Working Paper Series

Congressional Budget Office

Washington, D.C.

Can Tax Rebates Stimulate 

Consumption Spending in a

Life-Cycle Model?

Jonathan Huntley 

(jon.huntley@cbo.gov)

Valentina Michelangeli 

(valentina.michelangeli@cbo.gov)

Macroeconomics Analysis Division

Congressional Budget Office

Washington, D.C.

July 2011

Working Paper 2011-02

Working papers in this series are preliminary and are circulated to stimulate discussion and critical comment. 

These papers are not subject to CBO’s formal review and editing processes. The analyses and conclusions 

expressed in them are those of the authors and should not be interpreted as those of the Congressional Budget 

Office. Citations in other publications should be cleared with the authors. Papers in this series are available at

www.cbo.gov/publications (select Working and Technical Papers).

We thank Orazio Attanasio, Juan Contreras, Christopher D. Carroll, Robert Dennis, Ken Judd, Deborah Lucas, 

Damien Moore, Larry Ozanne, William Randolph, Felix Reichling, José-Víctor Ríos-Rull, Matthew D. Shapiro, 

Marika Santoro, Martino Tasso, and David Weiner for helpful comments and suggestions. An earlier version 

was circulated with the title “Tax Rebates and Fiscal Multipliers” and was presented at the 2010 NBER 

Summer Institute. 
ABSTRACT

This paper presents a life-cycle model with earnings risk, liquidity constraints, and portfolio 

choice over tax-deferred and taxable assets to evaluate changes to household consumption in 

response to transitory, anticipated income shocks, such as the 2001 federal income tax rebate. 

Households optimally hold a large share of savings in tax-deferred assets, which are encumbered 

by withdrawal penalties, and choose to relinquish some taxable precautionary savings in 

exchange for higher after-tax returns. The model predicts an increase of several percentage 

points in the marginal propensity to consume out of the tax rebate compared with results from a 

standard frictionless life-cycle model. Moreover, the model’s liquidity-constrained households—

which include households with very few financial assets and those whose portfolios are 

expensive to reallocate because they contain sizeable tax-deferred investments—consume a 

higher fraction of the tax rebate than do other types of households. The results presented are in 

keeping with empirical findings published in the literature concerning the 2001 federal tax 

rebate. 
1 Introduction

This paper presents a life-cycle model with asset choice and financial frictions that reproduces,

more accurately than standard life-cycle models can, observed household responses to anticipated, transitory shocks to income. One-asset life-cycle models with borrowing constraints

generate clear predictions about which households will respond to anticipated transitory income

shocks and about how strongly households respond to those shocks. Some predictions of oneasset models, however, are inconsistent with the household responses identified in recent empirical studies. In our two-asset model, households choose between investing in lower-return taxable

accounts or higher-return, tax-deferred accounts with withdrawal penalties. Some households

may optimally choose to invest in tax-deferred accounts, leading to a higher average consumption response that is closer to empirical observation. In this two-asset framework, the households

that respond to anticipated transitory income shocks are not limited to very young or very old

households. We evaluated our model by simulating the 2001 federal tax rebate and comparing

the predicted with the observed response in terms of household consumption.

Tax rebates, such as those under the Economic Growth and Tax Relief Reconciliation Act

of 2001 (EGTRRA) and the Economic Stimulus Act of 2008 are used to stimulate household

consumption and economic activity. Such rebates also have been used by researchers as tests of

households’ willingness to use anticipated transitory income shocks to fund consumption. Policymakers have included tax rebates in fiscal stimulus programs in keeping with the assumption

that households consume a significant fraction of temporary anticipated income, and the result

will be greater economic activity and the mitigation of the effects of economic downturns.

The assumption that households spend a significant share of transitory shocks to income has

been studied extensively since Modigliani and Steindel (1977), and validated by several empirical

studies using household-level data on the 2001 and 2008 tax rebates. Agarwal, Liu, and Souleles

(2007); Johnson, Parker, and Souleles (2006); Parker, Souleles, Johnson, and McClelland (2011);

and Shapiro and Slemrod (2003a, 2003b, 2009) show that temporary fiscal stimulus, in the form

of a check sent from the Internal Revenue Service (IRS), can boost household consumption.

Moreover, Johnson et al. and Agarwal et al. report larger responses for household consumption

1
of nondurable goods among liquidity-constrained households. They report that the permanentincome hypothesis (PIH) does not always hold, especially for households with low holdings of

liquid assets or households that have significant constraints on borrowing. Furthermore, neither

group reports a particular distinction in the household response conditioned on age.

The empirical findings are difficult to reconcile with the traditional Aiyagari (1994) model

or with a one-asset life-cycle model with a borrowing constraint without adding something like

the Campbell and Mankiw (1990) “rule of thumb”consumers. In the Aiyagari model, infinitely

lived agents face borrowing constraints, but most households optimally choose to save enough

to smooth their consumption over time. The share of liquidity-constrained households under

common parameterizations of the Aiyagari model is generally too low to generate the estimated

consumption responses to the tax rebate. One-asset life-cycle models with heterogeneous households and borrowing constraints also cannot reproduce observed household consumption out of

expected transitory shocks to income. Most households optimally set aside large precautionary

or life-cycle savings and therefore are unwilling to spend a significant share of any expected

and transitory increase in income, such as the 2001 tax rebate. The agents with the highest

marginal propensity to consume (MPC) out of an additional unit of income tend to be those

who are very impatient.1 Some of the more impatient agents in a life-cycle model may have

lower-than-typical survival probabilities, while others face very steep income profiles. Therefore,

such models predict that most of the response to transitory shocks to income will occur among

agents who are either very young or very old.

Our model offers a different mechanism to capture the observed MPC. We assume that the

forward-looking agents in our model are subject to idiosyncratic income shocks and have access

to separate tax-deferred and taxable assets, which provide different rates of return. Households

can freely increase their holdings in either account; however, only the taxable accounts offer the

household costless withdrawals. Some households might be liquidity constrained after a negative

shock occurs because of the previously optimal choices to invest in illiquid tax-deferred assets.

Those liquidity-constrained households are willing to spend a larger share out of anticipated

1Carroll (2001) notes that impatient households, such as those with steep income profiles, are more likely to

have a higher MPC than are those with shallower income profiles.

2
transitory shocks to income, thus driving the increase in the overall MPC. In addition to raising

the MPC out of the transitory income shock, the friction in the model induces a change in

the distribution of households that respond to anticipated income shocks. In this framework,

more middle-aged households respond to the tax rebate than they would in a single-asset lifecycle model. By introducing the tax-deferred and taxable assets, we can reconcile some of the

empirical observations with a life-cycle model in which households are forward-looking.

The importance of the distinction between investment in tax-deferred and taxable accounts

has been studied by Amromin (2008); Bergstresser and Poterba (2004); Gomes, Michaelides, and

Polkovnichenko (2009); and Shoven and Sialm (2004), among others. The distinction is important for household financial decisions: Although the tax-deferred accounts produce returns that

are superior to those from identical assets held in taxable accounts, the investments in the taxdeferred assets cannot be liquidated without cost. Households make investments in tax-deferred

retirement accounts through individual accounts like individual retirement accounts (IRAs) or

employee-sponsored retirement accounts (such as Keogh, 401(k), and defined-benefit pension

plans). And they can save in other tax-deferred instruments—such as the state-sponsored 529

education savings plans—that share many features with tax-deferred retirement accounts. The

value of other tax-deferred assets, such as life insurance, can increase without triggering tax

liabilities. Although there are a few exceptions, withdrawing assets from such accounts generally requires that the holders pay substantial penalties or income taxes. Marginal contributions

to the accounts and marginal purchases of the assets costs little. Reversing the transactions,

however, comes at a high marginal cost.

Despite the obstacles to withdrawing assets from the tax-deferred accounts, household-level

data from the Survey of Consumer Finances (SCF) show that, in fact, most chose to hold a

large share of total wealth in tax-deferred assets. The empirical data suggest that households

optimally trade higher returns for lower liquidity, reducing their precautionary savings against

short-run negative income shocks.

By incorporating similar transaction costs for withdrawing money from tax-deferred accounts

into our life-cycle model with idiosyncratic income shocks, we endogenously generate a segment

of households that has a higher MPC out of their anticipated tax rebates, particularly among

3
middle-aged households in the flatter portions of their lifetime income profile. We find, under a

variety of scenarios, that incorporating a withdrawal penalty typically increases by 4 percentage

points the marginal propensity to consume. In the benchmark scenario of a model incorporating

the transaction cost, households consume about 21 percent of the tax rebate upon receipt.

However, the ability of tax rebates to stimulate household consumption depends significantly on

households’ patience, rates of return on savings, and tax rates on capital income. Higher tax

rates on capital income and higher interest rates cause households to boost their investments in

illiquid tax-deferred assets as the returns from those assets increase substantially. Households

with zero or very small holdings of liquid assets, regardless of total savings in illiquid assets,

are more likely to behave like borrowing-constrained households when facing negative income

shocks; specifically, they tend to spend a substantial fraction of the tax rebate. These predictions

are more consistent with observed household-level data than are the predictions generated by a

single-asset life-cycle model.

In this paper, we consider household portfolio choice over financial assets, and we choose

not to model housing as a separate state and choice variable. Even though real estate shares

some characteristics with other tax-deferred assets, the structure of any transaction costs differs

substantially from most tax-deferred assets. Frictions in the housing market may generate similar

results; agents making infrequent transactions in real estate may find it optimal to constrain

themselves, producing a similar increase in the MPC out of anticipated transitory income shocks.

Our goal for this paper is to show how widely available, commonly held, and easily transferable

tax-deferred financial assets can affect optimal household savings and generate a higher MPC

out of a transitory income shock.

In the next section, we review the literature in this area. In Section 3, we describe the

empirical experiment that is replicated in our model. In Section 4, we describe the SCF data

that motivate our modeling assumptions. In Section 5, we present the model and the calibration.

In Section 6, we report the results. Section 7 reports our conclusions.

4
2 Literature Review

Many authors have analyzed predictable changes to household income and used data on household expenditures to test the permanent income hypothesis. Parker (1999) and Souleles (2002)

consider changes in tax withholding to measure how changes in income affect consumption.

Specifically, to demonstrate that changes in household-level consumption behavior are correlated

with changes in withholding, Parker examines predictable changes in Social Security withholding. Souleles (1999, 2002) uses changes in withholding that resulted from income tax refunds

and from the Reagan-era tax cuts. Instead of using changes in taxes, Stephens (2003) uses the

timing of the issuance of Social Security checks—well-known by recipients—to evaluate whether

the recipients’ spending patterns are based on the timing of benefits. Stephens reports that

households tend to increase spending, and “the increase is sharpest on the day of the check

arrival and is concentrated amongst households for whom Social Security is the primary source

of income.”

Shapiro and Slemrod (2003a, 2003b, 2009) take a different approach. They rely on questions

in the University of Michigan Survey Research Centers Monthly Survey, also known as the Survey

of Consumers, to identify the effects of the 2001 and 2008 tax rebates. In almost all cases,

households reported that they expected to spend a statistically and economically significant

fraction of the tax rebate on consumption.

More recently, Agarwal et al., Johnson et al., and Parker et al. examined the effects of the

2001 and 2008 tax rebates by exploiting the random timing of the rebate. In each case, the

rebate check was mailed on a date determined by the second-to-last digit of a tax filer’s Social

Security number. All three studies use the random timing of the rebates to identify household

spending, allowing them to separate the effects of the tax rebates from other economic conditions

or unobserved variables. Johnson et al. and Parker et al. examine household spending patterns;

Agarwal et al. examine credit card balance data. All three groups reject the PIH. Many studies

also indicate that liquidity-constrained households tend to spend a greater share of the tax rebate

on nondurable goods.

Our life-cycle model builds directly on those empirical studies. Frictionless life-cycle models

5
and infinite-horizon models of the kind described by Aiyagari cannot replicate the positive contemporaneous household response to a tax rebate. Moreover, life-cycle models with standard

borrowing constraints and typical income profiles cannot reproduce empirical results. Carroll

(2001) notes that households with higher income profiles are more impatient and more likely to

spend out of a temporary anticipated shock. When frictions are not introduced in our model, we

find that this group of consumers drives most of the observed response. However, in reality, this

is not the only group that responds positively to a tax rebate when it is received. By introducing

a household choice between taxable and tax-deferred assets, we can show that different groups

of consumers spend some fraction of the rebate.

The choice to explicitly model tax-preferred and taxable assets builds on the literature on

portfolio choice. Gomes et al. evaluate the impact of tax-deferred retirement accounts on lifecycle decisions over consumption, wealth accumulation, and savings. Longstaff (2009) evaluates

the asset-pricing implications of having different types of illiquid and liquid assets that share

some characteristics with our taxable and non-taxable assets. Dammon, Spatt, and Zhang (2004)

study the problem of optimal asset location and allocation in the presence of taxable and taxdeferred accounts. Bergstresser and Poterba use the SCF to analyze households’ allocation of

funds between taxable and tax-deferred accounts.

3 Empirically Evaluating the 2001 Tax Rebate

The Economic Growth and Tax Relief Reconciliation Act of 2001 was enacted to counteract the

2001 economic recession. One of the law’s provisions altered the lowest nonzero marginal tax

rate: The 15 percent tax rate was reduced to 10 percent for the first $6,000 of taxable income

for a single filer, for the first $10,000 of income for a head of household, and for the first $12,000

of income for joint filers.2

In lieu of applying the 10 percent marginal tax rate at filing time for the 2001 taxes, over

the course of a 10-week period ending in September 2001, the IRS mailed checks to taxpayers

2Other provisions of EGTRRA included the gradual reduction of other income tax rates. The 28 percent, 31

percent, 36 percent, and 39.6 percent brackets were reduced between 3 and 4 percentage points. The limitation

on itemized deductions was similarly phased out, being removed completely for the tax year 2010.

6
to stimulate consumption and economic recovery.3 As described by Johnson et al., each week

the checks were mailed to a different cohort of households on the basis of the second-to-last

digit of the filer’s Social Security number. Hence, the timing of the tax rebate was random and

orthogonal to other household features.

The checks were worth a maximum of $300, $500, or $600 for filers falling into the categories

of single, head of household, or married filing jointly, respectively. As described by Shapiro and

Slemrod (2003a, 2003b), the IRS determined the amount of the rebate on the basis of the 2000

tax return, and any rebate in excess of that to which the filer was entitled on the 2001 return

was forgiven. Any shortfall in the rebate could be claimed when the 2001 return was filed in

2002. According to Johnson et al., the total value of the rebate was a little under $40 billion,

roughly equal to 1.5 percent and 2.2 percent of total contemporaneous quarterly gross domestic

product and personal consumption expenditures, respectively.

In their experiment, Johnson et al. exploit the exogenous and random timing of the tax rebate

to estimate household consumption responses. Using questions in the Consumer Expenditure

Survey, they identify the period in which households received the rebate and the amount of

the rebate. They estimate linear relationships in which the change in various categories of

consumption are regressed against the amount of the rebate and several other variables, and

they report that the average fraction spent on nondurable goods by the household equals 0.373

(see Table 1).4 A second regression includes only those households that received the rebate;

the fraction that group spent upon receipt is 0.247. That smaller value is closer to empirical

results found for the 2008 tax rebate. In addition to the benchmark model, Johnson et al. also

introduce other interactions, including asset holdings, income, and age. They report that the lowincome and low-asset groups tended to spend more of the rebate. Households in the low-income

group, with annual income below $34,298, spent a statistically and economically significant 76

percent of the rebate on nondurable goods. The fraction of the rebate spent by middle-income

3See Joint Committee on Taxation, www.jct.gov/publications.html?func=startdown&id=2003, for a summary

of EGTRRA’s provisions.

4The nondurables category differs from the strictly nondurables by the inclusion of some semidurable goods,

such as apparel, household health care expenditures, and reading materials. See Lusardi (1996) for more information on how Johnson et al. construct spending categories.

7
or high-income groups cannot be statistically differentiated from zero.5 Households with low

taxable assets, defined as less than $1,000, spent more of the rebate in the quarter of receipt

than did either the middle- or the high-asset group. The low-asset group increased spending on

nondurable goods by about 63 cents for every dollar of the rebate received.6

Hamilton (2008) reexamined the findings of Johnson et al. and shows that small changes

in the analysis cause significant changes in the results. First, he modified the sample by dropping the top and the bottom 10 observations, which show excessively large changes in durable

consumption. The new estimates resulting from that restricted sample are significantly smaller.

The average fraction of the rebate spent on the consumption of nondurable goods upon receipt

drops from 0.373 in Johnson et al. to 0.226 in Hamilton.7 Moreover, the estimates also are sensitive to the regression technique used. Hamilton carried out a median regression on the entire

sample, and the fraction of the tax rebate spent on nondurable goods dropped to a statistically

significant 0.126 in the quarter in which the rebate was received and a statistically significant

0.292 for the cumulative, two-quarter response.

We extended the previous analysis in several ways. Following Hamilton, we restricted the

sample by dropping the outlier observations. When only those households that receive the rebate

are included, the average fraction of the rebate spent drops from the 0.247 reported in Johnson et

al. to 0.093. Removing the outliers has little impact on any other results, however. The results in

samples restricted by income or assets are effectively identical to those reported by Johnson et al.

Households with low income and low liquid assets still exhibit a strong, statistically significant

consumption response upon receipt of the rebate.

3.1 Household Response to Tax Rebate by Age

Johnson et al. tested whether the marginal propensity to consume varies with age. A life-cycle

model would predict that young households, with low liquid wealth and high income growth,

5The middle-income group contains households above the $34,298 cutoff for the low-income group and below

$69,000. The high-income group contains households with incomes over $69,000.

6Agarwal et al. suggest that consumption increases, upon tax rebate receipt, mostly for liquidity-constrained

households. Instead, liquidity-unconstrained households prefer to increase their savings. Agarwal et al. conclude

that liquidity constraints are the main driving force of the consumer response to “lumpy” changes in income.

7Hamilton reports that this estimate is relatively insensitive to the number of outliers removed.

8
are relatively impatient and are expected to have a high MPC. Older households with lower

survival probabilities also are impatient and may also have a higher MPC relative to middleaged households. Johnson et al. tested those hypotheses by estimating regressions in which they

interacted an age variable with the rebate to discern differences in the MPC out of a transitory

shock among three subsamples of households, segregated by age. They report no distinction in

response to the tax rebate among the ages, however, and thus fail to reject the null hypothesis

that age is not a determining factor in household response to the income shock.

However, after comparing the fraction of the rebate consumed by households of different

ages, Johnson et al. report that young (younger than 40) and old (older than 55) households

spend a little more of the tax rebate relative to middle-aged households but that the differences

are not statistically significant. Parker et al. evaluated the consumption response in a similar

experiment with the 2008 tax rebate. Their results were similarly inconclusive, and they were

unable to reject the null hypothesis that the households exhibit similar behavior.

Examining the same set of issues, Shapiro and Slemrod (2009) evaluated responses from

the Survey of Consumers. That survey includes direct questions about whether a household

would mostly spend, mostly save, or use the rebate to pay off debt. The authors identify a

correlation between the age of the household and a report that they would mostly spend the

rebate. Specifically, older households are more likely than the other age groups to plan to spend

the rebate. About 28.4 percent of older-than-65 households reported that they would mostly

spend the rebate; only about 17 percent of the younger-than-65 households said that they would

mostly spend the rebate.

Using the credit card data, Agarwal et al. evaluated the consumption response to the 2001 tax

rebate. They report that the long-run cumulative response is a monotonic decreasing function

of age. Younger households consume a larger fraction of the rebate relative to middle-aged and

older households. The marginal and cumulative coefficients for spending are jointly significant

if the three age groups (young, middle, old) are considered at the same time.

Despite those results, however, it is difficult to find agreement among the studies about

whether the MPC varies with age. The one feature common to all, however, is that middle-aged

households do not report uniformly spending less of their transitory and expected income than

9
the young and old households.

4 Data on the Household Portfolio

To evaluate our model and assess its implications, we explored household financial data in the

SCF. We focused on the household-level decision to hold assets in tax-deferred, as opposed to

taxable, accounts.

Taxable assets include those on which a yearly tax is assessed on the income earned. Following

the work of Bergstresser and Poterba, we classify taxable assets as those including money held

in savings, money market, and call accounts; certificates of deposit; directly held mutual funds;

stocks; bonds; and savings bonds. Cash and checking accounts are not included because neither

earns significant income or incurs more than minimal taxes. Alternatively, assets can be held in

tax-deferred accounts, which are characterized by growth and income that does not trigger yearly

taxes on capital gains, income, or distributions. Withdrawing assets from tax-deferred accounts,

however, typically triggers a large income tax assessment, substantial financial penalties, and

potentially large transaction costs.8 Some penalties and transaction costs are written into the

U.S. tax code. Others, such as those involving life insurance, are written into the contracts

signed at the time of investment. Tax-deferred assets include defined-contribution retirement

accounts, defined-benefit pension plans, whole-life insurance plans, and other nonfinancial assets

(see Table 2).

We classified the sample of households in the 2001 SCF into four groups. The first, consisting

of 16.2 percent of the population, holds neither tax-deferred nor taxable assets. The second, 8.9

percent of the U.S. population, includes households that invest in tax-deferred assets only. Asset

holdings by the second group are described in Tables 3 and 4. The third group, 18.6 percent of

8Some employers allow employees to borrow money from a 401(k) account. But according to Li and Smith

(2008), in 2004, only 16 percent of eligible households-those with access to 401(k) plans that permit withdrawalshad taken out such loans. Obstacles to borrowing money from the 401(k) accounts, such as origination fees,

timeliness, minimum and maximum loan sizes, and other expenses could account for the relatively low usage

rate. Typically, there is some significant, fixed loan origination fee that, for small loans, could represent a

substantial fraction of the principal. Furthermore, the household must pay interest on the loan, which ends up

being taxed twice (see Li and Smith). The term of such loans typically is short-between one and five years-and,

if the borrower defaults, the household must still pay any taxes and penalties that would have been assessed for

an early withdrawal.

10
the population, invests only in taxable assets and has no tax-deferred accounts (see Tables 5 and

6). The majority of the U.S. population, 56.3 percent, includes households that hold taxable

and tax-deferred assets (see Tables 7, 9, 8, and 10).

In each group, there are many households with small holdings of assets—the kinds of households identified by Johnson et al. as more likely to spend a tax rebate. Among those with

tax-deferred assets only, young households that belong to the first quartile in earnings hold very

few assets (Table 3). Middle-aged households hold larger amount of assets; however, those assets

are not easily accessible, because penalties are charged upon withdrawal. A large share of households with taxable assets alone has almost no assets saved (see Table 5). Through age 45, half

of that group has less than $1,200 in taxable assets, although the availability of taxable assets

increases substantially as the households approach retirement. The distribution of assets in that

group shows that a large fraction of the population of households that save only in taxable assets

has very little savings to draw from in the event of a negative income shock.

The households that hold taxable and tax-deferred assets tend to have more assets than the

other groups that hold only one type of asset (see Tables 7 and 9). Nonetheless, the value of

taxable assets held by a sizable fraction of that group is still quite low early in life. More than

one quarter of the households that are younger than 30 years old have less than $1,000 in taxable

assets. Of the households observed in the empirical studies of the 2001 federal tax rebate, those

are the types that were more likely to spend a tax rebate.

The SCF also provides more detailed disaggregated data on household asset holdings. (See

Table 11 for the U.S. average share of each asset by age, weighted by the sample weight. The taxable assets are the first six categories listed; the tax-deferred assets are the others). Households

hold a smaller quantity of taxable than tax-deferred assets as a share of total assets. (Table 12

shows average asset holdings as a function of total household financial assets.) For households

with total assets of less than $10,000, the share of total wealth held in taxable assets varies

between 1 percent and 99 percent. The share decreases for households with total assets between

$10,000 and $300,000. For households with the most financial assets, $300,000 and higher, the

share increases slightly as the households increase holdings of stocks and bonds.

11
5 The Model

We built a life-cycle model with portfolio choice over taxable and tax-deferred assets. The

economy is populated by a continuum of households, and the state variables are household age

j, stock of tax-deferred assets k

d

, stock of taxable assets k

t

, and labor productivity status θ. We

also include an additional exogenous and determined state variable s that represents a Social

Security number and does not change over time. Including this additional state allows us to

simulate the experiment in Johnson et al. in which multiple cohorts of households receive tax

rebates at exogenous, predetermined times. Then, we constructed moments that are close to

those estimated by Johnson et al. using data from the Consumer Expenditure Survey, and we

evaluated the effect of tax-deferred assets on household elasticity with respect to predetermined

transfers.

After receiving news of their labor productivity status, households purchase consumption

and invest in the two assets. The household’s problem is as follows:

V (j, kd, kt, θ, s) = max

c,kd0,kt0

u(c) + βEtS(j + 1)[V (j + 1, kd

0

, kt

0

, θ0, s)] (1)

The household’s choices are consumption c, next-period tax-deferred assets k

d

0

, and next-period

taxable assets k

t

0

. The function S(j) equals the likelihood that the individual survives to age j

conditioned on survival at age j − 1. The utility function u(c) takes the following form:

u(c) = c

1−γ

1 − γ

The household is subject to the borrowing constraints

κkd

0

≥ 0; k

t

0 ≥ b

12
Furthermore, the household is subject to a budget constraint

c(1 + τ

c

) + k

d

0

+ k

t

0 + φt(j, kd

0

− k

d

) ≤

(1 + r(1 − τ

kt))kt + (1 + r)kd + wθ(1 − τl

) + tr(j, e)

(2)

The after-tax consumption, next-period tax-deferred assets, next-period taxable assets, and

penalties must be no greater than the rental earned on capital, net of taxes, stock of capital,

wage income, and government transfer. We assume that government transfers, such as tax

rebates, are a function of age and earnings tr(j, e). Labor earnings are taxed at the effective tax

rate τ

l

, which includes labor income and the payroll tax rate. Taxable assets are taxed at the

effective tax rate τ

kt

. r is the real return on household investments.

The lifetime labor earning process evolves as follows:

ln θt = f(t) + zt + t

where

zt = ρzt−1 + ηt

f(t) is an age-polynomial, which represents the typical life-cycle earning profile and is entirely

known to the individual at the time of entry into the labor force. In addition to this deterministic

component, we introduce two random components: a persistent component zt that evolves as

an autoregressive process with persistence ρ, and a transitory component t. ηt and t are

i.i.d. with variance σ

2

η

and σ

2



, respectively. We follow Cocco, Gomes, and Maenhout (2005) for

the parameterization of the deterministic component f(t) and Storesletten, Telmer, and Yaron

(2004) for the parameterization of the stochastic idiosyncratic labor earning shocks.

5.1 Tax-Deferred and Taxable Assets

Taxable and tax-deferred assets differ principally in the tax treatment and in households’ ability

to liquidate their positions (Table 2). We model those differences by specifying different tax

rates for the two types of assets and by including a penalty function in the household budget

13
constraint.

In the model, any income earned on funds invested in taxable assets is taxed in the year it

is received. Even though interest, dividends, and capital gains might be subject to different tax

rates, for simplicity, we apply a uniform linear tax rate τ

kt to the income derived from this class

of assets.

Funds invested in tax-deferred assets k

d grow at an after-tax rate of return r. Taxes on

the income earned on those assets are not paid as long as the earnings remain in the accounts.

Instead, the household pays taxes on earned income at the time of withdrawal. However, some

companies pay taxes on their profits after deducting interest payments on new investments and

depreciation, therefore reducing overall household earnings. That feature of the model reflects

the U.S. tax treatment of those assets. For example, an IRA can generate a stream of dividends

and capital gains that, as long as they remain in the account, are untaxed at the individual level.

Although the tax treatment differs slightly across the set of tax-deferred assets listed in Table

2, we impose a linear tax τ

kd on the income derived from such assets.

Tax-deferred assets are substantially more illiquid than taxable assets. We address the distinction by applying an asymmetric penalty function φt(j, kd

0−k

d

), which depends on household

age j and the household’s change in the tax-deferred asset holdings. The age affects the penalty

function in that many of the penalties for drawing down tax-deferred assets decline substantially

upon retirement. Before age 59-1/2, individuals can contribute to their IRAs and 529 education savings accounts or increase their amount of life insurance with practically no transaction

costs and in increments that are approximately continuous. Even though there are limits in the

maximum amounts that households can contribute to some tax-deferred assets, those limits vary

among assets. We choose not to explicitly model the maximum amount that can be invested in

those assets, as this constraint is never binding in the model at the high end of the range. Even

though households may reach the maximum that can be invested in retirement accounts, they

still have the option of investing in other tax-deferred accounts, such as life insurance. On the

other side, there is generally no penalty for increasing one’s liquid position before retirement.

Working-age people who want to divest their portfolios of tax-deferred assets confront several

costly obstacles. Retirement funds and education savings accounts are taxed and penalized for

14
early withdrawal. Thus, we consider a penalty function that imposes a significant marginal cost

when reducing the stock of these assets. The benchmark penalty equals 30 percent of the decline

in assets and represents the transaction costs, the 10 percent penalty for early withdrawal, and

taxes accompanying a decline in assets. After retirement, households do not pay the 10 percent

early-withdrawal penalty, and the overall cost of withdrawal equals 20 percent of the decline in

assets. In the sensitivity analysis (Section 6.2), we select penalties that range from 20 percent

to 40 percent of the asset change. On the other side, after retirement, households often must

withdraw funds from tax-deferred assets. For example, after age 70, a person with an IRA

must start withdrawing funds to avoid paying a penalty. We do not explicitly model that afterretirement penalty because, in our model, it is optimal for a household to start withdrawing

assets. It follows that this simplification does not affect the results.

As described above, the penalty function is continuous but not differentiable. Therefore, we

approximate it with a Chen-Mangasarian smoothing function that approximates the properties

of the penalty, but allows the function to remain continuous, differentiable, and convex:

φt(j, ∆k) = ξ(j)[(−∆k − ζ) + log(1 + exp[a(∆k − ζ)])/a] (3)

The parameter a defines the curvature of the function around the kink. Higher levels of a push

the Chen-Mangasarian smoothing function closer to the value of the true “kinked” function.

The point at which the penalty begins to take effect is defined by ζ. Because many retirement

accounts and other tax-deferred assets appreciate in value but cannot be withdrawn, we set the

value of ζ to be slightly positive.

5.2 Parameterization

The model’s parameters are chosen to mirror the main features of the U.S. economy, and they

are consistent with the related literature (Table 13). In the benchmark experiment, the subjective discount factor β equals 0.957, the coefficient of relative risk aversion γ equals 2, and the

minimum net asset position equals 0. The parameters for the age-polynomial that describes the

life-cycle component of the income shock are taken from Cocco et al. and are obtained using a

15
sample of the population with a high school diploma. The persistence and variance of the persistent shock to the income process is taken from Storesletten et al. and subsequently employed

by Fernandez-Villaverde and Krueger (2011). The after-retirement income is assumed equal to

the product between the average income in the last working period and the replacement ratio.

The replacement ratio is assumed to equal 0.6 (Bernheim, Skinner, and Weinberg, 2001).

In the model, we assume that the pretax real gross return on assets is 5 percent. However,

corporations and businesses pay taxes on net profit, before the household receives any investment

income. To take into account the taxes on corporate and noncorporate business income, we

parameterize the interest rate r to about 4.3 percent, reflecting an average effective business tax

rate of 13.4 percent in 2001. That rate is obtained as the weighted average of the tax rates on

corporate and noncorporate business income, under the simplification that all capital income

from businesses can be placed into tax-deferred accounts. It follows that the household receives

about a 4.3 percent return on its investments. The return on capital income is slightly below

the interest rate calculated by taking the reciprocal of the time value of money, 1

β

. The tax rate

on the taxable assets, τ

kt, is chosen such that, together with the interest rate r, net of corporate

taxes, the implied total tax burden on taxable assets is 35.2 of capital income.9

In the following simulations we compare the model with taxable and tax-deferred assets

against a model with a single liquid asset. In the latter model, the tax rate on liquid assets has

been computed to match the total savings in the two-asset model.

6 Simulation Results

To conduct our tax rebate experiment, in the initial period we simulated a sample of two cohorts

of households that differ by Social Security number. One cohort receives the rebate in period

three; the other receives it in period four. In period one, each household believes with certainty

that current economic conditions will persist. In period two, each household updates its beliefs

to include the timing of the receipt of the tax rebate; its amount; and any future economic

conditions, such as reduced wages, increased likelihood of negative income shocks, and reduced

9Source: Staff estimates from the Congressional Budget Office.

16
return on capital.

We generated a random sample of households using the 2001 U.S. population distribution

by age and survival probabilities, so that our experiment is comparable with that performed by

Johnson et al. We estimate the following equation:

∆ci,t = γ0 + γ1rebatei,t + ζi,t (4)

where rebatei,t is the amount received by household i during period t. The dependent variable

is the change in consumption ∆ci,t for household i during period t. We use this equation

to identify the simulated consumption response to the tax rebate, which corresponds to the

real consumption response to the 2001 tax rebate as identified by Johnson et al. We also can

separately identify other effects, such as an identification effect, by constructing a counterfactual

simulation in which households do not receive a rebate.

We modeled the tax rebate as a temporary increase in lump-sum transfers, the timing of

which depends on the household state s. Specifically, a tax rebate is distributed to households

with state s = 0 in period t and to households with state s = 1 in period t + 1. We chose

the benchmark value of the simulated tax rebate to be approximately the same as the 2001

tax rebate. Specifically, about 0.5 percent of total annual personal consumption expenditure in

2001. In the next section we present the benchmark experiment. In the sensitivity analysis, we

show how the results are modified after changing the benchmark assumptions and the model’s

parameterizations.

6.1 Benchmark Case

We evaluated our simulated model against two features of household behavior over the life

cycle.10 Figure 1 shows a simulated profile of consumption, labor income, and total assets of

10We considered five grid points for the persistent shock and three grid points for the transitory shock. The

former were selected using the Tauchen (1986) approach, the latter using a quadrature. The household’s value

function is interpolated using 49-degree, 50-node Chebychev polynomials in each dimension. The number of

nodes was selected so that the MPC from the simulated tax rebate was largely invariant to a change in grid

points. The household policy functions are interpolated using bilineal interpolation. To solve the household’s

problem, we employed Fortran optimization software, dpsol, written by Yongyang Cai and Kenneth L. Judd.

17
an average household, without income shocks. For young households, between the ages of 20

and 30, consumption moves in tandem with current income, and little precautionary savings are

accumulated to buffer against negative income shocks.11 The rate of income growth slows as the

household approaches 30, and the household begins to accumulate assets. Total assets peak a few

years before retirement. As the household retires at age 65, income drops, but consumption does

not fall suddenly at retirement; households use funds accumulated in retirement accounts and

other tax-deferred assets to finance living expenses. Under our assumption that the household

lives until 90 and does not leave any bequest, the household’s assets are drawn down to zero

before dying.

To illustrate how different households behave when they receive a tax rebate, we studied the

consumption responses of selected households. Specifically, we charted the difference between

the simulated consumption of a household that receives a tax rebate in period three and the

simulated consumption of a household that does not receive a rebate (Figure 2). The difference

is expressed as a share of the tax rebate.

The upper-left panel of Figure 2 shows the different responses of three 25-year-old households.

The solid line shows consumption of a household with no financial assets. That household expects

fairly steep income growth in the near future and its borrowing is constrained; it spends nearly

all of the rebate upon receipt. The long-dashed line shows consumption of a household with

tax-deferred assets but no liquid taxable assets. In contrast to the first household, the second

household exhibits a small anticipatory response; namely, spending some of the rebate before

receiving it. Moreover, the fraction of the rebate spent upon receipt is smaller than that spent

by the first household. The dotted line shows consumption of a household with taxable and

tax-deferred assets. That household has a small consumption response, which includes a small

increase in consumption before the rebate arrived—the anticipation effect—and a small increase

in the year after receiving the rebate to smooth consumption. Clearly, liquidity constraints make

a significant difference in a young household’s willingness to spend its rebate.

The upper-right panel of Figure 2 shows the different consumption responses of three 30-

11Labor income, consumption, labor taxes, and capital taxes account for the wedge between income and

consumption on the figure.

18
year-old households with financial characteristics similar to those described for the upper-left

panel. Their consumption follows similar patterns; however, in contrast to the younger group,

all households display some anticipatory response. Moreover, the fraction spent upon receipt is

smaller. The first household, with no assets by the time the tax rebate is received but with some

accumulated savings in previous years, chooses to spend almost all of the rebate upon receipt.

The small anticipatory effect in period 2 in the graph is from a very small amount of money

that would otherwise have been invested but that is now spent in anticipation of having the

rebate in period three. The third household, with taxable and tax-deferred assets, spends a tiny

amount of the rebate upon receipt. Instead, it increases consumption, mostly before receiving

the rebate.

The bottom-left panel of Figure 2 shows consumption among 50-year-old households; the

bottom-right panel shows consumption among 70-year-old households. The panels display similar

behavior. The tax rebate makes a small positive impact on household consumption. Nearly all

households choose to spend a small fraction of it, either before or after receipt. For those

households, the anticipated growth in income is small, and they choose to smooth consumption

over time. The penalty for selling tax-deferred assets drops substantially after age 65, as the

obstacles to closing retirement accounts and other illiquid assets fall, and the distinction between

households with taxable and tax-deferred assets narrows.

The first panel in Table 14 presents the estimation results for γ1 obtained from Equation 4. In

the benchmark experiment, households receive a rebate either in period three or in period four,

and no change in wages, interest rates, or any other variables affects the household decision. The

coefficient γ1 can be interpreted as the average share of the tax rebate that households spend on

consumption upon receipt. We compared the results obtained from a standard life-cycle model

with only one liquid asset against those from a model with two assets, one taxable and one

tax-deferred, when households face a penalty to withdraw their funds invested in the taxable

asset.

In the standard life-cycle model with a single liquid asset, the coefficient γ1 implies that

households spend contemporaneously about 16.9 percent of the tax rebate on consumption.

When households can invest in tax-deferred and taxable assets and also confront a penalty for

19
withdrawing funds from tax-deferred account, that share increases to about 20.7 percent. The

increase of nearly 4 percentage points puts the simulated consumption response in line with that

documented by Hamilton, Johnson et al., and Parker et al. However, the simulated consumption

response is highly responsive to the model’s parameterizations.

In the context of our model, we can compare household consumption against a benchmark

scenario in which there is no tax rebate. The difference between identical households in the

two models can provide an alternative estimate of the MPC in the period of the rebate. Many

households spend the tax rebate smoothly. They take that transitory income and, particularly if

they are older and not liquidity constrained, spend the money over several periods. Comparing

households in the tax rebate scenario against this benchmark, the no-rebate scenario allows us

to identify additional spending generated by the tax rebate that may be generated but that

is otherwise not identified by the regression. The total consumption response in the period of

the tax rebate for a household receiving it compared with a comparable household that does

not jumps 4 percentage points to about 20.7 percent. The regression identifies a household’s

propensity to consume out of a contemporaneous transitory income shock, but as a policy, a tax

rebate can generate a still larger increase in household consumption.

Figure 3 presents the percentage of households in each age group that contemporaneously

respond to a tax rebate over the life-cycle, in the one-asset and in the two-asset model. We

identify such households with the indicator function:

I((c1,t > c2,t + )

\

(c1,t+1 +  < c2,t+1)) (5)

The first subscript of the household’s consumption c is the Social Security number and the

second is the period of time. We typically set  to be about 5 percent of the tax rebate to

screen out small positive responses that may result from the approximation methods used to

solve the model. The solid bars in the figure indicate the percentage of each cohort of households

that respond contemporaneously to tax rebates by increasing their consumption, as indicated

in Equation 5, in the one-asset model. The open bars display the same information but for the

two-asset model when the penalty to withdraw tax-deferred assets is positive.

20
The early-life and late-life behavior among households is fairly similar, both in the one-asset

and in the two-asset model. Most households at the beginning of their life cycles spend nearly

all of the rebate. At age 20, between 65 and 70 percent of households increase consumption

upon receiving the rebate. Those households are liquidity constrained and can look ahead to

steep improvements in their projected earnings.

By age 23, however, the household response to the tax rebate under the two models begins

to diverge.

In the one-asset model, households begin to accumulate assets. Those holdings, although

initially quite small, imply that households are no longer income constrained. Furthermore, the

slope of the expected future earnings curve begins to decline, indicating that households have a

smaller incentive to spend a large share of the tax rebate. By age 30, about 10 percent of the

households respond contemporaneously to a tax rebate. That proportion drops nearly to zero

by age 40, when almost none of the households contemporaneously respond to the tax rebate.

By contrast, in the two-asset model, even middle-age households are willing to spend some

of the tax rebate. The open bars show that approximately 50 percent to 70 percent of young

households (between 23 and 30 years old) and approximately 10 percent to 30 percent of middleaged households (between 30 and 60 years old) spend some fraction of the tax rebate upon

receipt.

Young households that have begun to invest in tax-deferred assets and have accumulated

small amounts of wealth are likely to be liquidity constrained if a negative income shock occurs.

Young households spend a significant fraction of the tax rebate also in the one-asset model, as

their accumulated wealth is low and they may be liquidity constrained if a negative event occurs.

However, in the one-asset model, the liquid assets accumulated are always larger relative to the

two-asset model, implying that a smaller fraction of the rebate is spent to buffer consumption

against negative income shocks.

Middle-aged households generally have accumulated more assets than the younger households have. However, some middle-aged households that have invested a large share of savings

in tax-deferred accounts may find themselves liquidity constrained if hit by a negative income

shock. Therefore, those households might find it optimal to consume some of a tax rebate

21
upon receipt rather than withdrawing funds from tax-deferred accounts and paying associated

penalties. Middle-aged households’ behavior in the two-asset model is very different from their

behavior in the one-asset model (see Figure 3). In the one-asset model, the middle-aged households have accumulated enough assets that can be withdrawn without penalty. The households

are almost never liquidity constrained, even when negative shocks occur.

After retirement, as the surviving households come closer to being liquidity constrained, they

begin to spend part of the rebate. Although the onset of a positive consumption response differs

slightly, the qualitative features of household behavior are similar in the two models.

Figure 4 shows the percentage of sample households that respond contemporaneously to a tax

rebate. The sample is constructed to match the U.S. household distribution by age in 2001. For

example, the 20-year-old households that respond to the tax rebate represent about 1.2 percent

of the total population. Although a large share of older households spend the tax rebates, that

group is not the main one that accounts for the increase in the observed γ1. Most of the increase

can be accounted for by households in the 20- to 60-year-old group.

The household response to the tax rebate, conditional on holding liquid taxable financial

assets, is illustrated by Figure 5. The households that spend a share of the rebate are predominantly those without liquid financial assets. This result mirrors the empirical findings by

Agarwal et al., Hamilton, and Johnson et al. The liquidity constraints are a driver of consumption dynamics. In the sample population, about 60 percent of households with zero liquid assets

spend the rebate upon receipt, whereas only about 10 percent of households with some liquid

assets do so.

Households with no financial assets display a high response to the tax rebate—nearly 100

percent spend the rebate (see Figure 6). The fraction spent declines slowly with the increase in

holdings of financial assets. The wealthiest households spend almost none of the rebate, although

in any decile there are households that respond to the rebate by increasing consumption. This

finding stands in sharp contrast to the results obtainable from the one-asset model.

Household consumption changes as income changes (see Figure 7), with most of the response

occurring in the first five quantiles. The lowest quantiles include the youngest households, who

are liquidity constrained and face an upward-sloping income curve. The consumption response

22
is high among those households. As income increases, the share of the tax rebate spent on

consumption decreases. Heterogeneity in current and expected labor income explains the nonmonotonic consumption response for the higher quantiles; nonetheless, for the higher quantiles

the consumption response is always smaller than for the lower quantiles.

6.2 Sensitivity Analysis

To identify possible responses of households to the receipt of a tax rebate, we compared three

scenarios: a one-asset model, a two-asset model without a withdrawal penalty, and a two assetmodel with a withdrawal penalty.

The results obtained in the one-asset model and in the two-asset model without withdrawal

penalty are similar, for two main reasons. First, in each model, the withdrawal penalty equals

zero. Second, in the one-asset model, the tax rate on liquid assets, even though slightly different

in the various experiments, is overall just a little higher than that applied to tax-deferred assets

in the two-asset model without a penalty. That rate is computed to match the total savings

obtained in the two-asset model with a withdrawal penalty. Therefore, in the two-asset model

without a penalty, households optimally choose to invest only in the tax-advantaged assets, as

those assets will result in lower tax assessments.

In the benchmark two-model asset with a positive withdrawal penalty, households consume,

on average, about 21 percent of the tax rebate upon receipt; additionally, their consumption out

of the tax rebate is about 4 percent larger than in the other models. In this section, we modify

some of the baseline assumptions and parametrization choices to evaluate the sensitivity of our

benchmark results and of the spread (see Table 14).

A household’s behavior is relatively insensitive to the size of the penalty; the MPC drops

slightly as the penalty increases, encouraging some households to substitute holdings in taxable

assets from the less-liquid tax-deferred accounts. The existence of a penalty above a threshold

is enough to generate the household’s response. In addition, the household response is almost

unchanged even if the 10 percent early-withdrawal penalty must be paid until age 50 instead

of age 60. This follows from the more active consumption exhibited by the young households

23
relative to the close-to-retirement households.

Instead, changes in the capital income tax rate make a larger difference in households’ willingness to hold taxable assets. Lowering the tax rate reduces the payoff for holding assets in

tax-deferred accounts. Households are more willing to hold savings in a liquid taxable form,

which allows them to smooth their consumption out of transitory shocks to their incomes more

easily. The consumption response to the tax rebate is less than in the benchmark case. Conversely, higher tax rates give a greater incentive to households to trade risk for tax-deferred

status, and both the households’ holding of tax-deferred assets and the consumption response

to the tax rebate are larger. If tax rebates are distributed in a period in which the Bush-era

tax cuts are not yet expired, the lower tax rate on capital income implies a smaller consumption

response.

The amount of the rebate affects the consumption response in the no-penalty and penalty

scenarios. In each case, increases in the rebate cause households to spend a smaller share upon

receipt. The desire to smooth consumption induces households to save part of a larger transfer

and spread consumption across several periods.

Changes in the discount factor β affect the penalty scenario and the no-penalty scenario.

When β increases to 0.98, without changing the interest rate, households are more patient and

prefer to hold more tax-deferred assets. They choose to consume less of the rebate in any model.

Without a penalty for withdrawing tax-deferred assets, the more patient households consume

only a negligible fraction of the rebate. In contrast, in the face of a withdrawal penalty, those

households consume a larger fraction of the rebate. The spread between the two scenarios is

significant, as the marginal propensity to contemporaneously consume out of a tax rebate almost

doubles. When β decreases to 0.94, households are more impatient, tilt their portfolios optimally

toward taxable assets, and choose to consume a significant fraction of the rebate, with or without

a penalty. The spread between the penalty and no-penalty scenarios is significant, even though

any additional response generated by the introduction of the penalty is mitigated somewhat by

the household’s impatience.

A larger replacement ratio implies that after-retirement income is a larger fraction of the lastworking-year income, compared with the benchmark case. Therefore, households would have to

24
save less for retirement and consume a larger share of the rebate upon receipt. On the other

side, a smaller replacement ratio implies a lower marginal propensity for consumption among

working-age households, which must save more to support consumption in retirement.

To test the effect of economic conditions on households’ MPC, we modeled a short and a

long recession as negative productivity shocks, causing a two-year or six-year decline in wages

and interest rates. In each experiment, the marginal propensity to consume decreased with

respect to the benchmark scenario. The longer the recession, the larger the decrease in the

MPC. Consistent with previous experiments, there is typically a spread of 3 to 4 percentage

points between the rate of spending out of those tax rebates, with and without penalties for

withdrawal.

6.3 Employee Matching Contributions

We evaluated one developing feature of employer-based retirement plans—the employer matching

contribution. We extended our penalty function to approximate this feature and identified the

difference between the standard penalty function and a hypothetical one reparameterized so that

positive contributions to tax-deferred accounts are matched to a small extent by the employer

(see Figure 8). In the benchmark scenario, the penalty paid by households is positive when

∆k is negative—when households withdraw assets from the tax-deferred accounts—and zero

otherwise. Instead, with matching contributions, the households receive a benefit from further

contributions to the tax-deferred accounts, up to some age-specific limit that we define as m(j).

In the figure, the upper dashed line reflects the originally formulated penalty function. The

vertical dashed lines delimit the approximate area in the scenario with employer matching in

which the household has access to the match. The matching-contribution limit is modeled by

allowing the penalty function to decrease up to some positive value for −∆k, and then the

function keeps a constant value.

The penalty function in Equation 3 is reformulated to include the maximum allowable match

m(j):

φt(j, ∆k) = ξ(j)[(−∆k − (ζ + m(j))) + log(1 + exp[a(∆k − (ζ + m(j)))])/a − m(j)]

25
(shown in Figure 8 as the solid line). In this function, m(j) is the maximum matching contribution, and the match depends on age.12 In the scenario with a 30 percent penalty, for

every dollar that goes into the matching-fund account, 30 cents is refunded to the household.

Therefore, every additional dollar in the tax-deferred account costs a net 70 cents. The effective

employer match would be ξ(j)

1−ξ(j)

, or about 43 percent of the household contribution up to the

matching-contribution limit.

Table 14 shows the effects of progressively larger values of m(j). The matching contributions

have two competing effects: They increase savings, and they increase the proportion of assets

held in tax-deferred accounts. From the smallest, m(j) = 0.0025, to the largest, m(j) = 0.03,

overall household savings increase by approximately 50 percent. The returns on investment in

tax-deferred accounts also increase substantially. The result is a decrease in the consumption

response to a tax rebate as the value of m(j) increases, because overall savings are so much

greater for the same tax and interest rates. However, savings in taxable accounts actually

decrease between the scenarios, suggesting that households invest an even larger fraction of

their money in tax-deferred accounts than would otherwise be invested in taxable assets.

6.4 Catastrophic Income Shocks

Amromin (2008) studied the household portfolio choice over taxable and taxable assets using

SCF data. Following Amromin (2003, 2008) and to provide an additional sensitivity test, we

introduced some amount of catastrophic income shock, which we modeled as a transitory income

shock, occurring with a 1 percent probability and generating a 90 percent drop in income.

When the shock is introduced, household MPC and household behavior change notably.

Specifically, the household MPC drops significantly, as households prefer to increase precautionary savings (see Table 14). In this scenario, the precautionary-savings motive drives the

dynamics of the model. Households face a trade-off between the wish to hold most savings

in tax-deferred accounts, to benefit from the tax advantage status, and their understanding of

having to make a costly withdrawal if a catastrophic income shock occurs. The precautionary12The match is set to zero during retirement because the employer’s matching ceases when the employee retires.

26
savings motive affects household behavior, especially at a young age. This group of households

prefers to save more of the rebate, instead of consuming it.

Similar to the benchmark scenario, our one-asset and two-asset models generate a measure

of young and middle-aged households that respond to tax rebates in the face of a catastrophic

shock (see Figure 9). In the one-asset model with catastrophic shocks, households between the

ages of 20 and 60 display almost no response, but in the model with taxable and tax-deferred

assets, those households demonstrate a modest response to the rebate. The largest response,

however, is exhibited by retirees, consistent with households surveyed by Shapiro and Slemrod

(2009). Retired households constitute the largest fraction of households in the population that

consume at least some of the tax rebate upon receipt (see Figure 10).

7 Conclusion

We developed a model of household behavior to explain the observed effects of the 2001 tax

rebate on household consumption. One component of tax reform in 2001 was a transfer that

accounted for about 0.6 percent of yearly personal consumption. We modeled this transfer,

disbursed through an exogenously staggered process, in a life-cycle model of portfolio choice

with liquid taxable and tax-deferred assets, labor earning risk, and liquidity constraints.

Our model is built on the distinction between tax-deferred and taxable assets. Tax-deferred

assets allow households to enjoy higher-than-normal returns in exchange for lower financial

flexibility because of the early-withdrawal penalty. Households may find it optimal to constrain themselves by holding few or no liquid taxable assets. When the tax rebates arrive,

those households—some holding substantial but few liquid assets—behave more like constrained

households in the standard life-cycle model. As their liquid wealth is small enough that it cannot

be used to respond to adverse income shocks, those liquidity-constrained households spend more

of the rebate on consumption.

In contrast to a standard one-asset life-cycle model, which does not generate any significant consumption response among middle-aged households, our model with two assets and a

withdrawal cost generates a consumption response for a wider range of age groups. Households

27
with a larger share of taxable assets are more likely to use those assets to smooth consumption

and not to spend a larger share of the tax rebate when received, behaving in accordance with

the standard life-cycle model. Given the overall higher consumption response for younger and

middle-aged households, relative to the one-asset model, our theoretical model generates a larger

marginal propensity to consume out of tax rebates. Specifically, households contemporaneously

consume part of their rebate and, on average, they spend almost 21 percent of it upon receipt.

Moreover, our theoretical model qualitatively replicates empirical choices about household consumption and holdings of assets, as well as the share of total wealth invested in taxable assets

over the life cycle.

This model can be used to help evaluate the effectiveness of policies designed to stimulate

household consumption under different scenarios. Specifically, we consider the impact of a tax

rebate during a recession, as well as in the presence of changes in interest rates and capital

income tax rates. Overall, our results indicate that transfer policies might be adequate to

provide economic stimulus, but targeted transfers may boost the MPC of a tax rebate more

than across-the-board transfers. To determine the aggregate economic effects of such a policy,

however, we would need to embed this model in a general equilibrium framework in which the

effectiveness of tax rebates in stimulating economic output can be evaluated under a variety of

economic scenarios.

28
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31
Table 1: Empirical Household Responses to Tax Rebates in Consumption of Nondura

Goods

Experiment Contemporaneous Cumulati

Response Two-Quarter Respon

Johnson et al. (2006)

All households 0.373 (0.135)1 0.691 (0.26

Only households getting rebate 0.247 (0.214)3 nLow income (< $34,298) 0.756 (0.223)4 1.380 (0.42

Low liquid assets (< $1,000) 0.633 (0.208)4 1.256 (0.42

Drop Outliers; Ordinary Least Squares

All households5 0.226 (0.105) 0.454 (0.21Only households getting rebate 0.093 (0.191) n

Low income (< $3,4298) 0.751 (0.220) 1.405 (0.42Low liquid assets (< $1,000) 0.634 (0.208) 1.316 (0.41

Median Regression

All households5 0.126 (0.048) 0.292 (0.12Only households getting rebate 0.002 (0.076) 0.075 (0.19

1 From Johnson et al., Table 2.2 From Johnson et al., Table 4.3 From Johnson et al., Table 3.4 From Johnson et al., Table 5. Standard errors for the contemporaneous respon

not published in original paper; authors estimated.

5 From Hamilton. We extended his analysis to every other category we reported 

this group.n/a = not applicable

32
Table 2: Taxable and Tax-Deferred Assets

Asset Example Tax Advantage Transaction CostTaxable Assets Bonds, stocks, CDs None; interest and dividends Very low to zero; must 

mutual funds subject to income tax any) on sales of mutual 

Defined-Contribution IRAs, 401(k)s, Tax-deferred, contribution Typically 10 percent peRetirement Accounts SEP-IRAs, Keoghs, may be tax-deductable if distribution occurs b

thrift-type acc’ts state and federal incom

Defined-Benefit Pensions Tax-deferred growth Limited withdrawals, sRetirement Accounts permitted; can be term

rolled into an IRA (see 

Life Insurance Tax-deferred growth Some have 10 percent p

if distribution occurs bstate and federal incomother penalties vary by 

Education Savings State-run education Tax-deferred; contribution Typically 10 percent peAccounts saving accounts may be tax-deductable used for education); sta(529 plans) federal income tax app

33
Table 3: Tax-Deferred Assets for Households That Hold Tax-Deferred Assets Only, by Ag

(Dollars)

Household Age Group

Percentile 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 25th 650 1,400 1,500 2,000 2,000 2,600 2,000 8,000 2,500 

50th 4,100 3,200 8,400 6,300 5,400 10,000 14,100 20,500 7,000 75th 9,500 5,000 28,000 28,000 21,000 47,000 69,000 29,000 15,000 

Table 4: Tax-Deferred Assets for Households That Hold Tax-Deferred Assets Only, by Tot

(Dollars)

Total Financial Assets

Percentile $ 0-10,000 $ 10,000-50,000 $ 50,000-100,000 $ 100,000-300,000 $ 300,0025th 900 15,200 64,000 129,000 385

50th 2,100 24,520 76,500 140,000 59575th 5,000 35,000 85,000 164,000 832

34
Table 5: Taxable Assets for Households That Hold Taxable Assets Only, by Age Gr

(Dollars)

Household Age Group

Percentile 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 25th 200 200 200 200 240 250 350 210 300 

50th 400 750 2,000 1,200 1,000 1,700 1,500 2,900 4,700 175th 1,470 4,000 11,800 5,000 5,600 8,800 10,000 15,500 29,000 6

Table 6: Taxable Assets for Households That Hold Taxable Assets Only, by Total A

(Dollars)

Total Financial Assets

Percentile $ 0-10,000 $ 10,000-50,000 $ 50,000-100,000 $ 100,000-300,000 $ 300,0025th 200 15,400 61,200 122,800 341

50th 780 24,500 73,000 156,000 550

75th 3,100 32,800 86,040 205,000 1,050

35
Table 7: Tax-Deferred Assets for Households That Hold Tax-Deferred and Taxable Assets, b

(Dollars)

Household Age Group

Percentile 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 25th 500 2,000 4,200 7,300 11,700 16,200 20,000 21,000 17,0050th 1,800 8,000 16,000 29,000 40,000 54,000 68,500 80,500 61,0075th 13,500 28,000 56,000 70,000 108,700 166,000 185,000 237,000 267,83

Table 8: Tax-Deferred Assets for Households That Hold Tax-Deferred and Taxable Assets, by 

(Dollars)

Total Financial Assets

Percentile $ 0-10,000 $ 10,000-50,000 $ 50,000-100,000 $ 100,000-300,000 $ 30025th 800 8,000 25,300 61,000 1

50th 2,000 15,100 50,000 103,200 3

75th 5,000 26,000 63,000 150,000 5

36
Table 9: Taxable Assets for Households That Hold Tax-Deferred and Taxable Assets, by A

(Dollars)

Household Age Group

Percentile 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 25th 500 810 1,200 1,500 2,000 3,000 5,000 7,600 3,500 50th 1,500 4,000 6,000 6,320 11,030 16,960 17,300 28,500 33,000 75th 7,050 13,000 22,800 32,200 52,000 82,460 68,300 132,800 160,800 

Table 10: Taxable Assets for Households That Hold Tax-Deferred and Taxable Assets, by T

(Dollars)

Total Financial Assets

Percentile $ 0-10,000 $ 10,000-50,000 $ 50,000-100,000 $ 100,000-300,000 $ 30025th 280 1,860 6,100 15,100 

50th 1,000 6,300 20,000 53,500 2

75th 2,500 14,300 44,000 113,000 5

37
Table 11: Mean Portfolio Share, by Age Group

Household Age Group

Asset 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 Saving, Money Market, Call Deposits 0.40 0.36 0.22 0.26 0.22 0.24 0.19 0.24 

Certificates of Deposit 0.03 0.02 0.03 0.02 0.02 0.04 0.06 0.05 Directly Held Mutual Funds 0.06 0.05 0.06 0.06 0.06 0.06 0.06 0.08 Stocks 0.09 0.08 0.08 0.07 0.07 0.05 0.04 0.06 Bonds 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 Savings Bonds 0.05 0.02 0.02 0.03 0.03 0.01 0.02 0.01 Quasi-Taxable Retirement Accounts 0.19 0.32 0.40 0.41 0.44 0.43 0.43 0.35 Life Insurance 0.09 0.08 0.13 0.10 0.11 0.11 0.14 0.12 Other Managed Assets 0.01 0.01 0.01 0.01 0.02 0.02 0.03 0.05 Other Financial Assets 0.07 0.05 0.05 0.04 0.03 0.04 0.03 0.04 

38
Table 12: Mean Portfolio Share, by Financial Assets

Total Financial Assets

$0-10,000 $10,000-50,000 $50,000-100,000 $100,000-300,000 $

Saving, Money Market, Call Deposits 0.48 0.22 0.15 0.14 

Certificates of Deposit 0.04 0.09 0.09 0.06 

Directly Held Mutual Funds 0.02 0.05 0.08 0.10 

Stocks 0.04 0.05 0.07 0.09 

Bonds 0.00 0.00 0.00 0.01 Savings Bonds 0.03 0.01 0.02 0.01 

Quasi-Taxable Retirement Accounts 0.23 0.38 0.44 0.41 

Life Insurance 0.11 0.15 0.10 0.11 Other Managed Assets 0.00 0.01 0.02 0.06 

Other Financial Assets 0.06 0.04 0.02 0.02 

39
Table 13: Parameterization

Parameter Variable Value

β Time rate of preference 0.957

γ Coefficient of relative risk aversion 2

b Minimum net asset position 0

ρ Autoregressive parameter of the persistent income component 0.935

σ

2

η Variance of the innovation to the persistent income component 0.061

σ

2

 Variance of the transitory income component 0.017

φ Replacement ratio 0.6

τ

kt Tax rate on taxable income 0.352

τ

kd Tax rate on tax-deferred income 0.134

τ

l Tax rate on labor income 0.30

τ

c Tax rate on consumption 0.05

w Wage rate 1.0

40
Table 14: Household Contemporaneous Consumption of Tax Rebate Under Alternative Scenarios

(Percent) One Two Assets, Two Assets,

Asset No Penalty Penalty

Benchmark Experiment 16.88 16.45 20.71

Changes in Penalty

40 percent nonretirement; 25 percent retirement n/a n/a 19.24

30 percent before age 50; 20 percent after age 50 n/a n/a 20.62

Alternative Tax Rates on Taxable Income

τ

kt = 0.25 n/a n/a 18.46

τ

kt = 0.45 n/a n/a 22.11

τ

kt = 0.299 (Bush) n/a n/a 19.30

Size of the Tax Rebate

Rebate = 0.3 percent of personal consumption 17.02 16.53 20.73

Rebate = 6.0 percent of personal consumption 13.19 12.80 16.40

Changes in discount factor

β = 0.98 5.06 4.87 7.64

β = 0.94 28.13 26.84 31.93

Replacement Ratio

φ = 0.7 21.10 20.57 24.30

φ = 0.5 13.46 13.10 16.47

Recession

Short (w, r fall by 5 percent for 2 years) 14.82 14.32 17.12

Long (w, r fall by 5 percent for 6 years) 14.77 11.48 15.18

Matching Contributions (30 Percent Match) Up To

0.0025 n/a n/a 19.98

0.005 n/a n/a 18.94

0.01 n/a n/a 17.42

0.03 n/a n/a 16.41

Catastrophic Shocks

1 percent chance of 90 percent drop in income 8.79 8.29 11.36

n/a = not applicable

[Table revised on July 22, 2011, from the original version, published on July 14, 2011]

41
20 30 40 50 60 70 80 90 

0

0.10.20.30.40.50.60.70.80.91

Age

Income

AssetsConsumptio

Figure 1: Income, Assets, and Consumption Path Over the Life Cycle

42
1

2

3

4

5

0

20406080

100

Age 25

Time Period (Rebate in Period 3)

Percent of Rebate Consumed

kd

 = 0, 

kt

 = 0

kd

 > 0, 

kt

 = 0

kd

 > 0, 

kt

 > 0

1

2

3

4

0

20406080

100

Age 30

Time Period (Rebate in Period 3)

Percent of Rebate Consumed

kd

 = 0, 

kt

 = 0

kd

 > 0, 

kt

 = 0

kd

 > 0, 

kt

 > 0

1

2

3

4

5

0

20406080

100

Age 50

Time Period (Rebate in Period 3)

Percent of Rebate Consumed

kd

 > 0, 

kt

 = 0

kd

 > 0, 

kt

 > 0

1

2

3

4

0

20406080

100

Age 70

Time Period (Rebate in Period 3)

Percent of Rebate Consumed

kd> 0, 

kt

 = 0

kd

 > 0, 

kt

 > 0

kd= tax-deferred assets, kt= taxable assets.

Figure 2: Household Consumption Response to Tax Rebate

43
20 30 40 50 60 70 80 

0

102030405060708090

100

Age Group

Percentage of Households

One Asset

Two Assets

Figure 3: Percentage of Households Responding Contemporaneously to Tax Rebate, by Age Group

the household probability of responding to a tax rebate upon receipt, conditional on age. The results model with one asset and for the model with two assets, tax-deferred and taxable, with withdrawal p

44
20 30 40 50 60 70 80 

0

0.20.40.60.811.21.41.61.82

Age Group

Percentage of Population

One Asset

Two Assets

Figure 4: Percentage of Total Population Responding Contemporaneously to the Tax Rebate, by Ag

reports the household joint probability of belonging to a given age group and responding to a tax The results are displayed for the model with one asset and for the model with two assets, tax-deferrwithdrawal penalty.

45
0 >0

0

102030405060708090

100

Taxable Financial Assets

Percentage of Households

Figure 5: Percentage of Households Responding Contemporaneously to Tax Rebate, by Holdings o

46
1

2

3

4

5

6

7

8

9 10

0

102030405060708090

100

Decile of Total Financial Assets

Percentage of Households

Figure 6: Percentage of Households Responding Contemporaneously to Tax Rebate, by Holdings 

47
1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

0

102030405060708090

100

Wage Income

Percentage of Households

Figure 7: Percentage of Households Responding Contemporaneously to Tax Rebate, by Inco

48
-0.1 -0.08 -0.06 -0.04 -0.02

0 0.02 0.04 0.06 0.08 

-0.01

-0.0050

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Penalty Function

∆

kd

 = Change in Tax Deferred Assets

With Matching Contrib.Without Matching Contrib

Figure 8: Cost of Changing Stock of Tax-Deferred Assets.

49
20 30 40 50 60 70 80 

0

102030405060708090

100

Age Group

Percentage of Households

One Asset

Two Assets

Figure 9: Percentage of Percentage of Households Responding Contemporaneously to Tax Rebate; in 

trophic Shocks, by Age Group. The figure reports the household probability of responding to a tax conditional on age. The results are displayed for the model with one asset and for the model with twand taxable, with withdrawal penalty.

50
20 30 40 50 60 70 80 

0

0.20.40.60.811.21.41.61.82

Age Group

Percentage of Population

One Asset

Two Assets

Figure 10: Percentage of Total Population Responding Contemporaneously to Tax Rebate; in a Mod

Shocks, by Age Group. The figure reports the household joint probability of belonging to a given age gto a tax rebate upon receipt. The results are displayed for the model with one asset and for the motax-deferred and taxable, with withdrawal penalty.

51