Tsukuba Economics Working Papers
No. 2014-004

Pareto-improving Consumption Tax When the Return from Capital is idyosyncraticand (Optimal or non-Optimal) Capital Income Tax is available
byHisahiro Naito
August 2014

UNIVERSITY OF TSUKUBA
Department of Economics
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPAN

Pareto-improving Consumption Tax When the Returnfrom Capital is idyosyncratic and (Optimal or
non-Optimal )Capital Income Tax is available
Hisahiro Naito ∗†Current version, August 2014

Abstract
In the standard multi-period model, the consumption tax and the wage tax are equivalent.
When a capital market is incomplete, such that the rate of return from capital is idiosyncratic,
the consumption tax, in contrast to the wage tax, can play a role in risk-sharing. However,risk-sharing may disappear if the government applies a linear or non-linear capital income
tax, because the source of risk is the return from capital. The present study shows that a
consumption tax, when instituted in the presence of a wage tax, increases welfare when thecapital market is incomplete, even if the government applies a non-linear capital income tax
for risk-sharing and subsidizes investments.

∗

email: naito@dpipe.tsukuba.ac.jp Address: Graduate School of International Political Economy, 1-1-1, Tennodai, Tsukuba City, Ibaraki Prefecture, Japan Postal Code: 305-8573.
†
The author is currently affiliated with the University of Tsukuba. This research was initiated during his affiliation with the Institute of Social and Economic Research, Osaka University. The hospitality of the Institute isappreciated, as are the comments of Professor Toshihiro Ihori at the University of Tokyo, Professor Daiji Kawaguchi
at Hitotsubashi University, the seminar participants of the Canadian Public Economics meeting 2004 and the International Institute of Public Finance 2006. The author is responsible for all remaining errors of this paper.

1

Introduction

Revenue derived from the consumption tax is substantial, and in some cases it is nearly the sameas revenue derived from direct income taxes in many developed countries. Accordingly, it would
be of interest to determine the optimal ratio of consumption tax to income tax, from the perspective of economic efficiency. In a simple standard multi-period model, the consumption tax andthe wage tax induce exactly the same consumption and labor-supply behavior and generate thesame discounted value of tax revenue, as long as the wedge caused by the two taxes is the same.
Although the two taxes induce different personal-savings behaviors, because of their different
effects on the timing of taxation, the consumption tax and the wage tax induce the same resource
allocation if government savings are taken into consideration. This means that the consumption tax becomes redundant in the presence of the wage tax. Thus, a simple multi-period modelcannot explain the optimal ratio of consumption and wage taxes.1
The optimal ratio between these two forms of tax can be evaluated with respect to the liquidity constraints of the economy. Judd and Hubbard (1987) analyzed the impact of fiscal policy
under liquidity constraints. When consumers face liquidity constraints due to the imperfectionof financial markets, they prefer to pay to tax in later periods than earlier periods. This implies
that the consumption tax is more efficient than the wage tax since the tax liability of the wage tax
occurs only when consumers are working while the tax liability of the consumption tax is spread
1This conclusion does not change even in the presence of a non-linear income tax system. In the famous papers of
Atkinson and Stiglitz (1976, 1980), it was shown that the introduction of price distortions between consumer goods
is optimal if the preference between consumption goods and leisure is not weakly separable in the presence of anon-linear income tax. However, consumption tax does not distort the relative price of consumption over multiple
periods. Thus, although the Atkinson and Stiglitz theorem is appropriate to explain the use of capital income tax, it
does not address the optimal combination of consumption and wage taxes.

1over the consumer’s life-time.
Tax-evasion behavior also offers insight into the optimal tax ratio. Boadway, Marchand andPestieau (1994) showed that, in the presence of tax-evasion behavior affecting labor-derived
income tax, an optimal combination of consumption tax and labor income tax is important.
The present study demonstrates that when a capital market is incomplete, such that the rateof return from investment is idiosyncratic among individuals and that there is no insurance toinsure stochastic returns, the introduction of a consumption tax, even in the presence of a wagetax, can increase efficiency although the government may also be using a capital income tax toabsorb idiosyncratic risks.
The importance of an incomplete capital market can be seen in the data in several ways. First,
the share of non-financial asset, whose stochastic rate of return are not likely to be insured, is
not low. For example, Bucks, Kennickell and Moore (2006) reported that, according to the US
Survey of Consumer Finance 2004, only 35.7 percent of all assets are held in the form of publicly
traded financial assets while the rest are held as non-financial assets, such as self-owned businesses, self-occupied homes, and real estate. Wolf (2002) also reported that for top 20 percent ofhouseholds in the Survey of Consumer Finances 1998, the share of residential homes, unincorporated businesses and real estate in the total assets is 52 percent. It should be noted that the shareof publicly traded financial assets becomes much lower if human capital, which is not includedin the Survey of Consumer Finances, is taken to be a component of consumer assets. Generally,
there is no insurance that insures the capital gains/losses of the value of residential home, realestate, unincorporated business and human capital due to moral hazard and adverse selection.

2

This implies that consumers buying those assets can be facing substantial risks, presumably withthe trade off of higher expected return.
Second, the variation of the rate of return of those assets is not trivial. Figure 1 and 3 show
the distribution of the rate of return from self-owned business and primary residential home,which is calculated from the Survey of Consumer Finances 1983-1989. As two graphs of the
distribution of the rate of return show, the rate of return varies substantially across individuals.
But one might argue that the substantial variation of the rate of return from those investments isdue to the heterogeneity of risk preference across individuals rather than idiosyncratic shocks.
To control preference heterogeneity of consumers, I regressed the rate of return on demographic
factors such as age and education and plotted the residuals in Figure 2 and 4. Those two graphs
show that, even after controlling the demographic factors that account for the heterogeneity ofpreference, the variation of the rate of return is substantial.
Although it is still possible to argue that the distribution of the residual rate of return stillreflects heterogeneity of unobserved risk preference, which cannot be controlled by the demographic factors, general anecdotal evidences also indicate that substantial part of the variation of
the rate of return comes from real idiosyncratic shocks and those shocks are not insured.
The ability of a consumption tax to increase efficiency when the insurance market for investors is incomplete can be understood by examining the basic economic logic behind the equivalence of the consumption tax and the wage tax. The key observation in this respect is that, in a
simple multi-period model, the budget constraint of each period can be merged into an intertemporal budget constraint by eliminating saving variables. As a result, individual decision-making

3can be defined as maximizing life-time utility, subject to an intertemporal budget constraint. In
this case, the intertemporal budget constraint together with the consumption tax can be re-written
as another budget constraint that also includes the wage tax, by dividing the intertemporal budget
constraint by one plus the consumption tax rate. This implies that, as long as the tax wedge ofthese two taxes is the same, the consumption tax and the wage tax induce the same allocation.
This logic does not change even when there is uncertainty, as long as the market can issue
state-contingent securities. For example, consider a two-period model in which there is uncertainty in the second period. When the market is able to issue state-contingent security, the budget
constraints of the two periods and of all states can still be merged into a single budget constraint.
Thus, the above logic can still be applied.
However, when the market is incomplete, budget constraints cannot be merged. In that case,individual decision-making cannot be characterized as maximizing life-time utility subject to asingle budget constraint. Thus, the logic supporting the equivalence of the two taxes cannot beapplied to a model that includes incomplete markets.
In the real world, there are many uncertain events for which state-contingent security is notavailable and the consumption tax could play an important role in risk-sharing. One example isinvestment in local real estate, local business, human capital or owner-occupied housing. Generally, the rate of return from those investments is different among individuals ex-post, but there isno insurance market that insures the stochastic return from those investments.
The basic idea of this paper is that the consumption tax may be able to help the government
share those risks that cannot be shared by the market. To illustrate, consider the case in which

4the return from investment is stochastic due to the incompleteness of the market. Assume that
there are two periods and that an individual works in the first period and she/he consumes in boththe first and the second periods. Also, assume that there is an investment opportunity for eachindividual and the return from this investment opportunity is stochastic. The second assumption
implies that consumption in the second period is also stochastic. When the government depends
entirely on the income tax from labor, government revenue comes only from the first period
and, as result, the government revenue from an individual is non-stochastic. However, when thegovernment relies entirely on the consumption tax, the government revenue from one individual
becomes stochastic. This implies that the government shares in some of the risk of the economy.
In that case, the consumption tax acts as a form of social insurance and it increases economic
efficiency unless the stochastic rate of returns that individuals invest are perfectly correlated.
Two questions arise from the above argument, however. The first is whether the private insurance market is able to insure individual investment risks and whether social insurance throughconsumption tax is redundant. The second is whether social insurance through a consumption
tax is redundant when the government can instead use a (non-linear) capital income tax, because
the source of risk is the return from capital and the non-linear capital income tax can absorb suchrisks.
To answer the first question, it is assumed that the capital market for investors is incomplete in
the sense that there is no insurance to insure the stochastic rate of return from investment. There
are two justifications for this assumption. First, empirically, the rate of return from an investment
is not the same among individuals. If all risks are insured, the rate of return from an investment

5

should be the same for all individuals. Second, to insure this type of investment risk would
involve both a moral hazard problem and an adverse selection problem. For example, if there
is no pressure to find a profitable investment opportunity because the insurance market insuresinvestment risks, people will not work hard to invest profitably. As for the second question, it isaddressed here in detail and is the focus of this paper.
The idea that taxation acts as a form of insurance is not new in the literature. Many authors
have pointed out this effect. For example, Eaton and Rosen (1980) showed that the wage tax acts
as insurance when the return from human capital is uncertain. Kopczuk (2006) demonstrated
that inheritance taxation provides social security for the rich. Krueger (2003) calculated the
optimal tax progression, focusing on the social-insurance role of taxation. Diamond (2003)
analyzed various social insurance and tax policies with respect to life-cycle when the market
is incomplete. The contribution of the present paper is the demonstration that the consumption
tax increases efficiency even if the government has instituted a non-linear capital income tax toabsorb risks arising from the idiosyncratic rate of return from capital.
The organization of this paper is as follows. In section 2.1, it is shown that an introductionof the consumption tax in the presence of the wage tax is welfare-improving. In section 2.2,thisconclusion is shown to hold even if the government uses a linear capital income tax. In section
2.3, the result is demonstrated to be robust even if it is assumed that the government has instituted
a non-linear capital income tax. In section 3, the conclusions are presented and issues for future
research are discussed.

62

Analysis

2.1

A simple illustration

Before offering a detailed model, this subsection demonstrates the effect on welfare of switchingfrom the wage tax to the consumption tax when a capital market is incomplete. Consider an
economy in which there is a continuum of agents whose preferences are identical and who live
for two periods. Those agents work only in the first period and they save for consumption in thesecond period. Each individual allocates his or her savings by investing in one safe asset and one
risky asset. In this economy, technology is linear and the rate of return from the safe asset is r andthe stochastic rate of return from the risky asset is re. In this paper,eis used to denote that the variable witheis a random variable. It is assumed that re = ε × (e + 1) , where ε is a random variable
and e ∈ [0, 1] is the individual’s effort level in searching for profitable investment opportunities.
It is also assumed that the random variable ε is independently and identically distributed (i.i.d.)
across agents and that the level of individual effort to look for a profitable investment opportunity
is not observable by the public. This lack of observability prevents private financial institutions
from sharing in investment risks. In addition, throughout this paper, the assumptions have been
made that the expected rate of return from risky investments is always greater than the rate of
return from the safe asset, as long as the agent makes a positive effort to look for a profitable
investments opportunities and that the expected rate of return from those investments is equal tothat of the safe asset if no such effort is made. Let E[er|e] be µ(e), where E[·|e] is the operator to
calculate the mathematical expectation for a given effort level e. Then, our assumption implies
that7µ(e) > rµ(e) = r

for any level of effort e ∈ (0, 1]
for e = 0

(1)
(2)

Thus, the agent can always invest in the safe asset with the rate of return r without any effort.
The preference of an agent is represented by the following additive and separable utility
function:u(c1 ) − h(l) − v(e) +

1
E[u(e
c2 )]
1+ρ

(3)

where u0 > 0, u00 < 0, h0 > 0, h00 > 0, v 0 > 0, v 00 > 0where (l, c1 , c2 ) are the labor supply, the consumption in period 1 and the consumption inperiod 2. It is assumed that the consumption in period 1 is a normal good. From the linear
technology assumption, the rate of return from investment and the wage rate are independent oflabor supply and the amount of investment.
As for government, it is assumed that the government can institute a linear consumption tax
and a linear wage tax.2 In this subsection, I assume that the government does not impose acapital income tax. In the next two subsection, I analyze two cases where the government can
2

The assumptions that the government institutes only a linear consumption tax and only a linear wage tax are
consistent with the literature on Ramsey-type optimal taxation. In this literature, it is assumed that the government
cannot directly control labor supply and investments, although it is aware of consumer preferences and the technology of the economy. The assumption about the government’s inability to directly control either the labor supply or
consumption can be easily justified by using the framework developed by Mirrlees (1971) and by assuming a modelin which there are multi-agents with unobserved abilities. However, to simplify the analysis, it is assumed that the
government uses a linear consumption tax, a linear wage tax, and either a quasi-linear or a non-linear capital income
tax.

8use a linear capital income tax and a more flexible non-linear capital income tax, respectively.3
Let s and I be the amount of investment for the safe asset and the risky asset, respectively. Let tw
be the rate of the wage tax, tc the rate of the consumption tax. Then, for the given wage tax rate
tw and the consumption tax rate tc , each agent will solve the following optimization problem:

max u(s,l,e(1 − tw )wl − s − I
1s(1 + r) + I(1 + re)) − h(l) − v(e) +E[u()](1 + tc )1+ρ
(1 + tc )(4)

If an interior solution is assumed, then the first-order conditions of l,s, I and e are(1 − tw )w 0
u (c1 ) = h0 (l),(1 + tc )1+r
u0 (c1 ) =E[u0 (ec2 )]1+ρ
1
u0 (c1 ) =E[u0 (ec2 )(1 + re)]
1+ρ
de
r I
1
]
E[u0 (ec2 )and v 0 (e) =1+ρ
de 1 + tc(1 − tw )wl − s − I
s(1 + r) + I(1 + re)where c1 =) and ec2 =(1 + tc )(1 + tc )

(5)
(6)
(7)
(8)

These first-order conditions result in the following relationship between the expected rates ofreturn of the risky investment and the safe asset.

−cov(u0 (ec2 ), re) = E[u0 (ec2 )]{E[er |e∗ ] − r}3

(9)

It is assumed that the consumption tax rate is constant over time. In the next two subsections, I allow a
linear capital income tax and a non-linear capital income tax. With capital income taxation, the assumption oftime-constant consumption tax is not restrictive since time-varying consumption tax can be decomposed into timeconstant consumption tax and capital income tax.

9

Note that re and ec2 are positively correlated. Thus, we must have E[er|e∗ ] − r > 0 and thisrequirement is satisfied by assumption(1). In other words, as long as an agent makes a positive
investment in the risky asset, the expected rate of return from that investment must be greaterthan the rate of return from the safe asset.
Now consider the effects of the consumption tax and the wage tax. From the above-describedfirst-order conditions, it is clear that neither the consumption tax nor the wage tax affect intertemporal conditions. Instead, the two taxes change only the relative price of consumption and leisure.
Thus, two situations can be considered: (1) the government imposes only the wage tax, and (2)the government imposes only the consumption tax. Furthermore, it is assumed that in both casesbb
the real wage rate, w(1 − tw )/(1 + tc ), is the same. Let {bl, eb, sb, I;
c1 , bec2 , } be the solution of theabove optimization problem and resulting consumption in periods 1 and 2 with (tw , tc ) = (τ, 0).
Also let and {l∗ , e∗ , s∗ , I ∗ ; c∗1 , ec∗2 , } be the solution and resulting consumption of periods 1 and 2with (tw , tc ) = (0, τ /(1 − τ )). Note that in either case the real wage, w(1 − tw )/(1 + tc ), is equalbb
to w(1 − τ ). From the above first order conditions, if {bl, eb, sb, I;
c1 , bec2 , } is the solution with theb + tc ).4
taxes (tw , tc ) = (τ, 0), then l∗ = bl, e∗ = eb, c∗1 = b
c1 , ec∗2 = bec2 , s∗ = sb(1 + tc ), I ∗ = I(1In other words, when the government switches from the wage tax to the consumption tax,
consumers’ labor supply and efforts are the same as before and individuals increase their investments in safe and risky assets 1 + tc times. Thus, the two cases induce exactly the same l,e,c1
and ec2 . This implies that the expected utilities of the agents under the two tax regimes are theexactly the same as long as the real wage rate, w(1 − tw )/(1 + tc ) , is the same. When there is4

b + tc ), we will have e∗ = eb.
Note that from the first order condition of e and I ∗ = I(110no uncertainty and, as a result, the µ(e) is equal to r, it is straightforward to show that the tax
revenue in both cases are the same. Thus, two taxes induce the same resource allocation. This is
the standard argument to support the equivalence of the consumption tax and the wage tax.
However, when there is uncertainty and the capital market is incomplete, the tax revenue of
two cases are different. For (tw , tc ) = (τ, 0), government revenue is:Tax revenue = τ wbl

(10)

In the second case, in which (tw , tc ) = (0, τ /(1 − τ )), the present value of the tax revenue is1
tc c∗1 + 1+rE[tcec∗2 ˙]. Note that for the second period tax revenue, the government discounts it with

1 + r since there is no macroeconomic shock regarding both capital income and consumption inc∗2 = s∗ (1 + r) + I ∗ (1 +the second period. Also note that c∗1 (1 + tc ) = wl∗ − s∗ − I ∗ and (1 + tc )eE[er|e∗ ]). Therefore, the tax revenue under the consumption tax is:tc
1tc(wl∗ − s∗ − I ∗ ) +{s∗ (1 + r) + I ∗ (1 + E[er|e∗ ])}(1 + tc )(1 + r) (1 + tc )(11)∗
1 + E[er|e ]= τ wl∗ + τ I ∗ {
− 1}
(12)
1+r

Tax revenue =Note that l∗ = bl and E[er|e∗ ] > r. Accordingly, the tax revenue generated by the consumption
tax alone is more efficient than that generated by the wage tax alone. Thus, as long as individuals
positively invest in risky assets, the consumption tax is more efficient. The intuition for thisresult is quite simple: when the government uses only the wage tax, government revenue from
one individual is certain. But when the government switches to the consumption tax, the tax
revenue from one individual becomes stochastic while individual consumption and the labor11

supply remain the same as before. This implies that, under the regime of the consumption tax,
the government starts to share some of the risk that individuals initially face. Since individual
risks are not perfectly correlated, risk-sharing through the consumption tax increases economic
efficiency.
However, there is another issue that needs more careful discussion regarding the above argument which is as follows: Uncertainty regarding the economy comes from uncertainty in therate of return from risky investments. Thus, one might argue that the government can institute a
capital income tax to absorb the risk of the stochastic rate of return from risky investments. If thegovernment institutes a non-linear capital income tax to absorb those risks, risk-sharing throughthe consumption tax may be redundant. To address this criticism, it is clear that a more formal
analysis is needed. Thus, in the next section, the effects of applying a consumption tax in thepresence of a wage tax and either a linear or non-linear capital income tax are examined in thecontext of a two-period model.

2.2

A model with a linear capital income tax

It was shown above that the consumption tax can help in risk-sharing in an economy in which
there are financial risks that the market cannot insure. However, one might speculate that a
consumption tax will be redundant in the presence of a capital income tax because the source ofrisks is the return on investment and because a capital income tax can absorb such risks. Here,it will be shown that this result does not change even if the government uses a linear tax (taxliability from random capital income is proportional to capital income plus some constant).
Let tks be the tax rate of capital income from safe assets and tkr be the tax rate on capital
12

income from risky assets. Let D be the constant that makes the tax liability on risky capitalincome affine, and tI the subsidy rate on risky investments. 5 In other words, the tax liability
from risky investments is tkr (erI)−D. Accordingly, the agent will maximize the following utility
function:max u({l,s,I,e}wl(1 − tw )s
(1 − tI )I
1(1 + r)s − tks rs + (1 + re)I − tkr reI + D−−)−h(l)−v(e)+E[u()]
1 + tc
1 + tc
1 + tc1+ρ
1 + tc(13)

Now let w(1−tw )/(1+tc ) and D/(1+tc ), s/(1+tc ), I/(1+tc ) be w and D, s and I, respectively.
Then, the above optimization problem can therefore be rewritten as follows:

V (w, D, tks , tkr , tI ; tc ) ≡ max u(wl − s − (1 − tI )I) − h(l) − v(e){l,e,s,I}

1
E[u( s(1 + r(1 − tks )) + (1 + re(1 − tkr ))I + D)]
+
1+ρ

(14)

The optimization problem (14) has an important implication. (14) shows that as long as w and D
do not change, a change of the consumption tax rate does not change the optimized value of (14)because tc does not directly enter (14). On the other hand, the government can control w and D
by controlling tw and D directly. This implies that the government can have policy instruments∗

to offset the effect of the consumption tax. Let the solution of (14) be (l∗ , e∗ , s∗ , I ). Also let theresulting consumption in period 1 and period 2 be c∗1 and ec∗2 . Since the maximized value does notdirectly depend on tc once w, D are defined, we have5

The availability of the subsidy on risky investments does not change the result. I allow the subsidy on investment
to demonstrate that the result is robust. Even if the subsidy on risky investments are not available, the result does
not change.

13∂V (w, D, tks , tkr , tI ; tc )/∂tc = 0(15)∗∂l∗ /∂tc = 0, ∂e∗ /∂tc = 0, ∂s∗ /∂tc = 0, ∂I /∂tc = 0(16)Then, the government’s problem is to choose tw , D, tks , tkr , tI and tc in order to maximize thevalue function subject to the government budget constraint. Let Q be the required government
revenue. The government budget constraint is

tw wl +

tks rs + tkr µI − D
tc E[ec2 ]+ tc c1 +− tI I − Q ≥ 0
1+r
1+r

(17)

Note that s(1 + tc ) = s and I(1 + tc ) = I. Thus, the government budget constraint can be
rewritten as

tw wl +

tks rs(1 + tc ) + tkr µI(1 + tc ) − D
tc E[ec2 ]+ t c c1 +− tI I(1 + tc ) − Q ≥ 0.
1+r
1+r

(18)

Thus, consider the following constrained maximization problem:
W (tc ) ≡s.t. tw wl∗ +max

V (w, D, tks , tkr ; tc ){tks ,tkr ,tI ,D,tw }∗tks rs∗ (1 + tc ) + tkr µI (1 + tc ) − D

1+r+ tc c∗1 +(19)

tc E[ec∗2 ]∗− tI I (1 + tc ) − Q ≥ 0
1+r

w = w(1 − tw )/(1 + tc ) and D = D/(1 + tc )tc is given
Note that W (tc ) is the maximized utility for a given level of the consumption tax when other taxes
are chosen optimally. Let λ be the Lagrangian multiplier of the government budget constraint14and let L be the associated Lagrangian function. Then, we can calculate the first order condition
with respect to tks , tki , tw and tI for a given level of tc . To save the space, the detailed firstorder conditions are calculated in Appendix. Our question is to calculate dW/dtc and evaluate
at tc = 0. From the envelope theorem, dW/dtc = ∂L/∂tc . Thus, dW/dtc is∂w ∂L ∂D ∂L ∂V++∂tc ∂w∂tc ∂D ∂tc
∗
E[e
c∗2 ]
tks rs∗ tkr µI∗+ λ{++ c∗1 +− tI I }
1+r
1+r
1+r

dW/dtc |tc =0 =(20)

Applying the first order conditions of tw and D and ∂V /∂tc = 0:1}+λdW/dtc |tc =0 = −λw(1 − tw )l + {−D}{λ
1+r
∗

(

tks rs∗ tkr µ(e∗ )I
+
1+r
1+r

∗

∗

[1 + r(1 − tks )]s∗ + [1 + µ(e∗ )(1 − tkr )]I + D∗− tI I+w(1 − tw )l − s − (1 − tI )I +
1+r
(21)


∗
µ(e ) − r
∗
=λ
I
(22)
1+r
∗

∗

∗

This result shows that introducing the consumption tax will improve welfare even when thegovernment applies a linear capital income tax to absorb idiosyncratic risks as long as the expected rate of return from risky assets is greater than the rate of return from the safe asset.6
6

Reader might think that the presence of constant part of the capital income tax implies that the government canuse a lump-sum tax and, as a result, optimal consumption tax should be zero. However, such intuition does nothold in our model. Because the market is incomplete, the availability of the lump-sum tax does not imply that otherdistortionary tax should be zero.

15

)

2.3

Risk sharing in the presence of the nonlinear capital income tax

The above-described results showed that, even if the government institutes a linear capital income
tax for the purpose of risk-sharing, it is still efficient to introduce a consumption tax when themarket is incomplete. However, it could be argued that the power of a linear capital income tax toshare in those risks is limited and that if the government instead uses a more flexible, non-linear
capital income tax the result will be different.
Here, it is assumed that the government imposes a tax on the net return from risky investments
and that the tax liability is a non-linear function of the net return from those investments. The
ability of the consumption tax to increase welfare under such circumstances is then determined.
Let z be the realized value of the random value of re. The non-linear capital income tax is afunction of R ≡ zI . Let T (R) be this non-linear capital income tax function. Then, for a given
T (R), tks , tw and tc , an individual will solve the following maximization problem:

wl(1 − tw )s
(1 − tI )I−−) − h(l) − v(e){l,s,I,e}
1 + tc
1 + tc
1 + tc1(1 + r(1 − tks ))s + (1 + reI) − T (e
rI)
+
)]
E[u(
1+ρ
1 + tcmax u((23)

The government budget constraint is
tw wl +

tks rs + E[T (erI)]
tc E[ec2 ]+ t c c1 +− tI I − Q ≥ 0
1+r
1+r

(24)

The government sets T (R), tks , tw and tI to maximize the indirect utility determined by equation(23) subject to the government budget constraint (24) for a given value of tc .

16

The concern here is to examine whether increasing tc from zero will increase the indirect
utility. To conduct such a thought experiment, the method first developed by Christiansen (1984)
is applied. Let T (R; η) be a non-linear tax function in which η is a parameter that shifts the tax
schedule. Also, let T ∗ (R) be the optimal nonlinear tax schedule that maximizes (23) subject tothe government budget constraint (24) when tc = 0. It is assumed that when η = 0 the nonlinear
tax function T (R; η) becomes T ∗ (R). More specifically, T (R; η) is defined as follows:
T (R; η) = T ∗ (R) + η × G(R)(25)

where G(R) is an arbitrary function. Accordingly, a representative agent will solve the following
optimization problem given tc , tw , tI and T (R; η) :wl(1 − tw )s
(1 − tI )I−−) − h(l) − v(e){l,s,I,e}
1 + tc
1 + tc
1 + tc(1 + r(1 − tks ))s + (1 + re)I − T (erI; η)
1
E[u(
+
)].
1+ρ
1 + tcmax u((26)

The objective of the government is to maximize the above indirect utility function subject tothe government budget constraint. The purpose here is to examine whether an introduction of
the consumption tax increases welfare. To consider this issue, it is useful to analyze the problem
through several steps. Accordingly, w, s and I are defined in the first period as w ≡ w(1 −sItw )/(1 + tc ), s ≡ 1+t, I ≡ 1+t
and the following sub-indirect utility function is considered:ccΨ(w, tI , s, I) = max u(wl − s − (1 − tI )I) − h(l)
{l}

(27)

Note that this sub-indirect utility function is concave with respect to s and I as long as theconsumption good is a normal good. Also, note that l is a function of w,tI ,s and I. Given this17

sub-indirect utility function, the following indirect utility function can be defined:
V (w, tI , tks , η; tc ) ≡ max Ψ(w, tI , s, I) − v(e){s,I,e}+1
T (erI(1 + tc ); η)E[u(1 + r(1 − tks ))s + (1 + re)I −)]1+ρ
1 + tc(28)

In the above optimization problem, s,Iand e can be considered as functions of tks , η, tI and tc .
Given this indirect utility function, the following constrained optimization problem for given tc
can be solved:
W (tc ) ≡max V (w, tI , tks , η; tc , )tc E[ec2 ] rtks s(1 + tc )T (erI(1 + tc ); η)+ λ wltw + tc c1 +++ E[
] − tI I(1 + tc ) − Q
1+r
1+r
1+r
(29)
{tw ,tI, η,tks }where λ is the Lagrangian multiplier of the government budget constraint.
W (tc ) is the indirect utility when the government chooses an optimal wage tax rate, a capitalincome tax rate for the safe asset and a nonlinear capital income tax schedule for a given consumption tax rate tc . Now, consider the effect of increasing the consumption tax rate from zero.
From the envelope theorem and the first order conditions defined above (see Appendix):dWµ(e) − r= λ{}I
dtc tc =0
1+r

(30)

The result can be summarized as follows:
Proposition
Increasing the consumption tax rate, tc , from zero in the presence of a optimal wage tax,
investment subsidy, nonlinear or linear capital income tax is welfare improving as long as
18

the expected rate of return from the risky investment is higher than the rate of return fromthe safe asset at tc =0
Economic interpretation of (30) is as follows. When the government increases the consumption rate by M tc from zero, the investment on the risky asset and the safe asset increases by M tc
in order to pay the consumption tax in the second period. Thus, in absolute terms, the amount ofinvestment in risky asset increases by the initial amount of investment on risky investment. For
each dollar of risky investment, since the expected rate of return from risky investment is higher
than that of the safe asset and there is no macroeconomic uncertainty. Thus, there is welfare gainby {µ(e) − r}I× M tc . In the present value, this is equal to {µ(e) − r}I× M tc /(1 + r).

2.4

Intuition and Discussion

At this point, readers are likely to have several questions. The first is ”Why does using the consumption tax together with the wage tax increase efficiency even if a non-linear capital income
tax has already been instituted to absorb idiosyncratic risks?” Based on the assumption that a
non-linear capital income tax can be applied very flexibly, one might wonder why a consumption tax still plays a role.
The basic intuition that a consumption tax increases efficiency comes from the interaction of
the risk-sharing effect, which is already discussed in this paper, and its timing effect, which was
first discussed by Summers (1981). In that paper, it was argued that the wage tax and the consumption tax have different timing effects and, as a result, different effects on individual savings.
As in many life-cycle models, in our model a switch from the wage tax to the consumption tax
shifts some of the tax burden from the first (early) period to the second (late) period. This implies
19

that consumers need to increase their investments in both safe assets and risky assets in order tobe able to pay more taxes in the second period, when the government switches from the wage
tax to the consumption tax. However, the expected rate of return from risky assets is higher thanthat from safe assets due to the assumption of an incomplete capital market, although there is no
macroeconomic risk for risky assets. Thus, from a social-planning perspective, it is desirable toincrease investments in risky assets while simultaneously sharing those risks. The shift from thewage tax to the consumption tax not only makes risk-sharing possible, it also compels individuals to increase investments in both risky assets and safe assets due to the timing effect of theconsumption tax. The effect of this shift does not disappear even in the presence of a non-linear
capital income tax.
Certainly, a non-linear capital income tax and investment subsidies can be used to increase
investments in risky assets and to enhance risk-sharing. However, those policy instruments also
introduce distortions. By contrast, a consumption tax can increase investments into risky assets
and improve risk-sharing without disincentive effects on investments because the consumption
tax does not change the relative price of consumption during different periods. Although theconsumption tax has limited risk-sharing power, the marginal cost of introducing this tax is zero.
Thus, even if the government applies a non-linear capital income tax to allow for risk-sharing,
introduction of a consumption tax in the presence of the wage tax improves public welfare.
The second question is the degree of combination of the two taxes. In the model discussed
here, as the logic of the analysis shows, a complete switch from the wage tax to the consumption
tax is optimal. Thus, one might argue that the analysis does not explain an optimal mix of the20

consumption tax and the wage tax. However, this shortcoming can be easily overcome by introducing ex-ante heterogeneity. When the wage tax and the consumption tax are compared, one of
the advantages of the wage tax is that it can be used to redistribute income. Although, theoretically, the consumption tax can also be used for income redistribution through the expendituretax, administration of the latter is quite difficult. Thus, if heterogeneity and the necessity of income redistribution are introduced ex-ante, a combination of the wage tax and the consumption
tax becomes optimal. In that situation, the role of the wage tax is limited to income redistribution
while the consumption tax, together with capital income tax, plays a role in raising revenue andrisk-sharing.

21

3

Implication and Conclusions

The present study has shown that an introduction of the consumption tax in the presence of thewage tax increases economic efficiency when a capital market is incomplete, even if the government has instituted either a linear or a non-linear capital income tax. The improved efficiency
comes from the fact that the consumption tax will shift the tax burden in later periods and, due tothe timing effect of the consumption tax (Summers 1981), increases investments in risky assets.
The rate of return on risky investments being higher than safe assets, but with no macroeconomicrisk on risky investments, increased investment on risky assets increases economic efficiency.
This intuition has an important implication for the privatization of public pension. In thestandard theory, it has been assumed that the expected rates of returns of the government fundand the individual fund are the same. As a result, it is argued that whether the fund is managed by
the government or individual does not matter and that the privatization of the public pension does
not bring the real welfare gain. However, when a capital market is incomplete, individuals might
be facing a higher expected rate of return, with higher variance of the idiosyncratic rate of return
due to incompleteness of capital market( but no macroeconomic risk), than that the government
faces. In such a case, the government might be able to improve welfare by privatizing social
security, let individuals invest in risky assets with a nonlinear tax schedule on the return fromthose investment to absorb idyosyncratic risks.

22

4Appendix

4.1

First order conditions under a linear capital income tax

The first order conditions are:∂w ∂L
+ λwl∗ = 0∂tw ∂w
∂D ∂L
1
=0
D:
−λ
∂D ∂D
1+r
∗
∂V
∂l∗
tks r(1 + tc ) ∂s∗
tkr µ(1 + tc ) ∂I
tks :
+ λw
+λ
+λ
∂tks
∂tks
1+r
∂tks
1+r
∂tks
∗
∂c∗
tc ∂E[ec∗2 ]∂Irs∗ (1 + tc )+λtc 1 + λ− λtI (1 + tc )
+λ
=0
∂tks
1 + r ∂tks
∂tks
1+r
∗
∂V
∂l∗
tks r(1 + tc ) ∂s∗
tkr µ(1 + tc ) ∂I
tkr :
+ λw
+λ
+λ
∂tkr
∂tkr
1+r
∂tks
1+r
∂tks
∗
∗
∗
∗
tc ∂E[ec2 ]∂IµI (1 + tc )∂c
− λtI (1 + tc )+λ=0+λtc 1 + λ∂tks
1 + r ∂tkr
∂tkr
1+r
∗
∂l∗
tks r(1 + tc ) ∂s∗
tkr µ(1 + tc ) ∂I
∂V
∂c∗1
+ λw
+λ
+λ
+ λtc
tI :
∂tI
∂tI
1+r
∂tI
1+r
∂tI
∂tI
∗
∗
tc ∂E[e
c2 ]
∂I
∗
+λ
− λtI (1 + tc )− λI (1 + tc ) = 0
1 + r ∂tI
∂tI
tw :

4.2

The effect of increasing consumption tax under a non-linear capital

23

income tax

The associated Lagrangian function is:
L = V (w, tI , tks , η; tc , )T (erI(1 + tc ); η)tc E[ec2 ] rtks s(1 + tc )++ E[
] − tI I(1 + tc ) − Q+ λ wltw + tc c1 +
1+r
1+r
1+r
(31)
The first order conditions regarding tw , tks , tI and η, are:∂w ∂L
+ λwl = 0(32)
tw :∂tw ∂w
1
tc ∂E[e
c2 ]
∂l ∂s
∂l ∂I
∂c1
0
tks : −
E[u (ec2 )rs] + λ wtw
+ wtw
+ tc
+
1+ρ
∂s ∂tks∂tks 1 + r ∂tks
∂I ∂tksrs(1 + tc )rrtks (1 + tc )∂s
T 0 (erI(1 + tc ); η)e∂I(1 + tc )++ E[
] − tI
1+r
1 + r ∂tks
1+r
∂tks

∂e
r ∂e
] =0
(33)
+E[T 0 (erI(1 + tc ); η)I(1 + tc )∂e ∂tks∂Ψtc ∂E[e
∂l
∂l ∂s
∂l ∂I
∂c1
c2 ]
tI :
+ λ wtw
+ wtw
+ wtw
+ tc
+
∂tI
∂tI
∂s ∂tI∂tI
1 + r ∂tI
∂I ∂tI
0rtks (1 + tc )∂s
T (erI(1 + tc ); η)er∂I(1 + tc )++ E[
] − tI
1+r
∂tI
1+r
∂tI

∂e
r ∂e
] =0
(34)
+E[T 0 (erI(1 + tc ); η)I(1 + tc )∂e ∂tI

1
1 ∂T
∂s
∂l
rtks
0
η:−
E[u (c2 )
]+λ
{wtw
+
}
1+ρ
1 + tc ∂η∂η∂s 1 + r
∂I
∂l
∂e
r ∂e
+ {wtw+ E[T 0 (erI(1 + tc ); η)(1 + tc )er] − (1 + tc )tI } + E[T 0 (erI(1 + tc ); η)I(1 + tc )
]
∂η
∂e ∂η
∂I

1 ∂T 1
+E[
] =0
(35)
1 + r ∂η 1 + tc

24

The effect of increasing the consumption tax from zero is equal to:dW
1
E[u0 (c2 ){T − T 0 reI}}=dtc tc =0 1 + ρ∂w ∂L+∂tc ∂w
∂l
rtks
∂I
∂l
T 0 (erI; η)e
r
∂s
{λwtw
+
}+
{λwtw
+ λE[
] − tI }+∂tc∂s 1 + r
∂tc
1+r
∂I
∂e
T 0 (erI; η)I ∂e
r
+
{λE[
]
∂tc
1 + r ∂eE[c2 ] rtks sT (erI; η)erI+ λ{c1 +++ E[] − tI I}(36)
1+r 1+r
1+r1=
E[u0 (c2 ){T − T 0 reI}]
1+ρ
∂L
∂s
∂l
rtks
∂I
∂l
T 0 (erI; η)er− w(1 − tw )
{λwtw
{λwtw
+
+
}+
+ λE[
] − tI }∂w ∂tc∂s 1 + r
∂tc
1+r
∂I
∂e
T 0 (erI; η)I ∂er
E[c2 ] rtks sT (erI; η)erI+λE[] + λ{c1 +++ E[] − tI I}(37)∂tc
1 + r ∂e1+r 1+r
1+r
By using the first order condition of tw , we have1dW=
E[u0 (c2 ){T − T 0 rI}]
dtc tc =0 1 + ρ− λw(1 − tw )lT 0 (erI; η)e
r
∂s
rtks
∂I
∂l
∂l
+ λE[
+
}+
] − tI }
{λwtw
{λwtw
∂tc
∂s 1 + r
∂tc
1+r
∂I
∂e
T 0 (erI; η)I ∂er
E[c2 ] rtks s
T 0 (erI; η)erI] + λ{c1 +++ E[] − tI I}+
λE[∂tc
1 + r ∂e1+r 1+r
1+r
+

(38)

Note that c1 = w(1 − tw )l− s − (1 − tI )I and E[c2 ] = (1 + r(1 − tks ))s + (1 + µ)I − E[T (erI; η)]

25

at tc = 0. Thus, we havedW
1
E[u0 (c2 ){T − T 0 reI}]=dtc tc =0 1 + ρ
∂s
∂l
rtks
∂I
∂l
T 0 (erI; η)e
r
+
{λwtw
+
}+
{λwtw
+ λE[
] − tI }∂tc∂s 1 + r
∂tc
1+r
∂I
∂e
T 0 (erI; η)I ∂e
r
+
λE[
]
∂tc
1 + r ∂eµ−r
T 0 (erI; η)erI
T (erI; η)+λ {}I + E[] − E[
]
1+r
1+r
1+r

(39)

At this point, we need to use the information on the first order condition on η. For this purpose,assume that the perturbation term G(R) = T ∗ (R) − T ∗0 (R) × R and that T (R; η) = T ∗ (R) +η × G(R) = T ∗ (R) + η[ T ∗ (R) − T ∗0 (R) × R]. This perturbation term has a special economic
meaning. This perturbation term is the increase of the tax burden due to the consumption tax.
When G(R) is defined in this way, we have ∂T= T ∗ (R)−T ∗0 ×R. From the first order condition∂ηof η , we have


1
∂s
∂l
rtks
0
∗
∗0
E[u (c2 ){T (R) − T × R}] = λ
{wtw
+
}
1+ρ
∂η
∂s 1 + r
∂l
∂e
r ∂e
∂I
+ {wtw+ E[T 0 (erI; η)er] − tI } + E[T 0 (erI; η)I
]
∂η
∂e ∂η
∂I

1
+E[
{T ∗ (R) − T ∗0 × R}]
1+r

(40)

Note that by the definition, R = reI. Therefore, we have
dW
∂s
∂s
∂l
rtks
∂I
∂I
∂l
T 0 (erI; η)er={+}{λwtw+}+{+}{λwtw+ λE[] − tI }dtc tc =0∂tc ∂η
∂s 1 + r∂tc ∂η
1+r
∂I
+{

∂e
∂e
T 0 (erI; η)I ∂e
r
µ−r
+
}{λE[
] + λ{}I∂tc ∂η
1 + r ∂e
1+r
26

(41)

∂s ∂I ∂e ∂s ∂I∂e
Now we need to calculate ∂t,, , and ∂η. The first order conditions of consumers forc ∂tc ∂tc ∂η ∂η

s, I and e for given tc are∂Ψ
1+
E[u0 (c2 ){1 + r(1 − tks )}] = 0
∂s
1+ρ
∂Ψ
1
+
E[u0 (c2 ){1 + re − T 0 re}] = 0
1+ρ
∂I
∂e
r
1
E[u0 (c2 ) {I − T 0 I}] = 0−v 0 (e) +1+ρ∂eNow, let SOC be the matrix that is derived from the second order condition. From the concavityof the objective function with respect to s, I and e, the determinant of SOC is strictly negative.
∂s ∂I∂eThen, ∂t,and ∂tare the solution of the following equations.
c ∂tc
c
 

  
1
∂s
E[u00 (c2 ){1 + r(1 − tks )}{T − T 0 reI}]
0
1+ρ
∂tc
 ∂I   1 E[u00 (c ){1 + re − T 0 re}{T − T 0 reI} − E[u0 (c )T 00 re2 I]   
SOC ×  ∂tc  +  1+ρ
2
2
= 0
2
∂e
r
r 00
1
0
E[u00 (c2 ) ∂e
{I − T 0 I}{T − T 0 reI} − E[u0 (c2 ) ∂e
T reI
∂tc
1+ρ
∂e
∂e
(42)
∂s ∂I
∂e
As for ∂η, ∂η and ∂η, we have ∂s 



  
1
00
−
E[u
(c
){1
+
r(1
−
t
)}T
]
2
ks
η
0
∂η
1+ρ
 ∂I   − 1 E[u00 (c ){1 + re − T 0 re}T ] − E[u0 (c ) ∂T 0 re]   
SOC ×  ∂η  +  1+ρ
2
η
2 ∂η
= 0
∂T 0 ∂e
1
∂e
r
r
∂e
0
0
00
0
− 1+ρ E[u (c2 ) ∂e {I − T I}Tη − E[u (c2 ) ∂η ∂e I]∂η0η]Note that Tη = T − T 0 reI and that ∂T= ∂[T= −T 00 R. Therefore,∂η∂R ∂s  1− 1+ρ
E[u00 (c2 ){1 + r(1 − tks )}{T − T 0 reI}]∂η ∂I  1− 1+ρ
E[u00 (c2 ){er − T 0 re}{T − T 0 reI} + [u0 (c2 )T 00 re2 I]
SOC ×  ∂η  + 

∂e
∂η




0
  
= 0
2
1
∂e
r
∂e
r 00
00
0
0
0
0
− 1+ρ E[u (c2 ) ∂e {I − T I}{T − T reI} + E[u (c2 ) ∂e T reI ]
(43)

∂s
∂s ∂I
∂I
∂e
∂e
= − ∂η
, ∂tc = − ∂ηand ∂t= − ∂η
Thus, dW/dtc is
From (42) and (43), we have ∂tccdWµ(e) − r= λ{}I
dtc tc =0
1+r
27



From the assumption, we have µ(e) > r. Therefore, introducing the consumption tax is welfareimproving.

References[1] Atkinson, A. and Joseph E. Stiglitz Lectures on Public Economics , McGraw Hill, 1980.
[2] Atkinson, A. and Stiglitz, Joseph E., “The Design of Tax Structure: Direct versus Indirect
Taxation,” Journal of Public Economics 6, July-Aug. 1976, pp 55-75.
[3] Boadway, Robin, Maurice Marchand and Pierre Pestieau, “Towards a Theory of the DirectIndirect Tax Mix,” Journal of Public Economics vol55, September 1994, pp71-88[4] Brian K. Bucks, Arthur B. Kennickell, and Kevin B. Moore ”Recent Changes in U.S. Family Finances: Evidence from the 2001 and 2004 Survey of Consumer Finances”, Federal
Reserve Bulletin, vol. 92 (February 2006), pp. A1-A38.
[5] Christiansen, Vidar“ Which Commodity Taxes Should Supplement the Income Tax?” Journal of Public Economics v24, n2 ,July 1984,pp 195-220[6] Diamond, Peter, Taxation, Incomplete Markets, and Social Security, Cambridge,MA, MIT
Press, 2003
[7] Eaton, Jonathan and Harvey Rosen, ”Labor Supply, Uncertainty, and Efficient Taxation”,
Journal of Public Economics v14, n3 (December 1980)28

[8] Hubbard, R. Glenn and Judd, Kenneth L. Title “Liquidity Constraints, Fiscal Policy, andConsumption”, Brookings Papers on Economic Activity v0, n1 (1986): 1-50
[9] Juan Carlos Conesa and Dirk Krueger ”On the Optimal Progressivity of the Income Tax
Code”, Journal of Monetary Economics, Vol. 53(7), pp. 1425-1450[10] Kopczuk, Wojciech ” The Trick Is to Live: Is the Estate Tax Social Security for the Rich?”,Journal of Political Economy v111, n6 (December 2003), pp1318-41[11] Summers, H. Lawrence ”Capital Taxation and Accumulation in a Life Cycle Growth
Model”, American Economic Review v71, n4 (September 1981), pp533-44[12] Wolf, Edward N. Top Heavy: The increasing inequality of Wealth in America and WhatCan be Done About It, The New Press, New York, United States.

29