Technical Paper Series

Congressional Budget Of¿ce

Washington, DC

CONSUMPTION TAXES AND ECONOMIC EFFICIENCY

IN A STOCHASTIC OLG ECONOMY

Shinichi Nishiyama

Congressional Budget Of¿ce

Washington, DC

E-mail: shinichi.nishiyama@cbo.gov

Kent Smetters

University of Pennsylvania & NBER

Philadelphia, PA

E-mail: smetters@wharton.upenn.edu

December 2002

2002-6

Technical 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 analysis and conclusions expressed in them are those of the authors and should not be

interpreted as those of the Congressional Budget Of¿ce. References in publications should be

cleared with the authors. Papers in this series can be obtained by sending an e-mail to

techpapers@cbo.gov.
Abstract

Fundamental tax reform is examined in a heterogeneous overlapping-generations lifecycle model in which agents face idiosyncratic earnings shocks and uncertain life spans.

Following Auerbach and Kotlikoff (1987), a lump-sum redistribution authority is used to examine ef¿ciency gains over the transition path. A progressive income tax is replaced with a

Àat consumption tax (for example, a value-added tax or a national retail sales tax). If shocks

are insurable (that is, no risk), this reform improves (interim) ef¿ciency, a result consistent

with the previous literature. But if, more realistically, shocks are uninsurable, this reform

reduces ef¿ciency, even though national wealth and output increase over the entire transition

path. This ef¿ciency loss, in part, stems from reduced intragenerational risk sharing that was

previously provided by the progressive tax system. Social safety net programs can substitute

for insurance to maintain ef¿ciency along with growth effects.

Journal of Economic Literature Classi¿cation Numbers: H0, H2, H3

Key Words: overlapping generations

Helpful comments were received from Ken Judd and participants at the Stanford Institute for Theoretical Economics, July 2002.
1 Introduction

The potential economic bene¿ts from replacing the current income tax system with a Àat

(proportional) consumption tax system have generated a considerable amount of attention

in recent years. Examples of a Àat consumption tax include a value-added tax (VAT) in

many European nations as well as a national retail sales tax that is gaining attention in the

United States as a possible substitute tax base. Replacing the current income tax system

with a revenue-neutral Àat consumption tax would (i) Àatten tax rates, (ii) tax consumption

rather than wage and capital income, and (iii) eliminate all tax-base reductions (preferences)

contained in current law. In all likelihood, this reform would signi¿cantly increase national

saving and output over the long run (Altig and others, 2001), a result that we con¿rm. This

paper examines whether this reform actually improves economic ef¿ciency.

Judging the economic ef¿ciency of a particular policy reform has always been important to economists, but it is important to remind ourselves why. Many different policy reforms can, for example, increase the welfare of people born in a long-run steady state, but

those gains might simply represent losses to intermediate generations. (Indeed, this point

has gained a considerable amount of attention in the recent debate on Social Security privatization.) If there is no economic gain after fully compensating intermediate generations

who otherwise lose from reform, then judgments over a reform must be made purely on a

philosophical a priori basis1 or based on subjective intragenerational and intergenerational

distribution choices.2 To be sure, economists contribute in important ways to these debates,

especially on distributional issues. For example, economists have estimated the impact that

a particular tax system has on the distribution of income or wealth within generations (Auerbach and Hassett, 2001) or between generations (Auerbach, Gokhale, and Kotlikoff, 1994).

Economists have also derived the implied optimal tax schedules under an assumed social

welfare function that weights the utility of different people in a particular way (Mirrlees,

1971). But economists are not particularly better quali¿ed in making philosophical or moral

1

For example, the philosopher Thomas Hobbes argued that it is a priori 3wrong ́ to tax estates on the basis

that wealth was already taxed when earned. Others have argued that it is a priori 3wrong ́ for the government to

take money from one generation to give to another. 2

For example, in terms of intragenerational distribution, the philosopher John Rawls argued that social welfare

must be judged on the basis of the utility of the worse-off person in society.

1
judgments than noneconomists. Ultimately, what economists bring to the table in policy

debates is our insight about ef¿ciency, interpreted here in the Pareto sense.

The point of departure of our paper is that previous analyses of fundamental tax reform

have not incorporated the intragenerational risk-sharing bene¿ts of the current progressive

income tax system. Our main ¿nding is that this risk sharing is important for determining the

ef¿ciency changes associated with tax reform, even at modest levels of risk aversion. When

idiosyncratic earnings shocks are assumed to be fully insurable so that each agent faces no

risk, moving to a Àat consumption tax increases ef¿ciency, a common result in the literature.

However, when wages are uninsurable, ef¿ciency is actually reduced by moving to a Àat

consumption tax, even though national wealth and output increase over the entire transition

path.3 Social safety net programs can substitute for insurance to maintain ef¿ciency along

with growth effects.

1.1 The Ramsey Model

The simplest way to analyze the impact that a revenue-neutral tax reform has on economic

ef¿ciency is with the Ramsey in¿nite-horizon representative-agent model, which assumes

that households are Ricardian. Since, in general, ef¿ciency changes are always calculated

relative to compensated changes, the presence of a single agent in the Ramsey model dramatically simpli¿es these calculations. In particular, calculating ef¿ciency changes does not

require redistributing resources across agents in order to compensate those who would otherwise lose from reform. As a result, one can often derive the most ef¿cient long-run tax

structure analytically (see the reviews by Judd, 1985, and Auerbach and Hines, 2001). But

the Ramsey model is less suitable for capturing the intragenerational risk-sharing bene¿ts of

a progressive income tax system, which is the focus of the current paper.4

3

This paper examines policy changes rather than attempting to derive optimal progressive tax schedules in

the Mirrlees tradition where a social welfare function must be assumed. However, our results on the importance

of risk sharing would also be relevant to that literature, which, thus far, has found little cause for progressive tax

schedules. Although computational considerations limit our ability to derive optimal progressive tax schedules

in our model, this extension could prove useful in the future as computers become more powerful. 4

Well-known complications arise when attempting to model multiple in¿nite-horizon agents. If agents have

identical time preferences (which, realistically, is a measure of zero), then there is an in¿nite number of wealth

distributions compatible with a steady state. If, more realistically, agents have nonidentical time preferences,

then the wealth distribution becomes trivial (one agent owns everything). Incorporating progressive tax rates

creates additional problems.

2
1.2 The Stochastic OLG Model with Finite Horizons

Instead, this paper uses a calibrated overlapping-generations (OLG) life-cycle model with

uninsurable idiosyncratic earnings shocks, uncertain life spans, and elastic labor supply to

examine the ef¿ciency gains associated with adopting a revenue-neutral Àat consumption tax.

The multiplicity of agents, though, complicates the ef¿ciency calculations since tax reform

redistributes resources across different households. To deal with this problem, we follow

the pioneering work done by Auerbach and Kotlikoff (1987), who developed the ¿rst largescale OLG model without uncertainty. Like theirs, our model incorporates a 3Lump-Sum

Redistribution Authority ́ (LSRA) that calculates the overall ef¿ciency gains or losses of a

policy change, by restoring the utility of the agents alive at the time of the reform to their

prereform levels through lump-sum redistributions both across and within generations.

A heuristic technique for calculating ef¿ciency gains that is more common in the literature would simply sum welfare changes across households (discounted if across time) a

policy is then deemed to increase [decrease] ef¿ciency if the net sum is positive [negative].

This latter approach implicitly assumes that lump-sum transfers are made across households

in order to compensate the losers of policy reform with some of the gains of the winners. But

this approach fails to capture the price effects associated with these lump-sum transfers.

1.3 Progressive Consumption Taxes

While Àattening tax rates receives the bulk of the attention in the tax reform debate, it is important to note that consumption-based taxes can also be progressive. In particular, allowing

¿rms to deduct their full investment expenses at the time of purchase from their tax payments

(3full expensing ́) would effectively produce a consumption tax.5 This approach would also

allow for some progressivity through either a standard deduction, as in the 3Àat tax ́ plan

(Hall and Rabushka, 1995), and/or progressive tax rates, as in the 3X tax ́ plan (Bradford,

1986). The 3Àat tax ́ and 3X tax ́ plans also protect housing wealth. By maintaining progressive tax rates, the 3X tax, ́ in particular, could increase ef¿ciency in our model.

The current paper, however, focuses on a Àat consumption tax, for several reasons. First,

we are more interested in creating an understanding of the risk-sharing aspect of the current

5

In fact, a Àat consumption tax produces the same outcomes as a Àat income tax with full expensing.

3
income system than in analyzing speci¿c reform proposals. Second, the computation requirements of our model are already signi¿cant. In order to capture the differential tax treatment

of various capital items under the 3Àat tax ́ and the 3X tax, ́ we would have to include housing wealth and other types of capital as separate capital categories. While this addition is

possible in deterministic models (Fullerton and Rogers, 1993 Jorgenson and Yun, 2001),

adding housing wealth to our stochastic model would alone increase the computation time

by two orders of magnitude, requiring about three months to solve a single simulation on a

two-gigahertz Pentium IV computer. We, therefore, leave this innovation to future research.

Third, as a practical manner, a national retail sales tax is gaining considerable attention in the

United States. Later in the paper, we do, however, consider a simple progressive consumption tax in the form of a national sales tax with a rebate of the ¿rst $20,000 of consumption

per household. This reform is similar to the 3Àat tax ́ proposal except that we don¶t protect

existing housing wealth our version of the 3Àat tax, ́ therefore, creates more potential for

ef¿ciency gains.

1.4 Outline

The outline of our paper is as follows. Section 2 gives a brief primer on the economic factors

associated with moving to a Àat consumption tax using a simple two-period model. Section

3 outlines our large-scale model that we use to simulate the introduction of a revenue-neutral

Àat consumption tax. Section 4 summarizes the calibration of the baseline economy. Section

5 explains the Lump-Sum Redistribution Authority and shows policy experiments. Section

6 concludes the paper. The appendix describes the computational algorithm.

2 A Primer on Tax Reform

This section uses a simple partial-equilibrium model to present a brief overview of tax reform

in an OLG economy. We ¿rst consider tax reform in a deterministic economy in which the

prereform income tax system is linear. We then expand the discussion to nonlinear taxes,

and then ¿nally to uncertainty in the form of idiosyncratic wage shocks and life spans. We

do not consider the effects of removing tax preferences this issue and others are addressed

in our larger-scale model, which is presented in the next section.

4
2.1 No Uncertainty, Linear Taxes

Consider a simple two-period model in which agents work during the ¿rst period and retire

in the second period. During the ¿rst period, an agent born in time w earns pretax wages

zc|, pays a wage tax at rate 

| , consumes fc|, and saves the remainder as assets, d2c|n,

in order to afford second-period consumption. During period two at time w . 4, the agent¶s

consumption, f2c|n, is equal to d2c|n plus net interest paid at time w . 4, u|n, after paying

a capital income tax at rate  o

| . The agent¶s budget constraints, therefore, are as follows:

fc| . d2c|n @ +4  

| , zc|>

f2c|n @ 

4 . 4   o

|n

u|nd2c|n=

A linear income tax,  +

| , is created by setting  o

| @ | @  +

| . Assuming that there are

no borrowing constraints (or, alternatively, d2c|n  3), the household¶s lifetime budget

constraint equals

fc| .

f2c|n



4 . 4   o

|n

u|n @ +4  

| , zc|= (1)

We can rewrite equation (1) as

fc|

4  

|

.

f2c|n



4 . 4   o

|n

u|n+4  

| , @ zc|

or



4. S

c|

fc| .



4. S

2c|n

f2c|n

+4 . u|n, @ zc|>

where S

c|  

3

|

 4 and S

2c|n  no|n

^n+3o

|n,o|n`E3

|   4. Hence, a system of linear

wage taxes, capital income taxes, and income taxes can be represented in terms of equivalent

age-indexed effective consumption tax rates, S. (The tilde [~] superscript is used to denote

effective rates.) Both tax systems collect the same lifetime present value of taxes from each

agent and offer the same incentives. Notice that if tax rates are stationary then S

2 A S when

 o

|n A 3, that is, a positive capital income tax increases the effective consumption tax rate

over the life cycle. By the inverse-elasticity rule of the optimal tax literature, this increasing

tax rate is a potential source of inef¿ciency unless the demand elasticity for second-period

5
consumption was lower relative to ¿rst-period consumption, a result that is not an implication

of most speci¿cations of household preferences.

Government revenue each period ¿nances a ¿xed level of spending, J:



| zc| .  o| u|d2c| @ J, (2)

where the population size is assumed to be stationary. Equation (2) could also be represented

in terms of the effective consumption tax rates shown above. However, when analyzing a

tax reform, which changes the present value of taxes paid by each generation, equation (2) is

needed in its current form because second-period agents alive at the reform did not actually

face the effective consumption tax S

 during their ¿rst period of life.

2.1.1 A Revenue-Neutral Consumption Tax

Now suppose that the government introduces a consumption tax,  S

| , at time w to replace the

income tax. The private budget constraints become

+4 .  S

| , fc| . d2c|n @ zc|>



4 .  S

|n

f2c|n @ ^4 . u|n` d2c|n>

which produces the following lifetime budget constraint:

+4 .  S

| , fc| .



4 .  S

|n

f2c|n

+4 . u|n, @ zc|=

The government¶s budget constraint equals

 S

| +fc| . f2c|, @ J= 

This tax reform has two major effects. First, capital income is no longer taxed, eliminating the intertemporal price distortion in a stationary economy. Second, as explained below,

this tax reform imposes a lump-sum tax on existing asset holders. We consider each of these

effects below.

This decomposition between these two effects is convenient because it allows us to interpret a move from a linear income tax to a Àat consumption tax in two steps: (i) replace a

linear income tax with a linear wage tax, thereby removing the tax on capital income and

6
(ii) replace this linear wage tax with a consumption tax, thereby imposing a lump-sum tax

on existing assets. Although we regard the second reform as occurring immediately after the

¿rst reform, the age-asset pro¿le in the initial income tax economy is the one that is relevant

for determining the impact of the lump-sum tax on existing assets.6

The Nontaxation of Capital Income. Notice that in a stationary economy, this tax reform

generates uniform consumption tax rates across the life cycle since capital income is no

longer taxed. As a result, the intertemporal price distortion is removed, encouraging saving.

If labor supply was also elastic, an increase in after-tax interest rates would encourage more

labor supply and saving earlier in the life cycle since asset values would accumulate more

quickly.

However, this good news does not come for free. The assets that have already been

accumulated by generation (w  4) at time w, which would have been taxed under the original

income tax, will not be taxed after the reform. So generation (w  4) receives a lump-sum

transfer (negative tax) equal to  o

| u|d2c|, which must be paid by future workers in the form of

wage taxes in order to make up the lost revenue.7 Because the wage base is smaller than the

income base, tax rates increase. Since tax distortion increases with the square of the tax rate, a

smaller tax base will produce more distortions. In the ¿nite-horizons OLG model, these new

distortions could, in theory, outweigh the gains associated with removing the intertemporal

price distortion. In fact, in their simulation analysis using a multiple-period deterministic

model with elastic labor supply, Auerbach and Kotlikoff (1987) ¿nd that replacing a linear

income tax with a linear wage tax reduces long-run output and welfare.8

A Lump-Sum Tax on Existing Wealth. As ¿rst demonstrated by Summers (1981), adopting a consumption tax also imposes a lump-sum tax on older people. In particular, the con6

In static models with one or two periods (and no bequests), one could alternatively distinguish between the

tax reform¶s 3substitution effect ́ and 3income effect. ́ However, this distinction is substantially more cumbersome in a model with more than two periods where some agents have accumulated wealth by the time of the

reform and will also live for more than one additional period after the reform. See Gravelle (2002) for a detailed

critique of intertemporal models. 7

Our simple two-period model, though, somewhat exaggerates this point by taxing capital income only at the

beginning of the second period. With multiple periods, asset holders will have already paid some taxes on capital

income before the tax reform. 8

In contrast, the optimal long-run tax rate on capital income is zero in the Ramsey model.

7
sumption by generation (w  4) in their second period of life is inelastic at time w since their

consumption is based on previous saving. Hence, when the government changes the tax

system at time w, these agents face a lump-sum tax equal to  S

| f2c|, which, under revenue

neutrality, accrues as reduced taxes paid by future workers.9

In fact, the lump-sum tax on generation +w  4, accrues as a reduced tax liability to all

future generations. To demonstrate this fact, consider the clever expositional simpli¿cation

used by Auerbach and Kotlikoff (1987, pp. 58 - 59): agents only consume during the second

period of their lives (that is, fc| @ 3) and taxes on interest are zero ( o @ 3) so that only

wages are taxed. Under these assumptions, wage and consumption taxes are lump sum for

all generations, allowing for easy illustration. Now suppose that the government switches

tax bases from wages to consumption at time w. Generation +w  4,, which is in their second

period of life when the tax reform occurs, is charged a wealth levy equal to J. Clearly,

they are worse off: they paid J during their ¿rst period of life under the previous wage tax,

and now they must pay it again during their second period under the new consumption tax.

Now consider generation w, which is in their ¿rst period of life when taxes are reformed.

Under the former wage tax, generation w would have paid J during their ¿rst period of life.

Instead, they now pay J in their second period, reducing the present value of their lifetime

taxes by 

J  C7

no



, or uJ@ +4 . u,. Similarly, every future generation v (vAw) receives a

present-value reduction in their tax liability equal to uJ@ +4 . u,, calculated with respect to

their generation index, v. The present value sum of tax saving across all future generations,

calculated at time w, therefore, equals S

"

'f

+oC*7 Eno,

Eno

 @ J. In other words, the present value

of the tax reduction to future generations exactly equals the loss to the initial elderly.

This intergenerational transfer has a very powerful impact on long-run output and welfare. In their multiple-period simulation model, Auerbach and Kotlikoff ¿nd that replacing

a linear income tax with a Àat consumption tax increases long-run output and welfare. Why

are the results so different in the AK model relative to the wage tax base discussed above?

The reason is the wealth levy on the existing capital held by generation +w  4, that occurs

with a consumption tax but not with a wage tax in fact, this wealth levy is the only differ9

As noted in the previous section, a Àat consumption tax produces the same outcomes as a Àat income tax

with full expensing. Under expensing, the lump-sum tax on existing wealth takes the form of a fall in Tobin¶s q,

as old capital becomes less valuable relative to new capital.

8
ence between those two tax bases. In the case of a consumption tax, this wealth levy extracts

enough resources to reduce future tax burdens, producing a long-run gain. In fact, over 100

percent of the long-run gain in the AK model stems from this wealth levy (Engen, Gravelle,

and Smetters, 1997).

2.1.2 Ef¿ciency Versus Redistribution

It is important, though, to distinguish between redistribution and ef¿ciency. In the previous

two-period example, replacing the linear wage tax with a consumption tax would produce a

sizable long-run increase in the capital stock and output. But this entire gain comes off the

backs of generation +w  4,. In other words, if generation +w  4, were to receive a lump-sum

rebate equal to J so that its utility is held ¿xed, future generations would no longer bene¿t

from tax reform. Hence, the ef¿ciency gain is exactly zero despite the large long-run gains.

Recall, though, two key assumptions that we made: (i) agents lived for only two periods, and

(ii) all taxes were effectively lump-sum for all generations. Not surprisingly, therefore, tax

reform produces zero ef¿ciency gains.

In a more realistic setting with more than two periods and with consumption in each period, replacing a linear income tax with a consumption tax would probably produce ef¿ciency

gains. To be sure, removing the tax on capital income alone has an unclear impact on ef¿-

ciency since the bene¿t from removing the intertemporal price distortion must be balanced

against the higher tax rate. However, the lump-sum tax is likely to lead to sizable ef¿ciency

gains inside a multiperiod model. The reason is that, after controlling for intergeneration redistribution, agents with assets who are alive at the time of the reform bene¿t from replacing

some of their own future distorting taxes with the lump-sum taxes that they pay today in the

form of a wealth levy.10 Accounting for these different effects requires simulation analysis.

Auerbach and Kotlikoff ¿nd that ef¿ciency is increased by replacing a linear income tax with

a linear consumption tax in a deterministic framework, a result that we verify later. Within

the Ramsey model, many previous papers have also found positive gains from adopting a

consumption tax (see the review in Stokey and Rebelo, 1995).

10We bene¿ted from a helpful conversation with Alan Auerbach on this point.

9
2.2 No Uncertainty, Nonlinear Taxes

With nonlinear tax rates in the prereform economy, the agent¶s budget constraints are

fc| . d2c|n @ zc|  W 

| +zc|,>

f2c|n @ +4 . u|n, d2c|n  Wo

|n +u|nd2c|n,>

where the W +, functions represent total taxes paid. The average wage tax rate equals

W 

| +zc|, @zc|, which is smaller than the marginal wage tax rate, CW | +zc|, @Czc|. Similarly, the average and marginal tax rates on capital income are Wo

|n +u|nd2c|n, @ +u|nd2c|n,

and CWo

|n +u|nd2c|n, @C +u|nd2c|n,, respectively.

With a stationary population size, the government¶s budget constraint is

W 

| +zc|, . Wo|n +u|nd2c|n, @ J= 

The introduction of progressive income taxes into the prereform economy alters the impact of the intertemporal price and lump-sum tax effects described above. It also means that

a shift to a consumption tax adds a third effect: Àattening tax rates. The adoption of a Àat

consumption tax now has three steps: (i) adopt a revenue-neutral progressive wage tax by

dropping taxes on capital income (ii) move from a progressive wage tax to a progressive

consumption tax, producing a lump-sum tax on existing assets

11 and (iii) move from the

progressive consumption tax to a Àat consumption tax.

2.2.1 The Nontaxation of Capital Income

Much of the debate about whether to adopt a consumption tax focuses on removing distortions caused by progressive tax rates. Indeed, progressive tax rates tend to magnify intertemporal distortions. First, even with inelastic labor supply, saving decisions are distorted more

with progressive taxes since the future marginal tax rate that a person faces on capital income is now directly affected by their saving decisions. Removing the tax on capital income,

therefore, encourages even more saving when the prereform income tax is progressive. Second, allowing for elastic labor supply tends to enhance this result. Marginal income tax rates

11In practice, this could be achieved with full expensing, discussed in Section 1. An equivalent progressive

VAT or sales tax could also be implemented, but would be substantially more cumbersome to administer.

10
tend to peak at middle age when labor productivity is high and after a fair amount of assets

have been accumulated for retirement. So agents tend to shift their labor supply away from

high-tax years in the middle years of life toward lower-tax years later in life. Since more labor income is earned later in life, less saving is needed earlier in life to smooth consumption.

Removing the tax on capital income, therefore, would eliminate those distortions, which tend

to be more signi¿cant when the prereform income tax is progressive.

As with the linear income tax considered before, however, one must also account for

the government¶s budget constraint. Increasing wage tax rates to make up lost revenue now

creates more distortions than in the linear tax case considered earlier. Distortions increase

the most if the progressive wage tax schedule is increased in a progressive manner in order to

protect the poor. But even if the additional tax burden is distributed in a proportional manner,

distortions will rise faster relative to the linear case considered earlier.

2.2.2 A Lump-Sum Tax on Existing Wealth

As noted earlier, in a multiple-period model, the ef¿ciency gains stemming from the lumpsum tax on existing wealth depend on the extent to which the lump-sum taxes replace the

future distorting taxes of those alive at the time of the reform. The more assets held by

younger and middle-aged workers at the time of the reform, the more likely the wealth levy

would produce ef¿ciency gains. Whether, for a given capital-output ratio,12 these cohorts

hold a larger share of capital under progressive taxes depends on the exact model parameters.13 Still, the ef¿ciency gains associated with moving to a consumption tax are likely to

be much larger when the prereform income tax is progressive since this tax system is more

distorting, increasing the value of the substitute lump-sum taxes.

12When comparing across models, one should always solve for the deep parameters that generate the same

observable economy, including the capital-output ratio. In this way, you ensure that the predictions are not being

generated by different calibrations. 13Younger workers tend to face lower tax rates under a progressive system, giving them more resources to save.

But they also face increasing marginal tax rates in the future as their human capital returns increase, decreasing

their incentive to save. Older people, except those who have accumulated lots of wealth, also have a few more

resources to reinvest under a progressive tax system. But the intertemporal shift in their labor supply described

earlier in the text tends to reduce their saving. Hence, the remaining share held by middle-aged workers also

depends on the parameters.

11
2.2.3 Flattening Tax Rates

The effect of moving from a progressive consumption tax to a Àat consumption tax produces

two competing major effects (and a couple minor effects that we¶ll ignore). First, it removes

an important price distortion across the life cycle. Speci¿cally, consumption increases over

the life cycle when the interest rate exceeds the rate of time preference.14 As a result, marginal consumption tax rates also increase over the life cycle, similar to the pattern produced

by a capital income tax considered earlier. A shift to a proportional tax, therefore, creates a

uniform tax rate on consumption, removing this intertemporal distortion. Second, this reform

gives many asset holders a lump-sum transfer (negative tax), reducing ef¿ciency.

2.2.4 Total Ef¿ciency Gains

On net, there are likely to be sizable ef¿ciency gains from adopting a Àat consumption tax

when the original income tax system is progressive. Both Auerbach and Kotlikoff (1987),

using the OLG model, as well as Jorgenson and Yun (2001), who use the Ramsey model

described in Section 1, ¿nd large ef¿ciency gains.

2.3 Wage and Life-Span Uncertainty, Nonlinear Taxes

Adding wage and life-span uncertainty inÀuences all three of the effects of tax reform discussed above. We consider each in turn.

2.3.1 The Nontaxation of Capital Income

The addition of wage uncertainty tends to reduce the importance of price distortions over

the life cycle. Even with linear tax rates and inelastic labor, agents will hedge their earnings

uncertainty by saving in a precautionary manner.15 As a result, household saving becomes

less responsive to an increase in the after-tax interest rate following a shift to a consumption

tax base.16 To be sure, with elastic labor supply, the need to save in a precautionary manner is

reduced somewhat since agents can, for example, work multiple low-paying jobs in order to

14This effect cannot happen in the in¿nite-horizon model since the interest rate equals the time preference rate

in a steady state. 15Precautionary saving is positive if the third derivative of the agent¶s felicity function is positive, a condition

which holds in our model. 16This point was emphasized in Engen and Gale (1996) and Engen, Gravelle, and Smetters (1997).

12
replace a former higher-paying job. But since utility is concave in leisure and the maximum

leisure time is bounded, the ability to vary working hours cannot eliminate precautionary

saving altogether. As a result, saving will be less sensitive to changes in the after-tax interest

rate relative to the case without uncertainty. Labor supply also becomes less sensitive to

changes in interest rates.

When fair annuities are not available, adding life-span uncertainty produces two competing effects. On the one hand, life-span uncertainty should lead to greater precautionary

saving, which decreases the interest elasticity of saving. On the other hand, the horizon of

agents is effective 3longer ́ since a prudent agent will plan for a time period longer than

average. This longer time period might enhance price sensitivity somewhat.17 The presence of a Social Security system in our large-scale model (Section 3) will tend to reduce the

importance of both of those effects.

2.3.2 A Lump-Sum Tax on Existing Wealth

While the lump-sum tax on existing wealth following the adoption of a consumption tax has

a large impact on ef¿ciency, the addition of uncertainty produces two competing effects. On

the one hand, for a given capital-output ratio, the asset-age pro¿le for the average person

is relatively less 3hump-shaped ́ due to greater precautionary saving both earlier in the life

cycle (due to earnings uncertainty) and later in the life cycle (due to life-span uncertainty).

Since labor supply is partly a self-insurance mechanism, people are also less likely in the

prereform economy to take advantage of falling future marginal tax rates by postponing their

labor supply, thereby generating more saving earlier in life. As a result, the wealth levy on

the young is higher, reducing their future distorting taxes by more than without uncertainty.

17The interaction of life-span uncertainty with wage uncertainty, though, complicates matters. For example,

if in the extreme, agents lived forever with certainty, we would be back in the in¿nite-horizons world where

we would want to focus on a single agent. In this case, Aiyagari (1995) demonstrates that the optimal tax rate

on capital income would actually be positive when the in¿nitely lived agent faces uninsurable indiosyncratic

earnings shocks. Intuitively, elastic labor supply prevents the government from employing con¿scatory wage

taxes to replicate full insurance. As a result, precautionary saving drives the interest rate below the agent¶s rate

of time preference, generating too much capital in the economy relative to the modi¿ed golden rule. A positive

capital income tax brings the economy¶s level of capital back to the ef¿cient level. The Aiyagari motive for a

positive capital income tax rate, though, is not present in a stochastic ¿nite-horizon OLG model2not even as an

approximation2unless precautionary saving produces enough capital so that the economy becomes dynamically

inef¿cient. Whereas dynamic inef¿ciency is guaranteed in Aiyagari¶s model (where the actual level of capital is

compared against the modi¿ed golden rule level of capital), it is not in a ¿nite-horizon OLG model (where the

comparison is made with the golden rule level of capital).

13
On the other hand, the value of this lump-sum tax is not as large as without uncertainty due

to the reduced importance of distortions, discussed above.

2.3.3 Flattening Tax Rates

When uncertainty is added to the model, Àattening the tax rates reduces risk sharing within

generations. The reason is that the original income tax system shares idiosyncratic earnings

shocks. To some extent, even a linear tax shares risks since people with higher earnings

realizations pay more taxes. But progressive taxes share those risks even more.

In fact, if labor supply was completely inelastic then all agents in identical states (that

is, same assets same current-period wage income same age and so on) would want to

fully share future wage shocks. In this case, the optimal tax on consumption would be extremely progressive: consumption levels above these individuals¶ expectation (that is, statecontingent expectation) would be taxed at 100 percent while consumption below average

would be taxed at a negative rate (subsidized) so that it equaled the expected outcome.

In reality, of course, the optimal progressive tax system will not fully share all future

risks. First, labor supply is obviously not completely inelastic. In essence, agents with

greater-than-expected wage realizations will distort their future labor supply in order to partially 3renege ́ on the previous risk-sharing 3agreement. ́ Since this incentive is understood

ex ante, the optimal tax schedule cannot fully share risks ex post. Second, in realistic tax

systems, taxes paid by any agent in a given year depend only on the agent¶s current state

and not on that agent¶s state in previous years. As a result, a progressive tax system will

redistribute resources in the future across agents at different states today. With elastic labor

supply, this 3extra ́ redistribution is a source of (interim) inef¿ciency.18

2.3.4 Total Ef¿ciency Gains

The remainder of this paper examines the importance of wage and life-span uncertainty when

analyzing tax reform. Since closed-form solutions are not possible in assessing these differ18In our analysis, we focus on 3interim ́ ef¿ciency, where the expected remaining lifetime utility of living

agents is calculated conditional on their current state at the time of reform, and the expected utility of future

generations is calculated conditional on the initial state into which they are 3born ́ as independent economic

actors. If we instead measured expected utility across all possible states (the so-called 3ex ante ́ position), our

results regarding the importance of risk sharing would only be strengthened.

14
ent competing forces, we use simulation analysis to help determine the impact on ef¿ciency

from replacing a progressive income tax system with a Àat consumption tax. The next section

lays out the computation model that we use.

3 Model

The economy consists of three main sectors: heterogeneous households with elastic labor

supply a perfectly competitive representative ¿rm with constant-returns-to-scale production

technology and a government with a full commitment technology.19

3.1 The Household Sector

Households are heterogeneous with respect to age l, working ability h (measured by its

hourly wage), beginning-of-period wealth holding d, and average historical earnings e that

is used to determine Social Security bene¿ts. Every year, a large number (normalized to

unity) of new households of age 20 enter into the economy.20 A household of age l observes

idiosyncratic working ability shock, h, at the beginning of each year and chooses its optimal

consumption f, working hours k, and end-of-period wealth holding dn, taking the government¶s policy schedule and a series of factor prices and the government¶s policy variables

as given.21 At the end of each year, a fraction of households die. Households are alive at

most up to 109 years old, and the mortality rate at the end of age 109 is one. Tables 1 and 2

show the main variables and functions used in the household¶s problem.

3.1.1 The Household¶s Problem

Let v denote the individual state vector of an age l household, let V| denote the aggregate

state vector at the beginning of year w, and let | denote the series of government policy

rules known at the beginning of year w,

v @ +l> h> d> e,> (3)

19As is standard in the optimal tax literature, we assume that the government can commit to future policies,

thereby ignoring time-consistency issues. 20The population of this economy is normalized by the constant population growth rate D. 21Because there are no aggregate shocks in the present model, the rational expectation of these policy variables

and factor prices are actually the ones of perfect foresight. But, they still do not know their future working ability

and mortality.

15
Table 1: Main Variables and Functions in the Model

Individual state: v @ +l> h> d> e,

l 5 L @ i53> ===> 43<j Age

h 5 H @ ^h4?> h4@ ` Working ability (hourly wage) (a)

d 5 D @ 

d4?> d4@  Beginning-of-period wealth (a)

e 5 E @ ^e4?> e4@ ` Average historical earnings (AIME12) (a)

Aggregate state: V| @ +{|+v,> Z}c|,

{|+v, Joint distribution of households (b)

Z}c| Beginning-of-period government wealth (c)

Policy schedule and rule: | @ iZ}crn> F}cr> Ucr+=,> cr+=,> cr> wu77cr+=,j"

r'|

Z}crn End-of-period government wealth (c)

F}cr Government consumption (c)

Ucr+=, Federal income tax function (d)

cr+=, Payroll tax function (d)

cr Consumption tax rate

wu77cr+=, Social Security bene¿t function (d)

Household decision rules: g+v> V|> |,@+f+=,> k+=,> dn+=,,

f+=, Consumption (a)

k+=, Working hours

dn+=, End-of-period wealth (a)

Main parameters and other variables

 5 Un Time preference (e)

! 5 ^3> 4` Survival rate at the end of age l

 5 U Labor augmenting productivity growth rate

 5 U Population growth rate

z| 5 Un Wage rate (1.0 in the baseline)

u| 5 U Interest rate

t| 5 Un Bequests per surviving working-age household (a)

(a) These variables are adjusted by the steady-state (per capita) economic growth rate.

(b) The measure of households is adjusted by the steady-state population growth rate.

(c) The government¶s net wealth and most aggregate variables (shown below) are adjusted by the steady-state

economic growth rate and population growth rate.

(d) The arguments of these functions are adjusted by the steady-state economic growth rate. Time invariant

tax and bene¿t functions imply that the actual schedules are adjusted so that there is no real bracket creep

whenever the economy is on the balanced growth path.

(e) The time preference parameter is adjusted by the steady-state economic growth rate. The adjustment

depends on the speci¿cation of utility function.

16
Table 2: Other Aggregate Variables in the Model

Z| 5 Un National wealth

O| 5 Un Total labor supply

N| 5 Un Capital stock

\| 5 Un Gross national product

WUc| 5 U Federal income tax revenue

Wc| 5 U Federal payroll tax revenue

Wc| 5 U Federal consumption tax revenue

W u77c| 5 U Total OASDI bene¿ts

Note: All aggregate variables are adjusted by the steady-state economic growth and population growth that is,

these variables in the model stay at the same level when the economy is on the balanced growth path.

V| @ +{| +=,> Z}c|,> (4)

| @ iZ}crn> F}cr> Ucr +=,> cr +=,> cr> wu77cr +=,j"

r'| = (5)

Then, the value function of a household is

y +v> V|> |, @ pd{ Scc@n

x +f> k, .  !H ^y +vn>V|n> |n, m h` (6)

subject to

dn @ 4

4 .  iz|hk . +4 . u|,d  Uc| +z|hk> u|d> wu77c| +l> e,, (7)

c| +z|hk, . wu77c| +l> e,  +4 . c|, fj  d4?>

and d2f @ 3> df  3.

22

Let cn +hn m h, be the conditional probability for the age l. 4 working ability being

hn when the age l working ability is h.

23 Then,

H ^y +vn>V|n> |n, m h` @ ]

.

y +vn>V|n> |n, cn +hn m h, d hn= (8)

At the beginning of the next period, the individual state, the aggregate state, and the

government policy rules become

vn @ +l . 4> hn> dn . t|> en, with cn +hn m h,> (9)

22Alternatively, we can use [| for [|n on the right-hand side of the objective function because [| includes

the information of [|n 23Since en is a random variable with conditional probability distribution Zcn Een  e, tn ' E n

c enc @nc Kn is a random vector. The present paper uses e as a realized number and t as a realized

vector.

17
V|n @ +{|n +=,>Z}c|n,> (10)

|n @ iZ}crn> F}cr> Ucr +=,> cr +=,> cr> wu77cr +=,j"

r'|n > (11)

where Z}c|n is determined by the government budget constraint, and e follows

en @

;

AAA?

AAA=

3 if l  57



32e i+l  58,e ||3 . min +z|hk@5> zhk4@ 

| ,j if 58  l  8<

+4 . ,3e if l  93>

(12)

where zhk4@ 

| is the threshold, which is $80,400 in 2001. For simplicity, we assume that the

highest 35 years of earnings correspond to those years of age between 25 and 59.24

The decision rule of an age l household in year w is a function of its individual state v,

the aggregate state V|, and the government policy rules |, and is shown as

g+v> V|> |, @ if +v> V|> |,> k +v>V|> |,> dn +v>V|> |,j = (13)

3.2 The Measure of Households

Let {| +v, denote the measure of households, and let [| +v, be the corresponding cumulative measure. The measure of households is adjusted by the population growth rate. The

population of age 20 households is normalized to be unity in the baseline economy on the

balanced growth path, that is,

]

.

g[| +53> h2f> 3> 3, @ 4= (14)

Let 4d@'+o be an indicator function that returns 1 if d @ | and 0 if d 9@ |= Then, the law of

motion of the measure of households is, for l 5 L @ i53> ===> 43<j >

{|n +vn, @ !

4 . 

]

.ff

4d@n'@nEtc5|([|n^|o (15)

4dKn'KnE|eEtc5|([cKo cn+hn m h, g[| +v,=

For simplicity, accidental bequests due to uncertain life span are captured by the government and distributed equally to all surviving working-age households in a lump-sum manner. Nishiyama (2002) presents a four-period model with altruistically motivated bequests

24Social Security bene¿ts in the United States are computed on the basis of the highest 35 years of earnings,

adding an additional state variable to our model. Earnings before age 60 are wage indexed, and earnings after

age 60 are price indexed. The approximation of AIME by the average historical earnings follows previous Social

Security literature, for example, Huggett and Ventura (1999) and Di Nardi and others (1999).

18
incorporating altruism into the current model in a realistic manner would make the model

computationally intractable at current computer speeds. The accidental bequests per household at the end of year w is

t| @ t+V|> |, @

Sfb

'2f+4  !,

U

.ff dn+v>V|> |, g[| +v,

SSe

'2f !

U

.ff g[| +v, = (16)

The steady-state condition is

V|n @ V| (17)

for all w and v 5 L  H  D  E=

3.3 The Firms¶ Problem

National wealth Z| is the sum of total private wealth and government net wealth Z}c|. Total

labor supply O| is measured in ef¿ciency units.

Z| @ Z+V|, @ [

fb

'2f

]

.ff

d g[| +v, . Z}c|> (18)

O| @ O+V|> |, @ [

fb

'2f

]

.ff

h k+v>V|> |, g[| +v,= (19)

There are a large number of perfectly competitive ¿rms in this economy. In a closed

economy, the capital stock is equal to national wealth, that is,

N| @ Z|> (20)

and gross national product \| is determined by a constant-returns-to-scale production function,

\| @ I+N|> O|,= (21)

The pro¿t-maximizing condition of the ¿rm is

u| .  @ Ig+N|> O|,> (22)

z| +4 .  

c|, @ Iu+N|> O|,> (23)

where  is the depreciation rate of capital and  

c| is the marginal payroll tax rate.25

25U.S. payroll taxes are divided equally between ¿rms and employees. While the incidence of the tax does

not depend on this division, our model explicitly includes the division for calibration purposes. In doing so,

we ignore the small fraction of the representative ¿rm¶s workforce whose wages exceed the payroll tax ceiling.

However, the ceiling is enforced on the worker¶s share, as shown earlier.

19
3.4 The Government¶s Policy Rules

Government tax revenue consists of federal income tax WUc|, payroll tax for Social Security

Wc|, and consumption tax Wc|. These revenues equal:

WUc| @ WU +V|> |, (24)

@ [

fb

'2f

]

.ff

Uc| +z|hk+v> V|> |,> u|d> wu77c| +l> e,, g[| +v,>

Wc| @ W +V|> |,@5  [

fb

'2f

]

.ff

Rc| +z|hk+v> V|> |,, g[| +v,> (25)

Wc| @ W+V|> |, @ [

fb

'2f

]

.ff

c|f+v> V|> |, g[| +v,=

Total Social Security bene¿ts W u77c| equals

W u77c| @ W u77+V|> |, @ [

fb

'2f

]

.ff

wu77c| +l> e, g[| +v,= (26)

The law of motion of the government wealth (normalized by productivity growth and population growth) is

Z}c|n @ Z}+V|> |, (27)

@ 4

+4 . , +4 . ,

i+4 . u|,Z}c| . +WUc| . Wc| . Wc|,  W u77c|  F}c|j >

where F}c| is government consumption.

3.5 Recursive Competitive Equilibrium

De¿nition Recursive Competitive Equilibrium (Steady State): Let v @ +l> h> d> e, be

the individual state of households and  be the time-invariant government policy rules,

 @ iZ}> F}> U +=,>  +=,> > wu77+=,j =

Factor prices +u> z, accidental bequests t the policy variables +Z}> F}> > wuu7, the

parameters * of policy functions +U +=,>  +=,> wu77+=,, the value function of households,

y +v> , > the decision rule of households,

g+v> , @ if+v> ,> k+v> ,> dn+v> ,j>

20
and the measure of households, { +v,, are in a steady-state recursive competitive equilibrium

if, in every period, each household solves the utility maximization problem (3) ± (7) taking 

as given the ¿rm solves the pro¿t maximization problem, and the capital and labor markets

clear, that is, (18) ± (23) hold the government policy rules satisfy (24) ± (27) the goods

market clears and the measure of households is constant that is, (17) holds.

De¿nition Recursive Competitive Equilibrium (Equilibrium Transition Path): Let v @

+l> h> d> e, be the individual state of households, V| @ +{|+v,> Z}c|, be the aggregate state

of the economy, and | be the government policy rules known at the beginning of year w,

| @ iZ}crn> F}cr> Ucr+=,> cr+=,> > wu77cr+=,j"

r'|=

A series of factor prices, accidental bequests, the policy variables, and the parameters of

policy functions,


 @iur> zr> tr>Z}crn> F}cr> cr> *rj"

r'|>

the value function of households,iy +v>Vr> r,j"

r'|> the decision rule of households,

ig+v>Vr> r,j"

r'| @ if+v> Vr> r,> k+v>Vr> r,> dn+v>Vr> r,j"r'|>

and a series of the measure of households, i{r+v,j"

r'|, are in a recursive competitive equilibrium if, in every period v @ w> ===>4, each household solves the utility maximization

problem (3) ± (7) taking | as given the ¿rm solves the pro¿t maximization problem, and

the capital and labor markets clear, that is, (18) ± (23) hold the government policy rules

satisfy (24) ± (27) and the goods market clears.

4 Calibration

Table 3 summarizes the parameter choices. For the baseline economy on a balanced growth

path, the degree of time preference  is chosen so that the capital-output ratio is 2.8 total

factor productivity D is chosen so that the wage rate equals unity, and the share parameter

of consumption  is chosen so that the average annual working hours of married couples

21
Table 3: Parameters

Time preference parameter  0.986

Share parameter for consumption  0.473

Coef¿cient of relative risk aversion  2.0

Capital share of output  0.32

Depreciation rate of capital stock  0.046

Long-term real growth rate  0.018

Population growth rate  0.010

Total factor productivity D 0.983

between the ages of 20 and 64 are consistent with U.S. data. As explained below, a CobbDouglas-CRRA utility function and a Cobb-Douglas production function are used for the

calibration.26

The following sections describe the choice of functional forms and parameter values, the

choice of four target variables and values.

4.1 Households

Utility Function. Like the recent important paper by Conesa and Krueger (1999) that focuses on Social Security reform, our model has elastic labor supply. We use the following

Cobb-Douglas utility function with constant relative risk aversion (CRRA), which is compatible with the existence of a steady state:

x+f> k, @

q

++4 . q@5,3l f,k+k4@ 

  k,3k

r3

4   >

where  is the coef¿cient of relative risk aversion, q is the number of dependent children, 

is the consumption adjustment parameter, and k4@ 

 is the maximum working hours.27 The

coef¿cient of relative risk aversion is assumed to be 2.0. The numbers of dependent children

by age cohorts are calculated from the Panel Study of Income Dynamics (PSID) 1993 Family

Data (see Table 4). The consumption adjustment parameter is assumed to be 0.6.

The annual working hours in the model are the sum of the working hours of a husband

26The calibration basically followed that of a four-period altruism model in Nishiyama (2002) but extended it

signi¿cantly because the present model is a 90-period model. 27In this setting, the growth-adjusted q becomes qE n >

kE3

, which is 0.977 in the calibration.

22
Table 4: Number of People Under 18 Years of Age in a Married Household

Age cohorts Number of people Age cohorts Number of people

under age 18 under age 18

20-24 0.895 45-49 1.011

25-29 1.149 50-54 0.445

30-34 1.617 55-59 0.188

35-39 1.905 60-64 0.094

40-44 1.649 65-plus 0.000W

Source: Authors¶ calculations from the Panel Study of Income Dynamics (PSID)

1993 Family Data.

WThe number 0.000 for ages 65-plus is an assumption and not from PSID data.

and a wife. The average working hours of married households between ages 20 and 64 are

3,368 hours in the 1998 Survey of Consumer Finances (SCF). The maximum working hours

are set to be 8,760, which is simply calculated from two persons times 12 hours times 365

days. In this calibration, the parameter  is chosen to be 0.473 so that average working hours

of age 20 and age 64 become 3,368 hours in the steady-state baseline economy.

Working Ability. The working ability in this calibration corresponds to the hourly wage

(labor income per hour) of each household in the 1998 SCF. The average hourly wage of a

married couple (family members #1 and #2 in SCF) used for the calibration is calculated by

Hourly Wage @ Regular and Additional Salaries (#1 . #2) + Welfare or Assistance

Working Hours (#1 . #2) =

To capture the earnings risk a household is exposed to more precisely, unemployment or

worker¶s compensation, TANF, food stamps, and other forms of welfare or assistance are

added to the salaries before calculating the hourly wage. Table 5 shows the eight discrete

levels of working abilities of ¿ve-year age cohorts.28 Taking a ¿ve-year moving average of

these numbers, we obtain the working ability of each age cohort. According to Bureau of

Labor Statistics data, the average hourly earnings of production workers have increased by

16.7 percent during the years from 1997 to 2001. In the calibration, the numbers in the table

are multiplied by 1.167 to convert the hourly wages in 1997 into those in 2001.

28Here, the hourly wage of a household that works less than 520 hours (10 hours a week per couple) is assumed

to be zero. In the real economy, some households have fairly high working ability but choose not to work (for

example, because of schooling). One observation of the age 20-24 cohort, which has an hourly wage of $193.01,

is ignored.

23
Table 5: Working Abilities of a Household (in U.S. Dollars per Hour)

Percentile Age cohorts

20-24 25-29 30-34 35-39 40-44 45-49

h 0-20th 3.83 5.42 5.42 6.93 6.12 6.59

h2 20-40th 7.07 8.64 9.76 11.28 11.36 12.70

h 40-60th 8.68 10.91 13.46 15.01 15.59 17.22

he 60-80th 10.67 14.01 18.08 19.96 22.09 23.22

hD 80-90th 14.05 17.52 27.17 25.27 30.89 31.58

hS 90-95th 18.20 22.48 33.71 33.38 48.59 44.31

h. 95-99th 28.43 32.64 54.11 52.16 76.13 86.50

hH 99-100th 36.81 46.09 167.15 186.47 221.34 301.99

Percentile Age cohorts

50-54 55-59 60-64 65-69 70-74 75-79

h 0-20th 5.48 3.52 0.00 0.00 0.00 0.00

h2 20-40th 11.53 10.06 4.54 0.00 0.00 0.00

h 40-60th 16.16 14.26 11.18 2.82 0.00 0.00

he 60-80th 23.44 21.28 18.16 10.37 1.81 0.00

hD 80-90th 32.14 30.93 28.56 19.48 12.57 0.00

hS 90-95th 43.01 44.10 59.36 27.68 29.03 1.96

h. 95-99th 78.61 85.29 96.22 59.34 64.91 14.25

hH 99-100th 314.59 379.44 421.55 299.25 195.73 146.14

Source: Authors¶ calculations from 1998 SCF data.

Markov Transition Matrix. The Markov transition matrix, , of working ability is calculated from the hourly wage of people ages 30-39 in 1991 in the PSID individual data. To

make the working ability process more persistent, the matrix is calculated as the transition

from the average of years 1989 and 1990 to the average of years 1990 and 1991.

 @

3

EEEEEEEEEEEEEEEEEEEC

0.7674 0.2049 0.0183 0.0045 0.0049 0.0000 0.0000 0.0000

0.1810 0.6033 0.1844 0.0129 0.0000 0.0086 0.0046 0.0052

0.0388 0.1517 0.6768 0.1220 0.0011 0.0046 0.0050 0.0000

0.0126 0.0361 0.1039 0.7210 0.0980 0.0139 0.0145 0.0000

0.0000 0.0081 0.0332 0.2360 0.6306 0.0676 0.0145 0.0100

0.0000 0.0000 0.0000 0.0582 0.3224 0.5303 0.0891 0.0000

0.0007 0.0000 0.0000 0.0354 0.0000 0.2827 0.6433 0.0379

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3553 0.6447

4

FFFFFFFFFFFFFFFFFFFD

>

24
Table 6: Survival Rates in the United States (Weighted Average of Males and Females of

Each Age)

Age Survival Age Survival Age Survival Age Survival Age Survival

Rate Rate Rate Rate Rate

20 0.999113 40 0.997978 60 0.989365 80 0.938048 100 0.676941

21 0.999066 41 0.997820 61 0.988361 81 0.931804 101 0.658846

22 0.999037 42 0.997654 62 0.987195 82 0.924980 102 0.639629

23 0.999028 43 0.997465 63 0.985840 83 0.917566 103 0.619216

24 0.999032 44 0.997267 64 0.984324 84 0.909481 104 0.597532

25 0.999043 45 0.997044 65 0.982631 85 0.900623 105 0.574495

26 0.999049 46 0.996797 66 0.980851 86 0.890904 106 0.550021

27 0.999041 47 0.996534 67 0.979101 87 0.880258 107 0.524022

28 0.999014 48 0.996258 68 0.977433 88 0.868650 108 0.496402

29 0.998970 49 0.995960 69 0.975763 89 0.856070 109 0.467066

30 0.998919 50 0.995626 70 0.973892 90 0.842518

31 0.998865 51 0.995247 71 0.971745 91 0.828007

32 0.998804 52 0.994823 72 0.969406 92 0.812554

33 0.998735 53 0.994352 73 0.966856 93 0.796181

34 0.998660 54 0.993826 74 0.964033 94 0.778913

35 0.998573 55 0.993231 75 0.960839 95 0.761457

36 0.998475 56 0.992570 76 0.957219 96 0.744011

37 0.998368 57 0.991857 77 0.953175 97 0.726790

38 0.998250 58 0.991094 78 0.948673 98 0.710031

39 0.998122 59 0.990263 79 0.943665 99 0.693980

Source: Authors¶ calculations from the Social Security Bulletin: Annual Statistical Supplement

(2001). In the calibration, the survival rate at the end of age 109 is set to zero.

where +m> n, @ +hn @ h&

n m h @ h



,=

Population Growth and Mortality. The population growth rate  is assumed to be 1.0

percent per year. The survival rates ! at the end of age l @ i53> ===> 43<j are the weighted

average of males and females in 1998 from Social Security Administration data (2001). The

survival rates at the end of age 109 are replaced by zero.

4.2 The Firm

Production Function. Production takes the Cobb-Douglas form,

I+N|> O|, @ D|Nw

| O3w

| =

25
To compute GNP, we use the sum of working hours in ef¿ciency units as total labor supply

O|. The capital share of output  is chosen by

 @ 4  Compensation of Employees . +4  ,  Proprietors¶ Income

National Income . Consumption of Fixed Capital =

The average of  in 1996-1998 is 0.32. The annual growth rate  is assumed to be 1.8 percent.

The annual population growth rate  is assumed to be 1.0 percent. Total factor productivity

D is chosen to be 0.983 so that the wage per unit of ef¿cient labor is normalized to be unity.

Fixed Capital and Private Wealth. The ¿xed capital N| for the calibration is obtained

by 3¿xed reproducible tangible wealth ́ minus 3durable goods owned by consumers ́ in the

Survey of Current Business (1997). In 1990-1996, ¿xed capital accounted for 89.7 percent

of ¿xed reproducible tangible wealth, and the capital-GDP ratio is approximately 2.8.

To connect the total private wealth with the ¿xed capital, it is assumed that all of the

private capital is owned by households and that part of the government-owned ¿xed capital

is effectively owned by households in the form of government bonds.

Private Wealth (Excluding Durables) @ Fixed Capital

 Government-Owned Fixed Capital

. Government Bonds Owned by Private Sector.

In the model, ¿xed capital is the sum of private wealth (excluding durables) and net

government wealth. Based on the data from 1990 to 1996, net government wealth in the

baseline economy is assumed to be 6.5 percent of total private wealth.

The Depreciation Rate of Fixed Capital. The depreciation rate of ¿xed capital  is chosen

by

 @ Total Gross Investment

Fixed Capital    =

In 1997-2000, gross private domestic investment accounted for, on average, 17.5 percent

of GDP, and gross government investment (federal and state) accounted for 3.2 percent of

GDP. When the capital-output ratio is 2.8, the ratio of gross investment to ¿xed capital is 7.4

26
percent. Subtracting the productivity and population growth rates, the annual depreciation

rate is assumed to be 4.6 percent.

4.3 Taxes and Transfers

Income Taxes. For federal income tax, the model uses the following tax function in Gouveia and Strauss (1994),

Federal Income Tax @ !f



|  +|3 . !2,

3*



 !@_>

where | is the taxable income (in thousands of dollars) of a household, which includes the

taxable portion of Social Security bene¿ts. In 2001, the standard deduction for a married

household was $7,600, and the exemption was $2,900 per person. When the parameters are

!f @ 3=74, ! @ 3=;8, !2 @ 3=348, and !@_ @ 4=3, this function replicates the statutory

income tax schedule. But because of itemized deductions, the effective tax rate of highincome households is much lower.29 Since in 2000 the ratio of total private income tax to

nominal GDP was 0.102, !@_ is assumed to be 0.604 so that income tax revenue is 10.2

percent of GDP in the steady-state equilibrium.

In addition to federal income tax, a 4.0 percent state tax is assumed for an income (excluding Social Security bene¿ts) above the same standard deduction and exemptions.

Social Security. The tax rate levied on both employers and employees for Old-Age, Survivors, and Disability Insurance (OASDI) is 6.2 percent, and the tax rate for Medicare (HI) is

1.45 percent. In 2001, employee compensation above $80,400 was not taxable for OASDI.

So, the ¿rm¶s pro¿t-maximization problem becomes

z  (4 . Marginal Payroll Tax Rate) @ DIu+N> O,,

where the marginal payroll tax rate is 0.0765 (equal to 0.062 + 0.0145).

Social Security bene¿ts are based on each worker¶s average indexed monthly earnings

(AIME), e@45, and the replacement rate schedule in the United States. The replacement

rates are 90 percent for the ¿rst $561, 32 percent for amounts between $561 and $3,381, and

15 percent for amounts above $3,381.

29See Gouveia and Strauss (1994) for effective federal tax rates.

27
The bene¿ts received by retired workers consisted of 69 percent of total OASDI bene¿ts

in December 2000.30 The calibration simply assumes that each elderly household receives

other bene¿ts2those for spouses, children, and disabled workers2proportionally. Bene¿ts

are multiplied by 1.543 so that total OASDI bene¿ts are equal to OASDI tax revenue in the

baseline economy.

5 Policy Experiments

We now consider the effects of reducing the marginal income tax rates by 10 percent proportionately and raising the consumption tax rate to replace the lost revenue. For this policy

change, we will show four different results2with or without the Lump-Sum Redistribution

Authority, and with or without wage uncertainty (life-span uncertainty is operative in all

the experiments). But before reporting these results, we ¿rst explain the construction of the

LSRA, followed by a discussion of our construction of the representative-agent OLG economy without wage uncertainty.

5.1 The Lump-Sum Redistribution Authority

We measure the welfare gain or loss from a policy change using a Lump-Sum Redistribution Authority, following Auerbach and Kotlikoff (1987), who used an LSRA in their

representative-agent OLG model. We extend their analysis to a stochastic OLG model with

heterogeneous agents.

Suppose that a new policy is announced at the beginning of period 1. The LSRA ¿rst

makes lump-sum transfers (taxes if negative) to each living household to bring their remaining expected lifetime utilities back to their levels in the baseline economy. Since the welfare

gain or loss depends on each household¶s own state, these lump-sum transfers (taxes)31 are

shown as wu- +v, where v @ +l> h> d> e, such that

y +l> h> d . wu- +v,> e>V> , @ y +v>Vf> f,=

Next, the LSRA makes lump-sum transfers (taxes) to each future household (that is, newborn

households in periods 2, 3, ...) to make them as well off in the baseline economy, conditional

30The number is from Social Security Administration (2001), Table 5.A4.

31These lump-sum transfers are negative of the compensating variations in wealth for current households.

28
on their initial state as independent economic actors. These transfers (taxes) are shown as

wu-2 +v2f> w, such that, for w @ 5> 6> ===>

y +53> h> d . wu-2 +v2f> w,> e> V|> |, @ y +v2f>Vf> f,=

Finally, the LSRA makes additional lump-sum transfers (taxes) to the newborn households

in periods 2, 3, ..., so that the net present value across all the LSRA transfers at the beginning

of period 1 is zero. We assume these additional transfers are uniform on a growth-adjusted

basis. When the steady-state growth rate is , the additional transfers are shown as wu- ,

where

[

t

wu- +v, { +v, .["

|'2

+4 . ,

|3 S

t2f iwu-2 +v2f> w, . wu- j {| +v2f,

T|3

&' +4 . u&,

@ 3=

When wu- A 3, all of the current households would be as well off as the baseline economy

and all of the future households would be strictly better off hence, the new policy is Pareto

improving after lump-sum redistributions. When wu- ? 3, the alternative policy is Pareto

inferior after lump-sum redistributions.

The wealth held by the LSRA (normalized by the productivity growth and population

growth), iZ@j"

|', is derived as

Z@c @ 3>

Z@c2 @ S

t

wu- +v, { +v,

+4 . , +4 . , >

Z@c| @ +4 . u|3,Z@c|3  S

t2f iwu-2 +v2f> w, . wu- j {| +v2f,

+4 . , +4 . ,

for w @ 6> 7> ===>4= National wealth Z| is the sum of total private wealth, government net

wealth Z}c|, and the LSRA wealth Z@c|. So, equation (18) is replaced with

Z| @ [

fb

'2f

]

.ff

d g[| +v, . Z}c| . Z@c|>

and the government policy rule is de¿ned as

| @ iZ}crn> Z@crn> F}cr> Ucr +=,> cr +=,> > wu77cr +=,j"

r'| =

29
Table 7: Working Abilities of a Representative Household (In U.S. Dollars per Hour)

Age Cohorts 20-24 25-29 30-34 35-39 40-44 45-49

h 9.87 12.44 17.58 18.78 21.81 23.80

Age Cohorts 50-54 55-59 60-64 65-69 70-74 75-79

h 22.98 22.33 20.67 11.34 7.62 2.13

Source: Authors¶ calculations from 1998 SCF data.

5.2 A Deterministic OLG Economy with Representative Agents

In order to investigate the importance of idiosyncratic wages, we also consider a deterministic

version of our model. We assume that the working ability schedule of the representative

household is that shown in Table 7. Those numbers are the weighted average of the values

shown in Table 5. The representative household does not receive any working ability shocks,

but there is still lifetime uncertainty.

Similar to the stochastic OLG economy with heterogeneous agents, the model is calibrated to the U.S. economy so that the capital-output ratio is 2.8 and average working hours

(per couple) of working age households (ages 20-64) are 3,368 hours. Also, the wage rate

(per ef¿ciency unit of labor) is normalized to unity. The obtained parameters are as follows:

Table 8: Parameters of the Representative Agent Model

D

1.019 0.431 0.983

5.3 A 10 Percent Income Tax Cut with an Increase in a Consumption Tax

In the baseline economy, the federal income tax function is

3=74 



|  +|3fHD . 3=348,3*fHD

 !@_ >

and the adjustment factor !@_ is set to 0.604. In this policy experiment, this !@_ is lowered

by 10 percent to 0.544. That change reduces marginal income tax rates proportionately by

10 percent, keeping deductions and exemptions unchanged.

From the government budget constraint (27), the consumption tax revenue needed to

30
¿nance the tax cut is

Wc| @ +4 . , +4 . ,Z}c|n  +4 . u|,Z}c|  +WUc| . Wc|, . W u77c| . F}c|=

When the aggregate private consumption is

F @ [

fb

'2f

]

.ff

f+v> V|> , g[| +v,>

and a balanced budget is assumed, that is, Z}c|n @ Z}c|> the consumption tax rate c| is

c| @ i++4 . , +4 . ,  +4 . u|,,Z}c|  +WUc| . Wc|, . W u77c| . F}c|j @ F=

5.3.1 In a Stochastic Economy with Heterogeneous Households

Table 9 shows the results of the tax change in a stochastic OLG economy with heterogeneous

households. (Tables 9 through 17 and Figures 1 through 3 are included at the end of this

paper.)

Without the Lump-Sum Redistribution Authority. In the long run, national wealth,

which is equal to capital stock in a closed economy, increases by 2.9 percent from the baseline economy, labor supply increases by 0.6 percent, and GNP increases by 1.3 percent (see

the bottom panel). The interest rate decreases by 0.18 percentage points and the wage rate

increases by 0.7 percent. To keep the government wealth (debt) at the same level as that

of the baseline, the consumption tax rate would have to be raised by 1.3 percentage points

in the long run if the income tax rates were reduced by 10 percent. Although GNP would

increase throughout the transition path, private consumption would decrease in the short run.

In the ¿rst period, a simple ad hoc measure of 3social welfare, ́ computed as a populationweighted average of the compensating variations of people alive at the time of the reform,

would decline by 1.54 percent of national wealth. When this measure is applied to new

households who enter the economy in year 2 and thereafter, they are worse off in the short

run but better off in the long run. (See also Figure 1.)

With the Lump-Sum Redistribution Authority. To evaluate the overall ef¿ciency effect

of the tax change, the top panel of Table 9 shows the results with the LSRA. In the ¿rst year,

31
there is no welfare change because the LSRA compensates for all of the welfare gains or

losses of the current households. In year 2 and thereafter, new households are worse off by

0.02 percent in total of national wealth. With the LSRA, the increase in private wealth is

slightly larger than that without the LSRA. But, national wealth would increase only by 2.1

percent because of the debt of the LSRA. Government wealth (including the LSRA) would

be reduced by 2.8 percent as a percentage of GNP in the long run.

Welfare Results by Cohorts. Table 10 shows the welfare changes faced by agents in a

particular birth cohort and the (stochastic) income class they are in at the time of the reform.

This disaggregation provides the most meaningful set of welfare measures because it does

not require assuming a social welfare measure that aggregates compensating variations across

households. The numbers are shown in units of $1,000.32 Without the LSRA, most of the

current households are worse off due to the lump-sum tax on existing wealth, especially

retired households with the most assets. One important exception is the high-working-ability

households who would bene¿t from the much lower income tax rates. Future households tend

to gain from the tax change. But this gain is not enough to compensate the loss of current

households. To see this fact, notice that after the LSRA is turned on, current households

gain exactly $0 by construction, but all future households are worse off by $3,200, which

represents a one-time loss per household at the beginning of age 20.33 (See also Figure 3(a).)

Keep in mind that these numbers correspond to replacing only 10 percent of the progressive

income tax system full replacement would produce even larger losses.

5.3.2 In a Deterministic Economy with Representative Households

In order to estimate the importance of earnings uncertainty, Table 11 shows the result of the

same tax reform within the deterministic version of our OLG economy.

Without the Lump-Sum Redistribution Authority. In the long run, national wealth would

increase by 3.7 percent from the baseline economy. This increase is larger than that in a sto32Unlike in the Auerbach and Kotlikoff model, our reported values cannot be expressed as a share of remaining

full lifetime income since that value is stochastic in our model. 33All dollar values in the model are growth-adjusted. A household born in year | is actually worse off by

$3,200fEn >

|3c where > is the per capita growth rate in the baseline economy.

32
chastic and heterogeneous economy. While precautionary saving increases in the stochastic

version of our model as the risk sharing in the progressive income tax system is reduced,

overall saving is more responsive to changes in after-tax interest rates in our deterministic

model with representative households. Labor supply is also more responsive in the deterministic model, increasing by 1.5 percent in the long run, causing GNP to increase by 2.2 percent.

The consumption tax rate increases by 0.8 percentage points in the long run. Contrary to the

stochastic and heterogeneous economy, private consumption would increase throughout the

transition path. So does our ad hoc social welfare measure, described earlier, which averages

across compensating variations in wealth. (See also Figure 2.) The bottom panel of Table

10 shows that, without the LSRA, older households lose several thousand dollars from the

reform while younger households gain. Households born in the long run gain over $22,000.

With the Lump-Sum Redistribution Authority. The top panel of Table 11 shows the results with the operative LSRA. Welfare of current households is unchanged by construction.

Although private wealth would increase by only 2.6 percent in the long run, total national

wealth would increase by 5.1 percent. The reason is that, in this version of the model,

younger households at the time of the reform gain more than the older households lose. (See

Figure 3 (b).) Holding their remaining lifetime utility ¿xed at the baseline level, therefore,

nets the LSRA a positive level of resources in the short run. As a result, future households

are better off by 0.26-0.27 percent of national wealth under our ad hoc social welfare measure. The bottom panel of Table10 shows the disaggregated results by birth cohort. Again,

by construction of the LSRA, households alive at the time of the reform gain $0. But future

households are better off by $25,100 per household, a remarkable difference relative to the

stochastic case considered earlier.

5.3.3 In a Small Open Economy

Table 12 shows the results of the tax change in a small open economy, in which the interest

rate and the wage rate are kept at the same levels as those in the baseline economy. As with

the closed economy case, this case assumes a stochastic OLG economy with heterogeneous

households.

33
Both with and without the LSRA, the long-run increase in national wealth is larger, and

the long-run increase in labor supply is smaller, than in a closed economy. The increase

in consumption is slightly smaller in the long run, as is the welfare gain of the newborn

households.

Without the LSRA, the average welfare loss of current households is smaller than it

would be in a closed economy. After the ¿rst several years, the interest rate in a small open

economy is higher, which attenuates the welfare loss of elderly households. The one-time

loss of current households totals 0.56 percent of national wealth.

With the LSRA, the welfare loss of current households is $0 by construction future

households will lose a relatively modest $1,000. (See Table 13 and Figure 3(c).)

5.3.4 A Progressive Consumption Tax

Table 14 shows the results of the tax change if a progressive consumption tax is introduced instead of the proportional consumption tax considered earlier. In this case, a Àat consumption

tax is levied on each household¶s annual consumption above $20,000. To keep the government¶s wealth at the same level as in the baseline, the consumption tax rate will have to be

higher by 1.9 percent in the long run.

Both with and without the LSRA, the long-run increase in national wealth is smaller than

it would be in the case of a proportional consumption tax. At 1.84 percent of national wealth,

the one-time welfare loss of current households is larger because the consumption tax rate is

higher.

With the LSRA, however, the welfare loss of future households will be $2,800, which

is smaller than the $3,200 loss in the proportional tax case without a deduction. The main

reason for the smaller loss is that the $20,000 deduction produces some progressivity in the

tax system, which has a positive risk-sharing effect. (See Table 15 and Figure 3(d).) However, future generations still face a loss, in part because a deduction is not as powerful a

risk-sharing device as the progressive marginal tax rate schedule found in the prereform system. For many households, income (and, hence, consumption) typically Àuctuates above the

deduction level, for two reasons. First, unemployment leads to a drawing down of previous

assets in order to support a level of consumption above the deduction. Second, unemploy34
ment insurance income (which is included in our income measure) provides some income

as well. For those households, therefore, a modest decline in normal labor income produces

only a small reduction in their average tax rates and no reduction in their marginal tax rates.

5.3.5 When Households Are More Risk Averse

Table 16 shows the results of the tax change when the coef¿cient of relative risk aversion,

gamma, equals 4.0 instead of 2.0.

Without the LSRA, the long-run increase in national wealth is larger because the precautionary savings of households increase. Although other aggregate variables2labor supply, GDP, and consumption2are roughly the same, the welfare gain of future households is

slightly smaller than the loss in the economy when the coef¿cient equals 2.0 because those

households face higher risk after the tax change. The welfare loss of current households is

larger, for the same reason.

With the LSRA, future households will face a one-time loss of $5,500 from the tax

change, which is larger than the loss when the coef¿cient is 2.0. (See Table 17 and Figure 3(e).)

6 Concluding Remarks

Tax reform has been analyzed in numerous papers. The calculation of the actual ef¿ciency

gains or losses stemming from tax reform, however, has been previously limited to models that assume an in¿nite-horizon representative agent, making it dif¿cult to analyze the

risk-sharing bene¿ts of progressive taxation in the prereform system. This paper presents a

new ¿nite-horizon OLG model with idiosyncratic earnings and longevity risks. Following

the deterministic OLG model developed by Auerbach and Kotlikoff (1987), we construct a

Lump-Sum Redistribution Authority that is used to ¿x the remaining lifetime expected utilities of agents alive at the time of the tax reform at their prereform level. The LSRA allows

us to report actual ef¿ciency gains associated with tax reform.

Our results point to the importance of incorporating the risk-sharing aspects of the prereform tax system into the analysis. With idiosyncratic shocks operative, tax reform produced

a $3,200 loss to future households under our benchmark paramater settings and model form.

35
But when these shocks were turned off and the model was recalibrated to match the same

initial economy, future households gained $25,100. Those results (at least the direction of

the gains) were robust to a fairly wide range of parameter and model assumptions.

The remarkable difference between the model with and without shocks has two sources.

First, the presence of idiosyncratic shocks reduces the intertemporal savings response to

a postreform increase in the after-tax interest rate. Second, the progressive nature of the

prereform tax system2which often leads to larger ef¿ciency gains after tax reform in models

without risk2reduces the bene¿t of moving to a Àat tax in the presence of idiosyncratic risks.

The reason is that fundamental tax reform removes an important source of risk sharing that

was present in the prereform progressive tax system.

Appendix

A The Discretization of the State Space

The state of a household is v @ +l> h> d> e, 5 L  H  D  E, where L @ i53> ===> 43<j,

H @ ^h4?> h4@ `, D @ 

d4?> d4@ 

, and E @ ^e4?> e4@ `. To compute an equilibrium, the

state space of a household is discretized as ev 5 L Ha DaEa, where Ha @ ih> h2> ===> he j,

Da @ id> d2> ===> d@j, and Ea @ ie> e2> ===> eK j=

For all these discrete points, we compute the optimal decision of households, g+ev> V|> |,

@ +f +=,> k +=,> dn +=,, 5 +3> f4@ `^3> k4@ 

 `D> and the marginal values, Y

Y@ y+ev>V|> |,,

given the expected factor prices and policy variables.

To ¿nd the optimal end-of-period wealth, we use the Euler equation and bilinear interpolation (with respect to d and e) of marginal value functions at the beginning of the next

period. In this paper, Qe, Q@, and QK are 8, 60, and 10, respectively. Since there are 90

different ages, the total number of discrete states is 432,000.

B A Steady-State Equilibrium

The algorithm to compute a steady-state equilibrium is as follows. Let  denote the timeinvariant government policy rules  @ +Z}> F}> U +=,>  +=,> > wu77+=,,=

36
1. Set the initial values of factor prices +uf> zf,, accidental bequests tf, the policy variables +Zf

} > Ff} >  f

,, and the parameters *f of policy functions +U +=,>  +=,> wu77+=,,

if these are determined endogenously.34

2. Given 
f @ +uf> zf> tf> Zf

} > Ff} >  f

> *f,, ¿nd the decision rule of a household g+ev>

> 
f, for all ev 5 L  Ha  Da  Ea.

35

(a) For age l @ 43<, ¿nd the decision rule g+evfb> > 
f,. Since the survival rate

!fb @ 3, the end-of-period wealth dn+evfb> =,@3 for all evfb. Compute

consumption and working hours +f+evfb> =,> k+evfb> =,, and, then, the marginal

values Y

Y@y+evfb> > 
f, for all evfb.

(b) For age l @ 43;> ===> 53> ¿nd the decision rule g+ev> > 
f, and Y

Y@y+ev> > 
f,for

all ev, using Y

Y@y+evn> > 
f, recursively.

i. Set the initial guess of df

n+ev> =,.

ii. Given df

n+ev> =,, compute +f+ev> =,> k+ev> =,,= Plug these into the Euler

equation with Y

Y@ y+evn> > 
f,.

iii. If the Euler error is suf¿ciently small, then stop. Otherwise, update df

n+ev> =,

and return to Step ii.

3. Find the steady-state measure of households {+ev> 
f, using the decision rule obtained

in Step 2. This computation is done forward from age 53 to age 43<. Repeat this step

to iterate t for t=

4. Compute new factor prices+u> z,, the policy variables+Z

} > F} ,, and the parameters

* of policy functions.

5. Compare 
 @ +u> z> t> Z

} > F} >  

> *, with 
f. If the difference is suf¿ciently

small, then stop. Otherwise, update 
f and return to Step 2.

34Actually, if we ¿nd the capital-labor ratio, both o and  are calculated from the given production function

and depreciation rate. 35In the steady-state economy, the decision rule of a household _Eet( [c lf is not a function of the aggregate

state of economy 5e ' E%Eetc `} The measure of household %Eet is determined uniquely by the steady-state

condition, and the government¶s wealth `} is determined by the policy rule [.

37
C An Equilibrium Transition Path

Let¶s assume that the economy is in the initial steady state in period 3, and that the new policy

schedule , which was not expected in period 3, is announced at the beginning of period 4,

where  @ iZ}c|n> F}c|> Uc| +=,> c| +=,> c|> wu77c| +=,j"

|'= Let Ve @ +{+ev,> Z}c, be

the aggregate state of the economy at the beginning of period 4. The state of the economy Ve

is usually equal to that of the initial steady state. The algorithm to compute a transition path

to a new steady-state equilibrium (thereafter, ¿nal steady-state equilibrium) is as follows.

1. Choose a suf¿ciently large number, W, such that the economy is said to reach the

new steady state within W periods.36 Set the initial guess, i
f

| jA

|', on factor prices

+uf

| > zf|,, accidental bequests tf| , the policy variables +Zf

}c|n> Ff

}c|>  f

c|,, and the parameters *f

| of policy functions for w @ 4> 5> ===> W.

2. Given 
f

A @ +ufA > zfA > tfA > Zf}cA > Ff}cA >  fc|> *fA ,> ¿nd the ¿nal steady-state decision

rule g+ev>VeA > A > 
f

A , and compute marginal values Y

Y@ y+ev>VeA > A > 
f

A , for all

ev 5 L  He  De  Ee. (See the algorithm for a steady-state equilibrium.)

3. For period w @ W 4> W 5> ===> 4> based on the guess, 
f

| > ¿nd backward the decision

rule g+ev> Ve|> |> 
f

|, and marginal values Y

Y@y+ev> Ve|> |> 
f

|, for all ev 5 L  He 

De  Ee, using the next-period marginal values Y

Y@y+ev>Ve|n> |n> 
f

|n, recursively.

(a) For age l @ 43<, ¿nd the decision rule g+evfb> Ve|> |> 
f

|, and compute the

marginal values Y

Y@ y+evfb>Ve|> |> 
f

|, for all evfb.

(b) For age l @ 43;> ===> 53> ¿nd the decision rule g+ev>Ve|> |> 
f

|, and compute

Y

Y@ y+ev>Ve|> |> 
f

|,for allev, using Y

Y@ y+evn> Ve|n> |n> 
f

|n, previously computed.37

i. Set the initial guess of df

n+ev> =,.

ii. Given df

n+ev> =,, compute +f+ev> =,> k+ev> =,,= Plug these into the Euler

equation with Y

Y@ y+evn> Ve|n> |n> 
f

|n,.

36For this to be the case, the government¶s policy rule has to be time-invariant suf¿ciently before period A,

that is, [r ' [A for  $ r	A. 37Note that this step does not use Y

Y@ Eetnc 5e|( [|( lf

|  recursively.

38
iii. If the Euler error is suf¿ciently small, then stop. Otherwise, update df

n+ev> =,

and return to Step ii.

4. For period w @ 4> 5> ===> W4, compute forward 


| @ +u| > z| > t| >Z

}c|n> F

}c|>  

c|> *

|,

and the measure of households {|n+ev,, using the decision rule g+ev>Ve|> |> 
f

|, obtained in Step 3 and using the state of economy Ve| @ +{|+ev,> Z}c|, recursively.

5. Compare i


| jA

|' with i
f

| jA

|'. If the difference is suf¿ciently small, then stop.

Otherwise, update i
f

| jA

|' and return to Step 2. (If the ¿nal steady-state equilibrium

is known, return to Step 3 instead.)

D The Lump-Sum Redistribution Authority

When the Lump-Sum Redistribution Authority (LSRA) is assumed, the following computation is added to the iteration process for an equilibrium transition path.

1. [Step 1 in the previous subsection 3An Equilibrium Transition Path ́ (hereafter ETP)]

Add the wealth held by LSRA iZf

@c|jA

|' to i
f

| jA

|' and set the initial value.

2. [Steps 2 and 3 in ETP] For period w @ W>W 4> ===> 5, compute the lump-sum transfers

to newborn households (age 20) wu-2 +ev2f> w, to make those households as much better

off as the baseline economy.

(a) Set the initial value of lump-sum transferswu-2+ev2f>Ve|> |> 
|,to newborn households.

(b) Given wu-2 +ev2f> Ve|> |> 
|,, ¿nd the decision rule of newborn households g+ev2f>

Ve|> |> 
|,=

(c) Find the compensating variation in wealth wu-2 +ev2f> Ve|> |> 
|, to make those

households indifferent from the baseline economy. (Initial wealth of newborn

households is assumed to be zero.) If the absolute value of wu-2 +ev2f>Ve|> |> 
|,

is suf¿ciently small, then go to Step (d). Otherwise, update wu-2 +ev2f> Ve|> |> 
|,

by adding wu-2 +ev2f>Ve|> |> 
|, and return to Step (b).

39
(d) Given wu-2 +ev2f> Ve|> |> 
|, and additional lump-sum transfers to newbornswu- ,

¿nd the decision rule of newborn households g+ev2f> Ve|> |> 
|,=

3. [Step 3 in ETP] For period w @ 4, compute the lump-sum transfers to current households (ages 20-109) wu- +ev, to make those households as much better off as the baseline economy.

(a) Set the initial value of lump-sum transfers wu- +ev>Ve> > 
, to current households.

(b) Given wu- +ev>Ve> > 
,, ¿nd the decision rule of current households g+ev> Ve>

> 
,=

(c) Find the compensating variation in wealth to make those households indifferent

from the baseline economy. Compute wu- +ev>Ve> > 
, as the difference

from current beginning-of-period wealth. If the absolute value of wu- +ev>Ve>

> 
, is suf¿ciently small, then stop. Otherwise, update wu- +ev>Ve> > 
,

by adding wu- +ev> Ve> > 
, and return to Step (b).

4. [Before Step 4 in ETP] Compute an additional lump-sum transfer wu- so that the

net present value of all transfers becomes zero. Compute the wealth held by LSRA,

iZ

@c|jA

|', which will be used to calculate national wealth.

5. [After Step 4 in ETP] Recompute wu- and iZ

@c|jA

|' using new interest rates iu|jA|'.

Compare iZ

@c|jA

|' with iZf

@c|jA

|'. If the difference is suf¿ciently small, then stop.

Otherwise, update iZf

@c|jA

|' and return to Step 2.

40
References

[1] Aiyagari, S. Rao (1995). 3Optimal Capital Income Taxation with Incomplete Markets,

Borrowing Constraints, and Constant Discounting. ́ Journal of Political Economy, 106:

1158-1175.

[2] Altig, David, Alan Auerbach, Laurence Kotlikoff, Kent Smetters, and Jan Walliser

(2001). 3Simulating Fundamental Tax Reform in the United States. ́ American Economic

Review, 91, 3: 574-595.

[3] Auerbach, Alan and Laurence Kotlikoff (1987). Dynamic Fiscal Policy. Cambridge University Press.

[4] Auerbach, Alan, Jagadeesh Gokhale, and Laurence Kotlikoff (1994). 3Generational Accounting: A Meaningful Way to Evaluate Fiscal Policy. ́ Journal of Economic Perspectives, 8, 1: 73-94.

[5] Auerbach, Alan and Kevin Hassett (2001). 3Tax Policy and Horizontal Equity. ́ In Kevin

Hassett and R. Glenn Hubbard, Eds., Inequality and Tax Policy. The AEI Press: Washington, D.C.

[6] Auerbach, Alan and James Hines, Jr. (2001). 3Taxation and Economic Ef¿ciency. ́

NBER WP #8181.

[7] Bradford, David (1986). Untangling the Income Tax. Cambridge, Mass.: Harvard University Press.

[8] Bureau of Economic Analysis (1997). 3Fixed Reproducible Tangible Wealth in the

United States: Revised Estimates for 1993-95 and Summary Estimates for 1925-96. ́

Survey of Current Business, September, 37-38.

[9] Chamley, Christophe (1986). 3Optimal Taxation of Capital Income in General Equilibrium with In¿nite Lives. ́ Econometrica, 54: 607-622.

[10] Conesa, Juan and Dirk Krueger (1999). 3Social Security Reform with Heterogeneous

Agents. ́ Review of Economic Dynamics, 2, 4: 757-795.

41
[11] De Nardi, Mariacristina, Selahattin ÚImrohoroglu, and Thomas J. Sargent (1999). Ù 3Projected U.S. Demographics and Social Security. ́ Review of Economic Dynamics, 2: 575-

615.

[12] Engen, Eric and William Gale (1996). 3The Effects of Fundamental Tax Reform on

Savings. ́ In Economic Effects of Fundamental Tax Reform, edited by Henry J. Aaron

and William G. Gale. Washington, D.C.: The Brookings Institution Press.

[13] Engen, Eric, Jane Gravelle, and Kent Smetters (1997). 3Dynamic Tax Models: Why

They Do the Things they Do. ́ National Tax Journal, 50, 3: 657-682.

[14] Fullerton, Don and Diane Lim Rogers (1993). Who Bears the Lifetime Tax Burden?

Washington, D.C.: Brookings Institution Press.

[15] Gouveia, Miguel and Robert P. Strauss (1994). 3Effective Federal Individual Income

Tax Functions: An Exploratory Empirical Analysis. ́ National Tax Journal, 47, 317-339.

[16] Gravelle, Jane (2002). 3Behavioral Responses to a Consumption Tax. ́ In United States

Tax Reform in the 21st Century, George Zodrow and Peter Mieszkowski, Editors. Cambridge University Press.

[17] Hall, Robert E. and Alvin Rabushka (1995). The Flat Tax, 2nd ed. Stanford, Calif.:

Hoover Institution Press.

[18] Huggett, Mark and Gustavo Ventura (1999). 3On the Distributional Effects of Social

Security Reform. ́ Review of Economic Dynamics, 2: 498-531.

[19] Jorgenson, Dale and Kun-Young Yun (2001). Lifting the Burden: Tax Reform, the Cost

of Capital, and U.S. Economic Growth, MIT Press.

[20] Judd, Kenneth (1985). 3Redistributive Taxation in a Simple Perfect Foresight Model. ́

Journal of Public Economics, 28: 59-83.

[21] Mirrlees, James (1971). 3An Exploration in the Theory of Optimal Income Taxation. ́

Review of Economic Studies, 38: 175-208.

42
[22] Nishiyama, Shinichi (2002). 3Bequests, Inter Vivos Transfers, and Wealth Distribution. ́ Review of Economic Dynamics, 5: 892-931.

[23] Social Security Administration (2001). Social Security Bulletin: Annual Statistical Supplement.

[24] Stokey, Nancy and Sergio Rebelo (1995). 3Growth Effects of Flat-Rate Taxes. ́ Journal

of Political Economy, 103: 519-550.

[25] Summers, Lawrence (1981). 3Capital Income Taxation and Accumulation in a Life

Cycle Model. ́ American Economic Review, 71: 533-544.

43
 Table 9: The Results of the Stochastic OLG Model with Heterogeneous Households (Closed Economy, Gamma = 2.0)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously)

Year

1 2 3 5 10 15 20 30 40 60 80 120

(With the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.2 0.3 0.7 1.2 1.6 1.8 2.0 2.1 2.1 2.1 2.1

%ch(Labor) 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6%ch(GNP) 0.5 0.6 0.6 0.7 0.8 0.9 1.0 1.0 1.1 1.1 1.1 1.1%ch(Consumption) 0.1 0.0 0.0 0.2 0.5 0.6 0.8 0.9 0.9 1.0 1.0 1.0%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3ch(Interest Rate%) 0.06 0.05 0.03 0.00 -0.05 -0.08 -0.09 -0.11 -0.12 -0.12 -0.12 -0.12%ch(Wage Rate) -0.2 -0.2 -0.1 0.0 0.2 0.3 0.4 0.5 0.5 0.5 0.5 0.5%ch(Private Wealth) 0.0 1.1 1.3 1.7 2.3 2.7 3.0 3.2 3.3 3.3 3.3 3.3ch(Government Wealth/GDP%) 0.0 -2.5 -2.5 -2.6 -2.7 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8%ch(Accidental Bequests) 1.6 1.8 1.9 2.1 2.6 3.0 3.3 3.7 3.9 4.0 4.0 4.0ch(Comp.Variation/Wealth%) 0.00 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02

(Without the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.3 0.6 1.0 1.8 2.2 2.5 2.8 2.8 2.9 2.9 2.9

%ch(Labor) 0.8 0.8 0.8 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6%ch(GNP) 0.6 0.6 0.7 0.8 1.0 1.1 1.2 1.3 1.3 1.3 1.3 1.3%ch(Consumption) -0.4 -0.3 -0.2 0.0 0.4 0.7 0.8 1.0 1.1 1.1 1.1 1.1%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3ch(Interest Rate%) 0.07 0.04 0.02 -0.02 -0.09 -0.13 -0.15 -0.17 -0.17 -0.18 -0.18 -0.18%ch(Wage Rate) -0.3 -0.2 -0.1 0.1 0.4 0.5 0.6 0.7 0.7 0.7 0.7 0.7%ch(Private Wealth) 0.0 0.3 0.6 1.1 1.9 2.4 2.7 3.0 3.0 3.1 3.1 3.1ch(Government Wealth/GDP%) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0%ch(Accidental Bequests) 0.2 0.5 0.7 1.0 1.7 2.3 2.7 3.1 3.3 3.4 3.4 3.4ch(Comp.Variation/Wealth%) -1.54 -0.01 0.00 0.00 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02

44
 Table 10: Welfare Changes in the Stochastic OLG Model and the Deterministic OLG Model (Closed Economy, Gamma = 2.0) 

Compensating Variations in Wealth ($1,000 per Household)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously)

 The Year When Age 20 (Top) / the Age in Year 1 (Bottom)

-79 -59 -39 -29 -19 -9 1 11 21 41 61 81

100 80 60 50 40 30 20 10 0 -20 -40 -60

(The Stochastic OLG Model with Heterogeneous Households, Closed Economy)

(With the Lump-Sum Redistribution Authority)

Before LSRA e1 -0.6 -6.4 -5.4 -3.4 -2.3 -1.5 -0.8 -0.1 0.2 0.3 0.3 0.3

 e2 -6.6 -4.1 -2.5 -1.7 -1.3 0.0 0.5 0.8 0.8 0.8 e3 -8.4 -4.7 -2.4 -1.5 -1.3 0.3 0.9 1.2 1.2 1.2 e4 -10.4 -4.8 -1.5 -0.4 -0.9 1.0 1.7 2.1 2.1 2.1 e5 -10.7 -4.0 0.2 1.3 -0.2 1.9 2.7 3.1 3.1 3.1 e6 -7.2 -1.6 4.2 3.6 0.9 3.2 4.1 4.5 4.5 4.5 e7 5.4 11.5 15.8 11.0 4.2 6.7 7.7 8.1 8.1 8.1 e8 97.4 90.5 70.0 42.6 9.2 12.1 13.2 13.7 13.7 13.7 Average -0.6 -6.4 -6.4 -2.5 -0.2 0.2 -0.6 1.0 1.6 1.9 1.9 1.9After LSRA Average 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -3.2 -3.2 -3.2 -3.2 -3.2

(Without the Lump-Sum Redistribution Authority)

Without LSRA e1 -0.7 -7.5 -6.9 -4.6 -3.0 -2.0 -1.0 -0.1 0.2 0.4 0.4 0.4

 e2 -8.4 -5.5 -3.5 -2.3 -1.7 0.1 0.7 1.0 1.0 1.0 e3 -10.6 -6.4 -3.8 -2.3 -1.7 0.4 1.1 1.4 1.4 1.4 e4 -13.4 -7.3 -3.6 -1.8 -1.4 1.1 1.9 2.2 2.3 2.3 e5 -14.6 -7.5 -2.8 -0.8 -0.9 1.8 2.8 3.1 3.2 3.2 e6 -12.9 -7.0 -0.7 0.5 -0.1 2.9 3.9 4.3 4.3 4.3 e7 -3.0 2.7 7.8 5.8 2.3 5.7 6.8 7.3 7.3 7.3 e8 76.2 68.4 50.8 30.6 6.1 10.0 11.3 11.8 11.8 11.8 Average -0.7 -7.5 -9.3 -5.0 -2.3 -1.2 -1.1 0.9 1.7 1.9 2.0 2.0

(The Deterministic OLG Model with Representative Households, Closed Economy)

(With the Lump-Sum Redistribution Authority)

Before LSRA -0.5 -3.8 -6.4 5.6 14.5 16.4 12.0 14.0 14.4 14.6 14.8 14.8

After LSRA 0.0 0.0 0.0 0.0 0.0 0.0 0.0 25.1 25.1 25.1 25.1 25.1

(Without the Lump-Sum Redistribution Authority)

Without LSRA -0.4 -3.2 -2.6 10.7 21.0 23.2 18.9 21.8 22.5 22.3 22.3 22.4

45
 Table 11: The Results of the Deterministic OLG Model with Representative Households (Closed Economy, Gamma = 2.0)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously)

Year

1 2 3 5 10 15 20 30 40 60 80 120

(With the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.5 0.9 1.6 2.9 3.7 4.3 4.9 5.1 5.2 5.1 5.1

%ch(Labor) 2.4 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.5 1.5 1.5%ch(GNP) 1.6 1.6 1.7 1.9 2.2 2.4 2.5 2.6 2.6 2.6 2.6 2.6%ch(Consumption) 0.3 0.6 0.8 1.1 1.6 1.9 2.2 2.3 2.4 2.4 2.5 2.5%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 0.9 0.9 0.9 0.9 0.8 0.7 0.7 0.8 0.8 0.8 0.8 0.8ch(Interest Rate%) 0.19 0.13 0.09 0.03 -0.08 -0.15 -0.20 -0.25 -0.27 -0.28 -0.28 -0.28%ch(Wage Rate) -0.8 -0.5 -0.4 -0.1 0.3 0.6 0.8 1.0 1.1 1.2 1.2 1.2%ch(Private Wealth) 0.0 -2.4 -2.0 -1.3 0.1 1.0 1.6 2.3 2.5 2.6 2.6 2.6ch(Government Wealth/GDP%) 0.0 6.2 6.2 6.2 6.1 6.1 6.1 6.1 6.1 6.0 6.0 6.0%ch(Accidental Bequests) 1.1 1.3 1.5 1.7 2.1 2.2 2.1 1.7 1.3 1.4 1.6 1.5ch(Comp.Variation/Wealth%) 0.00 0.26 0.26 0.26 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27

(Without the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.4 0.7 1.4 2.4 3.0 3.4 3.7 3.7 3.7 3.7 3.7

%ch(Labor) 2.1 2.0 2.0 1.9 1.7 1.6 1.6 1.5 1.5 1.5 1.5 1.5%ch(GNP) 1.4 1.5 1.6 1.7 1.9 2.1 2.1 2.2 2.2 2.2 2.2 2.2%ch(Consumption) 0.5 0.6 0.8 1.0 1.5 1.9 2.0 2.2 2.3 2.3 2.3 2.3%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 0.8 0.8 0.8 0.7 0.7 0.6 0.6 0.7 0.7 0.8 0.8 0.8ch(Interest Rate%) 0.16 0.13 0.09 0.04 -0.05 -0.11 -0.14 -0.16 -0.16 -0.16 -0.16 -0.16%ch(Wage Rate) -0.7 -0.5 -0.4 -0.2 0.2 0.4 0.6 0.7 0.7 0.7 0.7 0.7%ch(Private Wealth) 0.0 0.4 0.8 1.4 2.6 3.2 3.6 3.9 3.9 3.9 3.9 4.0ch(Government Wealth/GDP%) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0%ch(Accidental Bequests) 0.3 0.5 0.8 1.2 2.0 2.7 3.2 4.0 4.2 4.1 4.1 4.1ch(Comp.Variation/Wealth%) 4.89 0.19 0.19 0.20 0.21 0.22 0.22 0.22 0.22 0.22 0.22 0.22

46
 Table 12: The Results of the Stochastic OLG Model with Heterogeneous Households (Small Open Economy, Gamma = 2.0)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously)

Year

1 2 3 5 10 15 20 30 40 60 80 120

(With the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.3 0.5 1.0 2.0 2.6 3.1 3.6 3.8 3.9 3.9 3.9

%ch(Labor) 1.1 1.1 1.0 0.8 0.6 0.5 0.4 0.3 0.3 0.3 0.3 0.3%ch(GNP) 0.8 0.8 0.8 0.9 1.0 1.2 1.2 1.4 1.4 1.4 1.4 1.4%ch(Consumption) 0.0 -0.1 0.0 0.1 0.3 0.5 0.6 0.8 0.9 1.0 1.0 1.0%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3ch(Interest Rate%) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00%ch(Wage Rate) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0%ch(Private Wealth) 0.0 0.7 1.0 1.5 2.5 3.3 3.8 4.3 4.6 4.7 4.7 4.7ch(Government Wealth/GDP%) 0.0 -1.0 -1.1 -1.1 -1.2 -1.3 -1.3 -1.4 -1.4 -1.4 -1.4 -1.4%ch(Accidental Bequests) 1.4 1.5 1.7 1.9 2.6 3.2 3.8 4.7 5.2 5.5 5.5 5.5ch(Comp.Variation/Wealth%) 0.00 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01

(Without the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.4 0.7 1.3 2.4 3.2 3.7 4.3 4.5 4.6 4.6 4.6

%ch(Labor) 1.2 1.1 1.0 0.8 0.6 0.4 0.3 0.2 0.2 0.2 0.2 0.2%ch(GNP) 0.8 0.9 0.9 1.0 1.2 1.3 1.4 1.5 1.6 1.6 1.6 1.6%ch(Consumption) -0.4 -0.3 -0.2 -0.1 0.2 0.5 0.6 0.9 1.0 1.0 1.0 1.0%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3ch(Interest Rate%) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00%ch(Wage Rate) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0%ch(Private Wealth) 0.0 0.4 0.7 1.4 2.6 3.4 4.0 4.6 4.8 4.9 4.9 4.9ch(Government Wealth/GDP%) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0%ch(Accidental Bequests) 0.2 0.5 0.7 1.1 2.1 3.0 3.8 4.8 5.2 5.5 5.5 5.5ch(Comp.Variation/Wealth%) -0.56 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01

47
 Table 13: Welfare Changes in the Stochastic OLG Model (Small Open Economy, Gamma = 2.0)

Compensating Variations in Wealth ($1,000 per Household)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously)

 The Year When Age 20 (Top) / the Age in Year 1 (Bottom)

-79 -59 -39 -29 -19 -9 1 11 21 41 61 81

100 80 60 50 40 30 20 10 0 -20 -40 -60

(The Stochastic OLG Model with Heterogeneous Households, Small Open Economy)

(With the Lump-Sum Redistribution Authority)

Before LSRA e1 -0.6 -6.3 -5.0 -3.0 -2.0 -1.5 -0.8 -0.4 -0.2 0.1 0.2 0.2

 e2 -6.0 -3.3 -1.9 -1.5 -1.4 -0.7 -0.1 0.4 0.5 0.5 e3 -7.3 -3.5 -1.3 -1.0 -1.4 -0.6 0.1 0.6 0.7 0.7 e4 -8.3 -2.5 0.8 1.0 -0.7 0.2 0.9 1.4 1.6 1.6 e5 -7.4 -0.2 3.9 4.1 0.3 1.3 2.0 2.6 2.7 2.8 e6 -1.4 4.7 10.8 8.0 2.1 3.1 3.8 4.4 4.6 4.6 e7 15.5 23.0 27.5 18.8 7.0 8.0 8.8 9.4 9.6 9.6 e8 127.1 122.7 99.2 62.2 14.4 15.4 16.2 16.9 17.0 17.1 Average -0.6 -6.3 -4.2 -0.1 2.1 1.6 -0.3 0.5 1.0 1.5 1.6 1.6After LSRA Average 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 -1.0 -1.0 -1.0 -1.0

(Without the Lump-Sum Redistribution Authority)

Without LSRA e1 -0.6 -6.8 -5.6 -3.5 -2.4 -1.9 -1.0 -0.5 -0.2 0.1 0.2 0.2

 e2 -6.6 -3.8 -2.3 -1.9 -1.7 -0.8 -0.1 0.4 0.5 0.5 e3 -8.0 -3.9 -1.7 -1.4 -1.7 -0.7 0.1 0.6 0.7 0.8 e4 -9.0 -2.9 0.4 0.6 -1.1 0.1 0.8 1.4 1.6 1.6 e5 -8.0 -0.6 3.5 3.7 0.0 1.2 2.0 2.6 2.7 2.8 e6 -1.8 4.2 10.1 7.5 1.7 2.9 3.8 4.4 4.5 4.6 e7 14.9 22.0 26.6 18.2 6.6 8.0 8.8 9.5 9.6 9.7 e8 125.1 120.7 97.5 61.4 14.0 15.5 16.3 16.9 17.3 17.3 Average -0.6 -6.8 -4.9 -0.6 1.7 1.2 -0.6 0.4 1.0 1.5 1.6 1.7

48
 Table 14: The Results of the Stochastic OLG Model with Heterogeneous Households (Closed Economy, Gamma = 2.0)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and a Progressive Consumption Tax Is Introduced Contemporaneously)

Year

1 2 3 5 10 15 20 30 40 60 80 120

(With the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.2 0.3 0.6 1.1 1.4 1.5 1.7 1.8 1.8 1.8 1.8

%ch(Labor) 0.5 0.5 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3%ch(GNP) 0.3 0.4 0.4 0.5 0.6 0.6 0.7 0.8 0.8 0.8 0.8 0.8%ch(Consumption) -0.3 -0.2 -0.2 0.0 0.2 0.4 0.5 0.6 0.6 0.6 0.6 0.6%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 2.0 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9ch(Interest Rate%) 0.04 0.02 0.01 -0.02 -0.06 -0.08 -0.09 -0.11 -0.11 -0.11 -0.11 -0.11%ch(Wage Rate) -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.4 0.5 0.5 0.5 0.5%ch(Private Wealth) 0.0 1.3 1.5 1.8 2.3 2.6 2.8 3.0 3.1 3.1 3.1 3.1ch(Government Wealth/GDP%) 0.0 -2.9 -2.9 -3.0 -3.1 -3.1 -3.2 -3.2 -3.2 -3.2 -3.2 -3.2%ch(Accidental Bequests) 1.9 2.0 2.1 2.3 2.6 2.9 3.1 3.5 3.7 3.7 3.7 3.7ch(Comp.Variation/Wealth%) 0.00 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02

(Without the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.3 0.5 0.9 1.6 2.0 2.3 2.5 2.6 2.6 2.6 2.6

%ch(Labor) 0.5 0.5 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3%ch(GNP) 0.4 0.4 0.5 0.5 0.7 0.8 0.9 1.0 1.0 1.0 1.0 1.0%ch(Consumption) -0.6 -0.5 -0.4 -0.2 0.1 0.4 0.5 0.7 0.7 0.7 0.7 0.7%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 2.0 2.0 2.0 2.0 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9ch(Interest Rate%) 0.04 0.02 0.00 -0.04 -0.10 -0.13 -0.15 -0.17 -0.17 -0.17 -0.18 -0.17%ch(Wage Rate) -0.2 -0.1 0.0 0.2 0.4 0.5 0.6 0.7 0.7 0.7 0.7 0.7%ch(Private Wealth) 0.0 0.3 0.5 0.9 1.7 2.1 2.4 2.7 2.7 2.8 2.8 2.8ch(Government Wealth/GDP%) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0%ch(Accidental Bequests) 0.2 0.4 0.6 0.9 1.5 1.9 2.3 2.8 3.0 3.0 3.0 3.0ch(Comp.Variation/Wealth%) -1.84 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02

49
 Table 15: Welfare Changes in the Stochastic OLG Model (Closed Economy, Gamma = 2.0)

Compensating Variations in Wealth ($1,000 per Household)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and a Progressive Consumption Tax Is Introduced Contemporaneously)

 The Year When Age 20 (Top) / the Age in Year 1 (Bottom)

-79 -59 -39 -29 -19 -9 1 11 21 41 61 81

100 80 60 50 40 30 20 10 0 -20 -40 -60

(The Stochastic OLG Model with Heterogeneous Households, Closed Economy)

(With the Lump-Sum Redistribution Authority)

Before LSRA e1 -0.1 -6.7 -4.3 -1.2 -0.1 0.4 0.6 1.0 1.2 1.3 1.3 1.3

 e2 -5.8 -2.5 -0.7 0.2 0.8 1.8 2.2 2.4 2.4 2.4 e3 -8.4 -4.2 -1.5 -0.1 0.8 2.0 2.5 2.8 2.8 2.8 e4 -12.9 -6.9 -2.5 -0.2 0.9 2.3 2.9 3.3 3.3 3.3 e5 -16.1 -8.6 -2.9 0.3 1.1 2.8 3.5 3.8 3.8 3.8 e6 -17.5 -10.1 -2.0 1.0 1.7 3.5 4.3 4.6 4.7 4.6 e7 -12.9 -4.0 4.7 5.7 3.8 5.9 6.7 7.2 7.2 7.2 e8 50.1 52.5 44.1 30.1 7.3 9.7 10.7 11.2 11.2 11.1 Average -0.1 -6.7 -8.8 -4.0 -0.7 0.7 1.0 2.2 2.7 3.0 3.0 3.0After LSRA Average 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -2.8 -2.8 -2.8 -2.8 -2.8

(Without the Lump-Sum Redistribution Authority)

Without LSRA e1 -0.2 -7.9 -5.9 -2.5 -1.0 -0.2 0.3 1.0 1.3 1.5 1.5 1.5

 e2 -7.7 -4.2 -2.0 -0.7 0.3 1.7 2.3 2.6 2.6 2.6 e3 -10.9 -6.3 -3.2 -1.2 0.1 2.0 2.7 3.1 3.1 3.1 e4 -16.5 -9.9 -5.1 -2.0 0.1 2.2 3.0 3.4 3.5 3.5 e5 -20.7 -12.8 -6.5 -2.4 0.1 2.5 3.4 3.8 3.9 3.9 e6 -24.2 -16.5 -7.6 -2.7 0.3 2.9 3.9 4.4 4.4 4.4 e7 -22.6 -13.9 -4.3 -0.1 1.4 4.5 5.6 6.1 6.1 6.2 e8 27.7 29.2 23.8 17.1 3.5 7.1 8.3 8.8 8.9 9.0 Average -0.2 -7.9 -12.1 -6.9 -3.2 -1.0 0.3 2.0 2.7 3.0 3.1 3.1

50
 Table 16: The Results of the Stochastic OLG Model with Heterogeneous Households (Closed Economy, Gamma = 4.0)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously)

Year

1 2 3 5 10 15 20 30 40 60 80 120

(With the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.1 0.2 0.5 0.9 1.2 1.4 1.7 1.8 1.8 1.9 1.8

%ch(Labor) 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6%ch(GNP) 0.4 0.5 0.5 0.5 0.7 0.8 0.8 0.9 1.0 1.0 1.0 1.0%ch(Consumption) 0.2 0.1 0.1 0.2 0.4 0.6 0.7 0.8 0.9 0.9 0.9 0.9%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 1.4 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3ch(Interest Rate%) 0.05 0.04 0.03 0.01 -0.03 -0.05 -0.07 -0.09 -0.09 -0.10 -0.10 -0.10%ch(Wage Rate) -0.2 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.4 0.4 0.4 0.4%ch(Private Wealth) 0.0 1.3 1.5 1.7 2.3 2.6 2.9 3.2 3.3 3.4 3.4 3.4ch(Government Wealth/GDP%) 0.0 -3.2 -3.2 -3.3 -3.4 -3.5 -3.5 -3.6 -3.6 -3.6 -3.6 -3.6%ch(Accidental Bequests) 1.6 1.7 1.8 2.0 2.4 2.8 3.1 3.7 4.0 4.1 4.1 4.1ch(Comp.Variation/Wealth%) 0.00 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04

(Without the Lump-Sum Redistribution Authority)

%ch(National Wealth) 0.0 0.3 0.5 0.9 1.7 2.2 2.5 2.9 3.1 3.1 3.2 3.2

%ch(Labor) 0.8 0.8 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6%ch(GNP) 0.5 0.6 0.7 0.8 1.0 1.1 1.2 1.3 1.4 1.4 1.4 1.4%ch(Consumption) -0.3 -0.2 -0.1 0.1 0.4 0.7 0.8 1.0 1.1 1.1 1.2 1.2%ch(Income Tax Rates) -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -10.0ch(Consumption Tax Rate%) 1.4 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1.2 1.3 1.3 1.2ch(Interest Rate%) 0.06 0.04 0.02 -0.02 -0.08 -0.12 -0.15 -0.18 -0.19 -0.19 -0.19 -0.19%ch(Wage Rate) -0.3 -0.2 -0.1 0.1 0.3 0.5 0.6 0.7 0.8 0.8 0.8 0.8%ch(Private Wealth) 0.0 0.3 0.5 0.9 1.8 2.3 2.7 3.1 3.3 3.3 3.4 3.4ch(Government Wealth/GDP%) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0%ch(Accidental Bequests) 0.2 0.4 0.6 0.9 1.5 2.1 2.5 3.2 3.5 3.6 3.6 3.6ch(Comp.Variation/Wealth%) -1.91 -0.01 -0.01 -0.01 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01

51
 Table 17: Welfare Changes in the Stochastic OLG Model (Closed Economy, Gamma = 4.0)

Compensating Variations in Wealth ($1,000 per Household)

 (When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously)

 The Year When Age 20 (Top) / the Age in Year 1 (Bottom)

-79 -59 -39 -29 -19 -9 1 11 21 41 61 81

100 80 60 50 40 30 20 10 0 -20 -40 -60

(The Stochastic OLG Model with Heterogeneous Households, Closed Economy)

(With the Lump-Sum Redistribution Authority)

Before LSRA e1 -0.8 -7.0 -5.3 -3.4 -2.8 -2.2 -0.8 -0.3 -0.1 0.0 0.0 0.0

 e2 -6.6 -4.3 -3.3 -2.9 -2.0 -0.8 -0.2 0.2 0.3 0.3 e3 -8.3 -5.1 -3.5 -2.9 -2.3 -0.9 -0.1 0.4 0.5 0.5 e4 -10.0 -5.5 -3.0 -2.2 -2.2 -0.5 0.4 1.0 1.0 1.0 e5 -10.2 -4.8 -1.6 -0.7 -1.8 0.1 1.1 1.7 1.8 1.8 e6 -5.5 -1.8 2.2 1.4 -0.9 1.2 2.3 2.9 2.9 2.9 e7 8.1 11.1 12.6 7.6 1.9 4.2 5.4 6.1 6.1 6.1 e8 99.4 86.8 62.7 35.3 6.5 9.1 10.3 11.1 11.1 11.1 Average -0.8 -7.0 -6.0 -2.9 -1.4 -1.4 -1.5 -0.2 0.5 1.0 1.0 1.0After LSRA Average 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -5.5 -5.5 -5.5 -5.5 -5.5

(Without the Lump-Sum Redistribution Authority)

Without LSRA e1 -0.9 -8.8 -7.3 -4.9 -3.6 -2.6 -0.8 -0.2 0.1 0.2 0.3 0.3

 e2 -8.9 -6.0 -4.4 -3.2 -2.2 -0.4 0.5 1.0 1.0 1.1 e3 -11.2 -7.1 -4.8 -3.4 -2.4 -0.2 0.8 1.4 1.4 1.5 e4 -14.2 -8.5 -5.0 -3.3 -2.3 0.2 1.4 2.0 2.1 2.1 e5 -15.6 -9.0 -4.7 -2.6 -2.1 0.8 2.1 2.8 2.9 2.9 e6 -13.5 -8.6 -3.0 -1.5 -1.4 1.7 3.1 3.8 3.9 4.0 e7 -3.5 -0.1 3.5 2.7 0.5 3.9 5.5 6.3 6.4 6.4 e8 70.1 57.5 39.6 22.3 3.7 7.6 9.3 10.2 10.4 10.4 Average -0.9 -8.8 -10.0 -6.1 -3.6 -2.5 -1.8 0.3 1.2 1.8 1.8 1.8

52
 Figure 1: The Results of the Stochastic OLG Model with Heterogeneous Households (Closed Economy)

(When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously, Gamma = 2.0)

GNP

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

National Wealth

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

Labor Supply

1.0

0.8

0.6

0.4

0.2

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

Private Consumption

-0.5

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

53
 Figure 1 (Cont.): The Results of the Stochastic OLG Model with Heterogeneous Households (Closed Economy)

(When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously, Gamma = 2.0)

Accidental Bequests

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

Interest Rate

-0.05

-0.10

-0.15

-0.20

0.10

0.05

0.00

0 20 40 60 80 100 120

Year

Ch from the Baseline

With LSRA Without LSRA

Wage Rate

-0.2

-0.4

0.8

0.6

0.4

0.2

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

Consumption Tax Rate

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0 20 40 60 80 100 120

Year

Ch from the Baseline

With LSRA Without LSRA

54
 Figure 2: The Results of the Deterministic OLG Model with Representative Households

(When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously, Gamma = 2.0)

GNP

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

National Wealth

6.0

5.0

4.0

3.0

2.0

1.0

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

Labor Supply

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

Private Consumption

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

55
Figure 2 (Cont.): The Results of the Deterministic OLG Model with Representative Households

(When the Income Tax Rates Are Reduced by 10 Percent Proportionately, and the Consumption Tax Rate Is Raised Contemporaneously, Gamma = 2.0)

Accidental Bequests

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

Interest Rate

-0.10

-0.20

-0.30

-0.40

0.30

0.20

0.10

0.00

0 20 40 60 80 100 120

Year

Ch from the Baseline

With LSRA Without LSRA

Wage Rate

-0.5

-1.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120

Year

%ch from the Baseline

With LSRA Without LSRA

Consumption Tax Rate

1.0

0.8

0.6

0.4

0.2

0.0

0 20 40 60 80 100 120

Year

Ch from the Baseline

With LSRA Without LSRA

56
 Figure 3: The Welfare Gains/Losses of Households Under Different Models and

 Assumptions

(a) Stochastic OLG Model with Heterogeneous Households

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

-100 -80 -60 -40 -20 0 20 40 60 80 100 120

Year When Age 20

$1,000 per Household

With LSRA Without LSRA -SUM(tr_r1+tr_r2)

(b) Deterministic OLG Model with Representative Households

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

25.0

30.0

-100 -80 -60 -40 -20 0 20 40 60 80 100 120

Year When Age 20

$1,000 per Household

With LSRA Without LSRA -SUM(tr_r1+tr_r2)

(c) Stochastic OLG Model (Small Open Economy)

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

-100 -80 -60 -40 -20 0 20 40 60 80 100 120

Year When Age 20

$1,000 per Household

With LSRA Without LSRA -SUM(tr_r1+tr_r2)

57
 Figure 3 (Cont.): The Welfare Gains/Losses of Households Under Different Models

 and Assumptions

(d) Stochastic OLG Model (Progressive Consumption Tax)

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

-100 -80 -60 -40 -20 0 20 40 60 80 100 120

Year When Age 20

$1,000 per Household

With LSRA Without LSRA -SUM(tr_r1+tr_r2)

(e) Stochastic OLG Model (Gamma = 4.0)

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

-100 -80 -60 -40 -20 0 20 40 60 80 100 120

Year When Age 20

$1,000 per Household

With LSRA Without LSRA -SUM(tr_r1+tr_r2)

58