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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Revisiting the Supply Side Effects of
Government Spending Under
Incomplete Markets
George-Marios Angeletos
Vasia Panousi
Working Paper 07-18
May 4, 2007
RoomE52-251
50 Memorial Drive
Cambridge,
MA 02142
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at
Revisiting the Supply-Side Effects of
Government Spending Under Incomplete Markets^
George-Marios Angeletos
MIT
and
Vasia Panousi
NBER
May
MIT
4,
2007
Abstract
This paper
revisits
the macroeconomic effects of government consumption in the neoclassical
growth model augmented with idiosyncratic investment
plete markets, a
permanent increase
interest rate, the capital-labor ratio,
in
government consumption has no long-run
and labor productivity, while
These
to the familiar negative wealth effect.
markets.
The very same negative wealth
ment. This
in
*We
MIT
now
increases
we
effect
on the
work hours due
allow for incomplete
causes a reduction in risk taking and invest-
and lower wages.
codes: E13, E62,
Keywords: Fiscal
at
effect
it
results are upset once
Under com-
turn leads to a lower risk-free rate and, under certain conditions, also to a lower
capital-labor ratio, lower productivity
JEL
(or entrepreneurial) risk.
policy,
government spending, incomplete
Mike Golosov, Ivan Werning and seminar participants
comments. Email addresses: angelet@mit.edu, panousi@mit.edu.
are grateful to Olivier Blanchard, Ricardo Caballero,
for useful
risk sharing, entrepreneurial risk.
Introduction
1
Studying the impact of government spending on macroeconomic outcomes
ebrated pohcy exercises within the neoclassical growth model;
the business-cycle imphcations of
Some
section of countries.
and Eichenbaum
all
is
fiscal
policy
important
cel-
understanding
for
Barro (1981, 1989), Aiyagari, Christiano
and King (1993), and McGrattan and Ohanian (1999, 2006).
maintained the convenient assumption of complete markets, abstracting
from the possibihty that agents' saving and investment decisions
changes in
one of the most
pohcy, the macroeconomic effects of wars, and the cross-
classics include Hall (1980),
(1992), Baxter
These studies have
fiscal
it
is
— may crucially depend on the extent of
This paper contributes towards fihing this gap.
— and
hence their reaction to
economy.
risk sharing within the
the macroeconomic effects of government
It revisits
consumption within an incomplete-markets variant of the neoclassical growth model.
Apart from introducing undiversified idiosyncratic
ingredients of our
model are the same
risk in
production and investment,
as in the canonical neoclassical
is
CRRA/CEIS
and markets are competitive. The focus on idiosyncratic production/investment
motivated by two considerations.
less
other
growth model: firms op-
erate neoclassical constant-returns-to-scale technologies, households have standard
preferences,
all
First, this friction
is
empirically relevant. This
obvious for
is
developed economies. But even in the United States, privately-owned firms account
one half of aggregate production and employment. Furthermore, the typical investor
rich household
—holds a very undiversified portfoho,
private equity.-'
And
second, as
we explain
more than one half
next, this friction upsets
of
which
risk
for
about
—the median
is
allocated tO
some key predictions
of the
standard neoclassical paradigm.
In the standard neoclassical paradigm, the steady-state values of the capital-labor ratio, productivity (output per work hour), the
wage
rate,
and the
interest rate, are all
equahty of the marginal product of capital with the discount rate
change in the
level of
government consumption, even
values of these variables.^
On
if it is
pinned down by the
in preferences.
permanent, has no
As
eflFect
a result, any
on the long-run
the other hand, because higher spending for the government means
lower wealth for the households, a permanent increase in government consumption raises labor
supply. It follows that
employment (work hours) and output increase both
in the short
run and
in
the long run, so as to keep the long-run levels of capital intensity and productivity unchanged.
The
picture
is
quite different once
that, in response to
an increase
in
we
allow for incomplete markets.
The same wealth
government consumption, stimulates labor supply
dard paradigm, now also discourages investment. This
is
effect
in the stan-
simply because risk taking, and hence
'See Quadrini (2000), Gentry and Hubbard (2000), Carroll (2000), and Moskowitz and Vissing-Jorgensen (2002).
Also note that idiosyncratic investment risks need not be limited to private entrepreneurs; they may also affect
educational and occupational choices, or the production decisions that CEO's make on -behalf of public corporations.
^This, of course, presumes that the change in government consumption
efficiency or redistributive considerations
behind optimal taxation
is
is
financed with
beyond the scope of
lump-sum
this paper.
taxes.
The
investment,
is
sensitive to wealth.
We
thus find very different long-run effects. First, a permanent
And
increase in government consumption necessarily reduces the risk-free interest rate.
unless the elasticity of intertemporal substitution
ratio, productivity,
The
of
effect
which
is
low enough,
also reduces the capital-labor
it
and wages.
on the risk-free rate
consumption
is
second,
an implication of the precautionary motive: a higher
is
government imphes a lower aggregate
for the
level of
A
possible in steady state only with a lower interest rate.
does not necessarily imply a higher capital-labor
ratio.
This
wealth
for the
level
households,
lower interest rate, however,
because market incompleteness
is
introduces a wedge between the risk-free rate and the marginal product of capital. Furthermore,
because of diminishing absolute
risk
risk aversion, the lower the level of wealth, the higher will
premium on investment, and hence
this
government consumption, the capital-labor
follows that, in response to an increase in
It
ratio can fall even
a sufficient condition for this to be the case
sufficiently high relative to the
wedge.
is
be the
if
the interest rate also
Indeed,
falls.
that the elasticity of intertemporal substitution
income share of capital
— a condition
is
easily satisfied for plausible
calibrations of the model.
Turning to employment and output, there are two opposing
with complete markets, the negative wealth
employment and output.
intensity, productivity,
on labor supply contributes towards higher
the other hand, unlike complete markets, the reduction in capital
elasticities of labor supply, either of the
deviation from the standard paradigm
titatively.
the one hand, as
and wages contributes towards lower employment and output. Depending
on the income and wage
The
On
effect
On
effects.
is
two
effects
can dominate.
significant, not only qualitatively,
but also quan-
For our preferred parametrizations of the model, the following hold. First, the elasticity
of intertemporal substitution
is
comfortably above the
government consumption to reduce the long-run
value that suffices for an increase in
critical
levels of the capital-labor ratio, productivity,
wages. Second, the negative effects on these variables are quantitatively significant: a
in
1%
and
increase
government spending under incomplete markets has the same impact on capital intensity and
labor productivity as a 0.5%
— 0.6%
increase in capital-income taxation under complete markets.
Third, these effects mitigate, but do not fully
offset,
the wealth effect on labor supply. Finally, the
welfare consequences are non-trivial: the welfare cost of a permanent
consumption
The main
is
1%
increase in government
three times larger under incomplete markets than under complete markets.
contribution of the paper
to financial frictions can significantly
is
how wealth
thus to highlight
modify the supply-side channel of
effects
on investment due
fiscal policy. In
our model,
these wealth effects emerge from idiosyncratic risk along with diminishing absolute risk aversion; in
other models, they could emerge from borrowing constraints. Also, such wealth effects are relevant
for
both neoclassical and Keynesian models.
In this paper
not because of any belief on which paradigm best
fits
we
follow the neoclassical tradition,
the data, but rather because this clarifies
the value of our contribution: whereas wealth effects are central to the neoclassical approach with
regard to labor supply, they have been mute with regard to investment.
To the best
of our knowledge, this
ernment consumption
particular framework
in Angeletos (2007).
saving,
in
the
is
we employ
for this
purpose
That paper studied how
and contrasted
this
is
Buera and Shin
idiosyncratic capital-income risk affects aggregate
with the impact of labor-income
and Smith,
(2007), Caggeti
novelty of our paper
The
rest of the
characterizes
its
effects of gov-
a continuous-time variant of the one introduced
1998).
paper
is
is
Bewley type models (Aiyagari,
risk in
Other papers that introduce idiosyncratic
invest-
model include Angeletos and Calvet
(2006),
or entrepreneurial risk in the neoclassical growth
ment
macroeconomic
to study the
first
an incomplete-markets version of the neoclassical growth paradigm.'^ The
1994; Huggett, 1997; Krusell
The
paper
and De Nardi (2006), Covas (2006), and
Meh and
Quadrini (2006).^
to study the implications for fiscal policy in such an environment.
organized as follows. Section 2 introduces the basic model, Section 3
equihbrium, and Section 4 analyzes
its
steady state. The basic model fixes labor
supply so as to focus on the most novel results of the paper, namely the steady-state
government consumption on the
interest rate
and the capital-labor
three extensions that endogenize labor supply.
economy
The
2
Time
is
basic
i
which
in
Section 6 examines the dynamic response of the
government consumption. Section 7 concludes.
t
e
[0,oo).
and distributed uniformly over
it
Section 5 then turns to
model
continuous, indexed by
indexed by
labor,
permanent change
to a
ratio.
effects of
There
[0,1].
is
a continuum of infinitely-hved households,
Each household
supplies inelastically in a competitive labor market.
is
endowed with one unit
Each household
also
of
owns and
runs a firm, which employs labor in the competitive labor market but can only use the capital stock
invested by the particular household.^
Households cannot invest
in other households' firms
cannot otherwise diversify away from the shocks hitting their firms, but can freely trade a
bond. Finally,
all
uncertainty
is
purely idiosyncratic, and hence
all
and
riskless
aggregates are deterministic.
is conducted in Heathcote (2005). That paper studies tlie impact of a cliange
Bewley-type model like Aiyagari's (1994), where borrowing constraints limit the ability
of households to smooth consumption, thus breaking Ricardian equivalence, but where market incompleteness does
not impact the production side of the economy. In our paper, instead, the key friction is on the production side.
Moreover, deviations from Ricardian equivalence are not considered: our model allows households to freely trade a
riskless bond, thus ensuring that the timing of taxes has no effect on allocations.
""Related is also Obstfeld (1994), which assumes a continuous-time Epstein-Zin specification as this paper, but
with an AK technology.
^We can think of a household as a couple, with the wife running the family business and the husband working
in the competitive labor market (or vice versa). The key assumption, of course, is only that the value of the labor
endowment of each household is pinned down by the competitive wage and is not subject to idiosyncratic risk.
''A related,
in the
but
different, exercise
timing of taxes
in a
Households and firms
2.1
The
financial wealth of household
and the
riskless
bond,
denoted by
i,
x], is
the
sum
of
its
holdings in private capital,
bj:
4-I4 + KThe
evolution of xj
is
Rt
dvrj is
=
diri
+
the household's capital income
+uJt-Tt-
[Rtbi
(i.e.,
the interest rate on the riskless bond, ut
is
(1)
given by the household budget:
dxi
where
the profits
Whereas the sequences
of prices
(2)
enjoys from the private firm
the wage rate, Tt
is
game
is
the
lump-sum
condition
is
tax,
it
owns),
and
cj is
also imposed.
and taxes are deterministic (due to the absence
of aggregate
firm profits, and hence household capital income, are subject to undiversified idiosyncratic
risk),
In particular,
d4 =
Here, n\
is
the
amount
[F{klnl)
-
ujtni
is
-
Sk^jdi
+
ak\dz\.
(3)
of labor the firm hires in the competitive labor market,
returns-to-scale neoclassical production function,
risk
ci\dt,
it
the household's consumption. Finally, the familiar no-Ponzi
risk.
fcj,
and 5
is
mean
the
introduced through dz], a standard Wiener process that
F
is
a constant-
depreciation rate. Idiosyncratic
is i.i.d.
across agents
and across time.
This can be interpreted either as a stochastic depreciation shock or as a stochastic productivity
shock, the key element being that
amount
it
generates risk in the return to capital.
of undiversified idiosyncratic risk
and can be viewed
as an index of
with higher a corresponding to a lower degree of risk sharing (and a
markets).
Finally,
without serious
the technology: F{k,n)
=
loss of generahty,
k°n}-° with a G
The scalar a measures the
we assume
=
market incompleteness,
corresponding to complete
a Cobb-Douglas specification for
(0,1).^
Turning to preferences, we assume an Epstein-Zin specification with constant
elasticity of in-
tertemporal substitution (CEIS) and constant relative risk aversion (CRRA). Given a consumption
process, the utility process
is
defined by the solution to the following integral ecjuation:
/oo
z{cs,Us)ds
(4)
where
„i-i/e
z{c,U)
®The characterization
production function;
it is
=
of equilibrium
1-1/0
and the proof
rT7^-(l-7)C/
of the existence of the steady state
(5)
extend to any neoclassical
only the proof of the uniqueness of the steady state that uses the Cobb-Douglas specification.
^
Here,
/3
>
is
>
the discount rate, 7
is
the coefficient of relative risk aversion, and
>
6*
is
the
elasticity of intertemporal substitution/
Standard expected
utility
is
nested with 7
=
XjB.
We
find
to clarify that the qualitative properties of the steady state
it
useful to allow d
depend
crucially
^
I/7
on the
in order
elasticity of
intertemporal substitution rather than the coefficient of relative risk aversion (which in turn also
guides our preferred parameterizations of the model). However, none of our results rely on allowing
6
^
it,
1/7.
A
who
reader
feels
uncomfortable with the Epstein-Zin specification can therefore ignore
assume instead standard expected
utility,
and simply replace 7 with \/9
(or vice versa) in all
the formulas that follow.
Government
2.2
At each point
is
deterministic,
it
is
financed with lump-sum taxation, and
The government budget
private consumption or production.
dBi
where Bf denotes the
a no-Ponzi
level of
game condition
is
-
[RtBl
government assets
imposed
+ Tt-
(i.e.,
does not
it
constraint
is
affect either utility
from
given by
Gt]dt,
minus the
(6)
level of
government debt).
Finally,
to rule out explosive debt accumulation.
Equilibrium definition
2.3
The
Government spending
time the government consumes output at the rate Gf.
in
initial position of
the economy
is
given by the distribution of
(/cq,
&o) across
households. House-
holds choose plans {Ci,nj,A;|,5J}jg [0,00)1 contingent on the history of their idiosyncratic shocks, and
given the price sequence and the government policy, so as to maximize their lifetime
iosyncratic risk, however, washes out in the aggregate.
We
utility.
Id-
thus define an equilibrium as a de-
terministic sequence of prices {i-Ot,Rt}te[0,oo)^ ^ deterministic sequence of policies {Gt,Tt}t£[o ^00)
a deterministic macroeconomic path {Cj,
plans ({cl,nl,
prices
fcj,
&t}t6[o,oo))ie[o,i]i
and pohcies, the plans
/fj, Itjjgro^oo)!
^^^ ^
collection of individual contingent
such that the following conditions hold:
are optimal for the households;
Bf =
(ii)
t;
(iii)
the
bond market
in all
t;
and
(v)
the aggregates are consistent with individual behavior,
yt
=
0, in all
t]
(iv)
given the sequences of
the labor market clears, J^nJ
in all
clears, J^ b\-\-
(i)
the government budget
Q = /^cj, Kt
is
=
=
1,
satisfied
/j^Ji
^md
XF(/cJ,nj),inalH.s
^To make sure that (4) indeed defines a preference ordering over consumption lotteries, one must establish existence
and uniqueness of the solution to the integral equation (4); see Duffie and Epstein (1992).
^Throughout, J. denotes expectation in the cross-section of the population.
Equilibrium
3
In this section
optimal plan
we characterize the equihbriuna
for
of the
We
economy.
We
given sequences of prices and policies.
solve for a household's
first
then aggregate across households and
derive the general-equilibrium dynamics.
Individual behavior
3.1
Since employment
is
chosen after the capital stock has been installed and the idiosyncratic shock
By
has been observed, optimal employment maximizes profits state by state.
scale,
optimal firm employment and profits are linear in own capital:
=
nl
where
n{ujt)
=
and
n{ujt)kl
argmax„[F(l,n) —
and
uitn]
the household's expectation of the return to
shock
mean
as well as the
zl,
interpretation apphes to
The key
how
constant returns to
result here
fit
is
=
cZttJ
f{iOt)
its
=
f{ijJt)k\dt
= max„
+ adzl,
[F{l,n)
—
totn]
(7)
—
d.
Here,
ft
=
f{u)t) is
capital prior to the realization of the idiosyncratic
of the reahzed returns in the cross-section of firms.
Analogous
fi{uJt)-
that households face risky, but linear, returns to their capital.
this translates to linearity of
wealth in assets,
human
future labor income net of taxes, a.k.a.
let ht
To
see
denote the present discounted value of
wealth:
/oo
e--^i'^''^^{ujs-Ts)ds.
sum
Next, define effective wealth as the
wi
It
of financial
=
x\
+
ht
=
and human wealth:
k\
+
lJi
+
ht.
follows that the evolution of effective wealth can be described
dw\
The
first
prices
linearity of
and
problem
+
\ftk\
term on the right-hand side of
effective wealth; the
The
=
Rt{\^t
(10)
+
(8)
(9)
by
ht)-dt\dt-Vak\dzl.
(10)
measures the expected rate of growth
second term captures the impact of idiosyncratic
in the household's
risk.
budgets together with the homotheticity of preferences ensures that,
policies, the
for given
household's consumption-saving problem reduces to a tractable homothetic
as in Samuelson's
and Merton's
rules are linear in wealth, as
shown
classic portfolio analysis. It follows that the
in the next proposition.
optimal policy
Proposition
1.
Let {u!t,Ht}t€[o,co)
"''^d
{Gt,Tt}te\o,oo) be equilibrium price
Then, equilibrium consumption, investment and bond holdings for household
cj
where
=
mtw\,
kl
^
and
(ptwl,
6J
=
(1
-
(pt)wl
-
and
i
policy sequences.
are given by
(11)
ht,
the fraction of effective wealth invested in capital, is given by
(pt,
n-Rt
,
(12)
70"^
while m-t, the marginal propensity to consume out of effective wealth, satisfies the recursion
^
= m^ +
mt
with
pt
mean
=
pt
—
})^4>1a'^
(0
-
l)pt
-
(13)
6(3,
denoting the risk-adjusted return to saving and pt
(fitft
+
(1
—
(i>t)Rt
the
return to saving.
Condition (12) simply says that the fraction of wealth invested
=
— Rt, and
the risk
premium fit
(13)
essentially the Euler condition:
to
=
is
consume
ff
in the risky asset
amount
decreasing in risk aversion 7 and the
is
increasing in
of risk a.^ Condition
describes the growth rate of the marginal propensity
it
as a function of the anticipated
Whether
path of risk-adjusted returns to saving.
higher risk-adjusted returns increase or reduce the marginal propensity to consume depends on the
elasticity of intertemporal substitution.
condition reduces to
eiTective wealth)
if
m = dp —
and only
if
{&
9
>
—
To
1) p, so
1.
This
see this
more
that higher p decreases
is
due
to the familiar
m
(i.e.,
increases saving out of
income and substitution
effects.
General equilibrium
3.2
Because individual consumption, saving and investment are linear
at
note that in steady state this
clearly,
any point
in
in individual wealth, aggregates
time do not depend on the extent of wealth inequality at that time. As a result,
the aggregate equilibrium dynamics can be described with a low-dimensional recursive system.
= F{K,1) as the production in
ci>{K, R)^:^ {f'{K) -S-R), and p{K, R) =
Define f{K)
intensive form
and
R+^ {f'{K) -
policy rules of the agents and imposing market clearing,
^Clearly, in
would
fail
any equilibrium nt must be
to exist.
positive, otherwise
we
5
let
u){K)
= f{K) —
- R)\ Aggregating
f'{K)K,
across the
arrive at the following proposition.
nobody would
invest in capital
and an equilibrium
Proposition
In equilibrium, the aggregate dynamics satisfy
2.
Kt
=
f [Kt) - SKt
-Ct-Gt
(14)
^ = e(A-/?)- (0-1) 17^2^?
Ht
=
Kt
with
LOt
=
ujiKt),
4>t
Condition (14)
=
is
and
(l>{Kt,Rt),
pt
=
(15)
RtHt -uJt + Gt
(16)
-^Ht
(17)
=
p{Kt,Rt)-
The
the resource constraint of the economy.
depend on the degree of market incompleteness.
It
follows
resource constraint does not
from aggregating budgets across
all
households and the government, imposing labor- and bond-market clearing, and using the linearity
employment
of individual firm
Yt
=
/,F(/cj,nj)
=
Condition (15)
to individual capital together with constant returns to scale, to get
= F{Ku
Fif^klJ,r4)
is
1).
the aggxegate Euler condition for the economy.
It
follows from aggregating
consumption and wealth across agents together with the optimahty condition
propensity to consume.
It also
As
has a simple interpretation.
aggregate consumption growth unambiguously increases with
unlike complete markets, aggregate consumption growth
adjustment term. Whether more
and hence whether
whether the
for this
risk contributes to
new term
this
is
the
same
mean
the
complete markets,
return to saving. But
depends on
also
^ja'^cp'^,
a risk-
a lower or higher marginal propensity to save,
contributes to lower or higher consumption growth, depends on
elasticity of intertemporal substitution, 9,
property
in the case of
pt,
now
(13) for the marginal
is
higher or lower than
The
1.
intuition
as the intuition for the impact of the interest rate in a deterministic
saving problem, namely the opposing income and substitution effects of a higher rate of return to
saving.
Condition (16) expresses the evolution of the present value of aggregate net-of-taxes labor
income
in recursive form.
It
follows
from the definition of human wealth combined with the
in-
tertemporal government budget, which imposes that the present value of taxes equals the present
value of government consumption.
Finally, condition (17) follows
from bond-market clearing.
More
holdings across agents and imposing bond-market clearing gives (1
gating investment gives Kt
=
(ptWt,
and combining the two
—
precisely, aggregating
(pt)^Vt
— Ht =
complete
ital).
((7
=
0), this is
both
cases, condition (17) ensures that
possible only if/' {Kt) — S
=
while aggre-
gives condition (17).
These conditions characterize the equilibrium dynamics of the economy with
or complete markets. In
0,
bond
cpt
G
(0, 1).
either incomplete
But when markets are
Rt (meaning arbitrage between bonds and cap-
Condition (15) then reduces to the more familiar Euler condition Ci/Ct
—
9
[/'
(Kt)
—
S
—
0],
and one can track the dynamics
economy merely on the
of the
(C, A') space, using the Euler con-
dition
and the resource constraint. When, instead, markets are incomplete,
only
f (Kt) — 5 >
rate.
if
-Rt-
£
(0, 1) is
possible
which proves that the marginal product of capital must exceed the
risk- free
(j)t
Moreover, the dimensionahty of the system now increases by one: along with (C, K), we also
have to keep track of H, using condition
Still, this is
(16).
a highly tractable dynamic system, as compared to other incomplete-markets
— an
dimensional object
—
mod-
els,
where the entire wealth distribution
for
aggregate equihbrium dynamics. Indeed, the equilibrium dynamics can be approximated with
infinite
a simple shooting algorithm: for any historically given Kq, guess some
use conditions (14)-(16) to compute the entire path of {Ct,Kt,Ht) for
then iterate on the
special case that
initial
9—1
guess
{Ct,Kt,Ht)
till
(unit EIS),
=P
mt
close
is
enough to
=
and hence Ct
(3{Kt
its
+ Ht)
is
a relevant state variable
initial
t
€
values {Co,Hq) and
[O,^], for
some
large T;
steady-state value. ^° In the
for all
t.
One can then drop
the Euler condition from the dynamic system and analyze the equihbrium dynamics with a simple
phase diagram
in the {K,
H)
space.
Steady State
4
In this section
we study the steady
in Proposition 2)
4.1
and
its
economy
the fixed point of the dynamic system
comparative statics with respect to the
an equilibrium would
that of labor income.
We
fail
level of
government spending.
Proposition
3.
(i)
the interest rate
The steady
R
to exist
if
the present value of government spending exceeded
thus henceforth parameterize government spending Gf as a fraction g of
<
aggregate output Yt and impose
g
<
1
—
state exists
a.
and
is-
<P{K,R)
Output
(1
is
unique, (ii) In steady state, the capital stock
K
are the unique solution to
l-<PiK,R)
C=
(i.e.,
Characterization
Clearly,
and
state of the
then given by
Y =
f{K), the wage
[\-a-g)f{K)
R
rate hy
uj
=
(1
^
— a)f{K), and
'
consum.ption by
- g)f{K) - 6K.
'°This presumes that a turnpike theorem apphes; this
continuity to the complete-markets case.
is
likely to
be the case at
least for
a small enough, by
Condition (18) follows from the Euler condition
from the bond market clearing condition
by
namely
(16),
H^
{u
- G) /R =
{1
an international market
R >
same
that has the
preferences, technologies
R
aggregate wealth at which the precautionary motive
instead,
G
(0,
Therefore,
rate.^^
the open economy to admit a steady state.
however, a unique
what condition
precisely
R
for
but
is
open to
rate.
If
G (0,1//?)
just offset
is
by the gap between the
both necessary and
which the net foreign asset position
of the
is
sufficient for
given by (18).
economy
is
zero,
which
(19) imposes.
graphical representation
K. For any given
g,
conditions (18) and (19) with respect
to, respectively,
the intersection of the graphs of these two functions identifies the steady
To understand how
state.
R
is
For any such R, aggregate capital
Let A'i(i?) and K2{R;g) denote the solutions
to
risks
l/p), then diminishing absolute aversion ensures the existence of a finite
and the discount
interest rate
A
and
bond, thus facing an exogenously fixed interest
for the riskless
level of
4.2
implied
of the steady state of our economy, consider for a
If,
is
H
- a - g)f{K)/R.
bound.
is,
Condition (19) follows
0.
then the precautionary saving motive implies that aggregate wealth increases without
1//5'
There
C —
(17), substituting for the steady-state value of
To better understand the determination
moment another economy
(15), setting
these graphs look
lemma examines
the next
like,
the monotonicities of
these two functions with respect to R.
Lemma
The
1.
dK\/dR >
(i)
if
intuition behind part
H
reduces both
and only
is
(ii)
if
6
> j^.
8X2/ dR <
(ii)
For given K, and hence given
simple.
and (f>{K,R), and thereby necessarily reduces the
R
then for (19) to hold with the lower
it
must be that
K
monotonically decreasing function, as illustrated in Figure
The
intuition
behind part
(i)
is
a bit
stationarity of aggregate consumption.
gregate wealth. Since
From
Ct
consume
is
given by
right
also falls.
an increase in
ui,
hand
It follows
[pt
—
nnt)
It
K2{R)
a
1.
(18)
comes from
in
turn
is
the same as imposing p
— m.
the steady-state value of the marginal propen-
follows that aggregate wealth
is
stationary
if
and
if
p+{9-l)p^9p,
where p
state
is
equivalent to imposing stationarity of ag-
Wt, this
we have that
m = 60 — {9 — l)p.
is
that
R
But
side of (19).
more convoluted. Recall that condition
Clearly, this
—
condition (13), on the other hand,
sity to
only
Wt = Pt^t —
always.
K
is
the
and
mean
(20)
return to saving and p the risk adjusted return (both evaluated at the steady
for given R).
Of
course, this condition
developing intuition.
'These intuitions are similar to those
in
Aiyagari (1994).
10
is
equivalent to (18), but
it is
more
useful for
K
K,{R)
l<2iR\gio
K2[R\ghigh)
R
',3
Figure
First, note that
in
is
an increase in
K reduces /' {K).
course, the optimal
The steady
1.
4>
state
and the
K necessarily reduces p+(0— l)p. To see this, note that an increfise
For given 0, this reduces p and p equally, thus also reducing
must
but this only reinforces the negative
fall,
shifted towards the low-return bond), while
and the
maximizes
fact that
higher government spending.
effects of
it
R
and p increase with R. But now the
and only
if
either
hkely to dominate
(f>
is
low or 9
has an ambiguous effect on p
fact that
cp
falls
works
is
if
9
is
high.
high. Indeed,
We
+
[9
completes the argument behind part
we
further
(i)
if (p
>
U-shaped curve,
as illustrated in Figure
in
R
risk
K
is
of
we prove that
Lemma
satisfied
if
and only
1.
if
—
was small
thus expect p
(since the portfolio
l)p. For given
(f),
both p
this
+
is
(9
—
to start with. Moreover,
l)p to increase with
the case
if
if
and only
and only
\f
9
R
if
> j^.
> y^, which
if
1.
show that the steady-state
the condition 9
j^
Of
in the opposite direction, contributing
Combining the above observations, we conclude that dK\/dR >
In the Appendix,
\)p.
does not affect p (because of the envelope theorem
to lower p. Intuitively, though, this effect should be small
is
on p
—
[9
p).
Next, note that an increase in
the impact of p
effect
p+
R
is
Intuitively,
<?!i
a decreasing function of R. Hence,
is
high enough.
when R
is
It
follows that
close to
/?,
Ki (R)
is
a
a marginal increase
has such a strong positive effect on steady state wealth, that the consequent reduction in the
premium more than
offsets the increase in the
opportunity cost of investment, ensuring that
increases with R.
As noted
earlier, the intersection of the
two curves
identifies the
economy. The existence and uniqueness of such an intersection
the proof of Proposition
an increase
in
3).
What we
next seek to understand
government spending.
11
is
is
steady state of the closed
established in the
how
Appendix
this intersection
(see
changes with
The long-run
4.3
government consumption
effects of
Because g does not enter condition
the
Ki
On
curve.
hence lower
H
as illustrated in Figure
lump-sum
in
R
unambiguously
falls,
the two curves intersect in the upward or the
Lemma
main
1
we know
result of the
Proposition
rate
and
{uj),
it
the saving rate (s
government consumption
= 5K/Y)
of capital to the discount rate (/'
effect
is
{K/N) —
on either
R
and only
if
state interest rate
state capital-labor ratio
consumption has no
of the
upward portion
K\
if
K
downwards,
effect of higher
depends on whether
curve.
and only
From
>
if
or
is
equated to the discount rate {R
is
6
=
,6).
K/N, Y/N,
It
lower.
It
On
fall
R
in
—
(3),
ui,
and
s.^^
risk,
if
financed
they have a precautionary mo-
risk aversion, this
motive
is
stronger
by reducing household wealth, higher government
follows that,
can be stationary has to be lower, which proves that
of this
the
follows that, in the long run, government
spending stimulates precautionary saving. But then the
The impact
The
determined by the equality of the marginal product
Because preferences exhibit diminishing absolute
the level of wealth
y^^.
of
if
with lump-sum taxation. Because households face consumption
when
(i)
{g) necessarily decreases the risk-
Here, instead, government consumption can have non-trivial long-run effects, even
tive to save.
part
{K/N), labor productivity (Y/N),
also decreases the capital-labor ratio
With complete markets, the steady
and the steady
downward portion
and
then immediate.
is
4. In steady state, higher
free rate (R), while
wage
whereas the impact on
that the intersection occurs in the
paper
shift
simply a manifestation of the negative wealth
is
affect
net-of-taxes labor income,
government consumption causes the K2 curve to
This
1.
Clearly,
taxes.
means lower
the other hand, because higher g
an increase
,
an increase in government consumption does not
(18),
R
risk-free rate at
falls
which aggregate saving
with g?^
on the capital-labor ratio now depends on two opposing
effects.
the one hand, because of diminishing absolute risk aversion, a lower level of wealth implies a
lower willingness to take risk, which tends to discourage investment.
earlier,
the other hand, a lower
opportunity cost of investment, which tends to stimulate investment.
risk-free rate implies a lower
As explained
On
the wealth effect dominates
when
long as the elasticity of intertemporal substitution
is
d
>
yrs-^^ Since
(j)
<
a, this
is
the case as
high relative to the income share of capital.
(as in Section 5). The only difference is that in the
changes and hence
and Y also change.
'^A similar intuition underlies the steady-state supply of saving in Aiyagari (1994).
^''in the Appendix we prove that the steady-state (^ is a decreasing function of the stead-state R, and hence an
increasing function of g. It follows that, whenever the steady-state
is a. non-monotonic function of g, it is a
[/-shaped function of g. Note, however, that a high enough 9 may suffice for d to be higher than <!>/ {1 — <t>) for all
feasible levels of g, and hence for
to be a globally decreasing function of g.
'^This result
is
latter case, while
true even
K/N
and
when
Y/N
labor supply
is
endogenous
continue to not change,
K
A'"
K
K
12
> j^ is easily satisfied. For
be 65% of GDP. Then H is about
For empirically plausible calibrations of the model, the condition 9
example, take the interest rate to be
GDP,
16 times
state j3j
=
^,
i?
or equivalently 4 times
= 4%
and labor income
K,
we assume a
if
capital-output ratio of
about 0.25. This
j^
this exercise gives a calibrated value for
to
is
of the recent empirical estimates of the elasticity of intertemporal substitution,
around
1
if
not higher. ^^
follows that a negative long-run effect of
It
4.
Since in steady
far lower
than most
which are typically
government consumption on
aggregate saving and productivity appears to be the most hkely scenario.
Numerical simulation
4.4
We now
numerically simulate the steady state of our economy, to get a
first
pass at the potential
quantitative importance of our results.
The economy
P
is
fully
is
parameterized by (a,
the discount factor, 7
rate, 9
is
/?,
7,
5, 9,
where a
a, g),
is
the coefficient of relative risk aversion, 5
is
the elasticity of intertemporal substitution,
return on private investment, and g
is
In our baseline parametrization,
a
the (mean) depreciation
is
the standard deviation of the rate of
is
the share of government consumption in aggregate output.
we take a
—
0.36,
— 0.96,
7 = 5, a
(3
For risk aversion, we take
are standard in the literature.
the income share of capital,
and
value
5
—
these values
0.08;
commonly used
in the
macro-finance hterature to help generate plausible risk premia. For the elasticity of intertemporal
substitution,
we take
9
=
1,
a.
value consistent with recent micro and macro estimates.-'®
the share of government, our baseline value
alternative
What
is
g
— 40%
remains
is
(as in
a. Unfortunately, there
the "typical" investor in the
risks are significant.
entry
is less
some European
US
is
= 25%
g
(as in the
United States) and a higher
countries).
is
no
direct
measure of the rate-of-return
risk faced
For instance, the probability that a privately held firm survives
five years after
than 40%. Furthermore, even conditional on survival, the risks faced by entrepreneurs
ment, not only there
is
also the volatility of the
large:
as
Moskowitz and Vissing-j0rgensen (2002) docu-
a dramatic cross-sectional variation in the returns to private equity, but
book value
as that of the index of public firms
of a (value- weighted) index of private firms
—one more indication that private equity
public equity. Note then that the standard deviation of annual returns
for the entire
pool of pubhc firms;
firm-specific risk);
be similar
and
it is
it is
about 40%
is
is
is
twice as large
more
risky than
about 15% per annum
over 50%o for a single public firm (which gives a measure of
for
a portfolio of the smallest public firms (which are likely
to large private firms).
'^ See, for example, Vissing-j0rgensen and Attanasio
(2003), Mulligan (2002), and Gruber (2005). See also
(2006) and Angeletos (2007) for related discussions on the parametrization of the EIS.
16
by
economy. However, there are various indications that investment
and private investors appear to be very
to
For
See the references in footnote
15.
13
Guvenen
Given
and
this suggestive evidence,
these
numbers are somewhat
— 20%
a
baseline parameterization and consider
want of a better
in
and a
= 40%
comparable
is
Consumer Expenditure Survey (CEX),
we take a = 30%
is
3%
to its empirical counterpart. For in-
Malloy, Moskowitz and Vissing-j0rgensen
Similarly, using data that include
for non-stockholders).
goods, Ait-Sahalia, Parker and Yogo (2001) get estimates between
tions,
per
8%
stockholders
for
consumption of luxury
6% and
15%. In our simula-
on the other hand, the standard deviation of individual consumption growth
annum
our
the volatility of
reassuring:
(2006) estimate the standard deviation of consumption growth to be about
(and about
for
Although
for sensitivity analysis.
arbitrary, the following observation
individual consumption generated by our model
stance, using the
alternative,
less
is
than
5%
(along the steady state).
Putting aside these qualifications about the parametrization of
titative effects of
a,
we now examine the quan-
government consumption on the steady state of the economy. Table
1
reports
the per-cent reduction in the steady-state values of the capital-labor ratio (K/N), labor productivity
{Y/N), and the saving rate
what
(s), relative to
CM
Complete markets are indicated by
Y/N
CM
IM
s
CM
IM
-10.02
-3.73
-1.14
17
-12.18
-1.21
20
-6.78
-4.57
-2.5
-0.88
12
-17.82
-6.82
-2.05
28
Table
J
1.
The
steady-state effects of the size of government.
In our baseline parametrization, the capital-labor ratio
when g =
0.
Similarly, productivity
point lower. These are significant
effects of
is
is
about
effects.
4%
They
= 25%
about 10% lower when g
lower and the saving rate
is
about
1
than
percentage
are larger (in absolute value) than the steady-state
eff'ects
of a marginal tax on capital
income equal to 17%
(The tax rate on capital income that would generate the same
effects
in the
complete-markets
under complete markets
given in the last column of the table, as r^guiv)
Not
surprisingly, the effects are smaller
because then risk matters
row): productivity
is
less.
almost
18%
On
if
a
is
lower (third row) or
'^Here, since labor supply
is
if
7
is
the other hand, the effects are larger
lower, the saving rate
on capital income that would have generated the same
is
is
precautionary saving reported in Aiyagari (1994). They are equivalent to what would be
the steady-state
case.
CM
= 40%
= 20%
= 40%
g
g were 0.^^
eqmv
IM
baseline
a
a
if
and incomplete markets by IM.
K/N
CM
would have been
their values
exogenously fixed, the changes
in
is
g
= 40%
and
Y
under complete markets
coincide with those in
not the case in the extensions with endogenous labor supply in the next section.
14
when
2 percentage points lower,
effects
K
lower (not reported),
K/N
(final
and the tax
is
28%.
and Y/N\
this
,
Table 2 turns from
increase government spending
first case,
productivity
marginal
level to
by
1
reports the change in
it
25%
percent, either from
by 0.19%;
falls
effects:
in the second,
have been under complete markets the
and about
g
g
Table
2.
= 25%
= 40%
Long-run
effects of a
equivalent to
is
what would
0.8 percentage points in the second case.
Y/N
CM
equiv
IM
CM
-0.52
-0.19
0.75
-0.71
-0.26
0.8
IM
^ 26%
^ 41%
we
the tax rate on capital income by about
K/N
CM
s as
26%, or from 40% to 41%. In the
to
by 0.26%. This
effect of increasing
0.75 percentage points in the first case,
K/N, Y/N, and
permanent 1% increase
government consumption.
in
Endogenous labor
5
In this section
we endogenize labor supply
economy.
in the
We
consider three alternative specifi-
cations that achieve this goal without compromising the tractability of the model.
GHH
5.1
One
easy
preferences
way
to
accommodate endogenous labor supply
model
in the
is
to
that rule out income effects on labor supply, as in Greenwood, Hercowitz and
particular, suppose that preferences are given
u{ctM)
where
It
denotes leisure and u
=
by Uq
iziz [ct
—
+
Eq
/q°° e~'^^u [ct,
where
I
{ui)
argmaxj {v
(l)
—
Huffman
k) dt, with
1-7
f22^
The
a strictly concave, strictly increasing function.^®
is
u>l}
(1998). In
v(Zt)]
then proceed as in the benchmark model, with labor supply in period
=
assume preferences
t
given hy Nt
analysis can
=
I
—
I
{uit)
.
This specification highlights an important difference between complete and incomplete markets
with regard to the employment impact of
in
fiscal
Under incomplete markets, an increase
government spending can have a negative general-equilibrium
This
is
never possible with complete markets, but
increase in g reduces the capital-labor ratio,
labor supply. Indeed, with
in
shocks.
GHH
it is
effect
possible with incomplete markets
and thereby the wage
preferences, 6
> j^
on aggregate employment.
sufiices for
rate,
both
which
K/N
in
and
when an
turn discourages
N
to
fall
with g
both the short run and the long run.
To
allow for d
^
I/7,
we
let
Ui
=
E(
J^ z{cr + v(It), V-r)dT,
15
with the function z defined as
in
condition
(5).
Although
it
is
unlikely that wealth effects on labor supply are zero in the long run, they
weak
well be very
positive shock to
In the light of our results, one
in the short run.
may
may
then expect that after a
government consumption both employment and investment could drop on impact
under incomplete markets. ^^
KPR
5.2
A
preferences
second tractable way to accommodate endogenous labor supply
homothetic preferences over consumption and
u{cult)
where
It
The
denotes leisure and
ip
G
(0, 1) is
benefit of this specification
comparable to previously reported
is
=
= Eg / e""^*u {ctJt) dt,
(7o
that
it
is
standard
is
now
=
I
and Ct
(for
given
Christiano and Eichenbaum (1992), we take
A
The only
essential novelty
is
that
T
—
.
-IpUJt
now captured by
is
the negative relationship
ip
—
with King, Plosser, and Rebelo (1988) and
line
0.75.
This value ensures that the steady-state
worked approximately matches the
are as in the baseline specification of the
5.3
The homo-
tot).
For the quantitative version of this economy, in
fraction of available time
with one another. ^^
,
neoclassical effect of wealth on labor supply
A''^
cost in tractability once
— L {ut, Ct) where
I
between
leisure
the 'benchmark model. ^^
given hy Nt
the literature (making our results
then preserved and the equilibrium analysis
is
L{ut,Ct)^The
in
comes with zero
also
it
problem
theticity of the household's optimization
aggregate employment
(23)
^°
a scalar.
results), while
in
with
Th^[cl~'''lt]'-',
augmented with the assumption that agents can trade
proceeds in a similar fashion as
King, Plosser, and Rebelo (1988). In
leisure, as in
by
particular, suppose that preferences are given
to assume that agents have
is
US
data.
The
rest of the
parameters
benchmark model.
Hand-to-mouth workers
third approach
is
to spht the population into
two groups. The
holds that have been modeled in the benchmark model;
we
first
group consists of the house-
will call this
group the "investors".
'^This discussion indicates that an interesting extension might be to consider a preference specification that allows
for
weak short-run but strong long-run wealth effects, as in Jaimovich and Rebelo
for 9 ^ I/7, we let Ut = Et J^ 5(c?/}.-*, Ur)dT, with z defined as in
^°To allow
^'Clearly, this last assumption
with individual wealth.
is
for
modeling convenience:
^^The proofs are available upon request.
16
it
(2006).
(5).
allows individual leisure to increase proportionally
The second group
consume
consists of households that supply labor but
their entire labor
income at each point
workers". Their labor supply
is
we
in time;
do not hold any
assets,
and simply
group the "hand-to-mouth
will call this
given by
Nhtm^^e^^fjhtmyc^
where
C/'*'"
denotes the consumption of these agents, e^
labor supply, and
Cc
>
parameterizes the wealth
This approach could be justified on
its
own
(24)
>
parameterizes the wage elasticity of
elasticity.^''
merit. In the United States, a significant fraction of
the population holds no assets, has limited abihty to borrow, and sees
income almost one-to-one. This
But
model.
is
what the
unclear
hand-to-mouth workers
preserving
A
is
model
"right"
a crude
way
for these
of capturing this
elasticities of
that
is
it
households
is.
Our
form of heterogeneity
The
point
labor supply. Whereas the
is
freedom
also gives
benchmark
specification with
in the
model while
parameterizing the wage and
in
KPR preference specification imposes e^ = — e^ =
the specification introduced above permits us to pick
evidence.
of heterogeneity than our
its
its tractability.
side benefit of this approach
wealth
model
fact calls for a richer
consumption tracking
its
much
lower elasticities, consistent with micro
not to argue which parametrization of the labor-supply elasticities
appropriate for quantitative exercises within the neoclassical growth model; this
long debate in the hterature, to which
we have nothing
1,
to add.
The
point here
is
is
is
more
the subject of a
rather to cover a
broader spectrum of empirically plausible quantitative results.
For the quantitative version of this economy, we thus take
are in the middle of
most micro
e^^
—
What then remains
estimates.^'*
income absorbed by hand-to-mouth workers. As mentioned above, a
population holds no assets. For example, using data from both the
(2006) reports that the lower
accounts for about
70%
consumption even when
of aggregate
economy
80%
of the wealth distribution
of aggregate consumption. Since
their net
worth
consumption accounted
is
for
zero,
70%
is
value of the relevant parameter that one would estimate
consumption data
—we can deduce
this
and
Cc
=
—0.25, which
the fraction of aggregate
significant fraction of the
PSID and
the SCF,
be an upper bound
We
able to
and
smooth
for the fraction
thus opt to calibrate the
of aggregate consumption. This
if
US
Guvenen
of aggregate wealth
some households may be
likely to
50%
is
owns only 12%
by hand-to-mouth agents.
so that hand-to-mouth agents account for
0.25
the model were to match
US
is
also the
aggregate
from Campbell and Mankiw (1989)."^
=
Ci" — n^", for appropriate CcCnand Blundell and MaCurdy (1999).
^^Note that the specification of aggregate consumption considered in Campbell and Mankiw coincides with the
one implied by our model. Therefore, if one were to run their regression on data generated by our model, one would
correctly identify the fraction of aggregate consumption accounted for by hand-to-mouth workers in our model. This
impHes that it is indeed appropriate to calibrate our model's relevant parameter to Campbell and Mankiw's estimate.
^'Preferences that give rise to this labor supply are Ui
^"See, for example,
Hausman
(1981),
MaCurdy
(1981),
17
The long-run
5.4
Our main
effects of
theoretical result (Proposition 4) continues to hold in
benchmark model:
the wage rate
a; if
it
also reduces the capital-labor ratio
and only
if
K/N,
above variants of the
labor productivity
the elasticity of intertemporal substitution 9
not clear anymore
is
of the
all
steady state, a higher rate g of government consumption necessarily reduces
in
the interest rate R; and
What
government consumption with endogenous labor
the effect of g on
is
i^"
and Y, because now
Y/N, and
higher than yrs-^^
is
is
A''
not fixed.
On
one hand, the reduction in wealth stimulates labor supply, thus contributing to an increase
This
the famihar neoclassical effect of government spending on labor supply.
is
as long as ^
> j^,
it is
is
elasticity of labor
GHH
supply
is
sufficiently high relative to its
where the wealth
specification,
permanent increase
effect
hand-to-mouth workers, where we have freedom
KPR preferences,
of
overall
unambiguously positive under complete markets.
to a reduction in long-run emploj^ment after a
in the
The
therefore ambiguous under incomplete
Other things equal, we expect the negative general-equilibrium
wage
the other hand,
the novel general-equihbrium effect due to incomplete markets.
is
government spending on aggregate employment
markets, whereas
A''.
the reduction in capital intensity depresses real wages, contributing towards a
reduction in N. This
effect of
On
in
the
where both
in
is
zero.
It
in
government spending,
income
elasticity.
can also be verified
choosing these
elasticities are restricted to
dominate, thus leading
effect to
elasticities,
This
is
if
the
clear
for
the case of
but not
in the case
equal one.
Given these theoretical ambiguities, we now seek to get a sense of empirically plausible quantitative effects.
As already
of pedagogical value.
economy with
KPR
We
discussed, the
GHH
case (zero wealth effects on labor supply)
is
merely
thus focus on the parameterized versions of the other two cases, the
(homothetic) preferences and the economy with hand-to-mouth workers.
Table 3 then presents the marginal
effects
on the steady-state
levels of
the capital-labor ratio,
productivity, employment, and output for each of these two economies, as g increases from
to
26%, or from 40%
to 41%.^^ The case of
with hand-to-mouth workers
by
CM
is
indicated by
KPR preferences
HTM.
is
KPR,
indicated by
25%
while the case
In either case, complete markets are indicated
and incomplete markets by IM.
Regardless of specification, the marginal effects of higher government spending on capital intensity
K/N
and labor productivity
is
wealth
effect of
so that
A''
is
for aggregate
employment
increases with higher g under either complete or incomplete markets.
is
weaker under incomplete markets, especially
true as long as the steady state
benchmark model.
is
in the
the
However, the
economy with hand-
unique, which seems to be the case but has not been proved as in the
Also, in the variant with hand-to-mouth agents,
we have
to be cautious to interpret
of private equity to effective wealth for the investor population alone.
^^We henceforth
A'',
higher g turns out to dominate the effect of lower wages under incomplete markets,
employment stimulus
^^This
are negative under incomplete markets (and are stronger the
whereas they are zero under complete markets. As
higher
g),
Y/N
focus on marginal rather than level effects just to economize on space.
18
4>
as the ratio
K/N
CM
g
g
= 25%
= 40%
N
Y/N
CM
IM
Y
CM
IM
CM
IM
equiv
IM
CM
KPR
-0.33
-0.12
1.4
1.27
1.4
1.15
0.52
HTM
-0.3
-0.11
0.38
0.38
0.38
0.27
0.46
^ 41% KPR
-0.52
-0.19
1.76
1.53
1.76
1.34
0.68
HTM
-0.36
-0.13
0.57
0.57
0.57
0.44
0.48
26%
-^
Table
to-mouth workers. The same
complete markets, but
less so
are on average equivalent to
3.
is
Long-run
effects
with endogenous labor.
true for aggregate output:
under incomplete markets.
what would have been the
it
increases under either incomplete or
Finally, the
incomplete-markets
effect of increasing
effects
the tax rate on capital
income by about 0.55% under complete markets.
Dynamic responses
6
The
results so far indicate that the long-run effects of
affected
by incomplete
risk sharing.
We now
impulse response of the economy to a
Starting from the steady state with g
in
government spending.
to g
=
26%).
=
examine how incomplete
shock.
fiscal
government consumption can be significantly
risk sharing affects the entire
^^
25%, we
hit the
We then trace its transition
to the
economy with a permanent 1%
new steady
state (the one corresponding
We conduct this experiment for both the economy with KPR
and the economy with hand-to-mouth workers, each parameterized
either case, the transitional
increase
(homothetic) preferences
as in the previous section; in
dynamics reduce to a simple system of two
first-order
ODE's
in
[Kt-Ht]
when 0=1?^
The
results are presented in Figures 2
and
3.
Time
in years
is
on the horizontal
axis,
deviations of the macro variables from their respective initial values are on the vertical axis.
interest rate
differences.
and the investment rate
The
As evident
different
solid lines indicate
ai"e
in
while
The
simple differences, the rest of the variables are in log
incomplete markets, the dashed lines indicate complete markets.
in these figures, the quantitative effects of a
permanent
fiscal
shock can be quite
between complete and incomplete markets. The overall picture that emerges
"^Note that the purpose of the quantitative exercises conducted here, and throughout the paper,
the abihty of the model to match the data. Rather, the purpose
is
is
is
that the
not to assess
to detect the potential quantitative significance
we took from the standard neoclassical growth model.
^Throughout, we focus on permanent shocks. Clearly, transitory shocks have no impact in the long run. As for
their short-run impact, the difference between complete and incomplete markets is much smaller than in the case
of permanent shocks. This is simply because transitory shocks have very weak wealth effects on investment as long
as agents can freely borrow and lend over time, which is the case in our model. However, we expect the difference
between complete and incomplete markets to be larger once borrowing constraints are added to the model, for then
investment will be sensitive to changes in current disposable income even if there is no change in present- value wealth.
of the particular deviation
19
10
5
(a)
Aggregate Output Yt
10
(c)
(b)
25
15
(e)
20
25
2.
Dynamic responses
26
Aggregate Employment Nt
(d)
Investment Rate It/Yt
30
Labor Productivity Yt/Nt
Figure
20
30
Capital-Labor Ratio Kt/Nt
15
15
(f)
to a
permanent
20
Interest
stiock witii
Rate Rt
KPR preferences.
30
(g)
(i)
(h)
Capital-Labor Ratio Kt/Nt
(k)
Figure
Aggregate Output Yt
(j)
Labor Productivity Yt/Nt
3.
Dynamic responses
Aggregate Employment Nt
Investment Rate It/Yt
(1)
to a
Interest Rate Rt
permanent shock with hand-to-mouth
21
agents.
employment and output stimulus
of a
permanent increase
now we
see that the
This picture holds for both the economy with
But there are
workers.
of incomplete markets
stronger in the
some
also
government spending
And whereas we
incomplete markets than under complete markets.
long-run response of the economy,
in
same
is
true for
already
its
is
weaker under
knew
this for the
short-run response.
KPR preferences and the one with hand-to-mouth
interesting differences between the two.
on the employment and output stimulus
economy with hand-to-mouth workers. As a
of
result,
The
mitigating effect
government spending
whereas the short-run
is
much
effects of
higher government spending on the investment rate and the interest are positive under complete
markets in both economies, and whereas these
the economy with
KPR preferences,
effects
remain positive under incomplete markets
in
they turn negative under incomplete markets in the economy
with hand-to-mouth workers.
To understand
this result, consider for a
to-mouth workers and labor supply
change
in
is
moment
the benchmark model, where there are no hand-
completely inelastic. Under complete markets, a permanent
government spending would be absorbed one-to-one
in private
consumption, leaving
investment and interest rates completely unaffected in both the short- and the long-run.
incomplete markets, instead, investment and the interest rate would
long run. Allowing labor supply to increase in response to the
and the
interest rate
labor supply
is
jump upwards under complete
weak enough, the response
fall
fiscal
on impact, as well
Under
as in the
shock ensures that investment
markets. However, as long as the response of
of investment
and the
interest rate
can remain negative
under incomplete markets.
As a
final point of interest,
associated with a permanent
1%
we
calculate the welfare cost, in terms of consumption equivalent,
increase in government spending.
drops by 0.2%, whereas under incomplete markets
cost of an increase in
government spending
is
it
Under complete markets, welfare
drops by 0.6%. In other words, the welfare
three times higher under incomplete markets than
under complete markets.^''
To
recap, the quantitative results presented here indicate that a
modest
level of
uninsured
idiosyncratic investment risk can have a non-trivial impact on previously reported quantitative
economy with
KPR preferences
classics in the related literature, Aiyagari, Christiano
and Eichenbaum
evaluations of fiscal policy. Note in particular that our quantitative
is
directly
(1992)
fiscal
comparable to two
and Baxter and King
(1993). Therefore, further investigating the
macroeconomic
effects of
shocks in richer quantitative models with financial frictions appears to be a promising direction
for future research.
''"Here we have assumed that government consumption has no welfare benefit, but this should not be tal<en hterally;
nothing changes if Gt enters separably in the utihty of agents.
22
Conclusion
7
This paper
revisits the
macroeconomic
effects of
government consumption
version of the neoclassical growth model. Incomplete markets
make
an incomplete-markets
in
individual investment sensitive
government spending can crowd-
to individual wealth for given prices. It follows that an increase in
out private investment simply by reducing disposable income. As a result, market incompleteness
can seriously upset the supply-side
even
if
wages
financed with
an increase
in
government consumption,
reduce capital intensity, labor productivity, and
ta:xation, tends to
both the short-run and the long-run. For plausible parameterizations of the model, these
in
results
lump-sum
effects of fiscal shocks:
appear to have not only qualitative but also quantitative content.
These
results might, or
might not, be bad news
for the ability of the neoclassical
explain the available evidence regarding the macroeconomic effects of
fiscal
paradigm to
shocks. However, the
goal of this paper was not to study the ability of an incomplete-market variant of the neoclassical
growth model to match the relevant data. Rather, the goal was to identify a mechanism through
which incomplete markets modify the response of the economy to
fiscal
shocks.
This mechanism was the dependence of individual investment on individual wealth.
In our
model, this property originated from uninsured idiosyncratic investment risk combined with diminishing absolute risk aversion. Borrowing constraints
to wealth (or cash flow). Also, this
in the neoclassical
would lead
to similar sensitivity of investment
mechanism need not depend on whether
paradigm) or sticky
Keynesian paradigm). The key insights of
(as in the
paper are thus clearly more general than the
specific
model we employed
importance of these insights within richer models of the macroeconomy
An
important aspect
left
outside our analysis
tures. In this paper, as in
much
spending
lump-sum
is
financed with
prices are flexible (as
is
is
— but
this
the quantitative
an open question.
the optimal financing of government expendi-
of the related literature,
we assumed that
taxation, only because
we wished
the increase in government
to isolate wealth effects from
tax distortions. Suppose, however, that the government has access to two tax instruments, a lump-
sun tax and a proportional income
as to
maximize ex ante
tax.'^^
Suppose further that the government chooses taxes so
utility (equivalently,
a utilitarian welfare criterion) subject to
no inequality)
constraint. Clearly, with complete markets (and
exogenous increase
however,
it
is
in
likely that
an increase
in
government spending
Further exploring these issues
Werning
is
to finance
any
financed with a mixture of both
lump-sum tax would disproportionately
agents, using both instruments permits the
in
would be optimal
budget
government spending with only lump-sum taxes. With incomplete markets,
instruments: while using only the
'As
it
its
is left
(2006), this might be a
government to trade
affect the utility of
off less efficiency for
for future research.
good proxy
for
more general non-linear tax schemes.
23
more
poor
equality.
Appendix: Proofs
8
Proof of Proposition
Let
1.
J{iL',t)
The
value function depends on time
{u^i,
Rt}telQ,oo)
t
denote the value function
for the household's
problem.
because of discounting as well as because the price sequence
used not be stationary. However, the value function does not depend on
because
z,
households have identical preferences, they have access to the same technology, and they face
the same sequence of prices and the same stochastic process for idiosyncratic
equation that characterizes the value function
= max{
Tn,4>
z{m.w,J{w,t))
+
—-{w,t)
at
I
is
risk.
The Bellman
given by
+ -— (w, t)[(/)fi +
aw
-
(1
(l>)Rt
- m]w + --—^{w,t)(frw'^a'
I
ow^
(25)
The
first
term
of the
Bellman equation
from current-period consumption; the
(25) captures utihty
second term takes care of discounting and the non-stationarity in prices; the third term captures
the impact of the
mean growth
in wealth;
CRRA/CEIS
specification of preferences, an educated guess
and the
last
term
(Ito's
term) captures the impact of
risk.
Because of the
is
that there exists
a deterministic process Bj such that
J{w,t)^Bt-
1-77
Because of the homogeneity of J
in
w, the Bellman equation then reduces to
dJ
dJ
= m^|z(7n, J(l,t)) + ^(1,0
+ ^(l,i)[^fi +
,,
so that the optimal
= max
m,0
The
first
m
and
(p
^
|——^[B.^m^-i/^
Iju
1]
+
i
order condition for
first
(1
-
,,„
(j)
+
f^
~
-l
[0n
7
+
and
(1
1^2
J
- m] + l^(l,i)<^2^2
,
4>)Rt
are independent of w. Using (5)
-
(26), the
<t>)Ri
I
^
(3?)
above becomes
- m] - \i4>''A
(28)
I
gives
0t
while the the
(26)
.
order condition for
m
= n-Rt
7^2
(29)
'
gives
;ives
i-e
rrit
=
(30)
24
Substituting this into (28), using the definition of
o
This
ODE,
(30), this
—
= -T:ri
is
is
^*-
(0
-
l)f>t
the Euler condition (13).
labor supply
Fk
137 +
equivalent to
Proof of Proposition
wage
+
together with the relevant transversahty condition, determines the process for Bt- Using
—
= m, +
mt
which
and rearranging, we get
pt,
—
[Kt, 1)
Combining
uJt
= Fi
this
if
and only
with Kt
—
is
=
n{u)t)Kt
if
{Kt, 1) and, similarly, the equilibrium
The bond market, on
5.
demand
Since aggregate labor
the labor market clears
is 1,
satisfies
2.
J^n\
1. It
=
n{u)t)Kt
follows that the equilibrium
mean return
the other hand, clears
if
and aggregate
and only
to capital satisfies
if
=
(1
—
(t>t)Wt
ft
+
=
Hf.
4>tWt gives condition (17).
Combining the intertemporal government budget with the
definition of
human
wealth,
we
get
(31)
Expressing this in recursive form gives condition
Combining the
-
[ptWt
Kt
—
latter
- [RtHt - Wt
+ LOt — Ct — Gt-
Ct)
ftKt
(16).
= pWt — Ct-, we have Kt = Wt — Ht =
+ Gt) Using ptWt - ft<i>tWt + Rt{l- (pt) Wt - nKt + RtHt, we get
Together with the fact, in equilibrium, ftKt + uJt = F {Kt,l) — 5Kt, this
with Kt
+ Ht = Wt
and Wt
.
gives condition (14), the resource constraint.
Finally, using Ct
Ct
=
[pt
—
f^t)
=
Wt and
rht/mt
+ Wt/Wt
(13), gives condition (15), the aggregate
Proof of Proposition
K and
=
and therefore Ct/Ct
rritWt,
3.
First,
we
together with
Wt
=
PtWt
—
Euler condition.
derive the two equations characterizing the steady state
R. In steady state, the Euler condition gives
=
(p
-/?)-
(0
-
1) 1^0-2 02,
where
Combining and solving
directly
from
(16)
and
for /'
{K) gives condition
(18).
Condition
(19),
on the other hand, follows
(17).
Next, we prove existence and uniqueness of the steady state.
25
Let n{R) and 4>{R) denote,
respectively, the risk
by
premium and the
weahh
fraction of effective
held in capital,
when
K
given
is
(18):
m^{^^{^)
Note that
^i'{R)
<
and
4>'{R)
<
0.
Next,
and
m^^l^^^^^f^)-
K {R)
let
denote the solution to (18), or equivalently
1
^{R)
K[R)
+
6
+R
Q-l
(32)
a
Finally, let
This represents the ratio of the net foreign asset position to domestic capital of an open economy
that faces an exogenous interest rate
Y—
and
G
f{K) = K", where a > 0,^ >
of the steady state (for the
solves
R
=
D{R:g)
(Note that we have used
(0, 0).
0,
dosed economy),
it
i+e PV^'^
is
finite
and hence both
/i(/3)
=
DiR;
0,
g)
^
implying
hrn D{R;g)
=
(1
this,
establish existence
show that there
=
—
0"*"
>
and K{P)
^lim^
=
p
D {R)
^
"
Kcompl
- a - g)K{Pr-']- -
— a)Y, G —
gY,
and uniqueness
exists a
unique
R
that
and
R
—^
P~ Note that
.
/i(0)
=
in
lim
R^0-
^
=
+
1
= +-•
(/')~^ (0)
J- +
1
=
is finite.
It follows
-oo.
(p{R)
R, ensure the existence of an
R
£
(0,/3)
D (R) = 0.
we now show that
D {R; g)
strictly decreasing in
is
R, then we also have uniqueness. To show
note that from (33),
dD
dR
-^ = {l-a-g)
"
Now
{\
A'(0) are finite. It follows that
These properties, together with the continuity of
If
To
I.)
as /2
- a - ,)i.(0)"-
(/>(/?)
(1
and
(j){0)
D
fi-»/3-
such that
<
=
0.
^lim^
Furthermore,
g
suffices to
Fix g henceforth, and consider the limits of
(
+
and a
ui
K{R) Q-l
'
i?2
..-.«ll^-:
+
^.
note that
a
'
K
a-lf'{Ky
26
02
^2
'
(34)
.
we suppress the dependence
/here
dD
K,
of
and
ji,
on
<p
R
for notational simphcity.
2
l-a-gf'{K)
a
~dR
R^
^
a
/i'
{R)
<
and
R < f (K
.
.
R
all
G
/
.
ja^
R^
{R)) for
follows that
+
l~a-gRfi' + R-f'{iq
Since
It
II-
(0, /?),
we have
that
dD/dR <
for all
R
e
(0,
0),
which completes the argument.
Proof of
Lemma
1.
Recall that (18)
is
equivalent to
9{p-p)-{9-l)jci>^a^ =
where p
we
—
(j)f'
{K)
+ {l —
(p)
R
and
cp
=
(/' (A')
/'jcr'^.
Applying the implicit function theorem,
get
dK
4>-e{\-<j))
OR
which proves
i)
g) A'",
we
'
"
'
'
"
= [f {!<) -
I
dK
which proves that 8X2/ dR <
Proof of Proposition
dD/8R <
0, this
g on the steady-state
on g and
is
f [K) =
R)/ja^,
aK'^'^ and
w - G -
get
4.
always.
From
(33),
we have that dD/8g <
imphes that the steady-state
K
<0
i^:
(i-«)^
'dR
that
/"
i^<oo^< ^.
the other hand, from (19), using
— a—
+
1
that:
dR
On
<^(^
(18)
81
(1
— R)
0,
R
R
if
and only
27
if
9
Together with the property
necessarily decreases with
then follows from the fact that Ki {R)
increasing/decreasing in
0.
is
,
g.
The impact
of
defined by (18), does not depend
higher/lower than
(p/ {\
—
(f))
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