frequently referenced by the Fed when they discuss
the intuition behind the QE policies undertaken in the United States. While this paper does not
seek to make conjectures about whether or not these QE policies have sped up the U.S. recovery,
16
we use the model specified in this study to explain the phenomena being observed and show
clearly that, given the current economic climate, it is likely that certain factors are driving agents
to save in a rate environment that is specifically designed to discourage this behavior. This
conclusion can only be reached through the model specified in this study, as it accounts for the
perceived risk and uncertainty present in a post-recession economy that drive such a contrarian
decision.
When researching U.S. QE policies, consideration of the factors affecting consumer
behavior is something that is frequently omitted from two-dimensional analyses of such
programs. This paper‟s greatest contribution to the body of research is that it gives significant
consideration to the specific consumer behavior tendencies during such QE policy periods.
Unless invoked for inflationary purposes, QE is typically utilized as a last resort policy during
periods when interest rates are depressed near zero and the economy remains unresponsive. This
“unresponsiveness” is likely because the economy experiences a substantial economic shock that
thereafter works to alter consumer behavior. The Fed, in their analysis of the projected efficacy
of QE, omits consideration of the heightened level of risk aversion and increased uncertainty that
pervade an economy following a significant economic shock. By conclusion, the theory of
monetary expansion, in the form of manipulated easing, yields an ambiguous outcome when
actually introduced into an environment in which risk aversion is elevated and expectations of
future income streams have become increasingly uncertain.
While the latter portion of Section 4 focused on explaining the relationship between
discrete variables in the model and the saving and consumption decisions, no comments were
made about the outcomes when all these parameters experience changes of varying signs and
magnitudes simultaneously. In the following sections, we will calibrate the model, based on
specific economic data and past research, and then continue by testing the model in various
experiments, finally explaining the results. The testing process will be explicitly defined later.
Ultimately, the results will represent our best interpretation of the U.S. economy and the effect of
QE policy on the agents and their consumption decisions, given their aversion to risk and their
increased uncertainty in future income, as is typically seen in a post-recession climate.
17
5.1
Model parameterization
The comprehensive analysis of the 2007 SCF by Wolff (2010) is used to conveniently find
values for the model‟s parameters for each agent type. For simplification, we have also elected to
only use groups whose agents possess a net worth of $100,000 or greater. This ensures focus on
groups that pay positive taxes and are able to engage in some form of discretionary consumption.
At net worth levels below this point, agents have such limited wealth that nearly all consumption
decisions are assumed to be necessary. Therefore, cash is spent only on necessary items with
very little evidence of variance in their consumption behavior. This election limits us to
considering only 60% of the U.S. population.6 Thus, since the model economy will possess three
different types of agents, the top three quintiles will be used in this study. The top 20% (top 1% top 20%) will be represented by those with values of 1, with the lower two quintiles (21% 40%, 41%-60%) represented by values of
and , respectively. From Table 4 in the appendix
of Wolff‟s piece, we find the net worth values,
, for each quintile under consideration in our
model. Additionally, we also find the relevant first period incomes,
, for each group. As
previously discussed, these values for net worth are used only to determine which quintiles have
the financial means to engage in discretionary consumption and that
will not actually be
contained in the experiments with the model. The differences in wealth are captured by the
differing incomes of each agent group.
The range of acceptable values for the discount factor, , is 0.95 to 0.98 (Zhang, 1997).
Since we established in Section 4.1 that the discount factor is positively associated with income,
we have assigned the lowest value in this acceptable range to the lowest income agents and the
highest value to the highest income agents. An average of the two numbers was used for the
middle income group.
As mentioned earlier, future period income expectations, ( ), are a function of both an
agent‟s anticipated second period income,
as well as an agent‟s perception of the economic
climate, . In our model, agents anticipate their income to remain the same from period to period
6
The contribution to the saving rate by the bottom 40% of the population is insignificant. For example, assume all
agents save the same percentage of their income, 5%. In this scenario, the contribution of the bottom 40%, using
income data from the 2007 SCF (Wolff, 2010), is approximately 5%. Thus, given their low level of income, unless
their saving behavior is considerably different from the other groups, exclusion of this portion of the population will
not significantly alter the results of the model.
18
and thus
function of
is set equal to
. Their expectation of second period income, however, is also a
and, therefore, while they may expect their income to remain the same, they also
recognize that there is a probability that their income may be different. The variance in the
expectations of their future income increases as
increases. To give a real example, higher rates
of default are associated with higher levels of unemployment. Hence, in a higher default climate,
the risk of an agent losing his job, and thereby a portion of his income stream, escalates (i.e., the
economic climate is more stable and predictable in a lower default rate climate) (Figure 5). In the
first quarter of 2007, the delinquency rate on all loans was reported as being 1.75%. Exactly
three years later, this rate had ballooned to 7.50%, a nearly 330% increase!7
Whereas unemployment rate only captures changes in an agent‟s wage income, default
rate encompasses this effect, as well as the impact of diminished yields on other investments. An
agent will most often default on a loan because he or she cannot afford the loan payments.
Simply being unemployed may not inhibit an agent from being able to afford his or her loan
payments. An agent may be receiving income from other sources, such as investments. We
therefore expect
to increase both when fewer people are employed and when investments
perform poorly. Increased unemployment and decreased investment yield will cause total income
that agents receive in a period to be diminished, leaving them with less cash to allocate to loan
payments and ultimately, higher rates of default.
In this model, agents assume that all people who default are unemployed, but all
unemployed may not necessarily default. Figure 5 affirms this assumption by showing that the
rate of delinquency on loans has never exceeded the unemployment rate. Unlike the probability
of default, the default rate is readily available from published materials. We therefore make the
simplifying assumption in our model that the probability of default is equal to the default rate.
While the rate of default is clearly different in each income group, this simplification captures
the effect we desire within the model: certainty in future income expectations decreases with
instability in the economic climate increases.8
7
Delinquency rate on all loans in 1Q07 and 1Q 10, respectively. We consider 1.75% to be our “base case” value for
and 7.50% to be our “recessionary case” Data from Board of Governors of the Federal Reserve System
8
Robustness checks for yielded results that did not differ significantly from those produced using the values
selected for the experiments. See Appendix II, Part C for a summary of these tests.
19
Given that the default rate is less than the unemployment rate, we proceed to make
another simplifying assumption: agents expect to be unemployed and receiving benefits if they
default. Thus, the default rate is used in the expected value calculation in order to determine the
magnitude of unemployment benefits and their effect on future income. The calculation of these
specific benefits for each group is also somewhat simplified. Unemployment benefits are
currently based on a percentage of weekly wages and this percentage varies from state to state.
To make things easier, we have selected the average size, relative to weekly wage, and duration.9
Agents collecting benefits receive, on average, 36% of their weekly wage rate for 20 weeks. For
higher wealth agents (the top 2 quintiles used in this study), however, their weekly wages exceed
the maximums offered in most states. For simplicity, we have given the two wealthier income
groups a ceiling, as determined by looking at various state mandates. The average maximum
benefit per week comes out to approximately $500. With all these factors accounted for, the form
of the expected value calculation can be presented for second period income
( )
(
(
)
)
(
)
Research has shown that the values for an agent‟s CRRA, , can empirically range in
value from 2 when normal to 10 when stressed (Kaplow, 2003). Although a CRRA value of zero
would make agents truly risk neutral, given that this value reduces the utility function to a linear
formula, Kaplow‟s research shows that a
of zero is not observed empirically. Since the lowest
value found to be significant and reliable based on economic intuition and empirical data in his
study was 2, we have selected this to be the baseline level of risk aversion in a normal climate.
While a number of values larger than 10 were reported in his piece, Kaplow ultimately concludes
that the high end of a value for CRRA, based on his extensive review of other published
material, is likely 10. In this study, a CRRA value of 10 will correspond to agents in a highly risk
averse, post-crisis climate. We rationalize using the highest available value for the CRRA based
on the characteristics of the recent recession. The Great Recession was the longest and most
severe, in terms of percent GDP decline, in recent history. Additionally, household values, a
9
Source: “Unemployment Insurance at 75:Assessing Benefit Eligibility, Adequacy and Duration”, National
Employment Law Project, http://www.nelp.org/page/-/UI/2010/UIat75wentworth.pdf?nocdn=1
20
historically “bullet-proof” asset class, saw declines in prices for the first time (Figure 3). Finally,
the analysis of recent consumer data by Bicker et al. (2011) provides concrete evidence that
Americans have exhibited significant increases in aversion to risk following the recession. Given
these facts, it is therefore reasonable to assume that risk aversion is at its highest level.
The manner in which personal saving are measured, as defined by the Bureau of
Economic Analysis (BEA), is somewhat more complicated than it appears on the surface:
Personal saving may also be viewed as the net acquisition of financial assets (such as
cash and deposits, securities, and the change in life insurance and pension fund reserves),
plus the net investment in produced assets (such as residential housing, less depreciation),
less the net increase in financial liabilities (such as mortgage debt, consumer credit, and
security credit), less net capital transfers received10
Upon reading this definition of which allocations are considered to be “saving”, it is clear that a
single interest rate will not accurately describe the returns agents perceive when making their
consumption/saving decision. In our study, we have decided to use the average money market
yield for each period being analyzed. Money market yield was selected because these funds are
composed of various instruments mentioned in the BEA definition for saving (cash, CDs,
government securities). Additionally, investments in money market accounts are highly liquid
and nearly risk-free, allowing them to closely approximate the usability of normal deposits and
liquid security holdings.
QE involves the buying of treasury notes, similar to those held by money market funds.
The significantly depressed money market fund yields that we have seen since the recession are
the result of the Fed‟s substantial QE bond purchasing activity. Since the money market yield is
directly affected by easing activity, we capture the effect of QE in our model. Additionally, the
yields of money funds are also sensitive to changes in interest rates, meaning that the effects of
the interest rate policies that preceded QE will also be felt as additional depressions in the yield
of these accounts. Money market yields for 1Q07 and 1Q10 were 3.608% and 0.858%,
10
Source: “A Guide to the National Income and Product Accounts of the United States”, BEA,
http://www.bea.gov/national/pdf/nipaguid.pdf
21
respectively.11,12 A summary of the values used for each parameter in the model, as outlined in
the preceding sections, can be found in Table 1 below.
Table 1
Summary of model parameterization data13
Type,
$2,278.9
$257.9
(
5.2
)
$291.0
$106.0
$74.7
$46.7
(
)
2-10, based on economic climate
0.980
0.965
0.950
1.75%, 7.50%
3.608%, 0.858%
Model experiment scenarios
With values established for each variable, the model can now be put through a series of
experiments. Four different trials will be performed, with one being a baseline and three being
experiments off that base. The baseline saving and consumption model will be run with “precrisis” values for all parameters. This scenario will be considered the intertemporal saving
decision in the time period before any type of shock is applied; this is our modeling of the
American economy in the first quarter of 2007. In this scenario,
1.75% and
takes the base case value of
is set to 2 to represent a normal economic climate. The interest rate, , will be set to
3.608%, since this was the value observed prior to the enactment of QE. The first experiment
allows us to inspect the effects of a QE program performed in a non-crisis environment. Since
we assume no changes in the economic climate,
respectively. To represent the effect of QE,
and
will still take the values of 1.75% and 2,
will be set to 0.858%. The goal of this specific
experiment is to isolate the effects of QE on the economy. Given the abundance of previous
research on the topic and the Fed‟s rationalization for the policies, this trial should yield results
that prove the policy is stimulative in a normal economic climate. The second experiment
11
Average Money Market Yields in 1Q 2007 and 1Q 2010, respectively. Data from Bankrate.com
Interest rate in the model, , will consider (1+Money Market yield) to be the rate. The “(1+” is omitted in order to
yield cleaner derivations and easier to follow formulas.
13
All income and net worth values are listed in dollars in thousands
12
22
observes the effect of the recent economic crisis on consumer saving decisions, assuming no Fed
policy intervention. By removing the policy effects, the goal of this trial is to isolate the
influence of increased risk aversion and higher uncertainty in future income expectations on the
saving decision. For this experiment, since we assume no policy action by the Fed,
to 3.608%. Assuming the economy is stressed, we see
and
will be set
take values of 7.50% and 10,
respectively. The final experiment combines the effects of a QE policy and an economic crisis.
In this experiment, levels for
influenced
and
again rise to 7.50% and 10, respectively. Post-crisis QE-
will be assumed to be 0.858%. This trial most accurately models the U.S. economy
and can offer real conclusions for what levers are driving the increase in saving being observed.
These parameters are summarized in Table 2 below.
Table 2
Summary of selected values for parameters in each experimental scenario
Experiment
5.3
Baseline
QE, no Crisis
Crisis, no QE
QE, Crisis
2
1.75%
3.608%
2
1.75%
0.858%
10
7.50%
3.608%
10
7.50%
0.858%
Quantitative results
Herein, we will present the results from each experimental scenario, given the parameter values
as outlined in Section 5.2. The lengthiest discussion will be undertaken with respect to the results
from the third proposed scenario. The results from the baseline scenario and the third experiment
will be compared to the actual values seen in the U.S. economy during the pre- and post-crisis
periods, defined as 1Q07 and 1Q10. In each scenario, we will run the model and calculate the
saving decision and compare these results to the baseline scenario‟s output. The experimental
scenarios will use their corresponding values from Table 2. For brevity, a single graph (Figure 6)
is presented in Appendix A, containing the aggregate data for all three agent types from all four
scenarios. Graphs for each agent type in the third experiment scenario will be included in
Appendix A. Also, the numerical results for each individual type in all scenarios can be found in
Table 3 in Appendix A. Finally, for the first two experiments, commentary will be made only on
the aggregate saving decision, rather than discussing each individual group specifically. The
23
primary goal of the first two experiments is to prove that agents in the model behave rationally.
Specifically, we are going to use the first two experiments to show that agents respond properly
to isolated shocks on economic climate and saving yields. For the third experiment, which
represents our attempt to model the enactment of QE in the post-crisis American economy, each
specific agent group will be discussed in isolation. In addition, a commentary will be provided
on the aggregate saving rate in comparison to observed data in the United States. With the
specifics of the experimental procedure defined, we can now discuss the outcomes.
QE in a non-crisis environment
As stated earlier, this experiment is meant to reflect the effect of enacting a QE policy during a
time of economic stability. With rates depressed as a result of the policy, the outcome is a saving
rate that is significantly lower in the second trial than in the first. Specifically, aggregate saving
rate is found to be 1.87% in the baseline scenario and only 0.56% in the experimental scenario.
Again, these results, in addition to all others, can be found in Table 3 in Appendix A. With an
interest rate 76.2% smaller in the experimental trial, we find saving to be 69.8% less than the
baseline value. The difference between these two figures results from the differing values for
in each wealth group. Since the wealthy group is responsible for more than 80% of the saving
activity observed and they only slightly discount second period consumption, the percent change
in the saving rate closely approximates the percent change in the interest rate. The conclusion
that can be drawn from this model is that, as expected, when offered a lower return on saving,
agents will rationally save less and thereby consume more. Holding other economic factors
constant, a QE policy can generally be thought to have a stimulative effect.
Crisis environment with no policy intervention
This experiment is utilized to isolate the effect of the recent economic shock, assuming there is
no Fed policy intervention. With far higher uncertainty as result of the economic shock and
elevated levels of risk aversion, holding the interest rate constant, we observe a substantial
increase in saving across all agent types. Aggregate saving rate is determined to be 4.98%; this
marks an increase, from the baseline scenario, of over 166%. Given the higher levels of
uncertainty in future income, agents essentially expect to receive less in future and thereby save
more in the current period, in order to smooth consumption. Since we assume agents have an
24
expected value for future period income, rather than having knowledge of this income with
absolute certainty, whether or not their assumption is correct does not matter. Regardless, agents
will perceive the increased uncertainty in expectations as a negative income shock in future
income and therefore, save more in the current period. This effect is compounded by the
increased level of risk aversion seen in post-crisis economies, which also induces higher levels of
precautionary saving.
QE in a crisis environment
In this experiment, we combine the features of the first two scenarios, in an attempt to accurately
model the enactment of QE in the post-crisis economic climate seen in the United States. The
results from the preceding experiments show that QE policies and the occurrence of an economic
crisis produce competing effects in isolation. Whereas the QE policy induces consumption, the
increased uncertainty in future income expectations and heightened levels of risk aversion which
result from the economic crisis lead to higher levels of saving. When these two events with
opposing consequences occur concurrently, the model specified in this study predicts an increase
in saving for all agent types, despite the severely depressed yield offered.14 Specifically, postcrisis aggregate saving rate is determined to be 3.75%, despite a yield more than 75% lower than
the baseline scenario results. This result is strong evidence for the original hypothesis presented
in this piece. Within the framework of the model presented, the effect of increased uncertainty in
future income, coupled with higher levels of risk aversion, overwhelm the impact of a significant
decrease in interest rates, thereby leading to an increase in the saving rate.
Each wealth group experiences a shock to their income expectations of nearly the same
magnitude. The high wealth group‟s expectations are reduced by 5.62% (Figure 7), whereas the
low wealth group‟s prediction is shrunk by 5.03% (Figure 8). The medium wealth group falls in
the middle, with a reduction of 5.06% (Figure 9). Given the similarity of the relative magnitude
of the shocks to income expectations, it is interesting that the groups differ significantly in terms
14
Although one of the simplifications of the model is that no agents borrow in any scenario, this result makes sense
in the context of the recent recession. According to Bicker et al. (2011), we have seen a net reduction in consumer
debts, rather than an increase in said burdens. While some may argue that debt has increased since leverage ratios,
(
), have risen, this is merely the result of decreases in asset values, rather than increases in debt. In light of this
fact, it is clear that the exclusion of borrowers from our model does not reduce its explanatory power significantly.
25
of the percent change in saving rate. Upon closer examination, we find that this difference can
mostly be attributed to the differing values of
for each group, as well as each group‟s level of
consumption. The utility function used in this model exhibits diminishing returns. As a result, a
negative income shock that causes agents to decrease consumption will have a different marginal
effect on each group, given each agent‟s position on the utility function. From Figure 10, we can
see that the low wealth agents are positioned such that a decrease in marginal utility is
experienced most significantly, relative to the other groups. The tangent lines on the graph
represent estimates of the marginal change faced by each group. Since the low wealth group will
experience the most significant shock to utility, it is quite logical that they are the group in the
model that exhibits the most significant corrective saving decision following the simulated
economic crisis. The change in saving observed in the low wealth agents, which has a greater
magnitude than all other groups, is simply due to their position on the utility function. This act of
precautionary saving is performed in order to preserve what little utility the low wealth group
has. Conversely, the high wealth agents exhibit the smallest change in saving. For high wealth
agents, a change in expectations will cause them to underreact, as compared to other groups,
since they serve to lose far less in terms of utility.
Through added analysis, we find that smaller values for the discount factor, , further
intensify this effect. As one may predict, agents with high values for
will generally exhibit
higher saving rates than those with lower values for . This is clearly shown in Figure 11,
although the gap between the saving rates of the groups shrinks as risk aversion increases. This
relationship is confirmed after further differentiation of Equation 7, with respect to
( )
(8)
(
From Equation 8, we see that as
)
increases, the corresponding change in saving, relative to a
specific change in income, decreases in magnitude. Since we have established that all agent
classes experience a relatively similar income shock, we can simplify this further and conclude
that as
increases in the model, the change in saving response to shocks will decrease in
magnitude. Now that we have explained the differences between the agent groups, a thorough
26
comparison between the aggregate results and real data from the U.S. can be undertaken in order
to see how the experiment output compares and what additional conclusions can be made about
the impact of QE policies during a time of economic crisis.
Comparison of experimental results to real data
The reported values for personal saving rate in the U.S. for 1Q07 and 1Q10 are 2.6% and 4.9%,
respectively.15 The percent increase in saving rate observed in the U.S. during said three year
period is found to be over 88%. Experimentally, the rates computed for each period were 1.84%
and 3.64%, respectively. The percent increase in saving yielded by the model is 101.06%. While
the experimental results closely approximate the real economy, they are different in ways that
can be rationalized through an intuitive argument regarding the differences between the types of
investable asset classes available in the real economy that are not accounted for in the model
specified in this study.
In the model, agents can only save in a risk-free asset with a known return. In actuality,
given the definition of saving by the BEA, agents can also purchase risky assets offering
potentially higher yields, albeit at the added cost of increased uncertainty in these returns. Since
the model economy only considers a risk free asset, it underestimates saving rate because it omits
an entire potentially higher yielding asset class utilized by various agents to preserve wealth
across periods. Given that the personal saving rate is essentially a weighted average of all assets
matching the BEA definition of “saving”, the model in this study will underestimate the
observed rate because it only accounts for a single component of the BEA‟s definition; the risk
free asset (cash, CDs, Government securities). If the model included a risky asset class, it is clear
that the output values would more closely approach the true values seen in the economy.
Whereas the data points produced by the model underestimate the values observed in the
economy, the percent change in saving rate yielded is greater than the true percent change
produced in the economy. This result, again, can be reconciled through an argument regarding
the omission of risky assets from the model. In an uncertain economic time, it is clear that agents
will shift their wealth holdings out of risky asset classes, such as equities, whose returns are
dependent on economic prosperity, and into risk free assets that offer returns which are certain
15
Seasonally adjusted monthly data was aggregated by averaging. Data from BEA.
27
and far less volatile. Therefore, in relative terms, one would expect an agent to increase holdings
of risk free assets more than they would increase their holdings of risky assets. This statement is
confirmed by the experimental results of this study; in a model which allows agents to only
invest in a risk free asset, we see a percent change in saving rate that is greater than the percent
change observed in the real economy. Decreased participation by individuals in the equity
markets, as discussed earlier in Section 4.1, further strengthens this claim. It is clear that there
has been a shift out of risky assets and into risk free assets following the recession. This is likely
the result of the increased uncertainty that is typical of a post-crisis environment.
With confidence that the model is properly specified and the results are realistic,
providing an approximation the real economy, we can make comments about the efficacy of QE
in a crisis environment. Given the choice between an unstressed economy with high interest rates
and a recessionary economy with low interest rates, agents have chosen to save more in the low
rate environment. With the inclusion of measures that account for risk aversion and changes in
future income expectations based on the economic climate, agents prefer to save more, despite
significantly lower interest rates. With Figure 6, we can see that simply following the individual
curves allows us to compare how saving rate differs in each rate environment, holding
expectations on future income constant. As agents become more risk averse, even in the
predictable, normal climate trial, agents will chose to save more and sacrifice some units of
utility, in favor of an increased amount of guaranteed wealth to be preserved for the next period.
This added saving reflects a more risk averse agent‟s desire to more significantly smooth
consumption between periods.
In addition to heightened risk aversion, agents also have more uncertainty about future
income expectations during a crisis. When these two parameters are accounted for, we see that
the combined effects overpower the impact of a lower saving yield and we find agents in the
economy saving more despite the lower rates offered. The conclusion that we can draw from this
model is clear; it is difficult for a QE policy to be stimulative in a period in which uncertainty
and risk aversion are also high. Although this conclusion was derived from a simple model,
given the soundness of, and reasonable economic intuition behind, both the specifications and
assumptions, the model developed in this piece can be used to draw real conclusions about the
28
U.S. economy. Notably, the dramatic increase in saving and the resultant lack of inflationary
pressure clearly run counter to the Fed‟s stimulative intent in implementing the QE program.
6
Conclusion
Given the magnitude of the liquidity injected by the Fed in recent years, it is clear that there are
some heretofore unforeseen variables preventing significant inflation from appearing in the
United States. From the empirical work of Nasca (2011), we find that the concurrent increase in
saving during the time period of monetary expansion may potentially rationalize the delay of
what would have appeared to be inevitable inflation. We clearly derive the links between saving,
supply of money and price level in this piece to confirm this empirical conclusion that price level
in a period of monetary expansion can be stabilized by a tandem increase in saving. The study
presented in this piece explores the potential underlying factors that are driving agents to save
more in the U.S., despite the institution of many stimulative policies by the Fed that otherwise
might have actively discouraged such behavior.
From the model specified in this piece, we closely approximate the results observed in
the U.S. economy. Given the conditions of our model, we find that the increase in saving rate can
potentially be attributed to the significant increases in uncertainty of future income expectations,
coupled with the considerably higher levels of risk aversion that are commonly present in postrecession economies. Although the specified model is quite simple, the results are powerful, in
that they offer a potential rationalization for why the recovery from this economic crisis has
proceeded at such a slow rate.
The conclusions of this model suggest that while the recovery remains slow, increased
confidence and decreased risk aversion behavior may lead to increased consumption and
decreased saving. In the event this occurs, given the large increases in money supply from
enactment of the QE programs, Equation 1 highlights the potential inflation the U.S. may
experience. Given the stabilizing relationship between saving and price level during a time of
monetary expansion, the Fed must be cognizant of consumer sentiment and risk aversion levels
in order to properly time an exit strategy from QE so that excessive inflation can be avoided.
While the model presented here is shown to be economically intuitive and parameterized
using proper historical data, an expansion of this framework could potentially yield even more
29
valuable results. Additional utility could be realized by including a bank in the model. The
inclusion of an intermediary actor would provide a more detailed opportunity to inspect the
mechanism through which QE is enacted. Insertion a bank into the framework would
additionally enable the model to more closely replicate the true, multidimensional economy. The
model in this piece only allows us to comment on, and draw conclusion about, the efficacy of
such programs in regard to individual agents. Such a modification would be a valuable extension,
capable of further increasing the explanatory power of the model derived in this piece.
If empirically possible, a study that adjusts the magnitude of the effect of uncertainty in
future income expectations, based on wealth groups, could also greatly add value. This would
make the model increasingly more accurate and also, give further confidence in the results it
provides. It would also be interesting to see if the conclusions reached in this study are different
than those in one utilizing a model that weights
differently, according to income.
As discussed in the analysis section, the values produced by this model predictably
underestimate saving rate and overestimate the change in the rate, as a result of the omission of
risky assets from the model. Given that risky assets are included in the definition of saving by the
BEA, a more close approximation of the U.S. economy could be achieved through a model that
includes this asset class. In order to include risky assets in a model containing heterogeneous
consumers, correct data must be found on the relative portfolio allocation of each group. The
inability to find accurate data on portfolio allocation is precisely the reason why different asset
classes were omitted from this model.
7
Acknowledgements
This piece would not have been possible without the constant support of, and commitment by,
my thesis advisors, Professor Felicia Ionescu, Ph.D. (Colgate University), and Professor Dean
Scrimgeour, Ph.D. (Colgate University). I would also like to thank Colgate University for
providing the resources necessary to pursue and successfully complete this research.
30
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32
Appendix A: Figures and tables
Part I: Motivational charts and graphs
Figure 1 Percent change in household debt in quarters following a recession
Percent Change in Household Debt
35%
Year
recovery
began
30%
1954
25%
1958
20%
1961
15%
1970
10%
1975
1982
5%
1991
0%
2001
-5%
2009
-10%
0
1
2
3
4
5
6
Quarter after the start of the recovery
7
8
Source: Hilsenrath and Simon, “Spenders Become Savers, Hurting Recovery,” WSJ, Oct. 22nd , 2011
Figure 2 Indices for present value of an investment in January of 2006 in housing, equities
(represented by S&P 500), and commercial real estate
33
Figure 3 House price index over the last decade
Figure 4 Effective federal funds rate and personal saving rate since 1990
34
Figure 5 Aggregate default rate and unemployment rate since 1985
35
Part II: Experimental charts and tables
Figure 6 Aggregate experimental results
6.0%
4.98%
5.0%
Savings Rate
3.75%
4.0%
3.0%
Crisis, no
QE
QE, Crisis
2.0%
QE, no
Crisis
Baseline
1.87%
1.0%
0.56%
0.0%
0
2
4
6
8
σ
10
12
Figure 7 QE in a crisis environment: high wealth agents
4.0%
3.5%
3.90%
Saving Rate
3.0%
2.5%
2.0%
QE, Crisis
2.06%
1.5%
(
)
Baseline
1.0%
0.5%
0.0%
0
2
4
6
σ
36
8
10
12
Figure 8 QE in a crisis environment: medium wealth agents
4.0%
3.5%
3.47%
Savign Rate
3.0%
2.5%
2.0%
QE, Crisis
1.5%
(
1.60%
)
Baseline
1.0%
0.5%
0.0%
0
2
4
6
σ
8
10
12
Figure 9 QE in a crisis environment: low wealth agents
4.0%
3.5%
Saving Rate
3.0%
3.38%
2.5%
2.0%
QE, Crisis
1.5%
(
)
Baseline
1.0%
1.22%
0.5%
0.0%
0
2
4
6
σ
37
8
10
12
Figure 10 Relative position of each wealth group on the utility curve
High Wealth
Med. Wealth
Utility
Low Wealth
Total Consumption
Saving Rate
Figure 11 Saving rate for different values of
β High
β Mid
β Low
𝜎
38
Table 3
Summary of experiment results16
Baseline
Results
Type,
(
)
257.90
252.59
5.31
2.06%
253.56
254.52
74.70
73.50
1.20
1.60%
73.57
73.50
46.70
46.13
0.57
1.22%
46.00
45.77
1.87%
-
QE, no Crisis
Experimental results
Type,
(
)
257.90
255.94
1.96
0.76%
253.56
254.45
74.70
74.48
0.22
0.30%
73.57
73.48
46.70
46.74
-0.04
-0.09%
46.00
45.76
0.56%
-
-69.79%
-
4.98%
-
166.70%
-
3.75%
-
101.06%
-
Crisis, no QE
Experimental results
Type,
(
)
257.90
244.68
13.22
5.12%
239.31
245.06
74.70
71.18
3.52
4.71%
69.85
71.18
46.70
44.54
2.16
4.62%
43.68
44.47
QE, Crisis
Experimental results
Type,
(
16
)
257.90
247.84
10.06
3.90%
239.31
247.55
74.70
72.10
2.60
3.47%
69.85
71.91
46.70
45.12
1.58
3.38%
43.68
44.93
All income, consumption and saving values are listed in thousands
39
Appendix B: Derivations
Part I: Lagrange optimization
Consumer Optimization Problem:
Endowments:
( )
( )
( )
( )
Choice Variables:
Utility Function:
(
)
(
)
(
Lagrange Method:
(
)
( )
(
)
( )
Subject to:
( )
( )
( )
( )
[
( )
]
( )
40
)
( )
( )
( )
Solve in terms of
( )
( )
( )
( )
(
(
(
(
)
)
)
( )
)
(
)
(
(
(
)
)
( )
( )
)
( )
( )
[
]
( )
41
( )
Part II: Selected partial derivatives
With respect to ( ):
(
)
(
( )
:
( (
( )
( )
With respect to , for
(
)
(
( (
)
(
)
)
(
With respect to :
With respect to :
)
( )
( ))
(
( (
(
)
)
(
(
( ))
)
)
(
)
)
( )
( (
)
(
( ))
( (
)
( (
)
)
(
( ))
(
)
42
)
( ))
)
)
( ))
Part III: Robustness checks for
Table 4
Summary of robustness check results
Experiment 1
-10%
Changes in
BASE
+10%
Experiment 2
-10%
Changes in
BASE
+10%
Experiment 3
-10%
Changes in
BASE
+10%
Baseline
1.79%
-4.2%
1.87%
1.95%
4.2%
Experiment
0.48%
-14.3%
0.56%
0.65%
16.1%
-73.0%
-4.6%
-69.8%
-66.8%
-4.3%
Baseline
1.79%
-4.2%
1.87%
1.95%
4.2%
Experiment
4.64%
-6.8%
4.98%
5.32%
6.8%
159.6%
-4.2%
166.7%
173.2%
3.9%
Baseline
1.79%
-4.2%
1.87%
1.95%
4.2%
Experiment
3.41%
-9.2%
3.75%
4.10%
9.2%
90.6%
-10.4%
101.06%
110.7%
9.5%