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

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 Source: Hilsenrath and Simon, “Spenders Become Savers, Hurting Recovery,” WSJ, Oct. 22

nd , 2011

2 Source: Light, Pilon and Silver-Greenberg, “The Young and the Riskless,” WSJ, Nov. 5

th, 2011

2

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

3

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.

4

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,

5

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)

6

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, is the

quantity of money held by an agent, is a consumer‟s income, and 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 which leaves us

with

( ( ) ( ))

We know that saving in each period is equal to

and thus we can simplify the former equation to

7

( ( ) ( ))

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

( )

8

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 greater than , with gross saving

unchanged, will lead to that is greater than 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

( )

( )

( )

(1)

where 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

9

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

10

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 low-

skill 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

11

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 of

a lower value and thereby appear to be “less patient”. The use of differing values for 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

12

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. 20

th, 2012

13

[

( )

] (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 will influence . A partial derivative of

with respect to is shown in Appendix B, Part II. Since is always less than 1, this partial

derivative is negative. This shows that as 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

14

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 in the form

of a partial derivative. A partial derivative of with respect to is, again, presented in

Appendix B, Part II. Since is always less than 1, this partial derivative is positive. This shows

that as 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 . With

this fact in mind, we can easily deduce the relationships between and the other variables. An

increase in any factor that is positively associated with will cause

to decrease and thus, will

have a negative relationship with saving. Thus, , , and 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.

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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

16

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,

17

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.

18

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.

19

and thus is set equal to . Their expectation of second period income, however, is also a

function of 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.

20

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

21

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

22

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 $291.0 $106.0

$257.9 $74.7 $46.7

( ) ( )

2-10, based on economic climate

0.980 0.965 0.950

1.75%, 7.50%

3.608%, 0.858%

5.2 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 “pre-

crisis” 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, takes the base case value of

1.75% and 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, and will still take the values of 1.75% and 2,

respectively. To represent the effect of QE, 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 12

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

23

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, will be set

to 3.608%. Assuming the economy is stressed, we see and 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 and again rise to 7.50% and 10, respectively. Post-crisis QE-

influenced 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 Baseline QE, no Crisis Crisis, no QE QE, Crisis

2 2 10 10

1.75% 1.75% 7.50% 7.50%

3.608% 0.858% 3.608% 0.858%

5.3 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

24

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

25

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, post-

crisis 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.

26

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

27

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.

28

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

29

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 post-

recession 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

30

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.

31

References

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33

Appendix A: Figures and tables

Part I: Motivational charts and graphs

Figure 1 Percent change in household debt in quarters following a recession

Source: Hilsenrath and Simon, “Spenders Become Savers, Hurting Recovery,” WSJ, Oct. 22

nd , 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

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

35%

0 1 2 3 4 5 6 7 8

Per

cen

t C

ha

ng

e in

Ho

use

ho

ld D

ebt

Quarter after the start of the recovery

1954

1958

1961

1970

1975

1982

1991

2001

2009

Year

recovery

began

34

Figure 3 House price index over the last decade

Figure 4 Effective federal funds rate and personal saving rate since 1990

35

Figure 5 Aggregate default rate and unemployment rate since 1985

36

Part II: Experimental charts and tables

Figure 6 Aggregate experimental results

Figure 7 QE in a crisis environment: high wealth agents

4.98%

3.75%

0.56%

1.87%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

0 2 4 6 8 10 12

Sa

vin

gs

Ra

te

σ

Crisis, no

QE

QE, Crisis

QE, no

Crisis

Baseline

3.90%

2.06%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

0 2 4 6 8 10 12

Sa

vin

g R

ate

σ

QE, Crisis

Baseline ( )

37

Figure 8 QE in a crisis environment: medium wealth agents

Figure 9 QE in a crisis environment: low wealth agents

3.47%

1.60%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

0 2 4 6 8 10 12

Sa

vig

n R

ate

σ

QE, Crisis

Baseline

3.38%

1.22%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

0 2 4 6 8 10 12

Sa

vin

g R

ate

σ

QE, Crisis

Baseline

( )

( )

38

Figure 10 Relative position of each wealth group on the utility curve

Figure 11 Saving rate for different values of

Uti

lity

Total Consumption

Sa

vin

g R

ate

𝜎

β High

β Mid

β Low

Low Wealth

Med. Wealth

High Wealth

39

Table 3

Summary of experiment results16

Baseline

Results

Type,

257.90 74.70 46.70 -

252.59 73.50 46.13 -

5.31 1.20 0.57 -

2.06% 1.60% 1.22% 1.87%

( ) 253.56 73.57 46.00 -

254.52 73.50 45.77 -

QE, no Crisis

Experimental results

Type,

257.90 74.70 46.70 - -

255.94 74.48 46.74 - -

1.96 0.22 -0.04 - -

0.76% 0.30% -0.09% 0.56% -69.79%

( ) 253.56 73.57 46.00 - -

254.45 73.48 45.76 - -

Crisis, no QE

Experimental results

Type,

257.90 74.70 46.70 - -

244.68 71.18 44.54 - -

13.22 3.52 2.16 - -

5.12% 4.71% 4.62% 4.98% 166.70%

( ) 239.31 69.85 43.68 - -

245.06 71.18 44.47 - -

QE, Crisis

Experimental results

Type,

257.90 74.70 46.70 - -

247.84 72.10 45.12 - -

10.06 2.60 1.58 - -

3.90% 3.47% 3.38% 3.75% 101.06%

( ) 239.31 69.85 43.68 - -

247.55 71.91 44.93 - -

16

All income, consumption and saving values are listed in thousands

40

Appendix B: Derivations

Part I: Lagrange optimization

Consumer Optimization Problem:

Endowments:

( )

( )

( )

( )

Choice Variables:

Utility Function:

( )

( )

( )

( )

Lagrange Method:

( ) ( ) ( )

Subject to:

( )

( ) ( ) [ ( )

]

( ) ( )

41

( )

( )

( )

Solve in terms of

( )

( )

( )

( )

( )

( )

( )

( )

(

)

(

)

( )

( )

( )

( )

( )

( )

( )

[

( )

]

( )

42

Part II: Selected partial derivatives

With respect to ( ):

( )

( )

( )

( )

( )

( )

( )

With respect to :

( ( ) ( ))

( )

( ( ) ( ))

(

)

( ( ) ( ))

(

)

With respect to , for :

( ( ) ( ))

( )

( ( ) ( ))

( )

( ( ) ( ))

( )

With respect to :

( )

( )

( )

( )

( )

( )

Part III: Robustness checks for

Table 4

Summary of robustness check results

Experiment 1

Baseline Experiment

-10% 1.79% 0.48% -73.0%

-4.2% -14.3% -4.6%

Changes in BASE 1.87% 0.56% -69.8%

+10% 1.95% 0.65% -66.8%

4.2% 16.1% -4.3%

Experiment 2

Baseline Experiment

-10% 1.79% 4.64% 159.6%

-4.2% -6.8% -4.2%

Changes in BASE 1.87% 4.98% 166.7%

+10% 1.95% 5.32% 173.2%

4.2% 6.8% 3.9%

Experiment 3

Baseline Experiment

-10% 1.79% 3.41% 90.6%

-4.2% -9.2% -10.4%

Changes in BASE 1.87% 3.75% 101.06%

+10% 1.95% 4.10% 110.7%

4.2% 9.2% 9.5%