Working Paper Series

Congressional Budget Office

Washington, D.C.

The Welfare Cost of Capital Taxation:

An Asset Market Approach

Marika Santoro

Macroeconomic Analysis Division

Congressional Budget Office

(marika.santoro@cbo.gov)

Chao Wei

Department of Economics

George Washington University

(cdwei@gwu.edu)

August 2011

Working Paper 2011-03

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

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

them are those of the authors and should not be interpreted as those of the Congressional Budget Office. Citations in 

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

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

We would like to thank Michael Golosov, Jonathan Huntley, Deborah Lucas, Larry Ozanne, William Randolph, and 

conference participants at the 2011 annual meeting of the American Economic Association for their helpful 

comments and suggestions. All remaining errors are our own. Chao Wei was a visiting fellow at the Congressional 

Budget Office while working on this project.
Abstract

We use an asset pricing perspective to provide a novel interpretation of

the marginal welfare cost of capital income taxes. We show that the marginal

welfare cost can be interpreted as the normalized present discounted value

of consumption distortions from capital income taxes. Such an interpretation emphasizes the importance of the discount rate used to value future

consumption distortions, especially in the presence of uncertainty. We Önd

that the discount rate decreases as the capital income tax rate increases, thus

increasing the welfare cost of taxes. The variations in the discount rate are

caused by the ampliÖed responses of consumption to exogenous shocks as

a result of capital income taxation. We Önd that the welfare cost may be

underestimated if variations in risky discount rates are ignored, especially

when tax rates are high.

1
1 Introduction

This paper studies the determinants of the marginal welfare cost of capital

income taxes in a stochastic production economy. Starting from a balancedgrowth equilibrium consistent with an arbitrarily given capital income tax

rate, we examine the welfare cost of a permanent shift to a marginally higher

tax rate, under the assumption of a lump-sum full rebate of tax revenues.1

We measure the welfare cost of taxes as the compensation required to make

the representative household indi§erent between consumption plans with and

without the marginal shift in tax rates. We relate this measure of welfare cost

to the market value of a security that is a claim to consumption distortions

resulting from the marginal shift in tax rates. Such an interpretation brings

to the forefront the importance of the discount rate used to value future

consumption distortions. We Önd that the discount rate decreases as the

capital income tax rate increases, thus increasing the welfare cost of taxes.

The variations in the discount rate are caused by ampliÖed responses of

consumption to exogenous shocks as a result of capital income taxation.

We Önd that the welfare cost may be underestimated if variations in risky

discount rates are ignored, especially when tax rates are high.

Interpreting the marginal welfare cost as the market value of a security

allows us to examine the welfare cost from an asset market perspective. Just

like prices of any other risky securities, the marginal welfare cost of taxes is

determined by three factors: the stream of consumption distortions caused

by ine¢ cient allocation of resources, the discount rate used to discount the

stream of consumption distortions, and the covariance between consumption

distortions and systematic risk.

We study the welfare cost of taxes in a general equilibrium production

economy with varying degrees of uncertainty. We show that the marginal

welfare cost of capital income taxes increases with the tax rate in both deterministic and stochastic environments. However, the marginal welfare cost

curve is higher and steeper in the stochastic case than in the deterministic

case. In the deterministic case, the upward slope of the marginal welfare cost

curve is mostly driven by increasing consumption distortions as the tax rate

increases. In the stochastic case, however, variations in the discount rate

and in the covariance between consumption distortions and systematic risk

1We assume lump-sum tax rebates to distinguish between the ine¢ ciencies resulting

from distorted investment decisions (substitution e§ects) and the ine¢ ciencies resulting

from the use of tax revenues (income e§ects).

2
also play important roles. We denote consumption distortions, variations

in the discount rate and variations in the covariance term as the distortion,

discounting and insurance e§ects of capital income taxes, respectively.

The discounting and insurance e§ects that capture the variations in the

discount rate and in the covariance term are unique for a stochastic environment. We Önd that as the tax rate increases, the discount rate used to

value future consumption distortions tends to decrease, thus raising the welfare cost. Santoro and Wei (2011) show that capital income taxes can lead

to ampliÖed responses of the marginal utility of consumption to exogenous

shocks. Those ampliÖed responses are the main reason behind the decrease

in the discount rate as the tax rate increases. At the same time, those ampliÖed responses alter the magnitude of the covariance term. The covariance

term becomes increasingly negative as the capital income tax rate rises, thus

mitigating the marginal welfare cost. We Önd that the discounting e§ect

increasingly dominates the insurance e§ect as the capital income tax rate

rises.

Since the discounting and insurance e§ects can only a§ect the shape of the

marginal welfare cost in the stochastic environment, the increasingly dominant impact of the declining discounting e§ect leads to a marginal welfare

cost curve not only steeper than but also above that in a deterministic environment. We Önd that as the degree of uncertainty increases, the gap between

the marginal welfare cost curves in deterministic and stochastic economies

widens, reáecting the strength of the discounting e§ect as the degree of uncertainty increases.

In addition to the degree of aggregate uncertainty, the marginal welfare

cost of capital income taxes depends on the preference and production speciÖcations. We need a production economy, in which both consumption and

investment are endogenously determined, to study the distortionary e§ect of

capital income taxes. Since we use an asset market approach to price consumption distortions, it seems important to have a production-based model

that is not only able to mimic some basic asset pricing features but also

tractable enough to make transparent the mechanisms introduced by capital

income taxes. Jermann (1998) and Boldrin, Christiano and Fisher (2001)

feature such kind of models. They show that the key ingredients for such a

model are habit formation in preferences and adjustment costs in production

technology. Based on this consideration, we assume moderately high habit

persistence and capital adjustment costs in the benchmark calibration.

We also conduct a sensitivity analysis in the same stochastic environment

3
but with neither habit persistence nor capital adjustment costs2

. Although

the marginal rates of substitution are less volatile, we Önd that both the

discounting and the insurance e§ects are still present in such a setting, albeit

at a lower magnitude. The discounting e§ect dominates the insurance e§ect

as the tax rate increases, just as in the benchmark case, resulting in a higher

welfare cost in the stochastic environment.

Our quantitative Öndings can be related to those of Chamley (1981) and

Lucas (1990), which use a deterministic dynamic general equilibrium model

to evaluate the welfare gain obtained by abolishing the capital income tax.

According to Lucas (2003), ìthe overall welfare gains amount to perhaps 2 to

4 percent of annual consumption, in perpetuity.îSince tax reforms typically

involve discrete changes in tax rates rather than abolition of a tax, we focus

speciÖcally on the welfare cost of a marginal shift in the capital income tax

rate, and integrate the marginal welfare cost over the given range of tax rates

to compute the total gain from discrete changes in tax rates. Our calculation

yields an overall welfare gain from abolishing capital income taxes that is

at the high end of Lucasís estimate. The impact of capital income taxes on

the discount rate, which is absent in a deterministic setting, contributes to

the higher welfare cost of capital income taxes in the presence of aggregate

uncertainty.

Judd (1987) examines the marginal e¢ ciency cost of various factor taxes

in a deterministic model. He states that ìany biases of the deterministic

approach relative to a more realistic model with uncertainty must arise from

decreasing returns in capital intensity and third-order properties of utility

functions.îDecreasing returns to capital and third-order properties of utility

functions are important for capital income taxes to have a strong e§ect on the

discount rate. It is the omission of the possible e§ect of capital income taxes

on the discount rate that biases the estimate of the deterministic approach.

Our results advance the insights gained from Gordon and Wilson (1989),

which examines the marginal welfare loss of capital taxation in a stochastic

production economy similar to ours.3 They argue that past measures that

2The risk-free interest rate can be overly volatile in models such as Jermann (1998)

and Boldrin, Christiano and Fisher (2001). One purpose of the sensitivity analysis is to

examine the robustness of our mechanism when the marginal rates of substitution, and

also the risk-free rates, are less volatile.

3Bulow and Summers (1984) and Gordon (1985) also study the welfare cost of taxing

risky capital income. An important limitation of their work is that they both employ a

two-period framework, which alters the risk characteristics of any long-lived securities.

4
ignore the negative covariance between consumption distortions and the stochastic discount factor ìlikely overstate the e¢ ciency costs of a rise in the tax

rate, perhaps dramatically.îThe negative covariance stressed by Gordon and

Wilson (1989) is also present in our framework. However, since Gordon and

Wilson (1989) only examine the marginal welfare loss at a single tax rate,

they do not study the declines in the discount rate accompanied by increases

in the capital income tax rates. We Önd that the declines in the discount

rate are signiÖcant enough to dominate the increasingly negative covariance

as the tax rate increases.

We also relate our Öndings to the literature that uses an asset pricing approach to study welfare and budgetary issues. Alvarez and Jermann (2004)

use an asset market measure to study the welfare cost of consumption áuctuations. Geanakoplos and Zeldes (2010) highlight the importance of proper

discounting in deriving the market value of Social Security claims.

We organize the paper as follows. In Section 2, we derive our measure

of the marginal welfare cost of capital income taxes and describe its asset

pricing interpretation. In Section 3, we use a stylized model as a laboratory to

examine the properties of the marginal welfare cost curves. In Section 4, we

decompose the marginal welfare cost to analyze the distortion, discounting

and insurance e§ects separately. Section 5 describes the sensitivity analysis.

Section 6 concludes.

2 Asset Pricing Interpretation of the

Marginal Welfare Cost

In this section, we deÖne the marginal cost of capital income taxes in a

stochastic economy and relate that cost to the normalized market value of

consumption distortions.

2.1 Measures of Welfare Cost

Let  be an arbitrarily given capital income tax rate. Assume that in each

period t; the history of events up to and including period t is denoted by

s

t = (s0; s1; :::st): Here st

is a vector that includes both exogenous and

endogenous variables which may depend upon  .

We consider an economy characterized by the given tax rate  and an

initial state s0: Let C

;0

t

	1

t=0 represent the consumption stream for a repre5
sentative household in an economy with the tax rate  at each period t: The

Örst superscript, ; stresses the dependence of consumption on the given tax

rate, and the second superscript, 0; indicates that the tax rate remains unchanged at : C;0

t

is a function of the history of events, s

t

. Now we consider

a permanent marginal change of " in the tax rate at period 0: After that,

the economy moves on a dynamic path from the initial state (s0) consistent

with a tax rate  to a long-run state compatible with the higher marginal tax

rate,  +". We use fC

;"

t g

1

t=0 to represent the alternative consumption stream

along this transition path resulting from the marginal tax rate increase. Here

the Örst superscript, ; stresses that the initial state of the economy is consistent with the tax rate  , and the second superscript, "; indicates that the

tax rate shifts up from  by ":

We measure the welfare cost of this marginal tax change in terms of the

compensation that renders the representative household indi§erent between

the consumption streams with or without the permanent marginal increase

in the tax rate. This compensation is measured as a fraction of consumption

for each period in the economy with the marginally higher tax rate.

Let  be the fraction that will serve as a compensating consumption

supplement, and deÖne the indirect utility function V for the representative

household by:

V (; "j ) = E0U (f(1 + ) C

;"

t g

1

t=0):

where U () represents lifetime utility from any consumption stream.

Here V (; "j ) is interpreted as the utility the consumer would enjoy from

the alternative consumption stream fC

;"

t g

1

t=0 and a consumption supplement

C;"

t

for each period:

We deÖne the unique, positive value of  that satisÖes the condition:

V (; "j ) = V (0; 0j ); (1)

as the welfare cost of raising the capital income tax rate from  to  +". This

measure reáects the welfare cost of the marginal increase in the tax rate in

the long run, as well as during the transition period. Here  is a function of

both " and :

Using the implicit function theorem, we can derive the derivative of  (";  )

with respect to " :

" (";  )j"=0 = 

V" (0; 0j )

V (0; 0j )

: (2)

6
The welfare cost of a permanent discrete change from  to  + " can be

approximated locally as4

 (";  ) ' "" (";  )j"=0: (3)

Here " (";  ) is the marginal welfare cost of a marginal tax rate increase

of " in an economy characterized by an initial capital income tax rate of :

We show in the next section that this traditional measure can be interpreted

as a normalized net present value.

2.2 Asset Pricing Interpretation

Assume that U () takes the following form of expected utility,

U (fCtg

1

t=0) = E0

X1

t=0

h



t

u



Ct; fCtjg

N

j=1i ; (4)

where u



Ct; fCtjg

N

j=1

can accommodate both time-separable and nontime-separable preferences. We have the following representation of the marginal welfare cost of taxes, based on equation (2):

" (";  ) =

E0

P1

t=0 



t 

 ;0

t



 ;0

0

@(C

 ;"

t )

@" 

W

; (5)

where 

;0

t

is the derivative of the lifetime utility U



C

;0

t

	1

t=0

; as deÖned

in equation (4) with respect to C

;0

t

; and

W = E0

X1

t=0





t

;0

t



;0

0

C

;0

t



:

The marginal welfare cost of capital income taxes is now the ratio of the

values of two securities. The numerator represents the value of a security

that is a claim to the di§erences in the consumption streams (consumption

distortions) with the marginal increase in the capital income tax rate. The

4The complete approximation takes the form of  (;  ) '  (0;  )+ (;  ) : According

to our deÖnition,  (0;  ) is equal to zero.

7
denominator, instead, represents the value of a security that is a claim to

the consumption stream in an economy without the marginal increase in the

current tax rate  .

The marginal welfare cost of capital income taxes is now e§ectively the

normalized market value of consumption distortions, with the denominator

serving as the normalizing factor.

2.3 Comparison with the Lucas Method

Lucas (1990) uses Bernheimís (1981) formula to measure the welfare cost of

eliminating capital income taxes. That formula involves a weighted sum of

the percentage changes in consumption in the initial period and in the new

long-run state of the economy characterized by a higher marginal tax rate.

The weights are mainly determined by the subjective time preference and

the speed of convergence.

Our measure of the marginal cost of taxation can be written in a form

similar to Bernheimís formula:

" (";  ) =

E0

P1

t=0 



t 

 ;0

t



 ;0

0

@(C

 ;"

t )

@" 

W

(6)

= E0

X1

t=0



!t

@ (C

;"

t

) =@"

C

;0

t



; where

!t = 

t

;0

t



;0

0

C

;0

t

W

:

In an environment without uncertainty, the variables 

;0

t

, C

;0

t

; and C

 ;0

t

W

are constant for all t: As a result, the weight !t becomes 

t

P1

j=0 

j

: It is clear

that additional mechanisms other than the speed of convergence come into

play in the presence of uncertainty. The time-varying marginal utility of

consumption and its covariance with consumption distortions now play important roles in valuing future consumption distortions, which may lead to a

di§erent welfare cost of capital income taxes than previously accounted for

in a deterministic environment.

Since consumption distortions resulting from capital income taxes extend

into the inÖnite horizon, the rate at which those distortions are discounted

8
is important for the measure of welfare cost. As a result, the possible impact

of taxes on the marginal utility of consumption, and hence on the stochastic

discount rate, may play an important role in determining the welfare cost.

In a model without uncertainty, capital income taxes have no impact on the

discount rate, which is a constant depending only upon .

3 Valuing Consumption Distortions

In this section, we Örst describe a model that features a stochastic production

economy. We then use it as a laboratory to compute the market value of

consumption distortions as a way of measuring the marginal welfare cost of

capital income taxes.

3.1 A Stochastic Production Economy

In this section, we sketch the major elements of a stylized production economy. The details are contained in Appendix A.

There is a continuum of inÖnitely-lived identical households that own

a representative Örm. The government levies taxes on capital income and

rebates them back in a lump sum to the households. The economy grows at

a constant rate g:

The representative household maximizes its lifetime utility subject to a

standard budget constraint. The speciÖcation of the instantaneous utility

function, u(Ct; Ct1), allows for time-separable and non-time-separable preferences.

The Örm produces output, Yt; using the Cobb-Douglas production technology:

Yt = ZtK

t



(1 + g)

tLt

(1)

; (7)

where Lt represents labor, Ktis the capital stock, and the logarithm of the

stochastic productivity level, Zt; follows

zt = zt1 + t; t  N(0; 1): (8)

Here  is the persistence parameter and  indexes the degree of aggregate

uncertainty. The capital accumulation process is given by:

Kt+1 = (1  ) Kt + 	(Kt; It); (9)

9
where It represents investment. We specify the functional form of 	 (Kt

; It)

as

	 (Kt; It) = "

(g + )



1  



It

Kt

1

+

 (g + )

  1

#

Kt; (10)

where the capital supply is inelastic when  approaches inÖnity.5 The concavity of this function captures convex costs of adjustment.

In equilibrium, output is equal to the sum of consumption and investment:

Yt = Ct + It: (11)

All the asset markets clear. Given that leisure does not enter the utility

function, households will allocate their entire time endowment to productive

work. Labor supply is constant and normalized to 1: It is important to note

that since labor supply is inelastic in our stylized model, capital income taxes

are the only source of distortion.

The Örst-order condition for investment is:

1

	I (Kt; It)

= Et



t+1

t



(1   )

Yt+1

Kt+1

+

(1  ) + 	K (Kt+1; It+1)

	I (Kt+1; It+1)

 ;

(12)

where 

t+1

t

represents the stochastic discount factor and tis given by

t = u

0

(Ct  bCt1)  bEtu

0

(Ct+1  bCt):

Here b is the parameter that indexes the degree of habit persistence in

the non-time-separable preferences. 	I () and 	K () are the derivatives

of capital adjustment costs with respect to investment and capital stock.

Capital income taxes distort investment decisions by reducing the marginal

beneÖt of investment.

Proposition 1 The deterministic steady state of the economy, where Zt =

1; 8t; can be characterized as follows,

(a) The equilibrium capital stock, K

ss; is given by

K

ss =



(1   )

1  



(1  )

 1

1

;

5Capital adjustment costs have been studied by Eisner and Strotz (1963), Lucas (1967)

and others. The particular speciÖcation we adopt is similar to that of Jermann (1998).

10
where 



; which is equal to  (1 + g)




, is the trend-adjusted subjective time

preference.

(b) The equilibrium consumption, C



ss; is given by

C



ss = (K

ss)

  (g + ) K

ss:

(c) The equilibrium investment-output ratio, given by (g + ) (K

ss)

1

; is

monotonically decreasing in : The opposite is true for the equilibrium consumptionoutput ratio.

Proof. Equilibrium allocations are derived from the Örst-order condition

for investment (12) and the aggregate resource constraint (11).

3.2 The Marginal Welfare Cost

In this section, we use the stylized model described above as a laboratory

to compute the marginal welfare cost: We describe the benchmark parameterization in Table 1. Since our focus is on valuing stochastic consumption

distortions, it is important to have a model that can reproduce key asset

market statistics. Jermann (1998) shows that a combination of high habit

persistence and capital adjustment costs help to generate reasonable asset

return statistics in a model like ours. Based on this consideration, we introduce moderately high habit persistence and capital adjustment costs by

setting b equal to 0:6 and  equal to 4:24:

6

For any arbitrarily given ; we consider two scenarios starting from the

initial state s0 characterized by fK

ss; C

ss; Zssg ; which represent, respectively,

the steady-state values of capital stock and consumption given  and the

steady-state level of the aggregate productivity. In the Örst scenario, the

marginal tax rate remains unchanged at ; and under the alternative scenario,

there is a permanent marginal shift in the capital income tax rate from  to

 +" starting in the initial period. We assume the same aggregate technology

process, fZtg

1

t=0 ; with or without the shift in the marginal tax rate. In the

deterministic environment, Ztis equal to 1 for all t:

We apply equation (5) to compute the marginal welfare cost for all of

the tax rates. The details of the computation are contained in Appendix

6The value of b lies within the range widely used by the literature (see Cochrane and

Hansen, 1992). The value of  is similar to that used by Jermann (1998).

11
B. Figure 1 plots the marginal welfare cost, " (";  ); for a range of initial

tax rates  in both a deterministic ( = 0) and a stochastic ( = 0:016)

environment.7;8 These two environments are otherwise the same except for

the di§erences in : Here we approximate " with a value of 0:01:

Figure 1 displays three salient features about the marginal welfare cost

curves:

1. The marginal welfare cost curves are convex and upward sloping in the

tax rate in both the deterministic and stochastic cases.

2. The marginal welfare cost curve is higher and steeper in the stochastic

case than in the deterministic case.

3. The gap between the marginal welfare cost curves in the stochastic and

deterministic cases widens as the tax rate increases.

Since the total welfare cost of a discrete change in the tax rate is the

integral of the marginal welfare cost for the given range of tax rates, steeper

slopes imply higher total welfare costs. The gap between the two marginal

welfare cost curves can thus be translated into the di§erence in the total

welfare gain of eliminating capital income taxes by integration.

We Önd that the total welfare gain of eliminating capital income taxes

with  = 0:36 is equivalent to 3:47 percent of lifetime consumption in the

deterministic case, but is equal to 4:51 percent in the stochastic case. The

part of total welfare gain attributed to uncertainty amounts to almost one

third of the total welfare gain in the deterministic case. Ignoring uncertainty

may lead to an underestimation of the welfare cost of capital income taxes.

In a deterministic environment, the upward slope of the marginal welfare

cost curve is mostly driven by increasing consumption distortions as the

tax rate increases. In a stochastic environment, however, variations in both

the stochastic discount factor and the covariances between the stochastic

discount factor and consumption distortions also play important roles.

7McGrattan and Prescott (2005) estimate the corporate tax rate to be 0:43 for periods

of high tax rates: For illustration, we pick this as one of the extreme values that the capital

income tax rate can take. We use an intermediate tax rate of 0:36 to relate our measure

of welfare cost to Lucas (1990).

8We set the standard deviation of the aggegate productivity shock at 0:016 in accord

with Bloom, Floetotto and Jaimovich (2010).

12
In the next section, we decompose the marginal welfare cost and distinguish the impact of variations in the discount rates from those in the

covariance terms.

4 Decomposition of the Marginal Welfare Cost

In this section, we decompose the numerator of the marginal welfare cost to

shed light on its determinants in a stochastic environment. The marginal

welfare cost of taxes is decomposed as follows:

" (";  ) =

P1

t=0 

E0





t 

 ;0

t



 ;0

0



E0



@(C

 ;"

t )

@"  +

P1

t=0 cov0





t 

 ;0

t



 ;0

0

;

@(C

 ;"

t )

@" 

W

(13)

In the decomposition, the numerator of equation (5), E0

P1

t=0 



t 

 ;0

t



 ;0

0

@(C

 ;"

t )

@" 

;

is decomposed into two elements. In the Örst element, the product of E0



@(C

 ;"

t )

@" 

and E0





t 

 ;0

t



 ;0

0



represents expected consumption distortions at period t valued at the price of the t-th period discount bond. The second element captures the covariance between the stochastic discount factor and consumption

distortions. For future reference, we deÖne

 =

P1

t=0 

E0





t 

 ;0

t



 ;0

0



E0



@(C

 ;"

t )

@" 

W

; 
 =

P1

t=0 cov0





t 

 ;0

t



 ;0

0

;

@(C

 ;"

t )

@" 

W

:

(14)

The decomposition reveals multiple forces determining the marginal welfare cost in the presence of uncertainty. In  , E0



@(C

 ;"

t )

@" 

represents the

expected consumption distortion that varies with the tax rate. We label this

the distortion e§ect of capital income taxes. A corresponding term is also

present in a deterministic environment.

Also in  , E0





t 

 ;0

t



 ;0

0



is the price of a tth period discount bond, which

determines the term structure used to value future consumption distortions.

It is important to note that in a stochastic environment the discount rate not

13
only varies with time, but also depends upon the capital income tax rate,  .

We call that the discounting e§ect of capital income taxes.

The presence of uncertainty brings in another e§ect of capital income

taxes on the marginal welfare cost, which is reáected in cov0





t 

 ;0

t



 ;0

0

;

@(C

 ;"

t )

@" 

:

This term denotes a covariance between the stochastic discount factor and

consumption distortions conditional on the information at time 0: A negative

covariance between the two implies that consumption distortions are large

in states when such distortions are valued less. Such desirable coincidence

reduces the welfare cost of taxes and acts as insurance in the presence of

uncertainty (see, for example, Gordon and Wilson, 1989). We call this e§ect

the insurance e§ect of capital income taxes.

In all, we identify three e§ects of capital income taxes on the marginal

welfare cost: the distortion, discounting and insurance e§ects. The distortion

e§ect is present in both stochastic and deterministic environments, while the

second two e§ects, which are closely related to second-order moments, are

present only in stochastic environments.

Since the latter two e§ects appear only in stochastic environments, the

di§erence between the marginal welfare cost curves in the stochastic and

deterministic environments reáects the strength of the discounting and insurance e§ects. In the next sections, we examine each of the three e§ects in

detail.

4.1 Distortion

To understand the distortion e§ect, it is useful to examine the path of consumption distortions for a particular tax rate. In Figure 2, we plot the percentage di§erences in the two consumption paths 

C

;0

t

	1

t=0 and fC

;"

t g

1

t=0 as

a fraction of C

;0

t

	1

t=0 for  = 0:36: As mentioned above, 

C

;0

t

	1

t=0 represents the consumption path under the Örst scenario, where the marginal tax

rate remains unchanged at ; while fC

;"

t g

1

t=0 represents the corresponding

path under the second scenario, where the tax rate shifts permanently to

 + " at period 0:

Although those two scenarios start at the same initial steady state fK0:36

ss ; C0:36

ss ; Zssg

and are subject to the same aggregate technology process, their consumption paths diverge as soon as the marginal tax change is implemented. The

resulting path of consumption distortions demonstrates the short-run and

long-run trade-o§s following a marginal shift in the tax rate, a point also

14
made by Bernheim (1981) and Lucas (1990).

Under the second scenario, consumption temporarily increases above what

it would be under the Örst scenario, resulting in negative consumption distortions. Increases in consumption in the initial periods accommodate the

convergence of the capital stock to a lower level consistent with a higher tax

rate in the long run. However, as time passes, distorted investment decisions

also lead to both lower capital stock and lower output. Eventually consumption declines and converges to a lower long-run equilibrium level relative to

that under the Örst scenario, where the tax rate remains unchanged at :

Consequently, the amount of consumption distortions áuctuates around a

positive value in the long run.

In Figure 2, we also plot a corresponding relative measure of consumption distortions in a deterministic economy. The same short-run and long-run

trade-o§ applies, and the amount of consumption distortions eventually converges to a positive value.

4.2 Discounting

The rate that is used to discount consumption distortions is a function of the

stochastic discount factor, 



 ;0

t



 ;0

0

; which in turn depends upon the given capital income tax rate  . The discount rate plays a crucial role in determining

the welfare cost of taxes. In this section, we examine how the discount rate

varies across di§erent capital income tax rates and determines the welfare

cost of taxes together with other forces.

In order to isolate the e§ect of the stochastic discount factor on the marginal welfare cost, we compare two methods of valuing consumption distortions by discounting the same path of consumption distortions di§erently. In

the Örst method, we use ; the discount factor in the deterministic case, while

in the second method, we use the stochastic discount factor 



 ;0

t



 ;0

0

; which reáects consumption áuctuations across di§erent states contingent upon the

tax rate :

The upper left panel of Figure 3 plots the present discounted value of

strips of deterministic consumption distortions, @(C

 ;"

t )

@" j=0; at  = 0:36 using the above two methods. The horizontal axis denotes the time period for

the strip, and the vertical axis denotes the market value of the corresponding strip at period 0: The market value of the n-th strip computed with the

15
Örst method, 

t

@(C

 ;"

t )

@" j=0; stays below the value computed with the second method, E0





t 

 ;0

t



 ;0

0

@(C

 ;"

t )

@" j=0

for almost all the time periods. This

indicates that the values of discount bonds in the stochastic environment are

higher than those in the deterministic case. We plot the di§erences between



t (C

 ;"

t )

@" j=0 and E0





t 

 ;0

t



 ;0

0

(C

 ;"

t )

@" j=0

in the upper right panel of Figure 3.

This di§erence measures the strength of the discounting e§ect resulting from

the presence of uncertainty. In the left panel of Figure 4, we plot the di§erences between those two terms for three di§erent tax rates: 0:001; 0:13; 0:36:

As the tax rate increases, the discounting e§ect becomes stronger.9

The intuition for the increasing discounting e§ect is as follows. As the

capital income tax rate increases, investment is further discouraged. As

a result, investment constitutes an increasingly smaller share of aggregate

output in the steady state. A lower investment-output ratio implies that

investment needs to respond more to exogenous shocks in order to smooth

consumption, which now constitutes a larger fraction of output. Santoro and

Wei (2011) show that such a mechanism can be responsible for amplifying

the responses of consumption, and consequently of the marginal utility of

consumption, ; to aggregate technology shocks in an economy with higher

tax rates. The ampliÖed responses of the marginal utility of consumption

typically lead to precautionary motives, which tend to reduce the discount

rate used to value future consumption distortions.

4.3 Insurance

If the amount of consumption distortions is procyclical, the negative correlation between distortions and the stochastic discount factor will likely reduce

the present discounted value of consumption distortions. This desirable correlation may alleviate the marginal welfare cost of capital income taxes.

The lower left panel of Figure 3 plots E0

h



t 

 ;0

t



 ;0

0

i

E0



@(C

 ;"

t )

@" 

and E0





t 

 ;0

t



 ;0

0

@(C

 ;"

t )

@" 

for all periods when  is equal to 0:36: As shown in equation (13), the di§erence between the two variables represents cov0





t 

 ;0

t



 ;0

0

;

@(C

 ;"

t )

@" 

; the covari9The humped shape of the curves is explained by both the rising and then stabilizing

of consumption distortions and the decaying value of the discount factor as t increases.

16
ance between the stochastic discount factor and consumption distortions for

all time periods. The lower right panel of Figure 3 plots this di§erence. The

di§erence remains largely negative. Such a negative covariance implies that

consumption distortions are higher in states when they are valued less, reáecting the insurance e§ect of capital income taxes. This desirable property

would reduce the welfare cost of taxes in the presence of uncertainty.

In the right panel of Figure 4, we plot cov0





t 

 ;0

t



 ;0

0

;

@(C

 ;"

t )

@" 

for all time

periods with a tax rate of 0:001; 0:13 or 0:36: As the tax rate increases, the

covariance becomes more negative, reáecting a stronger insurance e§ect. The

increasingly negative covariance can be explained by both ampliÖed responses

of the marginal utility of consumption and higher consumption distortions

as the tax rate increases.

4.4 Distortion, Discounting and Insurance Across Tax

Rates

In this section, we use the decomposition analysis above to examine the

gap between the marginal welfare cost curves in stochastic and deterministic

environments shown in Figure 1. We follow the decomposition equation (13)

and plot the two decomposed elements,  and 
 . The sum of the two

elements represents the marginal welfare cost for each tax rate.

The left panel of Figure 5 plots  for both the deterministic ( = 0) and

stochastic ( = 0:016) cases for all tax rates. The curve  corresponding

to the stochastic case stays above that of the deterministic case. The fact

that  is increasing even without uncertainty shows that the distortion from

capital income taxes increases with the tax rate. Since the distortion e§ect

on consumption is present in both the deterministic and stochastic cases, and

the discounting e§ect is present only in the presence of uncertainty, the gap

between the curves across di§erent degrees of uncertainty mainly reáects the

importance of the discounting e§ect.10 The widening gap between the two

curves reáects stronger discounting e§ects as the tax rate increases.

The right panel of Figure 5 plots 
 for both the deterministic ( = 0) and

stochastic ( = 0:016) cases for all tax rates. The curve 
 corresponding to

10Our simulation shows that the di§erence between the paths of the deterministic consumption distortions and the expected path of the stochastic consumption distortions is

fairly small.

17
the stochastic case is increasingly negative as the tax rate increases, whereas

it is equal to zero in the deterministic case.

To further demonstrate the e§ect of capital income taxes on the discount

rate as the tax rate increases, we plot E0

P1

t=0 h



t 

 ;0

t



 ;0

0

i

; which is e§ectively

the market value of a perpetual bond (the sum of the values of all t-th period

discount bonds) and its reciprocal, which is equal to the long-term risk-free

interest rate. As shown in Figure 6, the price of the perpetual bond increases,

and the long-term risk-free discount rate declines as the tax rate increases.

As the capital income tax rate increases, the decline in the discount rate

used to value consumption distortions raises the marginal welfare cost of

taxes, while the increasingly negative covariance between consumption distortions and the stochastic discount factor reduces it. Two opposite forces are

at work. We Önd that the discounting e§ect from declines in the discount

rate increasingly dominates the insurance e§ect as the tax rate increases.

This dominance explains the widening gap between the marginal welfare

cost curves in the stochastic and deterministic environments, as displayed in

Figure 1.

5 Sensitivity Analysis

In this section, we conduct a sensitivity analysis to examine the impact that

preference and production speciÖcations have on the marginal welfare cost.

We use the same economic environment but assume that there is neither

habit persistence nor capital adjustment costs by setting both b and  to 0.

Figure 7 plots the marginal welfare cost curves for both the stochastic and

deterministic environments under this speciÖcation. The marginal welfare

cost curve in the stochastic environment stays above that in the deterministic

environment for medium-high and high tax rates and has a steeper slope

throughout. A decomposition of the marginal welfare curves into  and 
 ,

as shown in Figure 8, indicates that both the discounting and the insurance

e§ects are present, albeit at a smaller magnitude. The discounting e§ect is

weaker relative to the insurance e§ect for relatively low tax rates. However, as

the tax rate increases, the discounting e§ect dominates the insurance e§ect,

just as in the benchmark case.

We compute the total welfare gain from eliminating capital income taxes

at the rate of  = 0:36 just as in the benchmark framework. The total

welfare gain is now 4:62 percent of perpetual consumption in the stochastic

18
environment, as opposed to 4:51 percent in the deterministic environment.

The part of the total welfare gain attributed to uncertainty amounts to only

2:5 percent of the total welfare gain in the deterministic case.

As shown in Santoro and Wei (2011), as long as investment plays a major

role in smoothing consumption and the investment-output ratio declines as

tax rate increase, there will be discounting and insurance e§ects at work.11

The strong discounting e§ect in the benchmark framework is due to the

presence of both habit formation and capital adjustment costs. This is consistent with Santoro and Wei (2011), which Önds that the impact of capital

income taxes on asset prices is the strongest for this combination of preference

and production speciÖcations.12

6 Conclusion

In this paper, we employ an asset market approach to analyze the determinants of the marginal welfare cost of capital income taxes. We show that

the marginal welfare cost can be thought of as the normalized market value

of consumption distortions. Such a perspective brings to the forefront the

importance of the discount factor and its covariance with consumption distortions in valuing future consumption distortions.

We Önd that the marginal welfare cost curve is convex in and upward

sloping with the tax rate in both stochastic and deterministic environments.

Everything held equal, the marginal welfare cost curve in the stochastic environment lies above and is steeper than that in the deterministic case. A

decomposition of the marginal welfare cost shows that such features can be

explained by three e§ects of capital income taxes: the distortion, discounting

and insurance e§ects. The upward slope of the marginal welfare cost curve

results from the increasingly ine¢ cient allocation of resources as the tax rate

11We also investigate the discounting and insurance e§ects in the case of elastic labor

supply. If labor elasticity is not very high in response to technology shocks (as indicated

by the empirical literature), the self-insurance through adjustment of labor supply would

be limited, and investment would remain the main vehicle to smooth consumption. As a

result, the discounting e§ect still dominates, especially when tax rates are high.

12We also conduct a sensitivity analysis using di§erent values of : We Önd that the

discounting e§ect strengthens as the degree of aggregate uncertainty increases. Detailed

results are available from the authors upon request.

19
increases, which represents the distortion e§ect common to deterministic and

stochastic environments. The widening gap between the marginal welfare

cost curves between the stochastic and deterministic cases is attributed to

the dominance of the discounting e§ect over the insurance e§ect. The magnitudes of these two e§ects are determined, respectively, by the decrease in the

discount rate and the increasingly negative covariance between the stochastic

discount factor and consumption distortions as the tax rate increases. We

Önd that the welfare cost of capital income taxes can be under-estimated if

variations in risky discount rates are ignored, especially when tax rates are

high.

20
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23
Appendix

A. Detailed SpeciÖcations of the Stylized Model

The representative household maximizes an expected life-time utility function subject to a standard budget constraint.

max

Bt+1;Ct

E0

X1

t=0



t

(Ct  bCt1)

1


1  


subject to:

Ct + It + Bt+1V

B

t = (1   ) RtKt + WtLt + Bt + t

:

Here  is the subjective time preference and Ctis real consumption at time

t. The coe¢ cient 
 measures the curvature of the representative householdís

utility function with respect to its argument Ct  bCt1: When b > 0; the

utility function allows for habit persistence based on the householdís own

consumption in the previous period.

In the budget constraint, It represents investment. Bt represents private

risk-free bonds held from period t  1 to t: V B

t

is the price of the risk-free

bond, and  is the proportional capital income tax rate on capital income

RtKt. The variable WtLt represents labor compensation. The variable tis

a lump-sum transfer of all the tax revenues from the government.

We assume that there exists a representative Örm owned by households.

The representative Örm chooses optimal capital and labor inputs given the

wage and rental rate at each period. In equilibrium, the real wage and the

gross rental rate are given, respectively, by

Wt = (1  )

Yt

Lt

:

Rt = 

Yt

Kt

:

24
B. Computing the Marginal Welfare Cost at 

We assume that the economy starts from the deterministic steady state

consistent with a given tax rate : Based on our stylized model, the initial

state can be characterized by fK

ss; Css; Zssg ; which represent, respectively,

the steady-state values of capital stock and consumption given  and the

steady-state level of aggregate productivity.

Now we consider two scenarios starting from the initial state. In the

Örst scenario, the marginal tax rate remains unchanged at : We denote the

optimal decisions under such a scenario as

Kt+1 = g

;0

(Kt; Ct1; Zt); Ct = h

;0

(Kt; Ct1; Zt)

where g

;0

() and h

;0

() represent respectively the optimal decision rules

for next period capital stock and current consumption. The Örst superscript,

; stresses the dependence of those decision rules on the given tax rate, and

the second superscript, 0; indicates that the tax rate remains unchanged at

: A dynamic path under this scenario is characterized by the initial state,

fK

ss; Css; Zssg ; and the optimal decision rules described above.

Now consider the alternative scenario, under which there is a marginal

shift in the capital income tax rate from  to  + " in the initial period.

Under this scenario, the transition path to the new long-run equilibrium is

characterized by the initial state fK

ss; C

ss; Zssg and the optimal decision

rules dictated by g

;" (Kt

; Ct1; Zt) and h

;" (Kt

; Ct1; Zt) for all t. Here the

Örst superscript  stresses that the initial state of the economy is consistent with the tax rate  , and the second subscript stresses that the optimal

decision rules are derived under the marginally higher tax rate,  + ":

We assume the same aggregate technology process fZtg

1

t=0 with or without the marginal shift in the tax rate. In computing the marginal welfare

cost, we approximate " with a small positive value of 0:01: Consequently, the

consumption distortion at period t;

@(C

 ;"

t )

@" ; is approximated by C

 ;0

t C

 ;0:01

t

0:01

for all : Given the approximated values of consumption distortions, it is

straightforward to use Monte-Carlo methods to compute the marginal welfare cost using equation (5).

25
Table 1: Benchmark Parameterization

Category Parameter DeÖnition Notation Value

Production trend growth rate g 0:005

depreciation rate  0:03

share of capital income  0:36

Preferences subjective time preference  (1 + g)

1


0:992

curvature of utility function 
 5

Technology Process persistence  0:995

standard deviation  0:016

26
Figure 1: The Marginal Welfare Cost Curves

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

tax rate (τ)

Marginal Welfare Cost

σ = 0.016

σ = 0

27
Figure 2: Paths of Consumption Distortions ( = 0:36)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 -0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Periods (quarters)

Percentage Change in Consumption

deterministic

stochastic

28
Figure 3: AmpliÖcation and Insurance E§ects ( = 0:36)

50 100 150 200 250 300 350 400 450 500 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

value of distortions

50 100 150 200 250 300 350 400 450 500

-0.05

0

0.05

0.1

amplification effect

50 100 150 200 250 300 350 400 450 500 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

time periods (quarters)

value of distortions

50 100 150 200 250 300 350 400 450 500

-0.05

0

0.05

0.1

time periods (quarters)

insurance effect

deterministic

determinisit with stochastic discount

stochastic no insurance

stochastic

29
Figure 4: Discounting and Insurance E§ects for Di§erent Tax Rates

50 100 150 200 250 300 350 400 450 500

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time periods (quarters)

discounting effect

Value of Distortions

50 100 150 200 250 300 350 400 450 500

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time periods (quarters)

insurance effect

τ = 0.001

τ = 0.13

τ = 0.36

τ = 0.001

τ = 0.13

τ = 0.36

30
Figure 5: The Decomposition of the Marginal Welfare Cost Curves

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

tax rate (τ)

Φτ

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

tax rate (τ)

Ωτ

σ = 0.016

σ = 0

σ = 0.016

σ = 0

31
Figure 6: Prices of Perpetual Bonds and Discount Rates for Di§erent Tax

Rates

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 185

186

187

188

189

190

191

192

193

194

tax rate (τ)

price of perpetual bond

stochastic environment

σ = 0.016

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 2.07

2.08

2.09

2.1

2.11

2.12

2.13

2.14

2.15

2.16

tax rate (τ)

long-term discount rate

stochastic environment

σ = 0.016

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 128

128.5

129

129.5

130

130.5

131

tax rate (τ)

price of perpetual bond

deterministic environment

σ = 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 3.06

3.07

3.08

3.09

3.1

3.11

tax rate (τ)

long-term discount rate

deterministic environment

σ = 0

32
Figure 7: The Marginal Welfare Cost Curves (b = 0,  = 0)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

tax rate (τ)

Marginal Welfare Cost

σ = 0

σ = 0.016

33
Figure 8: The Decomposition of the Marginal Welfare Cost Curves (b = 0,

 = 0)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

tax rate (τ)

Φτ

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

tax rate (τ)

Ωτ

σ = 0

σ = 0

σ = 0.016

σ = 0.016

34