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Catching up with the Keynesians
Lars Ljungqvist and Harald Uhlig
Working Papers Series
Macroeconomic Issues
Research Departm ent
Federal Reserve Bank of Chicago
October (W P -96-15)
■...»
M m Hn
FEDERAL RESERVE B A N K
OF CHICAGO
Catching u p
with
the Keynesians*
Laxs Ljungqvist
Re s e a r c h D e p a r t m e n t
Federal Reserve B a n k of C h i c a g o
Chicago, IL 6 0 6 9 0
USA
ljimg@frbchi.org
and
H a r a l d Uhlig
C e n t E R for E c o n o m i c R e s e a r c h
Tilburg University
Postbus 90153
5 0 0 0 L E Tilburg
THE NETHERLANDS
e-mail: uhlig@kub.nl
September 1996
*The views expressed herein are those of the authors and not necessarily those of the
Federal Reserve Bank of Chicago or the Federal Reserve System.
A bstract
This paper examines the role for tax policies in productivity-shock
driven economies with “catching-up-with-the-Joneses” utility func
tions. The optimal tax policy is shown to affect the economy countercyclically via procyclical taxes, i.e., “cooling down” the economy with
higher taxes when it is “overheating” in booms and “stimulating”
the economy with lower taxes in recessions to keep consumption up.
Thus, models with catching-up-with-the-Joneses utility functions call
for traditional Keynesian demand management policies. Parameter
values from Campbell and Cochrane (1995) are also used to illustrate
that the necessary labor taxes can be very high, in the order of 50
percent. However, Campbell and Cochrane’s nonlinear version of the
aspiration level in the catching-up-with-the-Joneses preferences has
the additional implication that consumption bunching can be welfare
enhancing.
1
Introduction
Envy is one important motive of human behavior. In macroeconomics, the
ories built on envy have been used in trying to explain the equity premium
puzzle as described by Mehra and Prescott (1985). Abel (1990) and most re
cently Campbell and Cochrane (1995) postulate utilityfunctions exhibiting a
desire to “catch up with the Joneses”,i.e.,ifothers consume more today, you
yourself will experience a higher marginal utility from an additional unit of
consumption in the future.1 In some ways, the idea of “catching up with the
Joneses” is a variation of the theme of “habit formation”,see Constantinides
(1990). The key difference is that “catching up with the Joneses” postulates
a consumption externality since agents who increase their consumption do
not take into account their effect on the aggregate desire by other agents
to “catch up”. Thus, this externality allows room for beneficial government
intervention. The optimal tax policy would induce agents in the competitive
equilibrium to behave in a first-best manner, which is given by the solution
to a social planner’s problem with habit formation.
While “catching up with the Joneses” has been the focus of quite some
research in the asset pricing literature, its implications with respect to pol
icy making have rarely been explored. The purpose of this paper is to do
exactly that. In particular, we examine economies driven by productivity
shocks where agents care about consumption as well as leisure, and there is
a “catching-up” term in the consumption part of the utility function. For
simplicity, the model abstracts from capital formation.23 In this framework,
we examine the role for taxing labor income. The optimal tax policy turns
1Gall (1994) explores an alternative assumption where agents’ preferences depend on
current instead of lagged per capita consumption ( “keeping up with the Joneses” as com
pared to “catching up with the Joneses”).
3As noted by Lettau and Uhlig (1995), the inclusion of capital formation in models
based on catching-up-with-the-Joneses utility functions have the implication that con
sumption becomes excessively smooth. For a similar observation and a possible remedy in
models with habit formation, see Boldrin, Christiano and Fisher (1995).
1
out to affect the economy countercyclically via procyclical taxes, i.e., “cool
ing” down the economy with higher taxes when it is “overheating” due to
a positive productivity shock. The explanation is that agents would other
wise end up consuming too much in boom times since they are not taking
into account the “addiction effect” of a higher consumption level. In re
cessions, the effect goes the other way around and taxes should be lowered
to “stimulate” the economy by bolstering consumption. Thus, models with
catching-up-with-the-Joneses utility functions call for traditional Keynesian
demand management policies. W e also use parameter values from Camp
bell and Cochrane (1995) to illustrate that the necessary labor taxes can be
very high, in the order of 50 percent. However, Campbell and Cochrane’s
nonlinear version of the aspiration level in the catching-up-with-the-Joneses
preferences has the additional implication that consumption bunching can
be welfare enhancing. As an example, we show how welfare can be improved
upon in their framework by inducing business cycles in an otherwise station
ary environment.
The paper is organized as follows. In section 2, we examine a simple oneshot model as well as an infinite horizon version, where agents care about
“keeping up with the Joneses”. The assumption being that contemporaneous
average consumption across all agents enters the utility function. In that
case, it turns out that there is a constant tax rate on labor, which delivers
the first best outcome independent of the productivity shock. In section 3, we
allow the agents’aspiration level to be a geometric average of p a s t per-capita
consumption, i.e., specifying a utility function which exhibits “catching up
with the Joneses”. This framework has Keynesian-style countercyclical policy
implications. In section 4, we examine the utility function used by Campbell
and Cochrane (1995), adding another layer of complexity. Here, we are in
particular interested in the quantitative tax implications of their parameter
values. Section 5 concludes.
2
2
K eeping up w ith th e Joneses
W e imagine an economy with many consumers, each with the same utility
function
1 - 7
where c
>
0 is the individual’s consumption, C > 0 is average consump
tion across all agents and n > 0 is labor supplied by the individual. The
parameters oc G [0,1), 7 > 0 and
A
> 0 determine the relative importance
of average consumption, the curvature of the consumption term and the
relative importance of leisure. This utility function captures the notion of
“keeping-up-with-the-Joneses”,i.e., average consumption decreases an indi
vidual’s level of utility and increases his marginal utility of an additional unit
of consumption. This specification is different from the formulation in Abel
(1990), who uses ratios rather than differences to aggregate consumption,
but is in line with the catching-up formulation in Campbell and Cochrane
(1995). No “keeping-up” is imposed on the leisure part of the utility func
tion. In other words, we assume that agents are competing in, say, having
the biggest car or the biggest house rather than having the most amount of
leisure. The utility in leisure is also assumed to be linear. This assumption is
partly done for convenience, but can also be motivated by indivisibilities in
the labor market and is an often used assumption in the real-business cycle
literature, see e.g. Hansen (1985) and the explanations therein. W e imagine
that the production function takes the form
c = On,
where 6 is a productivity parameter. Thus, there is no capital, and output
is simply linear in labor.
The government levies a flat tax r on all labor income and the tax rev
enues are then handed back to the agents in a lump-sum fashion. Let
v
be the lump-sum transfer to each agent. Since all agents are identical, the
3
government’s budget constraint can be written as
=
rd n
v
.
A competitive equilibrium is calculated by having an agent maximize the
utilityfunction above with respect to c and n subject to his budget constraint,
c = (1 —
A
r)6 n
+
v.
consumer’s optimal consumption is then found to be
c= aC+^(l-r)^
where average consumption
C
,
(1 )
is taken as given by the individual agent.
However, in an equilibrium it must be true that c =
C ,
so the equilibrium
consumption level is
1
c = C =
1
— a
The government’s optimal choice of r can be deduced from the solution
to the social planner’s problem. The social planner would take the exter
nality into account be setting C
in the utility function above, and then
= c
maximize with respect to consumption and labor subject to the technology
constraint. The first-best outcome is then given by
c* =
C
Comparing the social planner’s solution to the competitive equilibrium, we
find:
Proposition 1 (“Keeping up with Joneses”)
T h e fir s t - b e s t c o n s u m p t io n a llo c a t io n c a n be a c h ie v e d w ith a ta x ra t e
t
=
4
a
.
This result is quite intuitive. A fraction a of any increase in the representative
agent’s consumption does not contribute to his utility since itisoffset through
the consumption externality. It is therefore socially optimal to tax away a
fraction or of any labor income so that the agent faces the correct utility
tradeoff between leisure and consumption. It can also be noted that the
optimal tax is independent of the productivity parameter 8. While the tax
can potentially be high depending on the value of a, it does not react to
current economic conditions. In particular, we do not get any Keynesian
effects in the sense of setting taxes procyclically.
Given the solution above, one can easily examine a dynamic model, in
which there are periods denoted by t
=
0,1,2,... and agents have the utility
function
'(c . - q C,)1-1't= 0
where E
q
\
- -
,
1-7
is the expectation operator conditioned upon information at time 0
and /? € (0,1) is a discount factor. The production function is the same as
before, and so are the budget constraints of the government and the agents.
There isnow also some stochastic process driving productivity 8t . Computing
the competitive equilibrium and the social planner’s solution amounts to the
same calculations as above, since this dynamic model simply breaks into a
sequence of one-shot models. The first-best solution is again achieved at
r = a, i.e., there are no cyclical consequences for the tax rate.
Finally, itisworth pointing out that the tax analysis here is closely related
to the literature on redistributive taxation when individual welfare depends
on relative income. Given a social welfare function, Boskin and Sheshinski
(1978) analyze how the standard results of optimal tax theory axe altered
when individuals care about relative income, and they demonstrate that
the scope for redistribution becomes much larger. Persson (1995) extends
their argument by showing that high taxation can even constitute a Pareto
improvement as long as individuals’ pre-tax incomes are not too different.
In fact, his discussion of the special case of identical individuals corresponds
5
directly to our treatment of “keeping up with the Joneses”.
3
C atching up w ith th e Joneses
3.1
T he m od el
W e now assume that the utility function does not depend on current aver
age consumption as assumed above, but rather on some measure
X t
of p a s t
average consumption,
(c* t=0
\
x tr * * -
1 -7
In particular, we let the aspiration level X
t
- -
.
(2 )
be a geometric average of past
per-capita consumption levels,
Xt
with 0 < ^ < 1 and 0 <
a
=
a ( l - <j>)Ct-,
+
,
(3)
< 1. Otherwise, the production technology and
the budget constraints of the consumers and the government are the same as
before. In addition, we now need to.be more careful about the productivity
process. W e postulate the following stochastic process,
where
xp
6 [0,1) and
et
is i.i.d, has mean zero and is bounded below by
€t > — 1 .®3
3The stochastic process (4) is approximately the same as postulating an AR(1) process
for the logarithm of 9t ,
log(tft) = (1 - V1)log(0) + xj>log(0<—i)+ c,.
Thus, our exact analytical results below pertaining to the stochastic process (4) can also be
interpreted as approximations to the corresponding formulas valid for the more commonly
used AR(1) process for the logarithm of 9t .
6
For the competitive equilibrium in this model, one finds analogously to (1)
that the agent will set consumption equal to
Thus, given a first-best path for consumption
c\
= C*, one can achieve this
outcome with a sequence of taxes rt satisfying
n = i - j ' ( c ; - x ly.
(6)
To characterize the optimal tax policy, we now turn to the social planner’s
problem.
3.2
S o l v i n g t h e social p l a n n e r ’s p r o b l e m
The social planner maximizes the utility function (2) subject to the produc
tion technology and the constraint (3), taking as given the process for
and the initial conditions
X
q
and
0q .
$t
Since this maximization problem is
a concave one, we can analyze it by using first-order conditions. Let
Xt
be
the Lagrange multiplier for the constraint (3). The two first-order conditions
with respect to C t and
X t+ i
can then be written as
( C .- X ,) " 1
=
£ + «(l-«A,,
(7)
A,
=
W [ ( C , „ - A , t, r ] + « [ A , til-
(8)
The first equation contains the additional third term a(l — <^)Atas compared
to the corresponding equation of the private agent’s optimization problem.
Here, the social planner takes into account the “bad” effect on future utility
of additional aggregate consumption today, since itraises the aspiration level
X t+ i
tomorrow and beyond. In particular, a fraction a ( l
— <f>)
of an increase
in today’s per-capita consumption spills over to -X’i+i,and the shadow value
of a higher X t + i is given by A*. Equation (8) shows in turn how the shadow
7
value A* is the sum of the expected effect on tomorrow’s discounted marginal
utility of consumption and its impact on still future periods. The latter
effect is captured by the discounted expected value of At+i multiplied by
where
$
<f>,
is the fraction of the aspiration level that carries over between two
consecutive periods.
Using the two first-order conditions (7) and (8) as well as the constraint
(3), the steady-state consumption level can be calculated to be
C* =
1
1—
a
Comparing this expression to the agent’s consumption rule in equation (5)
and noting that
X
— a C
, we see that the first-best steady-state allocation
is supported by a tax of
-
-
<t>)
For example, ifthe aspiration level is simply a times the level of yesterday’s
per-capita consumption (<f> = 0), we get f =
a/3.
This formula is rather intu
itive compared to the simple model above of “keeping up with the Joneses”,
where we got r =
a.
Since the consumption externality now enters the utility
function with a one-period lag, the adverse future effect of being “addicted”
to today’s consumption is discounted by /? so the optimal steady-state tax
rate is also scaled down by /?.
In order to characterize the optimal consumption and taxation outside
of a steady state, we can actually solve the dynamic equations in closed
form. The substitution of equation (7) into (8) yields a first-order difference
equation in the shadow value At, which can be solved forward in the usual
manner,
OO
At
= p A E t
j =0
where
8
1
With the law of motion for 0* in (4), one can then calculate X t to be
A
‘
, / W ( l
8 )6 + i - 6$ U
M
(i -
l\
(10)
0) '
After substituting this expression into the first-order condition (7), the opti
mal consumption level is found to be
Cf
=
Xt
+
( a i- w
\7 i - « +
/j__ iu
U
e)
-
w
r'
i- 8 x i> )
(ii)
The tax necessary to support this optimal consumption allocation is then
given by equation (6).
Rather than calculating the tax rate rt,it is more appealing to calculate
the ratio of taxes to after-tax income. Using equations (6) and (7), we get
<*(1 ~
rt
1-
<f>)
(12)
O tX t.
A
rt
With the productivity process in (4), A4 is given by (10) and the tax ratio
can then be rewritten as in the following proposition.
Proposition 2 (“Catching up with Joneses”)
T h e ta x ra te Tt s u p p o r t in g the fir s t - b e s t c o n s u m p t io n a llo c a t io n c a n be s o lv e d
fro m
Tt
=
1 — r<
<*/?(! ~ 4>) ( ,
1-8 $
V
,1
I-
M
8
(13)
6 ) '
w ith a s t e a d y - s ta t e v a lu e o f
- _ aft(i T
3.3
l- f i*
4)
‘
Tax p o licy im p lication s
Corollary 1 (“Catching up with Joneses”)
T h e o p t im a l ta x p o l ic y a ffe c ts th e e c o n o m y c o u n t e r c y c lic a lly v ia p r o c y c lic a l
ta x e s.
9
This corollary follows directly from equation (13), the tax ratio (and thus
the tax rate itself) varies positively with productivity
0t.4
Thus, we get
Keynesian-style policy recommendations. A government that maximizes wel
fare should “cool down” the economy during booms via higher taxes because
agents would otherwise consume too much as compared to the first-best so
lution. Likewise, the government should “stimulate” the economy during
recessions by lowering taxes and thereby bolstering consumption. Of course,
these optimal fiscal policies axe here driven by a rather unorthodox argu
ment. Taxation is needed to offset the externalities associated with private
consumption decisions. One individual’s consumption affects the welfare of
others through agents’desire to “catch up with the Joneses”.
To shed light on how different parameters affect the cyclical variations of
optimal taxation, let w t be the relative deviation of the tax ratio r t/ ( l —
r t)
from its steady-state value. That is, u t tells us, how the ratio of taxes to
after-tax income responds to productivity shocks relative to its steady-state
value. PYom equation (13), we can calculate
n
(
f
1 — rt \1 —
_
t
1-0
1—
)
8rj>
-9
(14)
$
Doing comparative statics on this expression, we see that the size of the cycli
cal tax effect in absolute terms varies negatively with 0 and positively with
a, /? and <f>. The intuition for this is straightforward by considering the tax
response to a positive productivity shock. A higher 0, i.e., a more persistent
productivity shock, means that future production and consumption oppor
tunities are also expected to be better than average. The anticipation of the
economy being able to sustain a higher consumption level for a prolonged
period of time mitigates the adverse effects of making people “addicted” to
4It is worth noting that this result holds for a much larger class o f stochastic processes
than given by equation (4). According to equations (9) and (12), the optim al tax rate
goes up with 9t as long as E t [SJLo ^ &7+i + j ] decreases less than proportionally with the
inverse o f 0 t -
10
higher consumption today. It is therefore socially optimal to take more ad
vantage of a persistent productivity shock, so the optimal tax hike is lower
with a higher ip. In contrast, preferences with a higher weight on yesterday’s
consumption (a higher
a ),
a higher degree of persistence in the aspiration
level (a higher <f>), or a higher emphasis on the future (a higher ft) give rise
to a larger cyclical tax effect. The reason is, of course, that the consumption
externality is more important for such preferences and the government must
consequently be more resolute in moderating agents’consumption behavior.
As a point of reference, the largest tax effect as defined by (14) is attained
for transient one-period productivity shocks
(ip
= 0). The percentage devi
ation of the tax ratio from its steady-state value responds then one-for-one
to the percentage change in the productivity from its steady state. However,
besides noting that the cyclical tax effect can be large relative to the mag
nitude of the productivity shock, it is also important to keep in mind that
most aggregate economic shocks are usually relatively small so the cyclical
tax changes considered here are really examples of extreme “fine tuning” of
taxes.
Finally, Figure 1 illustrates the consumption dynamics in response to a
productivity shock. After a one-percent initial shock to 9 t at time t = 0, the
hump-shaped dashed line traces out the response of consumption from the
steady state when taxes are adjusted optimally and the solid line displays the
consumption response when the tax rate is not changed but kept constant at
its steady-state value. As a parameterization, we used ip = 0.9, a = 0.8, (3 =
0.97 and varied 7 € {0.5,1.5}. Not surprisingly, the consumption response
becomes muted with a higher 7 ,since a more rapidly diminishing marginal
utility of consumption reduces the attractiveness of increasing consumption.
It is interesting to note that for both values of 7 in Figure 1 the deviation
of consumption from steady state is reduced by around 25
%
under optimal
tax adjustment as compared to keeping the tax rate constant at its steadystate value. The figure also contain the change in the tax ratio u>t needed to
accomplish this “cooling down” of the economy.
11
PanalA: Gamma-0.5
Panel B: Gamma -1.5
Figure 1: Consumption dynamics in response to a one-percent productivity
shock from the steady state. The dash-dotted line depicts the optimal re
sponse in the tax ratio rt/(l— rt). The parameters are 7 € {0.5,1.5} (panel A
and panel B, respectively), ^ = 0.9, a = 0.8, and
12
0
= 0.97.
4
T h e
C a m p b e l l - C o c h r a n e utility f u n c t i o n
W e now turn to the utility function proposed by Campbell and Cochrane
(1995) extended with a linear disutility term for labor. These preferences
are then also given by our expression (2), but the aspiration level X
t
is now
a complex nonlinear function of current and past per-capita consumption as
shown below.56 A useful concept when studying this model is the “surplus
consumption ratio” defined for an individual as
< h -X t
st
= ----- ,
Ct
and the upper case letter S t will be used to denote the economy-wide value
of s t. In an equilibrium, S t will of course be equal to st since all agents are
identical.
Campbell and Cochrane postulate an implicit law of motion for X
t
by
A
writing a law of motion for
St-
Let
St
= log S t be the logarithm of the
economy-wide surplus consumption ratio, and likewise C t = log C t is the log
arithm of average consumption across all agents. Abstracting from economic
growth, it is then assumed that S t evolves according to
st
where
S
= (1 -
<t>) log S + <f>St-t
+ A(5t_0
(C t -
C t- i j
,
(15)
is the steady-state value of S , and the function A (S') is given by
A( S )
-{
S - 'y / l
-2(5 — log 5) — 1,
0,
S < S „
(16)
S _
> S^max
,
5Our notation differs from Campbell and Cochrane (1995) in order to stay consistent
with the notation above. In particular, we use hats rather than sm all letters to denote
logs, while sm all letters still denote the individual’s choice variables. We use
instead of
for the discount factor and we abstract from growth, i.e., their parameter g is here set
equal to zero.
6
13
with
Sm ax =
log 5 + (l — S2) /2.6 Given equation (15), one can back out
the implied law of motion for X
t.
imation shows that the log of X
t
Near the steady state, a log-linear approx
is a moving average of past consumption in
logs and it does not depend on contemporaneous consumption,
x
t
= iog(i -
s
) + (i -
<f>)
£;
p e t - ;.!
.
i —0
For our purposes, the steady-value
S
can be thought of as a parameter in
this model. By picking a value of S , we are effectively choosing a particular
preference specification.7
Taking
X t
as given, the agent maximizes utility subject to the usual
budget constraint. Analogously to the previous sections, the agent’s optimal
consumption is found to be
°t = X t +
^ ( 1 — rt)^
,
which can also usefully be written as
(^ “(T=W
(17)
Instead of solving the social planner’s problem, we now turn to a more
modest question. In a steady state, we ask what tax rate is needed to support
the best possible constant consumption level. (The word ‘constant’will soon
be shown to be restrictive in terms of maximizing welfare).
6The purpose of Campbell and Cochrane’s rather complicated preference specification
is to assure that c* — X t> 0, and that the risk-free rate is constant when is a random
walk with drift.
7To understand the correspondence between S and the preference specification, let us
consider the model in Section 3 where S = (<?— X )/ C = (C —aC )/C = 1 — a. That is, S
maps directly into a and is unaffected by the tax rate. (The only exception being a 100 %
tax rate which would close down all economic activity, and the surplus consumption ratio
would no longer be defined.)
14
Proposition 3 (Campbell-Cochrane)
I n a s t e a d y s t a te w ith
7 > 1,
th e re e x is ts a u n iq u e c o n s u m p t io n le v e l t h a t
c a n n o t be im p r o v e d u p o n th ro u g h a o n c e - a n d - f o r - a ll c h a n g e to a n o t h e r c o n
s u m p t io n le v e l.
T h e s t e a d y - s ta t e t a x ra te s u p p o r t in g t h is best p o s s ib le c o n s t a n t
c o n s u m p t io n le v e l is g iv e n by
(l-S).
i - M
The derivation of this proposition is deferred to the appendix, and here we
only note that the tax rate is the same
the steady-state tax rate for the
as
usual linear version of the aspiration level in Proposition 2. To see this, we
only have to use the observation in footnote 7 which is that the parameter
a and the steady state surplus consumption ratio satisfy the relationship
S
= 1 — a.
It is also interesting to take a look at the quantitative tax implication
of Proposition 3. Campbell and Cochrane use the parameters
<j>=
0.97 and
S
0
= 0.973,
= 0.049. For these parameters, we obtain
f = 0.494,
i.e., almost 50 percent of labor income should be taxed away in steady state
in order to support the best possible constant consumption level. Taken
seriously, this would indicate that current labor taxes are too low in the
United States, but about right in, say, the Netherlands or Sweden.
Finally, we have refrained from using the word ‘first-best outcome’sim
ply since we have not presented the optimal solution to the social planner’s
problem. The nonconcave character of this maximization problem makes it
analytically intractable, so here we rather use an example to demonstrate that
consumption bunching can improve upon a constant consumption allocation.
In particular, Figure 2 explores the welfare consequences of a temporary oneperiod increase in consumption starting from a steady state with the best
possible constant consumption level C ,
15
as
described in Proposition 3. The
x-axis in Figure 2 shows the size of the one-period consumption deviation as
percentage of C , and the y-axis depicts the life-time utility associated with
that policy. It is clear from the figure that there are one-time consumption
deviations that can increase life-time utility. The intuition is that a tempo
rary consumption increase acts as an ‘investment’in the surplus consumption
ratio S because of the persistence parameter <f> in equation (15). But let us
first consider what happens in the first period when the consumption hike
takes place. Equation (15) shows how the log of S increases by the logdeviation in consumption multiplied by the steady-state value of the A-function.
Since the A-function is decreasing in the surplus consumption ratio, it follows
that the negative impact on
S
is smaller in the next period when consump
tion reverts back to C . In fact, there is no effect at all ifthe log of the surplus
consumption ratio has reached S mas in equation (16) when the A-function be
comes zero. (This critical point shows up as a kink on the curve in panel B of
Figure 2.) In consecutive periods, welfare is positively affected by the slowly
decaying surplus consumption ratio (while consumption is kept constant at
C ).
Concerning the parameterization in Figure 2, we have used Campbell
and Cochrane’s values mentioned above and their parameter 7 = 2.372, and
we have set A =
$
= 1.
Figure 2 suggests that a first-best outcome for the Campbell-Cochrane
utility function will involve consumption cycles even in an otherwise station
ary environment. The social planner would like to exploit the law of motion
for the surplus consumption ratio in order to increase the well-being of in
dividuals. The rationale for this is that the dynamics of the law of motion
for the surplus consumption ratio can be said to exhibit increasing returns
to scale. For a related argument on welfare-improving cycles in models with
increasing-returns-to-scale production technologies, see Murphy, Shleifer and
Vishny (1989).
16
PanelA
Panel B
Figure 2: Impact on life-time utility of a one-period consumption deviation
from the steady state in Proposition 3. Panel B is a magnification of the
left-hand portion of panel A. The parameters are /? = 0.973,
S
= 0.049, 7 = 2.372, and
A = 9
= 1.
17
<f> =
0.97 and
5
Conclusions
This paper examined the role for tax policies in simple productivity-shock
driven economies with “catching-up-with-the-Joneses” utilityfunctions. These
utility functions give rise to consumption externalities, but taxation can be
used to get back to the first-best solution. The optimal tax policy turns
out to affect the economy countercyclically via procyclical taxes. When the
economy is “overheating” due to a positive productivity shock, a welfaremaximizing government should raise taxes to “cool down” the economy. Like
wise, taxes should be cut in recessions to “stimulate” the economy by bol
stering consumption. Thus, models with catching-up-with-the-Joneses utility
functions call for traditional Keynesian demand management policies. W e
also used parameter values from Campbell and Cochrane (1995) to illustrate
that the necessary labor taxes can be very high, in the order of 50 percent.
However, Campbell and Cochrane’s nonlinear version of the aspiration level
in the catching-up-with-the-Joneses preferences has the additional implica
tion that consumption bunching can be welfare enhancing. An example was
used to illustrate how welfare can be improved upon in their framework by
inducing business cycles in an otherwise stationary environment.
18
Appendix
P r o o f o f P r o p o s itio n 3
Let us start our argument from an arbitrary initial steady state with
consumption
0
(and a surplus consumption ratio of S ) . At time t
consider an alternative future consumption allocation of C t
= Co
=
0, we
for all t
>
0.
According to equation (15), the associated sequence of surplus consumption
ratios can be expressed in log form as
So
=
log S + A(Co — log C ),
St
=
(1 -
<j>)
log5 +
(j>St-i
= (1 - ^ ) log5 +
tfS o ,
fort > 1 ,
where A = A(log S ) = 5 -1 — 1. That is, the sequence of surplus consumption
ratios is given by
for t
>
0.
The life-time utility associated with such an alternative consumption alloca
tion is
For 7 > 1, this expression is concave in C o and the first-order condition with
respect to C q is
(18)
19
For any initial steady state C , equation (18) can be used to solve for the
best C o when we are constrained to only consider once-and-for-all changes in
the consumption level. To find the unique constant consumption level that
cannot be improved upon in this way, we solve for C in equation (18) such
that C o =
0 .
The best possible constant consumption level is then found to
be
(19)
To support this consumption allocation in a competitive equilibrium, we first
solve for the tax rate in the agent’s first-order condition in equation (17),
_
A ( c t s t)7
with a steady-state value of
After substituting equation (19) and A = 5 -1 — 1 into this expression, we
arrive at the steady-state tax rate in Proposition 3.
20
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