Consumption Smoothing with State-Dependent Loss

Consumption Smoothing with State-Dependent Loss-Aversion Preferences
Ofer Cohen∗
JOB MARKET PAPER
Latest version: http://gradstudents.wcas.northwestern.edu/~oco924/LossAversion.pdf
04/24/2015
Abstract
Empirical evidence regarding households’ consumption sensitivity to income cannot be fully
explained by existing models. The prevalent model for consumption over time holds that the
sensitivity of consumption can be explained by liquidity constraints; however, this model is
unable to account for the consumption sensitivity of households that have sufficient wealth or
experience a declining income. I present a new model based on reference-dependent preferences,
which resembles an endowment effect for dynamic and stochastic income processes. This model
is better able to explain existing data and also provides a unified explanation for employees’
passive saving behaviors and for a version of “mental accounting”. I show that this model can
generate consumption sensitivity that is in line with the estimates in the literature. I also apply
the model to a risk-sharing network, and using a panel dataset from Thailand, I estimate a structural preferences parameter of loss aversion. This model can therefore provide an explanation
for households’ consumption behavior for a wide range of scenarios that cannot otherwise be
sufficiently explained by a single model. JEL codes: D03, D14, D91, E21, O12
∗
Department of Economics, Northwestern University: ofercohen2014@u.northwestern.edu I would like to thank
Lori Beaman, Cynthia Kinnan, Lee Lockwood, Marciano Siniscalchi, David Berger, Seema Jayachandran, Lorenz
Kueng, Bridget Hoffmann, Juan Prada and seminars participants at Northwestern University and Notre Dame University for their helpful comments and their contributions to the paper.
1
1
Introduction
In this paper, I present a new behavioral model of household’s consumption based on the idea of
loss-aversion. Consumption is evaluated relative to a stochastic and dynamic reference point that
is equal to the household’s endowment process. The standard expected utility model, a special
case of the model in this paper, is not able to explain incomplete consumption smoothing by
risk averse households without additional assumptions such as commitment problems, information
asymmetries, or exogenous borrowing constraints. My model extends the standard model, maintains
dynamic consistency and rational expectations, and generates a new version of the endowment effect,
applied to income over time and across states of nature. This version of the endowment effect can
explain sensitivity of consumption to changes in income, particularly when liquidity constraints are
an insufficient explanation, without using any further assumptions about the economic environment.
Furthermore, this new model provides a unified explanation for workers’ passive saving behavior
and for differential marginal propensities to consume out of different “mental accounts”, within the
framework of utility maximizing behavior, time consistency and rational expectations. Finally, I
show that the same behavioral model can explain incomplete risk sharing, as documented in many
studies in developing countries.
The key assumption for the model in this paper is that preferences are state and/or time dependent,
which breaks the direct relationship between marginal utility and the level of consumption. I set
the state (or time) dependent reference point to be equal to the realized household’s income or
endowment at every state. This random (or dynamic) endowment process is the status-quo for the
household and, therefore, seems like the appropriate choice for a reference point. By choosing the
stochastic reference point equal to income, I make the utility function of consumption dependent
on the income realization. Consumption lower than income will be associated with a loss feeling
and consumption greater than income will be evaluated as a gain. Therefore, under a loss aversion
assumption, states of nature or periods with higher levels of income will be associated with higher
marginal utility of consumption, and with the household optimally choosing to consume more. I
use this feature of the preferences to explain consumption’s sensitivity to income.
The behavioral feature emphasized in this paper is the idea of loss aversion that was first introduced as part of Prospect Theory (Kahneman and Tversky (1979)). One of the main contributions
2
of prospect theory is the idea that preferences may not depend directly on the absolute level of
outcomes (how many oranges do I have?), but instead compares outcomes to some reference point
(how many more/less oranges do I have relative to the reference point?). Loss aversion is based on
the idea that losses relative to the reference point are more painful than the pleasure associated
with relative gains. Since its introduction, loss aversion has been supported by many laboratory
experiments (Kahneman and Tversky (1979); Kahneman et al. (1990); Carmon and Ariely (2000))
and has been widely used in modeling (see Barberis (2013) for a survey).
I develop a model that combines features of two behavioral models of loss aversion in a risky
environment. The first model is Koszegi and Rabin’s model of Reference Dependent Preferences
(Koszegi and Rabin (2006, 2007)). From Koszegi and Rabin’s model, I adopt the idea that the utility
function depends on both the actual (or absolute) level of consumption and the level of consumption
relative to the reference point. The second model is the Reference-Dependent Subjective ExpectedUtility or Third Generation Prospect-Theory model of Sugden (2003) and Schmidt et al. (2008).
The key feature of this model is that it allows the reference point to depend on the realization of the
state of nature. In other words, the reference point is allowed to depend on outcomes, and should
not be deterministic (as in prospect theory) or common in all states as in Koszegi and Rabin. To
the best of my knowledge, this is the first model that combines these two features.1 I go further
and extend Sugden’s model to dynamic choice problems, and use it to study consumption over
time. While allowing preferences to be state-dependent, I maintain the assumptions of expected
(state-dependent) utility maximization, forward looking agents, rational expectations, and time
consistency. This allows me to apply the preferences in commonly used economic models and to focus
on the effect of the behavioral assumptions. I refer to these preferences as State-Dependent LossAversion (SDLA) Preferences, a name that emphasizes both of the main features of the preferences.
First, I apply the SDLA model to study the allocation of households’ consumption over the lifecycle. Following the contributions of Modigliani and Brumberg (1954) and Friedman (1957), the
basic economic model, of a household maximizing discounted expected utility, typically referred
to as the permanent income hypothesis, predicts small consumption response to unpredictable,
1
A model of similar fashion, though conceptually different, is Orhun (2009). In an Industrial-Organization application, Orhun models consumer’s utility of a good as the sum of utilities from different dimension of the product (for
example, the amount of sugar or the price). The utility of any good in any dimension is a function of the actual level
in this dimension and in reference to the relevant level of some reference good.
3
transitory changes in income and no response to predictable changes in income. However, this is
not what is reflected in the empirical data. There is a vast literature that tries to close this gap and
suggests several classical and behavioral explanations for the observed co-movements of households’
income and consumption over time. In this paper, I contribute to this literature by providing a novel
explanation to the consumption-income tracking, and I show how the model can explain rejections of
tests based on the Euler equation (Hall (1978); Hall and Mishkin (1982)). Intuitively, under SDLA
preferences, decision makers will optimally choose not to completely smooth their consumption,
since doing so will be associated with feelings of loss in periods when consumption is below their
income or their reference point. Instead, households will only partially smooth their consumption,
balancing risk aversion, which makes consumption smoothing desirable and loss aversion which
makes deviations from the endowment undesirable. Therefore, consumption will be responsive to
changes in income beyond their effect on lifetime resources, also known as “excess sensitivity” of
consumption (Flavin (1981)).
Moreover, since the SDLA model can explain consumption’s response to predictable and transitory
shocks to income, it can explain the results in Johnson et al. (2006) and Parker et al. (2013) where
two episodes of federal government economics stimulus programs from 2001 and 2008 were studied.
The stimulus payments to the households, which were announced in advance, appear to have a
significant effect on contemporary consumption, in contradiction to the prediction of the traditional
life-cycle permanent-income hypothesis model. Models of liquidity constraints, such as Kaplan and
Violante (2014), are the leading explanation in the literature for this result. However, liquidity constraints models generate predictions that are not in line with some empirical evidence. In particular,
liquidity constraints can only affect households who experience temporary low income and that hold
no liquid assets. If liquidity constraints were the sole force generating consumption’s response to the
rebate’s arrival, these measures should be strongly related to consumption’s sensitivity. However,
the results are mixed, suggesting that additional explanations are needed. I, therefore, present the
SDLA model as a possible alternative explanation to consumption’s response to predictable changes
in income and I show that it is able to explain consumption’s sensitivity also for households’ whose
income is declining and/or hold sufficient liquid wealth. To examine the quantitative ability of the
model in explaining the response to a predictable shock to income I solve a stochastic-dynamic life
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cycle model, and I show that the model can match the estimates of the change in consumption
obtained from the economic stimulus programs of 2001 and 2008.
Additionally, I explain how the SDLA model can explain results in the literature that were previously
attributed to other factors. First, I show that the SDLA preferences can explain large group of
workers’ “passive” behavior of not adjusting their savings in response to changes in automatic
contribution rates or in response to changes saving’s subsidy rates (Chetty et al. (2014)). Similarly,
the model explains the tendency of employees to stick to default long term savings contribution
rates (Madrian and Shea (2001); Choi et al. (2004)). In addition, I show that the model can
rationalize differential marginal propensities to consume out of different “mental accounts” (Shefrin
and Thaler (1988); Thaler (1990)). Finally, I show how the SDLA preferences can generate the drop
in consumption at retirement (Bernheim et al. (2001)). The ability to explain this set of results
regarding households’ consumption over their life cycles proves the usefulness of this model and
supports its possible validity as a description of preferences.
As a second application, I examine households’ behavior in choosing contingent consumption when
facing risk. I use the framework of risk sharing between risk averse expected utility maximizers
based on the seminal works of Bernoulli (1954), von Neumann and Morgenstern (1944) and Debreu
(1959). This model was rejected was rejected by Cochrane (1991) and Townsend (1994), who found
that households’ consumption responds when shocks to their individual incomes are experienced,
and I show that the SDLA model can explain why households do not fully share idiosyncratic risk.
I use the SDLA preferences in this model of a village economy, studied intensively in the economic
development literature, and derive a closed form solution for the ex-ante contract. I focus on the
role of preferences in the absence of any frictions and show how SDLA preferences can explain
the rejections of the tests for complete risk sharing. If households feel a loss when giving their
own income to the risk sharing network, the ex-ante contract will prescribe only partial sharing of
idiosyncratic risk. To the best of my knowledge this is the first attempt to apply a loss aversion
model in this framework.2 To estimate a structural parameter that captures loss aversion, I use
a panel of households’ data in Thai villages and obtain estimates that I compare to experimental
2
Bryan (2011) used and tested a model of ambiguity aversion to explain incomplete risk sharing in a village
economy when people have incomplete knowledge of the environment, or a non-unique prior about the distribution of
states of nature.
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evidence on loss aversion.
The paper proceeds as follows: In section 2, I briefly describe the concept of reference dependent
preferences and loss aversion and lay out the assumptions of the behavioral model that I use in this
paper. Then, in section 3, I turn to the analysis of household decisions over the life-cycle. Section 4,
describes an efficient risk-sharing among households with SDLA preferences. I conclude in section
5.
2
Reference Dependence and State Dependent Loss Aversion (SDLA)
Preferences
The model of a decision maker maximizing her expected utility of consumption is the leading
modeling choice by economists in all fields studying decision making when facing risk. In this
model, the utility that the decision maker assigns to an act3 or a lottery is the expected value of
some utility function of the consumption realizations, typically called a Bernoulli utility function.
More formally, let S be the set of possible states of nature. A realization of the state of nature is
donated s ∈ S. The outcome or the consumption realization, if state s is realized, is denoted c (s).
The utility function 4 of consumption will be state independent, and is traditionally donated
u (c (s)) .
(1)
Preferences are defined such that a lottery with the higher expected utility is preferred:
c (s1 ) , ..., c s|S|
c0 (s1 ) , ..., c0 s|S|
⇐⇒
X
s
p (s) u (c (s)) >
X
p (s) u c0 (s)
s
where p (s) is the probability that state s occurs.
The expected utility maximizing model is tractable, intuitive, and appealing but has been shown to
have limited ability in predicting human behavior. In their seminal work, Kahneman and Tversky
(1979) presented a series of experimental evidence inconsistent with expected utility maximization
3
An act is a mapping from states of the world to outcomes / consequences.
The level of consumption will depend on the realized state, but the function operated will be independent of the
state.
4
6
and the idea that decision maker only care about final outcomes.5 They suggested an alternative
to expected utility theory, which they called Prospect Theory. Prospect theory introduced four new
ideas that are not captured in traditional expected utility theory: (i) reference dependence, (ii)
loss aversion, (iii) diminishing sensitivity, and (iv) probability weighting (Kahneman and Tversky
(1979); Barberis (2013)). While each of these has significant and potentially useful behavioral implications, this paper focuses on the first two elements and their possible implications for consumption
smoothing.
Reference Dependent Preferences
The idea of reference dependence is that decision makers may evaluate outcomes in reference to
some value, and not just by their absolute level. In prospect theory, Kahneman and Tversky
allowed preferences to depend on changes rather then on final levels of wealth, typically referred
to as “terminal consequences”. While this sounds very appealing, this assumption contradicts the
traditional economic models in which individuals only care about final outcomes. For example, in
an exchange economy, people’s preferences depend only on their consumption (number of apples
and oranges consumed). Dependency of preferences on the initial endowment (number of apples
I brought to the market) or anything else (number of apples my neighbor consumed, number I
consumed in the previous day, number I was expecting to consume) is not allowed. When modeling
choice when facing risk, reference dependence may contradict the idea that decision makers always
reduce complex lotteries to simple lotteries. If the outcome in a first stage of a complex lottery
affects the second stage preferences, reduction to a simple lottery is inappropriate and therefore
the behavior is inconsistent with the joint assumption of expected utility and consequentialism. In
this paper, I maintain the assumption of a decision maker with rational expectations that values
lotteries by their expected utility. However, I allow for utility functions that are not only a function
of the level of consumption, but also reference dependent.
I develop an expected utility model with state-dependent reference-dependent preferences. A model
of similar fashion is Sugden’s (2003) Reference Dependent Subjective Expected Utility model. This
model was also used by Schmidt et al. (2008), and called Third Generation Prospect Theory (P T 3 ).
5
Rubinstein (2012) refer to the view the view that decision makers only care about final outcomes or the “terminal
consequences” as the “doctrine of consequentialism”. See also Rubinstein (2001, 2006).
7
In this model, the reference point is a random variable, and called a reference act. The reference
point is state dependent, so the reference act is denoted r (s) = r (s1 ) , ..., r s|S|
∈ R|S| . That
is, if state s is realized, the reference point will be some real number r (s). The state s referencedependent utility from the consumption level c (s) given the reference point r (s) is given by
U [c (s) , r (s)] .
(2)
The utility function in (2), should be thought of as the state-dependent reference-dependent extension of (1). It is the utility level the household will obtain if state s is realized and consumption is
c (s). Note that U [·, ·] nests traditional expected utility as a special case. Moreover, it generalizes
Sugden’s reference dependent preferences by releasing the requirement that U [x, x] = 0 for all x
that Sugden has in his paper.
Since state dependent utility can generate any behavior, in order to get predictions from the model
additional structure is required. Following Koszegi and Rabin (2006, 2007) I allow the modified
Bernoulli utility to depends both on (i) the realized level of consumption, and (ii) the value of the
outcome in relation to the state-specific reference point. I follow Koszegi and Rabin and assume that
the two terms are additive in the decision maker’s reference dependent Bernoulli utility function.
Assumption 1. The SDLA Bernoulli utility function is given by
U [c (s) , r (s)] = u (c) + µ [u (c (s)) − u (r (s))]
Also assume that u (·) is twice continuously differentiable, strictly increasing, strictly concave and
satisfies the Inada conditions.
The SDLA Bernoulli utility function captures two possible effects on the level of utility. The first
term captures utility from the level of consumption c (s). The second term, µ [·], is equivalent to
the value function in prospect theory and captures utility in relation to the state-specific reference
point r (s).
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Figure 1: Kinked Utility Function
Reference-dependent Bernoulli utility function under assumptions 1 and 2.
Loss Aversion
Hand-in-hand with reference dependence, Kahneman and Tversky argued that “losses loom larger
than gains”. After observing that people treat losses differently than gains, they suggested that
the marginal drop in utility from a loss relative to the reference point is larger than the marginal
increase in utility from a similar sized gain, an assumption that is now known as loss-aversion.
Applied to utility over consumption, the marginal disutility from consuming one dollar below the
reference point is strictly greater than the marginal utility of consuming a dollar above the reference
point.
Assumption 2. Assume that
µ [x] = γ · max {x, 0} + λ min {x, 0}
and that λ > γ, corresponding to the assumption of loss aversion.
The reference dependent part of the SDLA Bernoulli function is assumed to be piece-wise linear.
The reference dependent part is allowed to have different slopes for gains and losses relative to the
reference point: a slope of λ for losses (c < r) and a slope of γ for gains (c > r).
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Figure 1 shows the reference dependent Bernoulli utility function under assumptions 1 and 2. The
utility function is kinked at the reference point, since the marginal utility is larger for losses relative
to reference point than for gains relative to the reference point. Note that, since the reference point
is state dependent, different states may be associated with different reference points.
Income Process as Reference Point
Kahneman and Tversky didn’t explicitly specify what the reference point should be or how it
should be determined, only related the reference point to the status-quo for the decision maker.
This allows for different interpretations and a variety of applications, and indeed models with
reference dependent preferences are used in many fields of economics and finance (Barberis (2013)).
Throughout this paper, I use the endowment as the reference point. Since households own a risky
and time varying endowment, the reference point is assumed to be stochastic and state dependent.
This choice of the reference point is consistent with Kahneman and Tversky view of the status-quo
being the reference point: for a household who owns a stochastic endowment process, consuming the
endowment at every period is the status-quo. If no actions will be taken, the endowment process will
also be the consumption process. The interpretation of this assumption in the risk sharing model is
that participation in an insurance network generates a sense of gain when some funds are received
from the insurance network and consumption is above income. When the household is required
to deliver some of its income to the network, it may feel a loss. In a life-cycle model, actively
saving out of a current paycheck or transferring money from a checking account to a saving account
may be evaluated as a loss and associated with higher pain, than the joy of consuming more than
the current paycheck. This idea can be seen in a nice quote by Brigitte Madrian, who explained
that people who set aside money see it as a loss out of their checking account. She recommends
automatic deductions from the paycheck, since
“People know they should save. But most of us still just don’t like writing a check to squirrel
money away for the future. Madrian says that feels like a loss from our checking account... The
money you don’t see is the money you don’t miss”,
which is similar to the SDLA idea that the paychecks often act as reference points.6
6
Madrian’s interpretation is that households can affect their reference point by changing their automatic deductions.
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Assumption 3. For any period t and any state st ,
r (st ) = y (st )
Assumptions 1-3 completes the description of the SDLA Bernoulli utility function.
SDLA Preferences
To study decisions in a risky endowment economy, I incorporate the SDLA Bernoulli utility function
in a standard expected utility decision rule. The ex-ante utility from the act c = {c(st )}st ∈S t , given
the reference act y = {y (st )}st ∈S t and the probability distribution p = {p (st )}st ∈S t is given by
V (c, y) =
X
p (st ) U [c (st ) , y (st )] .
(3)
st ∈S t
Preferences between consumption acts given the endowment are defined by
c c0 |y ⇐⇒ V (c, y) > V c0 , y .
This is an extension of the traditional expected utility model to a model that features a stochastic
reference point. Similar to traditional expected utility theory, the ex-ante utility is the average of
the states’ levels of utility weighted by their probabilities.7 The difference is that the Bernoulli
utility is allowed to be state dependent and to differ across states of nature. The dependency of the
utility function, and in particular the marginal utility of consumption, on the state of nature or the
realized endowment, is what generates the result of incomplete consumption smoothing.
Finally, to study consumption over time, I extend the model following the convention in the literature. That is, households seek to maximize the expected, discounted sum of future (SDLA) utility
In the SDLA, I refer to the post default auto-deductions income as a reference point. I discuss this point more when
I show how the SDLA model can explain tendency of employees to maintain their default pension contribution rates.
7
Unlike prospect theory, in this paper I do not consider probability weighting, focusing on the effect loss-aversion
around the non-deterministic reference point.
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realizations, which is written as
T
X
j=0
X
βj
h i
p st+j U c st+j , y (st+j ) .
(4)
st+j ∈S t ×...×S t+j
where c st+j is the household’s consumption if history st+j = (st , ..., st+j ) occurred. It is impor
tant to note that the household is evaluating consumption in a dynamically consistent way. The
household is not evaluating consumption in future history st+j based on the current reference point
(y (st )), but knows what the reference point will be if this history will occur (y (st+j )) and uses this
future reference point to evaluate contingent consumption in this future history. An alternative
model, in which distant outcomes are evaluated by the current reference point may generate time
inconsistency and are typically studied as a model of dual-selves.
The assumptions of reference dependence and loss aversion can generate differences between willingness to pay (WTP) for a good and the willingness to accept (WTA) for giving up on the same
good. This is known as the endowment effect: people assign different values to objects that they
own and can sell or trade, or that they don’t own and can buy or trade for (the mug experiment
of Kahneman et al. (1990), and NCAA tickets of Carmon and Ariely (2000). The term endowment
effect was first used by Thaler (1980)). For lotteries, the endowment effect was documented by
Bar-Hillel and Neter (1996). The SDLA preferences that are used in this paper generate a version
of the endowment effect that applies to a dynamic and/or stochastic endowment. In this case, the
tendency not to deviate from the endowment will be the force generating incomplete consumption
smoothing.
In the next section, I use the SDLA preferences in a rational expectations, additively separable, discounted (state dependent) expected utility model, as in (4), to show how it can explain documented
households’ behavior of consumption smoothing over time.
3
Life Cycle Behavior
Economic models of households’ consumption over their life cycle were developed as early as
Modigliani and Brumberg’s Life Cycle Model (Modigliani and Brumberg (1954)) and Friedman’s
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Permanent Income Hypothesis (Friedman (1957)). Subsequently, these models have been refined
to a benchmark model of a household with rational expectations, maximizing its expected discounted lifetime utility from consumption. Tests of this model were typically based on the model’s
prediction that lagged (or predicted by lagged) values, particularly lagged income, cannot predict
current consumption when lagged consumption is taken into account. A large empirical literature
looking at these predictions found that, in contrast to the theory, consumption was sensitive to
past information. For example, Hall (1978) used aggregate data and found that even though lagged
values of income had no predictive power for consumption, taking into account lagged consumption,
lagged stock performance had a significant predictive power. Flavin (1981) found that consumption
responded to current income beyond what is implied by the change in permanent income. Hall
and Mishkin (1982) used household data and rejected the rational expectations permanent income
hypothesis because consumption was too sensitive to transitory changes in income. Campbell and
Mankiw (1990) instrumented current changes in aggregate income by lagged changes and concluded
that about 50% of households are using a rule-of-thumb which prescribes consuming their current
income. Carroll and Summers (1991) documented that consumption and income grow at similar
rates over the life cycle.
More recently, Parker (1999) found consumption responds to a paycheck increase after crossing the
annual Social-Security payment cap, and Souleles (2002) documented a consumption response to
pre-announced tax cuts. Johnson et al. (2006) and Parker et al. (2013) looked at households’ consumption response to temporary and predictable changes in income from the United States economic
stimulus programs of 2001 and 2008. Both papers found a significant response of consumption to
the arrival of the stimulus payment, even though these payments were announced ahead of time
and households knew of these payments in advance. For long surveys of the literature see Attanasio
and Weber (2010) and Jappelli and Pistaferri (2010).
The rejections of the benchmark model called for theories that could explain the co-movements of
consumption and income. The most common explanation for consumption’s sensitivity to income
changes is a model with borrowing constraints. If households are liquidity constrained, they may
wish to borrow in order to smooth consumption, but cannot do so. In this case, consumption will
respond to shocks to income. Tests of liquidity constraints as the cause for consumption and income
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tracking go back to Flavin (1985) who found that the unemployment rate affects the sensitivity of
consumption to income. In another study, Zeldes (1989a) splits the sample of households by their
wealth and finds stronger sensitivity of consumption to income changes for low wealth households
who are expected to be liquidity constraint. Similarly, Baker (2014) documented a differential
responsiveness of consumption to income shocks at different levels of debt, wealth, or liquid assets.
The current state-of-the-art in models of liquidity constraints is Kaplan and Violante (2014), who
present a model with two types of assets: a liquid asset with low returns and low or no adjustment cost, and an illiquid asset with higher returns and higher adjustment costs. In their model,
households may hold illiquid assets (like housing) and still be liquidity constrained in the short run.
These consumers own illiquid wealth and nevertheless live hand-to-mouth, generating co-movements
of consumption and income. They use their model to simulate a response to a predictable change
in income, and show that the model can generate a response in magnitude similar to the estimate
from the economic stimulus programs in Johnson et al. (2006) and Parker et al. (2013).
However, as is evidenced by two main findings in the literature, models with liquidity constraints are
insufficient in explaining consumption’s response to income changes. First, consumption appears
to be sensitive to income even among households that hold sufficient levels of liquid assets. Parker
(1999) looked at the effect of crossing the annual Social-Security payment limit, and found that this
predictable increase in net paycheck is associated with an increase in consumption expenditures.
Since the households reaching the payments cap are ones with relatively high income, liquidity constraints may not be the most obvious explanation. Souleles (1999) found that consumption responds
to tax rebates even for households that are not liquidity constrained, and Souleles (2002) found no
relationship between liquid assets and sensitivity of consumption to the Reagan tax cuts. Johnson
et al. (2006) and Parker et al. (2013) examined the consumption response to the 2001 and 2008 federal stimulus programs across households and found no conclusive evidence for a differential effect
for young households, households with low income, or households with low levels of liquid assets.
Second, consumption is sensitive to predictable income changes, even when income is declining. In
this case, liquidity constraints should have no effect since households can always save in anticipation
for the income drop. Shea (1995), using union labor contracts, found that consumption’s response
to predictable declines in income was stronger in comparison to the response to predictable income
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increases. This directly opposes the prediction of a liquidity constraints model. Bernheim et al.
(2001) documented a drop in consumption at retirement, which also cannot be explained by liquidity constraints. Recently, Parker (2014) found that consumption’s response to the 2008 economic
stimulus program was not affected by whether the household’s income was increasing or decreasing.
This is, again, inconsistent with a liquidity constraints model in which temporary episodes of low
income are the source of consumption’s sensitivity.
Insufficiency of liquidity constraints in explaining consumption and income tracking calls for alternative explanations. While the liquidity constraints model is the most widely used explanation for
the rejections of the tests of the permanent income hypothesis, some of the alternative explanations
in the literature include:
• Precautionary Savings - Carroll (1997); Deaton (1991).
• Non-Separable Utility of Consumption and Leisure - Heckman (1974).
• Habit formation - Harl E. Ryder and Heal (1973).
• Durables consumption - Eichenbaum and Hansen (1990).
• Hyperbolic Discounting / Dual Self models - Laibson (1997); Fudenberg and Levine (2006)
• Expectations Based Loss-Aversion / News Utility - Koszegi and Rabin (2009); Pagel (2014)8
Overall, the literature doesn’t provide a singular explanation to observed households life-cycle behavior, and this line of research is still very active (Pagel (2014); Kaplan and Violante (2014)).
Many questions are still open, and moreover, the behavioral approach seems to be gaining more
attention (Parker (2014)). As I show below, the SDLA is a new explanation to consumption’s sensitivity to predictable and unpredictable changes income. In particular, the SDLA model can explain
consumption’s sensitivity to income changes, even when households hold enough liquid wealth and
8
A work that should be emphasized is Pagel (2014), which presents a life-cycle model based on the preferences in
Koszegi and Rabin (2009). The model is based on “news utility” preferences, in which utility depends on the level
of consumption, on consumption relative to expectations, and on changes in expectations about consumption in the
future. Pagel shows that the model can generate excess sensitivity of consumption to unpredictable changes in income.
Households would like to “surprise” themselves and consume positive shocks to income, because if saved for a future
period this extra consumption will no longer generate a sense of gain. However, the news utility model is limited in
its ability to explain consumption’s response to predictable changes income, since households can’t be surprised by
predictable shocks. Finally, these preferences are dynamically inconsistent which is somewhat undesired.
15
in periods when income declines, in which liquidity constraints models have previously failed to
explain the data. Thus, the results above should be seen as evidence in favor of the SDLA model
when compared to a liquidity constraints model. I go further and study the quantitative ability of
the SDLA model to generate economically meaningful consumption responses. I simulate a version
of the model and find results in line with the literature.
3.1
SDLA Life-Cycle Model
The household is maximizing its discounted, expected state-dependent loss-aversion utility, (4),
under assumptions 1-3 subject to a standard intertemporal budget constraint. I start with a simple
model with two periods and a deterministic income process that captures the intuition of how the
predictions of the SDLA preferences differ from standard expected lifetime utility. Then I present
the extension of the model to T -periods and show that the SDLA can explain several stylized facts
about income, consumption, and savings. Finally, I describe a numerical solution to a T -periods
dynamic stochastic model and use it to simulate consumption’s response to a stimulus program.
2-Periods, Deterministic Income Process
As a starting point, in order to obtain intuition, consider the simplest dynamic decision problem,
with two periods and a deterministic income process with no discounting. That is, suppose y1 and
y2 are known, the subjective discount rate β = 1, and the net interest rate is r = 0. The household
therefore solves
max U [c1 , y1 ] + U [c2 , y2 ]
s.t.
c1 + c2 ≤ y1 + y2
The following proposition shows that income growth/decline across periods will be translated into
imperfect consumption smoothing. A decision maker with SDLA preferences will choose a consumption path such that consumption is increasing (decreasing / constant) if and only if income is
increasing (decreasing / constant).
16
Proposition 1. Suppose yi > yj , where i, j ∈ {1, 2}. Then, ci > cj .
Proof. Suppose to the contrary, that ci ≤ cj . From the budget constraint, it follows that ci < yi
and cj > yj . Therefore,
∂U [ci , yi ]
∂U [cj , yj ]
= u0 (ci ) [1 + λ] > u0 (cj ) [1 + γ] =
∂ci
∂cj
violating the first order conditions. The decision maker would substitute consumption from time j
to time i until the inequality above turns to equality.
Proposition 1 illustrates the mechanism that generates the co-movement of consumption and income.
The income dependent Bernoulli utility functions generate different marginal utility of consumption,
for the same level of consumption, at different levels of income. A decision maker will have different
marginal utility of consumption at some consumption level c0 if income is larger or smaller than
c0 . The assumption of loss-aversion implies that, for any level of consumption, higher levels of
income are associated with (weakly) higher marginal utility. In this case, standard optimization
arguments that equate the marginal utility of wealth across periods will imply higher consumption
when income is higher.
Figure 2 illustrates the result. At any point on the certainty line (45◦ line or the complete consumption smoothing line) the household is not indifferent between (a marginal units of) consumption in
period 1 and in period 2. At any such point, south-east of the endowment, consumption in period
1 is evaluated as a gain, and in period 2 as a loss.9 Therefore, on the margin, consumption in
period 2 will be more valuable to the household. The household will require more than one unit
of consumption in period 1 in exchange for one unit of consumption in period 2. The slope of the
indifference curve crossing the certainty line is shallower than the subjective discount rate (β −1 ).
As long as β (1 + r) = 1, optimal consumption will occur, on the budget line, north-west of the
certainty line. In figure 2, this will be at the point c∗ .
Another way to understand how the SDLA model implies imperfect consumption smoothing, is to
note that two forces in the household’s preferences are affecting its optimal consumption path. First,
9
Points in which both periods are evaluated as a loss are considered irrational, and points in which both periods
are evaluated as a gain are not feasible.
17
Figure 2: Partial Consumption Smoothing
Figure 3: No Consumption Smoothing
18
similar to the traditional expected-utility model, concavity of the Bernoulli utility function implies
willingness to equate consumption across periods of time / states of nature, “pushing” towards
the certainty line. The state dependent loss-aversion feature, on the other hand, “pulls” optimal
consumption toward the endowment. When consumption is completely smoothed, the marginal gain
of smoothing consumption goes to zero, similar to the well known result of “local risk neutrality at
the certainty line”. However, at the certainty line, there is a strictly positive potential benefit from
setting consumption closer to the endowment process, since the loss-aversion effect is positive.
The previous proposition established that with SDLA preferences, perfect consumption smoothing
will not be optimal. The following proposition shows that if income growth (decline) is fast enough,
some smoothing may be optimal.
Proposition 2. Suppose, without loss of generality, that yi > yj where i, j ∈ {1, 2}. Then, there
exists a strictly increasing function k (·) such that
1. If yi > k (yj ) then yi > ci > cj > yj .
2. If yi ≤ k (yj ) then ci = yi and cj = yj .
Proof. Let k (x) = u0
−1
yi > k (yj )
n
u0 (x)
h
1+γ
1+λ
io
. Then,
0−1
1+γ
u (yj )
1+λ
0
⇐⇒
yi > u
⇐⇒
u0 (yi ) [1 + λ] < u0 (yj ) [1 + γ]
⇐⇒
The DM would like to move consumption from ito j.
If yi > k (yj ), continuity and monotonicity of u0 (·) implies that there exists yi − yj > ∆ > 0 such
that
−
∂U [yi − ∆, yi ]
∂U [yj + ∆, yj ]
= u0 (yi − ∆) [1 + λ] = u0 (yj + ∆) [1 + γ] =
| {z }
| {z }
∂∆
∂∆
ci
cj
0−1
yi − ∆ = ci = u
19
1+γ
u (cj )
1+λ
0
> cj = yj + ∆
If yi ≤ k (yj ), the decision maker cannot improve by shifting consumption. For any ∆ > 0
u0 (yi − ∆) [1 + λ] > u0 (yi ) [1 + λ] ≥ u0 (yj ) [1 + γ] > u0 (yj + ∆) [1 + γ]
and
u0 (yi + ∆) [1 + γ] < u0 (yi ) [1 + γ] < u0 (yj ) [1 + λ] < u0 (yj − ∆) [1 + λ]
so consuming at the endowment is optimal (ci = yi , cj = yj ).
Finally, k (·) is increasing since u00 (·) < 0.
The function k(x) relates consumption levels evaluated as gains and losses. The marginal utility of
consumption at a consumption level x that is evaluated as a gain will be the same as the marginal
utility of consumption at a consumption level k (x) that is evaluated as a loss. If the endowment
is increasing fast enough, such that yi > k (yj ), consuming time i’s endowment is associated with
very low marginal utility (relative to the marginal utility when consuming the endowment at time
j), even when time i’s consumption is evaluated as a loss. In this case, the household will consume
less than its income at time i and more than its income at time j. Thus, the optimal consumption
path will be such that the two forces (smoothing due to concavity and staying at the endowment
due to loss aversion) perfectly balance each other. Figure 2 shows a case of partial consumption
smoothing.
Alternatively, if the endowment is fairly stable, yi ≤ k (yj ) and yj ≤ k (yi ), the optimal consumption path will be equal to the endowment. In this case, the benefit of smoothing consumption is
small, since marginal utilities (net of gain-loss feelings) at the endowment don’t differ much, and is
outweighed by the gain-loss feelings of deviating from the endowment. Therefore, no consumption
smoothing will occur. Figure 3 shows this case.10
Together, propositions 1 and 2 illustrate the intuition of how the SDLA model can explain incomplete
consumption smoothing. The following result is a direct corollary of proposition 2.
10
The optimal consumption profile is never more volatile than the income profile. While concavity calls for smooth
consumption and loss-aversion calls for consumption close to the endowment, in any consumption profile steeper than
the income profile both forces are working in the same direction of smoother consumption. Overall, the SDLA model
can generate the well documented empirical evidence of partial consumption smoothing over the life cycle.
20
Passive Saving Behavior and Tendency to Stick to Default Contribution
Chetty et al. (2014) use data from Denmark and found that a large share of workers behave as
“passive savers”. These workers don’t change their voluntary contributions in response to changes
in subsidies or automatic contributions. Therefore, these workers’ consumption does not response to
changes in pension subsidies, but is responsive to changes in automatic contributions. The SDLA
model predicts the same results for households with stable income profile satisfying yi ≤ k (yj )
and yj ≤ k (yi ).11 For these households, a small shift in the automatic contributions is a shift in
income, in the reference point, and in their consumption choice which is set equal to income. On
the other hand, a small enough price change will not violate the inequality conditions and thus
these households will optimally choose the same consumption and savings out of current income!
Closely related results are in Madrian and Shea (2001) and Choi et al. (2004) who find that employees
tend to stay with their default option for pension contribution rates and only few opt out of the
default option. This kind of stickiness to the default option is usually interpreted as a switching cost,
either a monetary cost, a time cost, or a mental cost. The SDLA model can explain the tendency
to stay at the default as a result of loss-aversion. If households see the post-default income as the
reference point and the implied income path is fairly stable, the result follows from proposition
2. If the household is enrolled in a default that implies that the post default income levels satisfy
yi ≤ k (yj ) and yj ≤ k (yi ), the household will choose to stick to the default.
The SDLA prediction of the tendency to stick to the default pension contribution rate does not
rely on any real or mental cost of calculating the optimal option or in adjusting the contribution
rates. It is instead a result of the desire to avoid feeling a loss in the period when income has to be
saved. Therefore, this mechanism may be a more appropriate explanation in certain situations when
switching costs seem to be low. In an important paper, Benartzi and Thaler (2004) conducted an
experiment and learned that people are willing to commit, in advance, to higher future contribution
rates. If we believe that the Save More Tomorrow program somehow allowed households to adjust
their future reference point, then the SDLA model is a possible explanation of their findings.
11
In a more general model, as the T -periods model presented later, the passiveness prediction will hold for households
in the middle region of their consumption functions, which are households that earn income that is intermediate relative
to their expected average life time income.
21
A T -periods model
I now consider the generalization of the 2-period model to an arbitrary (finite) horizon. I still
restrict attention to a deterministic income process in order to obtain a closed form solution. The
decision maker’s objective, given the (exogenous and deterministic) income process is
maxc0 ,...cT
PT
t=0 β
tU
[ct , yt ]
s.t.
A0 +
PT
1
t=0 (1+r)t
[ct − yt ] ≤ 0
where A0 is the beginning of life non-human wealth.
Proposition 3. Suppose β (1 + r) = 1, and assume that there exists (t, s) such that yt > k (ys ).
¯ such that the optimal consumption path
Given the income process y0 , ..., yT , there exist k and k,
satisfies:
1. If yt ≤ k, then ct = k. “borrowing periods”
¯ then ct = k.
¯ “saving periods”
2. If yt ≥ k,
3. If k¯ ≥ yt ≥ k, then ct = yt .
Proof. Since preferences are monotonic, at the optimum the budget constraint must hold with
equality. If the household never saves, the budget constraint implies that it also never borrows:
@t : ct < yt ⇐⇒ @t0 : ct0 > yt0 . In this case, the household always consumes its endowment,
∀t : ct = yt .
Suppose there is at least one period in which the household borrows. The budget constraint implies
that there must be at least one other period in which the household saves. Then, it must be the
case that for any borrowing period b and any saving period s,
u0 (cb ) [1 + γ] = u0 (cs ) [1 + λ]
cs = u0−1 u0 (cb )
22
1+γ
1+λ
= k (cb )
Figure 4: Consumption Function in a T - Periods Model
and consumption in any saving (borrowing) period must be equal cs = k¯ (cb = k). The function
¯
k (·) relates any k to a unique k.
Next, it is optimal to set consumption equal to income (ct = yt ) if and only if there is no beneficial
adjustment of increasing or decreasing consumption.



¯ [1 + λ]
u0 (yt ) [1 + λ] ≥ u0 k


u0 (y ) [1 + γ] ≥ u0 (k) [1 + γ]
t
so,
k¯ ≥ yt ≥ k
Finally, note that since total discounted expenditure is increasing in k. Optimality then implies that
the pair k, k¯ that solves the optimization problem is the pair that satisfies the budget constraint
with equality.
Proposition 3 illustrates how the SDLA model can explain the results in the literature that document
consumption tracking income. In particular, the SDLA model explains Campbell and Mankiew’s
(Campbell and Mankiw (1990)) observation that 50% of consumers behave as rule-of-thumb con-
23
sumers and Parker’s (Parker (2014)) result that some households behave as “spenders” and tend to
spend their current income. The SDLA model explains this result without assuming any exogenous
rule for behavior but as an optimization result with rational expectation and dynamically consistent
preferences.
The consumption function, from proposition 3 is plotted in figure 4. When income is very low,
relative to lifetime earning, the household will choose to borrow to ensure a minimal standard of
¯ Finally, for intermediate
living k. When income is high, the household will save income beyond k.
levels of income, the household will simply live hand-to-mouth or according to a rule-of-thumb.
Changes in initial assets or lifetime earnings (that are not immediately consumed) will move the
consumption function in a pseudo-parallel fashion, as shown in the figure. In the case of no loss
aversion, the consumption function will be a flat line that will be move up and down as a result of
any change in the household’s permanent income.
Beyond explaining consumption’s response to predictable changes in income, proposition 3 and the
corresponding figure 4 show how the SDLA model can explain the two examples from the literature
where the empirical evidence cannot be explained by liquidity constraints. First, households with
intermediate income will consume their endowment, even if they hold sufficient liquid wealth. Second, households’ consumption may be sensitive to predictable changes in income even when income
declines. Consider, for example, a household that lives for three periods with an income process that
¯ It will decrease
satisfies y1 ≥ k¯ > y2 > k > y3 . The household will save in period 1, consuming k.
its consumption going into period 2 in which it consumes its income y2 . Finally, the household will
further decrease its consumption between period 2 and period 3, despite holding positive level of
wealth. The next result will build on the same intuition.
Consumption Drop at Retirement
Bernheim et al. (2001) document a sharp drop in income and consumption at retirement. In the
SDLA model, the consumption drop at retirement is a direct corollary from proposition 3. The
drop in income and in the reference point implies that consumption will optimally be reduced.12
12
A sufficient condition for the result to hold is that income pre-retirement y P re is greater than k, and income post
retirement is y P ost ≤ k.
24
Pagel (2014) shows that the news utility model can explain the consumption drop at retirement
under the assumption of a decline in risk and in precautionary savings motives following retirement.
Note that for the SDLA model the result is robust to assumptions about the different risks faced by
retired households. Finally, it is worth noting that the liquidity constraints model cannot explain
the drop in consumption at retirement: even if households cannot borrow, they can always save for
retirement!
SDLA As A Mental Accounting Model
Shefrin and Thaler (1988) and Thaler (1990) develop a Behavioral Life-Cycle Model of consumption
and saving, based on the idea of Mental Accounting. In their words,
“...the key property of the model is...the introduction of the assumption that the marginal
propensity to consume additions to wealth depends on the form in which this wealth is received”
In mental accounting models, people hold various mental accounts in their minds, and these accounts
are not perfectly fungible. Money in one account (“kid’s college”) is viewed differently from money
in another account (“vacation money”). In particular, Thaler and Shefrin distinguish between a
current income account, an assets account, and a future income account. They observe that people
tend to spend greater percentage of additions to their current income (“current income account”)
than additions to their assets (“assets account”) or additions to their future income (“future income
account”). They provide a model that is based on temptation, self control, and rules that constrain
future selves. The SDLA model can generate the same prediction with a mechanism that is not
based on self control or dynamic inconsistency.
h
i
Consider two periods, s > t. In either period, income may be in the region k, k¯ or out of it.
In periods when income falls within this region, consumption is responsive to current income and
only to current income. In periods when income falls outside this region, consumption is equally
responsive to innovations to current income, changes in current assets, or changes in future income
that will not be fully consumed when received. Table 1 summarizes the four possible combinations
for periods s and t. For each combination, I obtain the marginal propensity to consume out of
changes in current (time t) income, previously accumulated assets (At ) and future income ys . In
25
the top row, current consumption equals current income, so the marginal propensity to consume out
of current income is one, and the marginal propensity to consume out of assets and future income is
¯ In this case, consumption is responsive
zero. In the second row, consumption is set equal to k or k.
¯ Thus, changes in current income and in accumulated assets has the
only to changes in k and k.
¯ The effect of future income on current
same effect on consumption, through changes in k and k.
h
i
consumption depends on whether this income will be consumed when it arrives (ys ∈ k, k¯ ) or not.
Table 1: Marginal Propensities to Consume
h
ys ∈ k, k¯
yt ∈ k, k¯
h
i
h
i
yt ∈
/ k, k¯
∂ct
∂yt
= 1,
∂ct
∂yt
=
∂ct
∂At
∂ct
∂At
=
>
i
h
ys ∈
/ k, k¯
∂ct
∂ys
∂ct
∂ys
=0
=0
∂ct
∂yt
= 1,
∂ct
∂yt
=
∂ct
∂At
∂ct
∂At
=
=
i
∂ct
∂ys
∂ct
∂ys
=0
>0
Overall, the SDLA prediction coincides with the Mental Accounting prediction:
1≥
3.2
∂ct
∂ct
∂ct
≥
≥
≥0
∂yt
∂At
∂ys
Simulated Dynamic Stochastic Model
The ability of the SDLA preferences to theoretically explain consumption’s response to income
changes doesn’t imply that it is capable of explaining an economically significant and meaningful
response. In order to investigate the ability of the SDLA model to quantitatively explain households’ consumption patterns, I use numerical methods to solve a stochastic, dynamic model of the
household’s life cycle under SDLA preferences.13 I then use the model to simulate the effect of
13
Except under very specific conditions, it is not possible to solve analytically for the optimal life cycle consumption
profile in a risky environment. It is possible to solve analytically for the consumption plan in a rational expectations,
LC/PIH model under strong assumption about the utility function and stochastic process. It is possible to obtain a
close form solution if the utility function is quadratic (“certainty equivalence”) or if the utility function is exponential
(Constant Absolute Risk Aversion) and the shocks are normally distributed. In order to allow for flexibility in the
functional form assumptions of the model, I solve the model numerically.
26
a transitory and predictable shock to income on households’ consumption. The simulations are
designed to illustrate that the SDLA model can sensibly explain consumption’s response to the
economic stimulus payments.
Most of the parameters used in the simulation exercise are taken from the literature, not directly
estimated. A complementary work will include estimation of the parameters of the environment and
the utility function using the Method of Simulated Moments as in Gourinchas and Parker (2002)
or Pagel (2014). However, since estimating values of the parameters is not evidence of the model’s
quantitative ability to explain the data or in favor of the model, I focus on examining the ability
of the model to match existing results in the literature. Exercises in this fashion can be found in
Angeletos et al. (2001) or Kaplan and Violante (2014).
Economic Environment
Following Gourinchas and Parker (2002), a household starts its working life at age 26 with some
stochastic wealth A˜0 . The household works for a stochastic wage for W = 40 years, corresponding
to the working ages 26 to 65. The household is free to save and borrow at the net interest rate
r. As in Pagel (2014), the household retires at the age of 66, and is retired for R = 12 years.
During the retirement period the household is assumed to earn some non-stochastic income (set to
zero), and consume all of its savings. The household lives until age 77. The possibilities of stochastic
retirement income, stochastic retirement age, stochastic age of death, and of bequest motives are left
for further research. I chose to do so in order to focus on the effect of state dependent loss aversion
on life-cycle decisions. However, I do not expect that including these possibilities or changing the
parameters above will generate significantly different results.
As in Zeldes (1989b); Gourinchas and Parker (2002); Pagel (2014), and others, I assume that the
working life income process of the household is composed of three components. The income of
household i, at age t is
Yi,t = Gt · Pi,t · Ui,t .
First, a deterministic trend or drift, Gt that is common to all households. This is the non-stochastic
27
part of household income, and is known to the households in advance. Second, Pi,t is a random
walk process that captures permanent shocks to the household income,
Pi,t = Pi,t−1 · Vi,t
where Vi,t is the innovation to the permanent component of income. Finally, Ui,t is an idiosyncratic
transitory shock to income. Following the rest of the literature, I assume that ln [Ui,t ] ∼ N 0, σU2
and ln [Vi,t ] ∼ N 0, σV2 . The household is assumed to know the random processes and the real
ization of each process (i.e., they know how to decompose unpredictable changes in income into
permanent and transitory shocks).
Parameters
Table 2 lists the parameters used in the simulations and their source. The interest rate and the
parameters governing the working life income distribution are obtained from Gourinchas and Parker
(2002).14 The post retirement income is set to zero. The subjective discount rate is the inverse of
the gross interest rate.15 The first two moments of the distribution of initial wealth are estimated
from the CEX data on the NBER website16 and the distribution is assumed to be normal.
The preference parameters, the coefficients of Relative Risk Aversion and of Loss Aversion, are
taken as typically appear in the literature. The coefficient of relative risk aversion is set to 2 and
the coefficient of loss aversion is set such that the marginal utility of a loss is twice the size of the
marginal utility of a gain.17,18
14
The life cycle income profile reported in Gourinchas and Parker (2002) is taken as the deterministic trend in
income. In order to focus on the effect of loss aversion, I did not include an absorbing state of zero income in all
future periods.
15
Buffer stock models must be combined with the assumption of “impatience” in order to generate the consumptionincome tracking over the life-cycle. Since the SDLA model does not rely on this assumption I set the subjective and
objective discount rates equal. See Carroll (1997).
16
http://www.nber.org/data/ces_cbo.html
17
The fact that expected utility preferences are identified only up to an affine transformation implies that only the
ratio of the marginal losses and gains is identified. I therefore normalize γ = 1.
18
These values are also in line with the estimates in the risk-sharing environment presented in the next section.
28
Table 2: Simulations Parameters
Parameter
Value
r
β
σ
λ
γ
2
σU
µU
σV2
µV
µA 0
2
σA
0
P0
65
(Gt )t=26
77
(Gt )t=66
3.44%
−1
(1 + r)
2
3
1
0.04
0
0.02
0
0.18G26
0.73G26
1
Source
Comments
Gourinchas and Parker (2002)
Literature convention
Tversky and Kahneman (1992), using
arbitrary / unidentified
1+λ
1+γ
=2
Interest rate
Subjective discount rate
Coefficient of relative risk aversion
Loss parameter
Gain parameter
Transitory shock distribution
Gourinchas and Parker (2002)
Permanent shock distribution
Own from CEX 1980-1994.
Gourinchas and Parker (2002)
(0, ..., 0)
Initial income mean
Initial income variance
Initial permanent shock variance
Deterministic income trend
The Economic Stimulus Payments Simulation
Facing two macro economic recessions, the federal government designed and executed the 2001
and 2008 Economic Stimulus Payments programs in order to boost the economic activity through
an increase in private consumption. The average marginal propensity to spend on services and
non-durable goods out of the stimulus program was estimated to be between 20 and 40 percent in
the quarter when the rebate was received, and over 60% during the quarter when the rebate was
received and the following two quarters (Johnson et al. (2006); Parker et al. (2013)). I use the model
to simulate the consumption response to an anticipated change in income, and compare my results
to these estimates.
To find consumption’s response to income received at age t ∈ [27, 65], I first solve the model under
two different deterministic trends (with and without the shock). Using the policy functions associated with the deterministic trend without the stimulus payments, I first simulate the household’s
life cycle consumption without the shock. I draw realizations of 5, 000 households’ vectors of lifecycle shocks from the transitory and permanent shocks distributions. I use the optimal policy to
simulate the first period consumption of a household that starts its working life with some level of
wealth and then experiences the first period transitory and permanent shocks realizations. I iterate
29
forward and solve for the household’s consumption and future assets in each period after observing
the shocks realizations. Appendix A shows the simulated population averages, by age, of income
and consumption over the life cycle.
I treat the stimulus rebate payments as a transitory shock, announced and known to households
in advance. I assume that the policy is announced and the households learn about the payment
one period before it takes place. That is, a policy that will take place at period t is announced
before consumption and saving decisions for time t − 1 are made. As a benchmark, I treat the
payment as a net change in lifetime income of the household.19 I assume that households assign
probability zero to the payment’s arrival prior to the the announcement and, therefore, until age
t−2 behaves according to the original policy of no income shock. Then, I re-simulate the household’s
consumption and saving decisions, under the same draws, from age t − 1 going forward, using the
policy that takes into account the stimulus payments. This will generate the consumption and
saving behavior of the household who learns, at time t − 1, about the change in income that will
happen at time t.
Comparing the household’s consumption at the time of the rebate’s arrival (t) across the two
simulated life consumption profiles captures the marginal propensity to consume out of the stimulus
program. That is, household i’s marginal propensity to consume out of a rebate received at age t
is calculated as
ESP
M P Ci,t
O ESP
cESP
− cN
i,t
i,t
=
ESP SIZE
(5)
O ESP
where cESP
is household i’s consumption under the time t economic stimulus program, and cN
i,t
i,t
is the consumption under the no stimulus policy. ESP SIZE is the size of the economic stimulus
payment to each household.
Similarly, the news response to the stimulus program is given by
N EW S
M P Ci,t
N O ESP
cESP
i,t−1 − ci,t−1
=
.
ESP SIZE
19
Alternatively, the rebate can be made “budget balanced”. In this case, I assume that the government is increasing
taxes during the 10 years following the rebate, so each household’s lifetime earnings remain unchanged. Under the
rational expectations permanent income hypothesis with complete markets, the change in the composition or timing
of lifetime earnings should not have a first order effect on the consumption profile (Barro (1974)). For the benchmark
parameters the difference in simulated rebate coefficients between the two assumptions is less than 0.4%.
30
Finally, I calculate the equivalent to the rebate coefficient calculated by Johnson et al. (2006) and
Parker et al. (2013), as20
h
Rebate Coef f icienti,t =
i
h
N O ESP − cN O ESP
cESP
− cESP
i,t
i,t−1 − ci,t
i,t−1
ESP SIZE
i
ESP
N EW S
= M P Ci,t
− M P Ci,t
. (6)
Results
Figure 5 shows the average simulated rebate coefficient (as in equation (6)) at every age and for
different sizes of payments. The simulation results are in line with the estimates obtained from the
economic stimulus programs of 2001 (Johnson et al. (2006)) and 2008 (Parker et al. (2013)) and
the simulations in Kaplan and Violante (2014). The average, across ages, of the rebate coefficient
is 0.37 for a rebate size of $500, 0.25 for the larger rebate of $5, 000, and 0.36 for the baseline specification rebate size of $1, 000. The simulations illustrate that the SDLA model is a quantitatively
plausible explanation to the high frequency co-movements of consumption and income: the contemporaneous response to the government rebate payment can be reasonably explained as a behavioral
phenomenon, without any assumptions of borrowing constraints. Similarly, the SDLA model can
explain other results in the literature, such as Campbell and Mankiw (1990), where a large portion
of households will live following a rule-of-thumb or hand-to-mouth. The ability of the SDLA model
to generate a sizable effect, under conservative parametrization, is the complementing piece to its
ability to generate theoretical results that fits the empirical evidence (and, for some readers, to its
appeal as a description of preferences).
The simulations also reveal several other predictions of the SDLA model regarding the marginal
propensity to consume out of a predictable, transitory shock. These results are important in evaluating existing tests in the literature regarding the sources of sensitivity of consumption to income.
However, it is important to note that these are the results of the simulations under the chosen
parameters, mostly the income profile, and are not necessary predictions of the model. First, as
seen in figure 5, the sensitivity of consumption to the stimulus payment is different at different ages.
Younger households tend to spend a higher fraction of the rebate payment upon arrival. This is
20
Kaplan and Violante (2014) devote a long discussion to the timing of learning about the stimulus payment. Their
preferred specification treat the rebate as a surprise to half of the households and as anticipated by the other half. I,
on the other hand, treat the payment as anticipated.
31
Figure 5: Consumption Response to Predictable Transitory Shock
because these younger households tend to earn income that falls within the middle region of their
predictable lifetime earnings (including retirement). Therefore, concluding that a stronger response
of consumption to income changes for younger households is evidence of liquidity constraints may
be inaccurate. Higher rebate coefficient is also obtained for older households. Consistent with this
prediction, both Johnson et al. (2006) and Parker et al. (2013) found stronger effect of the rebate for
older households (older than 56 in 2001 or older than 58 in 2008) compared to the middle age group
in the sample, although these differences between the age groups are not statistically significant.
Second, the simulations of the SDLA model can be used to estimate the differential sensitivity of
consumption to income across households with different levels of wealth or income. Table 3 column
(1) shows the results of regressing the simulated rebate coefficients on households’ assets holdings.
The SDLA simulations predict that households who hold more wealth tend to adjust consumption
less in response to an income shock. In this sense, the convention of comparing sensitivity across
households with different wealth is not a valid way to distinguish between a model of liquidity
constraints and a behavioral model of SDLA. Similarly, column (2) shows the differential sensitivity
of consumption to income shocks across households with different levels of income. For households
with higher income, consumption is less sensitive to income. Once again, conclusions regarding
the importance of liquidity constraints based on differential sensitivity across income levels may be
inaccurate. See Johnson et al. (2006); Parker et al. (2013); Baker (2014); Parker (2014) for such
32
Table 3: Differential Response by Assets and Income
(1)
−0.73∗∗∗
(0.013)
Assets
(2)
Constant
0.41∗∗∗
(0.001)
−0.11∗∗∗
(0.001)
0.75∗∗∗
(0.006)
N
195, 000
Simulations
195, 000
Simulations
Income
∗∗∗
: p < 0.01,
∗∗
: p < 0.05,
∗
: p < 0.10
In column (1), the effect of wealth is for 1 million dollars change in wealth.
In column (2), the effect of income is calculated for $10, 000 change in annual income.
Regressions are for ESP = $1, 000 with no repayment.
tests.
Third, as can be seen in figure 5, the average rebate coefficient diminishes with the size of the rebate.
This is because as the rebate size increases, households with intermediate income who consume their
endowment will eventually view their contemporary income as high enough to start saving despite
the associated loss feelings. This qualitative result was presented in Kaplan and Violante (2014)
as a possible explanation for the difference in effectiveness between the 2001 and 2008 stimulus
programs, and as an evidence in favor of their wealthy-hand-to-mouth model. Since the SDLA
model generates the same prediction regarding the relationship between the payment’s size and the
effectiveness of the program, this can also be seen as evidence in favor of the SDLA model.
Figure 6 shows the age profiles of the simulated rebate coefficients for different values of the loss
aversion parameter λ. First, as expected, consumption sensitivity at any age is increasing with
stronger loss aversion. The stronger the loss aversion feelings, the more likely it is that the household
chooses not to save out of current income and to avoid the sense of loss. Second, shutting down loss
aversion implies a rebate coefficient close to zero, as predicted by the permanent income hypothesis.
If there is no loss aversion, consumption will increase at the time of the news arrival, and not at the
time of the rebate arrival. The figure shows that adjusting the loss aversion parameter can generate
a significant response in sensitivity of consumption to income. The simulated effect associated with
“main stream” loss aversion coefficients from the literature are in line with the estimated effects of
33
20% − 40% and can thus be seen as further evidence supporting the SDLA model’s applicability.
Figure 6: The effect of loss aversion
In conclusion, the SDLA model seems to provide a qualitatively and quantitatively good fit to
empirical results about households’ consumption sensitivity over their life cycles. In the next section
I study a model of risk sharing between SDLA households and provide an estimate of the loss aversion
parameter using income and consumption data from Thailand.
4
Risk Sharing
Insurance and risk sharing have been studied by economists in many fields. In this section, I apply
the SDLA preferences to a general equilibrium model of ex-ante risk sharing and show that the
equilibrium outcome predicts incomplete risk sharing or imperfect diversification of idiosyncratic
risk. I then use data from Townsend’s Thai project to structurally estimate the parameters of the
utility function.
34
4.1
Risk Sharing
Economic theory implies that expected utility maximizers with common beliefs will fully diversify
all idiosyncratic risk. I will refer to this prediction as complete risk sharing. In an ex-ante equilibrium, households optimally equate the marginal rates of substitution of consumption between any
two states to the price ratio of the contingent claims in these states. In that case, under standard
expected utility, consumption levels move together and all idiosyncratic risk is diversified. Moreover, individual household’s consumption is not a function of individual household’s income when
correctly conditioning on aggregate income and the household’s time invariant characteristics.21
Townsend (1994) tested these theoretical predictions in three ICRISAT villages in India. By looking
at the effect of a household’s income on its consumption, taking into account aggregate shocks and
the household’s relative weight, he rejected the hypothesis of full risk sharing. Townsend’s rejection
of the test for full risk sharing led to a large literature aiming to explain this result. In this part of
the paper, I’m providing a new explanation to incomplete risk sharing, that is based on the SDLA
preferences and does not require further assumptions about the environment.
A partial list of explanations that appear in the literature includes:
21
The First Welfare Theorem allows us to characterize properties of the market equilibrium by solving for the efficient
allocation. The planner’s problem is
PN
max{ci (s)}s∈S
i=1
i=1...N
δi
P
s∈S
p (s) u (ci (s))
s.t.
PN
i=1
ci (s) ≤
PN
i=1
for all s ∈ S
yi (s)
The efficiency condition follows from the first order conditions of the planner’s problem
0
u1 c1 (s)
0
0
u1 (c1 (s0 ))
=
ui ci (s)
0
ui (ci (s0 ))
and therefore each household i’s consumption in any state can be written as a function of household 1’s consumption
in that state and the household’s Pareto weight:
ci (s) = u0−1
u01 c1 (s)
i
δ1 δi
The sum of households’ consumption at any state can’t exceed the aggregate endowment (resource constraint).
Since individual income only appear in the planner’s problem as an element of total income, any household’s optimal
consumption is only a function of the aggregate income and the household’s relative weight.
35
• Limited commitment - If people have the option to take their endowment, leave the insurance
network, and go into autarky, then the full risk sharing allocation may not be possible (Ligon
et al. (2002)).
• Moral hazard - If the probability distribution of income levels depends on people’s actions,
the complete risk sharing allocation is likely to be unattainable.
• Hidden income - If individual income and consumption are unobserved, truth-telling constraints would imply that the first-best and second-best allocations might not coincide (see
Kinnan (2011) for more detailed description of these three models).
• Unit of insurance provision - Risk may be shared at the family level or caste level, and the
village is not necessarily the appropriate unit to apply Townsend’s test (Deaton (1997)).
• Heterogeneous attitude toward risk - different coefficients of relative risk aversion can generate
a failure of a test that assumes identical risk attitudes for the households (Mazzocco and Saini
(2012)).
• Coarser partition of the state space - If contracts cannot be contingent on every state of
nature, but instead only on a coarser partition, full insurance would not occur.
• Costly enforcement of contracts can also lead to incomplete risk sharing and failure of the test
(see Dubois et al. (2008), for the last two points).
• Ambiguity aversion - If people’s beliefs depend on the contract signed (i.e., agents believe that
the states where they have to give away some of their income are more likely) full insurance
would not be an ex-ante first best (Bryan (2011)).
Since different models could have different policy recommendations, it is important to learn what
models are a good description of reality. Thus, laying out a new mechanism that can explain partial
insurance take-up, partial risk diversification and incomplete risk sharing is a valuable contribution.
Risk Sharing Model
Consider an endowment economy. In any period, there are S t possible states of nature drawn
randomly and independently over time from a set S t . Each state st ∈ S t arrives with probability
36
p (st ) > 0. There are N infinitely lived households indexed by i ∈ {1, . . . , N }. Households have
SDLA preferences over consumption and income. The households interact in a mutual insurance
network: after each history st = (s1 , ..., st ), adjustments are made such that each household i
consumes ci st , the consumption prescribed by the insurance contract. Since preferences are
monotonic, the first welfare theorem holds implying that the market equilibrium allocation will be
an efficient allocation. Therefore, I restrict attention to solving the planner’s problem of maximizing
a weighted sum of households’ utilities. Each property of all efficient allocations will then hold in
equilibrium.
A Static Model of 2 Households, 2 States and no Aggregate uncertainty
I start the analysis with the simplest example of two households and two states of nature. A very
nice feature of this example is the ability to plot it in the well-known Edgeworth box. For this
example, I assume that there is no aggregate uncertainty. Fortunately, the intuition also applies to
the static N × S model and to the dynamic model.
Consider an economy with two households with identical preferences and two states of nature. Let
θ∈
1
2, 1
be a constant. Consider the random variable
s=



θ
with probability 1/2


1 − θ
with probability 1/2 .
Endowments are y 1 (s) = s and y 2 (s) = 1 − s. The aggregate endowment is constant and equal to
one. Each household has preferences that depend on consumption and the endowment and display
loss-aversion.
For this particular example the efficient allocations can be found by maximizing the expected SDLA
37
utility of household 1, under the constraint of providing household 2 with utility of at least v 2 .
max
c1 (θ),c1 (1−θ)
1 1 1 u c (θ) + µ u c1 (θ) − u (θ) +
u c1 (1 − θ) + µ u c1 (1 − θ) − u (1 − θ)
2
2
s.t.
1 u 1 − c1 (θ) + µ u 1 − c1 (θ) − u (1 − θ)
2
1 + u 1 − c1 (1 − θ) + µ u 1 − c1 (1 − θ) − u (θ) ≥ v 2
2
where I used the fact that c2 (s) = 1 − c1 (s).
Proposition 4. Suppose T = 1 (static model), S = 2, N = 2, θ ≥ 12 , y 1 (s) = s , y 2 (s) = 1 − s
and assumptions 1-4 holds. Then,
1
2
c (s) = c (1 − s) =



θ
if


θ ∗
if θ∗ ≤ θ ≤ 1
1
2
≤ θ < θ∗
for some unique θ∗ satisfying
u0 (θ∗ )
1+γ
=
.
u0 (1 − θ∗ )
1+λ
Proof. It is never optimal to set c1 (s) < c1 (1 − s), since the full insurance allocation will dominate
this allocation (Since y 1 (s) > y 1 (1 − s) both risk aversion and loss aversion works towards more
risk sharing).
It is optimal to set ci (s) = y i (s) if and only if
u0 y 1 (s) [1 + λ]
u0 (y 1 (1 − s)) [1 + γ]
u0 (θ)
u0 (1 − θ)
u0 y 2 (s) [1 + γ]
u0 (y 2 (1 − s)) [1 + λ]
1+γ
1+λ
≥
≥
θ ≤ θ∗ .
Otherwise, using the first order condition, the optimal allocation is
u0 c1 (s) [1 + λ]
u0 c2 (s) [1 + γ]
=
u0 (c1 (1 − s)) [1 + γ]
u0 (c2 (1 − s)) [1 + λ]
38
and using the symmetry of the problem
u0 ci (θ) [1 + λ]
u0 ci (1 − θ) [1 + γ]
=
u0 (ci (1 − θ)) [1 + γ]
u0 (ci (θ)) [1 + λ]
or
ci (θ) = θ∗ .
Figure 7 illustrates the different cases graphically. The intuition is the same as in the life cycle
application. Risk aversion induces consumption smoothing, while loss aversion induces the statusquo effect of staying closer to the endowment. First, if
1
2
< θ < θ∗ , the endowment allocation is
not very risky, loss aversion dominates, and thus it is optimal to have no risk sharing. This case is
shown by the dotted indifference curves. The indifference curves are kinked at the endowment and no
alternative allocation is a Pareto improvement. Second, represented by the solid indifference curves
are the households’ preferences for the case θ = θ∗ . In this case, the indifference curves are tangent
in the direction of full insurance. Household, again, will optimally choose to set consumption equal
to the endowment. For θ∗ < θ ≤ 1, represented by the dashed indifference curves, the endowment is
risky, the marginal utility of consumption when receiving the high endowment is very small relative
to the marginal utility in the other state, and therefore, some consumption will be smoothed. The
optimal consumption allocation will be c (θ) = θ∗ , the point where the risk sharing and the loss
aversion forces are balanced. Note that the dashed level curves are tangent at (θ∗ , 1 − θ∗ ) following
assumption 2 that the function µ [·] is piece-wise linear and gain-loss feelings depends only on if
consumption is greater or lower than income. Thus, moving north-west of θ∗ , the dashed and the
solid indifference curves coincides.
This example captures the intuition for the more general results that are presented below. Perfect
consumption smoothing is not optimal due to loss aversion relative to the state dependent endowment. Individual household’s consumption will be higher in the state when the household’s income
is higher.
39
Figure 7: Edgeworth Box
40
4.2
A Static Model With N Households and S States, With Aggregate Uncertainty
Next, I generalize the result to a model with arbitrary numbers of households and states of nature.
The contract ci (s) is set in an efficient way before the state of nature is revealed. A Pareto efficient
allocation is a solution to the planner’s problem
X
max
{ci (s)}i∈I
s∈S
h h h h ii
p (s) u c1 (s) + µ u c1 (s) − u y 1 (s)
s∈S
s.t.
δi :
X
p (s) u ci (s) + µ u ci (s) − u y i (s)
ii
≥ v i i = 2, . . . , N
s∈S
η (s) :
n
X
i=1
i
y i (s) ≥
n
X
ci (s) , s ∈ S
i=1
c (s) ≥ 0, i = 1, . . . , n; s ∈ S
The first N − 1 constraints are the promise keeping constraints of providing household i with utility
v i . In addition, the resource constraint and the non-negativity of consumption must hold state by
state. The Pareto frontier is parametrized by the values v i . As in the 2X2 example above, I obtain
the result that consumption is an increasing, piece-wise linear function of the endowment.
Proposition 5. Suppose that there exists a household (wlg, household 1) such that c1 (s) 6= y 1 (s)
for all s ∈ S. For each household i, if there exists a state s0 ∈ S such that ci (s0 ) 6= y i (s0 ), then the
marginal utility of consumption (net of gain-loss feelings) of household i in state s is:



η(s)



 δ i (1+γ)


u0 ci (s) = u0 y i







 i η(s)
δ (1+λ)
if y i (s) < u0
h
−1
if y i (s) ∈ u0
if y i (s) > u0
−1
−1
η(s)
δ i (1+γ)
η(s)
δ i (1+γ)
η(s)
δ i (1+λ)
−1
, u0
η(s)
i
δ i (1+λ)
Proof. It is optimal to set ci (s) = y i (s) if there is no beneficial trade between household i and
41
household 1, or iff
u0i y i (s) [1 + γ]
Ui0 (ci (s0 ) , y i (s0 ))
U10 c1 (s) , y 1 (s)
u0i y i (s) [1 + λ]
≤
≤
U10 (c1 (s0 ) , y 1 (s0 ))
Ui0 (ci (s0 ) , y i (s0 ))
1
1
Ui0 ci (s0 ) , y i (s0 )
Ui0 ci (s0 ) , y i (s0 )
0
1
1
0
i
0
1
1
×
U
c
(s)
,
y
(s)
≤
u
y
(s)
≤
× 0 1 0
×
×
U
c
(s)
,
y
(s)
1
i
1
1 + λ U1 (c (s ) , y 1 (s0 )) |
1 + γ U10 (c1 (s0 ) , y 1 (s0 ))
{z
}
|
{z
}
δi−1
η(s)
η (s)
η (s)
0
i
≤
u
y
(s)
≤ i
i
i
δ (1 + λ)
δ (1 + γ)
η (s)
η (s)
0−1
i
0−1
u
≥ y (s) ≥ u
.
δ i (1 + λ)
δ i (1 + γ)
Otherwise, the optimality condition determines the value of the marginal utility of consumption:
Ui0 ci (s) , y i (s)
Ui0 (ci (s0 ) , y i (s0 ))
=
U10 c1 (s) , y 1 (s)
U10 (c1 (s0 ) , y 1 (s0 ))
=
U10
Ui0
i
i
c (s) , y (s)
|
u0i
i
c (s)
=
U 0 ci (s0 ) , y i (s0 )
c (s) , y (s) × 0i 1 0
) , y 1 (s0 ))
{z
} |U1 (c (s{z
}
1
1
η(s)
δi−1



 η(s)
if ci (s) < y i (s)


 η(s)
if ci (s) > y i (s) .
δ i (1+λ)
δ i (1+γ)
Finally, it is optimal to ci (s) < y i (s) iff
U10 c1 (s) , y 1 (s)
U10 (c1 (s0 ) , y 1 (s0 ))
U10 c1 (s) , y 1 (s)
U10 (c1 (s0 ) , y 1 (s0 ))
u0i y i (s) [1 + λ]
>
Ui0 (ci (s0 ) , y i (s0 ))
η (s)
−1
y i (s) > u0
δ i (1 + λ)
and ci (s) > y i (s) iff
u0i y i (s) [1 + γ]
<
Ui0 (ci (s0 ) , y i (s0 ))
η (s)
i
0−1
y (s) < u
.
δ i (1 + γ)
42
Figure 8: Consumption and Income in a Risk Sharing Environment
For the cases of CRRA and CARA utility functions, consumption can be written as22
c˜ivt =




η˜vt + δ˜i + γ˜





y˜ivt






˜
η˜vt + δ˜i + λ
if y˜ivt < η˜vt + δ˜i + γ˜
h
˜
if y˜ivt ∈ η˜vt + δ˜i + γ˜ , η˜vt + δ˜i + λ
i
(7)
˜
if y˜ivt > η˜vt + δ˜i + λ
where x
˜ = x if preferences are CARA and x
˜ = ln (x) if preferences are CRRA. Figure 8 shows
consumption as a function of income and the parallel shift of the two bounds as an effect of a
change in δˆi or in ηˆvt . Proposition 5 implies that consumption is increasing in individual income
even when taking into account the household’s Pareto weight and the aggregate shock, which is
consistent with the empirical evidence. Note that this result is not driven by any constraints
on information, commitment or contracting, but is the first-best solution. Therefore, if SDLA is
indeed the cause of incomplete risk sharing, policy interventions aimed at easing constraints may
be ineffective and policies that affect the allocation of consumption directly will decrease welfare!
22
I use sub-script v to denote village as well as normal v for the promised utility in the optimization problem. The
relevant interpretation should be clear.
43
Comparison To Alternative Models
The static model’s predictions are closely related to the results in Bryan (2011). In his model,
households hold multiple priors regarding the distribution of states of the world, and assign a
higher probability to states when consumption must be given away. Bryan then shows that under
some conditions, there exists a solution to the ex-ante risk sharing contract that features partial
risk sharing and a piece-wise linear contract. The fact that the same behavior can be explained by
allowing for subjective beliefs or state dependent expected utility is not surprising. It is equivalent to
the observation that subjective beliefs and state dependent preferences cannot be jointly identified.
The SDLA model differs from Bryan’s ambiguity aversion model in a few respects. First, the SDLA
model, unlike Bryan’s ambiguity model, allows for the flexibility of an environment that is not
symmetric (i.e., an environment with S states and N households) and does not require pairs of
symmetric states of nature (i.e., composition of several 2X2 models to a symmetric SXN model).
Second, the SDLA extends easily to a dynamic model and maintains dynamic consistency. The
extension of the ambiguity aversion model to dynamic settings, in particular in a dynamically
consistent way, is non-trivial. Finally, the assumption that households’ preferences are such that
households assign higher probability to states of nature in which consumption is smaller than income
resembles a notion of loss aversion and not of traditional ambiguity aversion regarding the level of
consumption.
As mentioned before, a commonly used model of loss aversion is the endogenous reference point
model of Koszegi and Rabin (Koszegi and Rabin (2006, 2007)). In Koszegi and Rabin’s model
the reference point is the most recent beliefs held about the distribution of consumption. It is
worth noting that, since the reference point (or reference distribution) in their model is not state
dependent, a model based on these preferences cannot generate incomplete risk sharing. In their
model, utility does not depend on income or the state of nature. Additionally, since in the risk
sharing model, only aggregate income appears in the planner’s resource constraint, the optimal
solution cannot depend on individual income. Under any beliefs, the optimal contract will be a
complete risk sharing, since this contract will (1) maximize regular consumption utility, and (2)
minimize the disutility of gain-loss feelings that is increasing when consumption is more volatile.
Furthermore, since the reference point is equal to the distribution of consumption implied by the
44
contract, the preferred contract will be a complete risk sharing as to avoid gain-loss disutility. A
model that incorporates Koszegi and Rabin’s preferences and explains incomplete risk sharing must
incorporate individual income into the utility function. The natural way to do so is by allowing the
reference point to be updated upon observing individual income.
4.3
Dynamic Model
In this part, I characterize properties of the dynamic problem of providing loss-averse households
with a required level of ex-ante utility. I derive predictions regarding the efficient allocation of
consumption across states and histories. I analyze the simplest model of a money lender and one
household. I show that the problem is stationary and therefore presents no dynamics. Nevertheless,
I am able to illustrate that the model predicts that tests based on the intertemporal optimality
conditions as in Kinnan (2011), will be rejected.
Suppose that there is a money lender and one household who interact over an infinite time horizon.
The money lender is risk neutral, exhibits no loss aversion and has a discount factor of β. The
household evaluates consumption plans according to equation (4). Since the value function is
concave, I use a recursive formulation to write the money lender’s problem as:23
P (vt ) =
max
t
X h
ct (s ),vt+1 (st )
p st
i
yt st − ct st + βP vt+1 st
s∈S
subject to the promise-keeping constraints in expectations (δt ):
X h
p st
U ct st , yt st
i
+ βvt+1 st
≥ vt for all t, st
s∈S
Since P (·) is strictly concave, and since there is no way to ease the constraints by manipulating
future promises, it is never optimal to promise different continuation utilities after different states.
Therefore, future promises don’t change and the model reduces to a static model. Strict concavity
of P (·) also implies that the cheapest way to provide the promised utility to the agent is by setting
a fixed continuation value. Therefore, vt+1 (st , vt ) = vt for all t and st or δt = δ for all t. The
23
To see that P (·) is concave note that the constraint set is strictly convex by assumptions on U [c, r]. The objective
is weakly concave (it is strictly concave in the risk sharing environment) and therefore the value function is strictly
concave.
45
solution to the static model, thus, is also the solution in the dynamic extension.
One point to notice is insufficiency of (any function of) lagged consumption in predicting current
consumption. Kinnan (2011) uses sufficiency of lagged inverse marginal utility (LIMU) in predicting
current inverse marginal utility to distinguish between hidden income, moral hazard and limited
commitment. In the middle region of the consumption function of the SDLA model, consumption is
set equal to the endowment. Therefore, it does not contain all the relevant information on the future
promise to the household (in figure 8, the location of the consumption curve cannot be determined).
In that sense, the SDLA model is a possible explanation of the rejection of sufficiency of LIMU in
Kinnan.
4.4
Empirical Work
I use the closed form solution to the model (equation (7)) to estimate the parameters of the utility
function. I focus on the case of u (·) being of constant relative risk aversion, which implies a piecewise linear relationship between the log of consumption and log of income. Only one preferences parameter can be estimated, so I normalize γ such that (1 + λ) (1 + γ) = 1 or ln (1 + λ) = − ln (1 + γ).
Under this normalization, the consumption function can be written as
ln (civt ) =



˜

η˜ + δ˜i − 21 · λ


 vt


if ln (yivt ) < η˜vt + δ˜i −
if η˜vt + δ˜i +
ln (yivt )







˜
η˜vt + δ˜i + 1 · λ
2
where η˜vt = − ln(ησvt ) , δ˜i =
normalization, implies that
ln(δi )
σ ,
1+λ
1+γ
1
2
2 ln(1+λ)
.
σ
˜
·λ
˜ ≥ ln (yivt ) ≥ η˜vt + δ˜i −
·λ
if ln (yivt ) > η˜vt + δ˜i +
˜ =
and λ
1
2
1
2
˜
·λ
1
2
˜
·λ
(8)
.
The last equation, combined with the chosen
˜ , which I’ll later use to compare my estimates to the
= exp λσ
results of loss aversion parameters in the literature.
A problem in estimating the model in (8) above arises from non-linearity. Differencing out the
latent fixed effects is not possible, and the model is prone to the incidental parameters problem.
In particular, consistent estimates of the households’ fixed effects, {δi }i=1,...,N , is not possible even
˜ cannot be consistently estimated. To deal
when N → ∞. Therefore, the structural parameter λ
with the incidental parameters problem, I make the assumption that all households are identical
46
or the assumption that the households’ fixed effects are a function of households’ characteristics.
These assumptions will allow for consistent estimates of the households’ fixed effects as N → ∞
even if T, the number of periods, is fixed. Since in my panel T = 6, I take this route. My estimation
procedure contains two steps. In the first step, I obtain estimates of the households’ and village-time
fixed effects using a linear regression. In the second step, I use these estimates in a Non-linear Least
Squares regression to obtain estimates of the structural parameter and to test the model.
The first step of the estimation is aimed to get the village-time and the households fixed effects. The
SDLA risk sharing model’s prediction is that each household’s consumption function is a piece-wise
linear function of the household’s income (equation (7)). The “location” of the function depends
on the household’s time invariant fixed effect and a village-time shock. I recover the fixed effects
by estimating
ln (civt ) = ηvt + Xi0 β
(9)
and use my estimates of ηˆvt as village-time fixed effects and predict the households fixed effects as
ˆ
δˆi = Xi β.
As a second step, I use Nonlinear Least Squares estimator to fit a piece-wise linear function relating
deviations of consumption and income from the estimated location parameter, as in (10).
ln (civt )− ηˆvt + δˆi =



˜

− 21 · λ





if ln (yivt ) < ηˆvt + δˆi −
ˆ
ln (yivt ) − ηˆvt + δi






˜
1 · λ
2
if ηˆvt + δˆi +
1
2
1
2
˜
·λ
˜ ≥ ln (yivt ) ≥ ηˆvt + δˆi −
·λ
if ln (yivt ) > ηˆvt + δˆi +
1
2
˜
·λ
1
2
˜ (10)
·λ
.
˜ using the MATLAB function fminunc, to minimize the sum
I search over the parameter values of λ
of squared residuals. To calculate the residuals, given the parameters, the range in which income
falls (i.e. low, high, or intermediate) and the implied expected consumption are determined. Then,
residuals are calculated given the predicted consumption.
Beyond obtaining estimates of the preference parameter, I test the model based on estimated slopes
parameters for the household’s log consumption as a function of log income. I allow for different
slopes of the household’s consumption functions, for different regions of income, and compare the
47
estimates to the predictions of the model. The test is based the prediction that, taking into account
fixed effects, households’ consumption will not respond to changes in income when income is very
low or very high, and will respond one-to-one to income when income is intermediate. TI estimate
the following continuous consumption function:24
ln (civt )− ηˆvt + δˆi =

i
i
h
h


1 ˜
ˆ

·
λ
−
(β
−
α)
α
ln
(y
)
−
η
ˆ
+
δ

ivt
vt
i

2


 h
i
ˆ
β ln (yivt ) − ηˆvt + δi



i
h
h
i



˜
α ln (yivt ) − ηˆvt + δˆi + (β − α) 1 · λ
2
if ln (yivt ) < ηˆvt + δˆi −
if
1
2
1
2
˜
·λ
˜ ≥ ln (yivt ) − ηˆvt + δˆi ≥ − 1 · λ
˜
·λ
2
if ln (yivt ) > ηˆvt + δˆi +
1
2
˜
·λ
.
˜ > 0. I use confidence intervals generated
The model’s testable predictions are α = 0, β = 1, and λ
using a boot strap procedure. In all the reported results confidence intervals are calculated using
boot-straps with 5, 000 simulations.
Data
I use the publicly available 72 month panel of the Townsend Thai Project Monthly Survey for the
years 2000 to 2006. The data include weekly and monthly surveys of a total of 778 households
in 16 villages. I recover measures for monthly consumption and income for each household. I
was granted access to the baseline and the demographic changes in households composition, and
I was able to obtain households compositions for 765 households. I normalize the consumption
and income measures to adult equivalents based on the approach and the weights in Townsend
(1994). A total of 709 households appear in both the consumption and income panel and in the
households’ composition data and has at least one month of non-zero income or consumption. I
aggregate the data to get annual measures of consumption and income that seems more plausible
given the reference dependent context (i.e., allowing the reference point to change on a monthly
basis with the agricultural seasons seems implausible). Aggregating the data to annual levels also
reduces measurement errors. To avoid the effect of outliers, I drop observations that fall outside the
middle 95% of the the observations of consumption and income. Finally, working in logs restricts
the data to observations with strictly positive income, which leaves a dataset of 613 households in
24
A test in this fashion can be find in Bryan (2011).
48
the annual panel and a total of 2, 598 observations.
In order to estimate the households’ fixed effects I use data form the baseline survey of the
Townsend’s Thai Project collected in 1997. This is the only publicly available annual survey that
can be matched to the publicly available monthly surveys. From the baseline survey, I’m able to
match a subset of 236 households corresponds to a total of 1416 annual observation.25 Working in
logs and dropping the extreme 5 of consumption observations leaves a sample size of 197 households
and 849 observations.
Results
I consider different specifications for the first stage estimation of the fixed effects (equation (9)).
First, I consider the case of no fixed effects. The assumption for this specification is that all
households in all the villages are the same, and that villages are not affected by aggregate shocks.
The results are presented in table 4 in columns 1-3. In column 1, I setα = 0 and β = 1. The
˜ is estimated to be equal to 0.52 and the boot strapped confidence interval
preferences parameter λ
˜ and
lays away from zero. In column 2, α is allowed to differ from zero and is also estimated. Both λ
α are estimated to be positive at the 1% significance level. Finally, in column 3, β is also estimated
˜ > 0. Therefore, under
and while the estimates are imprecise, I cannot reject α = 0, β = 1 or λ
the assumptions of identical households and no aggregate uncertainty I find evidence in favor of
˜ > 0) and I cannot reject the sufficiency of the SDLA model (α = 0 or β = 1).
the SDLA model (λ
Appendix B plots the second stage estimations.
Next, I consider the case of village-time fixed effects. In this case, households within each village
are assumed to be identical but variation across villages and over time is taken into account.26
The results are presented in columns 4-6. As before, in all three specifications (with α and β set
equal to their theoretical values or estimated) the effect of loss aversion is estimated to be strictly
˜ are roughly half of the estimates from the specification without
positive. The point estimates for λ
village-time fixed effects. However, in columns 5 and 6, sufficiency of the SLDA model can be reject
25
The baseline survey included 15 households in each of the 16 villages that were later used in the monthly survey.
Out of these 240 households, 236 also appear in my panel.
26
The assumption of identical households is imposed after normalizing households’ consumption and income to
adult equivalents. That is, expected income and consumption per adult equivalent is the same across households.
49
Table 4: Estimation and Testing of the SDLA Model - Identical Households
(1)
∗∗∗
0
α
0.087
(3)
0.019
[0.059, 0.110]
[−0.032, 0.100]
1
0.268
1
β
(4)
0
(5)
∗∗∗
0.053
[0.032, 0.072]
1
1
[0.230, 1.011]
(6)
0.053∗
[−0.016, 0.072]
0.417
[0.133, 6.363]
0.523∗∗∗
0.284∗∗∗
2.462∗∗∗
0.235∗∗∗
0.111∗∗∗
0.292∗∗∗
[0.472, 0.573]
[0.207, 0.387]
[0.258, 3.602]
[0.184, 0.266]
[0.037, 0.166]
[0.013, 2.866]
R2
0.166
0.184
0.198
0.257
0.265
0.266
N
2598
2598
2598
2598
2598
2598
No
No
No
Yes
Yes
Yes
˜
λ
Village-Time FEs
∗∗∗
(2)
: p < 0.01,
∗∗
: p < 0.05,
∗
: p < 0.10
Boot-straped 95% confidence intervals are in parentheses.
at the 5% and 10% significance levels respectively, as the SDLA model cannot explain the result of
α > 0 (see further discussion below).
Table 5 restricts attention to the partial sample that can be matched to the baseline survey. Columns
1-3 of table 5 presents the results for this subsample for the same village-time fixed effects specification as in table 4. The point estimates are very similar to the results in columns 4-6 of table
4, suggesting that restricting attention to this subsample does not affect the results. However, due
˜ in the specifications with flexible α are no longer
to the smaller sample size the estimates of λ
significantly different than zero.
Columns 4-6 of table 5 presents the results for the specification that includes estimates of the
households’ fixed effects. To get a consistent estimate of the fixed effects I use households’ baseline characteristics as deterministic predictors of the fixed effects. I estimate equation (9) with
households’ characteristics being the households total cultivated land and the households’ value of
agricultural assets in the 1997 baseline survey.2728 Overall, the results are close to the results in
columns 1-3. This can be because agricultural assets and land cultivated are not good predictors of
the households fixed effects, in particular several years in advance. At the same time, if households
27
Agricultural assets include both livestock and equipment.
Estimates with other combinations of covariates, including the households business assets, non-productive assets,
and households income in the year before the baseline survey does not change the results.
28
50
within village are indeed the same there is no surprise that these baseline variable cannot capture
the variation across households. Moreover, since I normalize income and consumption by adults
equivalent the fact that households appear to be identical may be sensible.
Overall, I find evidence in favor of the SDLA model, as is reflected in the positive estimates of
˜ Restricting α = 0 and β = 1, the estimates of λ
˜ are slightly over 0.2 which corresponds to
λ.
reasonable effect of loss aversion when recovering a parameter that can be compared to previous
results in the literature (see table 6 and the discussion below). When allowing the slope parameters
˜ are somewhat lower but remain positive.
to differ from their theoretical values, the estimates of λ
˜ > 0 and, therefore, the SDLA model cannot
I cannot reject the hypothesis that β = 1 or that λ
be rejected. However, since the hypothesis that α = 0 is rejected, I conclude that the SDLA model
is insufficient in explaining the data. Additional explanations or a more flexible functional form
of loss aversion are needed to explain a positive α (corresponds to a positive effect of income on
consumption for high or low levels of income).
Table 5: Estimation and Testing of the SDLA Model
(1)
∗∗∗
0
α
0.042
[0.009, 0.067]
1
β
(3)
∗∗∗
0.038
(4)
0
[0.005, 0.066]
1
0.401
(5)
(6)
∗∗
0.039
[0.008, 0.065]
1
1
[0.209, 12.50]
0.040∗∗
[0.005, 0.065]
0.968
[0.216, 12.51]
0.206∗∗∗
0.094
0.292
0.205∗∗∗
0.106∗
0.107∗
[0.116, 0.247]
[−0.013, 0.187]
[−0.013, 0.667]
[0.118, 0.246]
[−0.009, 0.188]
[−0.008, 0.635]
R2
0.349
0.353
0.354
0.353
0.358
0.358
N
849
849
849
849
849
849
Yes
Yes
Yes
Yes
Yes
˜
λ
Village-Time FEs
∗∗∗
(2)
: p < 0.01,
∗∗
Yes
: p < 0.05,
∗
: p < 0.10
Boot-straped 95% confidence intervals are in parentheses.
Recovering Loss Aversion Parameters
˜ and values for the coefficient of relative risk aversion to obtain the value
I can use my estimate of λ
of
1+λ
1+γ .
As derived above, under the used normalization of (1 + λ) (1 + γ) = 1, the ratio of marginal
51
utilities of consumption associated with losses and gains can be recovered as
1+λ
1+γ
˜ . Table
= exp σ λ
6 shows the implied estimates of the loss aversion parameter for different values of the coefficient
˜ The implied ratio should be compared with
of relative risk aversion and different estimates of λ.
experimental results in the literature (e.g. Tversky and Kahneman (1992)) of
1+λ
1+γ
≈ 2.
˜ between 0.10 and 0.25 predict a slightly low although reasonable ratio
Overall, my estimates of λ,
of marginal utilities of gains and losses. This may be a result of imprecise estimates of the fixed
˜ due to other forces affecting the consumption function.
effects or imprecise estimates of λ
Table 6: Implied coefficient of relative risk aversion
5
σ=
0.5
1
2
5
˜ = 0.1
λ
1.05
1.10
1.22
1.64
˜ = 0.2
λ
1.10
1.22
1.49
2.71
˜ = 0.3
λ
1.16
1.34
1.82
4.48
Conclusions
In this paper, I developed a new dynamic behavioral model that extends Sugden’s reference dependent subjective expected utility theory (Sugden (2003)). The model is based on the idea of loss
aversion, and generates a version of the endowment effect (Thaler (1980)) for dynamic and stochastic
income processes. I showed that the state-dependent loss-aversion model provides a unified explanation for several findings regarding households’ income, consumption, saving, and risk-diversification
behavior in developed and in developing countries that cannot be otherwise explained using a single
model. First, the model can explain consumption’s sensitivity to income changes, and the consumption and income tracking over the life cycle. In order to avoid or minimize loss feelings in periods or
states of nature when some of the contemporaneous income is “lost” for past of future consumption
households will optimally choose consumption that is sensitive to temporary income shocks. The
52
model can, therefore, explain insufficiency of lagged consumption in predicting current consumption
(Hall (1978)), the existence of consumers who behave as according to a rule-of-thumb (Campbell
and Mankiw (1990)) and consumption and income tracking over the life cycle (Carroll and Summers
(1991)). Moreover, the model explains consumption’s response to predictable changes in income
such as the drop in income at retirement or the pre-announced federal economic stimulus payments
programs of 2001 and 2008. Using a simulations exercise, I showed the SDLA preferences’ ability
to match the documented average marginal propensity to consume out of rebate payments and,
therefore, provides an alternative to existing models of liquidity constraints that are insufficient in
explaining the data. In addition, the SDLA preferences provides a dynamically consistent explanation to the results presented in Shefrin and Thaler (1988) about differential marginal propensities
to consume out of different mental accounts, and in Madrian and Shea (2001), Choi et al. (2004),
and Chetty et al. (2014) regarding employees’ passive saving behavior and their tendency to stick to
default pension contribution rates. I, therefore, conclude that the SDLA model provides a general
explanation to households’ income and consumption behavior in developing countries.
The second application of the state-dependent loss-aversion preferences that presented in this paper
is to risk sharing in developing economies. The responsiveness of households’ consumption to
idiosyncratic risks, that cannot be explain under standard expected utility, is explained by the
SDLA model as optimal behavior of avoiding loss feelings in states of the world when the household’s
income high and has to be shared with the rest of the risk sharing network. The model is presented
as an alternative to existing neo-classical models of limited commitment of asymmetric information
that are insufficient in explaining the data (De Weerdt and Dercon (2006); Kinnan (2011)). Using
data from Thai villages I obtain an estimate of the structural behavioral parameters that is close to
the estimates obtained in laboratory experiments. This result suggests that (1) the SDLA model is
a plausible explanation to incomplete risk sharing in developing countries, and (2) attitudes towards
risks and losses in developed and in developing countries may be quite similar.
The ability of the SDLA model to tie together a wide variety results regarding households’ behavior,
together with its robust behavioral foundations and its analytical clarity make it an important
contribution to the economics literature. I believe that the model can explain additional results
in various fields, such as the demand for insurance, technology adoption, low migration, and may
53
have interesting applications for asset pricing. The policy implications derived from models with
SDLA preferences may suggest nontraditional policy designs, for example in the fashion of Benartzi
and Thaler’s (2001) “Save-More-Tomorrow” to induce savings or communal production in a village
economy to improve idiosyncratic risk diversification.
54
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, “Optimal Consumption with Stochastic Income: Deviations from Certainty Equivalence,” Quarterly Journal of Economics, 1989, 104(2), 275–298.
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A
Life Cycle Profiles
Figure 9 shows the simulated life cycle profiles of consumption and income. These profiles are
qualitatively close to the results in the literature.
Figure 9: Life Cycle Profiles (Simulated Population Averages)
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B
Risk Sharing Estimation Plots
Figure 10: Table 4 Estimation Plots
Figure 11: Table 5 Estimation Plots
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