Massachusetts
Institute of
Technology
Department of Econonnics
Working Paper Series
SPECULATIVE GROWTH:
HINTS FROM THE US ECONOMY
Ricardo
J.
Caballero
Emmanual
Mohamad
L.
Farhi
Hammour
WORKING PAPER
02-45
November 27, 2002
REVISED: May 10, 2004
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E52-251
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Iittp://www.archive.org/details/speculativegrowt0245caba
Speculative Growth:
US Economy
Hints from the
Ricardo
J.
Emmanuel
Caballero
May
This draft:
Mohamad
Farhi
10,
L.
Hammour*
2004
Abstract
We propose
a framework for understanding recurrent historical episodes of vigorous
economic expansion accompanied by extreme asset valuations, as exhibited by the U.S.
in the 1990s.
We
interpret this
phenomenon
as a high-valuation equilibrium with a
low effective cost of capital based on optimism about the future availability of funds
The key
for investment.
to the sustainability of such an equilibrium
We
increased growth to an increase in the supply of effective funding.
feedback from
is
show that such
feedback arises naturally when an expansion comes with technological progress in the
capital producing sector,
when
fiscal rules
generate sustained
the rest of the world has lower expansion potential, and
relaxed by the expansion
itself.
when
fiscal surpluses,
financial constraints are
Arguably, these ingredients were
present in the U.S. during the 1990s.
We also show
improving but they can crash. The
latter is
when
all
simultaneously
that such expansions can be welfare
more
likely if
bubbles develop along the
expansionary path. These (rational) bubbles can emerge even when the interest rate
exceeds the rate of growth of the economy.
•Respectively:
seajch unit).
Briigemann
MIT and NBER; MIT; DELTA
CEPR (DELTA
is
a joint
E-mail: caball@mit.edu; efarhi@mit.edu; hammour@delta.ens.fr.
for
CNRS-EHESS-ENS
We
re-
are grateful to Bjorn
outstanding research assistance and to the European Central Bank Research Department
and the International Institute
for
Economic Studies
Angeletos, Gadi Barlevy, Olivier Blanchard,
Portier, Pietro Reichlin,
DELTA, ESSIM,
for useful
and
We
thank Franklin Allen, Marios
Peter Diamond,
Thomas
Philippon, Frank
Jean Tirole, Jaume Ventura and seminar participants at Boston University,
Harvard, IIES, MIT, the
comments. Caballero thanks the
transformed version of
for their hospitality.
Thomas Chaney,
NBER WP
NBER EFG
NSF
meetings. Northwestern, Toulouse, and
for financial support.
#9831, December 2002.
This
is
CORE,
Wharton
an extensively revised and
Introduction
1
Economic history has witnessed many stark "speculative growth" episodes
of extreme stock
market valuations accompanied by brisk economic growth. The most notable recent expe-
was that
rience
the
NASDAQ
of the United States in the 1990s.
in the 1990s, followed
by valuations shown
in figure lb,
Figure la illustrates the sharp rise in
by the collapse of 2000-2001. The extremes reached
and
their collapse in the absence of obvious changes in
fundamentals, are suggestive of widespread speculation.
Figures Ic and Id illustrate the
growth and investment boom-and-bust that accompanied the market's gyrations^.
a:
Stock
Mvkat Indices
b: Plica /Earnings
Y*anfro(n Ptik
c:
Real
GDP Growth
Rsllo
YsaiBfroin Pa all
{%)
d:lnvasUranKGOP(%)
Yaiisfrom Paak
YaarafromPaik
Figure
1:
Speculative
Growth
in the
US.
The nature and policy dilemmas of speculative expansions have attracted much attention
(e.g.,
International
Monetary Fund 2000,
Shiller 2000, Cecchetti et al. 2000),
but our formal
understanding of the macroeconomic mechanisms that underlie the relation between stock
market speculation and
real
economic activity remains quite limited.
to attribute such episodes to irrational exuberance,
'Note: Panel b: the numerator
inator
Panel
b:
is
a:
is
and indeed
always possible
highly likely that
some
the real (inflation-adjusted) Datastream Total Market Index; denom-
moving average over preceding ten years
of real earnings corresponding to the index.
Nasdaq Composite Index from The Nasdaq Stock Market,
Datastream Total Market Index
it is
It is
for
US. Panel
c:
Annual report on national accounts (CD-ROM) 1998
Inc,
Sources:
SP 500 from Datastream. Panel
Bureau of Economic Analysis (BEA). Panel
d:
BEA.
10 years treasury rate
(real
and nominal)
^—
^—
"real rate: pre 1990 a\«rage
-real rate' post 1990 average
real rate
-nominal rate
- - -
Figure
of that
is
invariably present.
Long Real and Nominal Rates.
2:
But what are the environments that
facilitate these
irrationality or, put differently, that facilitate the confusion of intelligent
by having a rational path that
In this paper
is
bouts of
economic agents
not too distant from observation?
we take an extreme approach
for rational expectations equilibria that
to answering the previous question
can mimic speculative growth episodes.
hint from the recent U.S. episode to formulate owe theory.
We
start
and look
We
take a
from the observation
that long run interest rates, and in particular the cost of capital faced by growing companies,
declined throughout the 1990s.
year
US Treasury-bond
rate, to
The dotted
line in figure 2 illustrates
which we subtract the University of Michigan Inflation
Expectations Index to obtain the corresponding real rate
during the 1990s
is
and 1990s (dashed
the path of the 10-
(solid line).
The
latter 's decline
apparent: the difference between the average real rates for the 1980s
lines) is
about 200 basis points.
Such a decline can account
for a significant share of the
decline in the real rates implicit in
US
observed
rise in asset prices:
The
Treasury bonds alone can explain over 50 percent of
the rise in broad market's valuations during the second half of the 1990s, and
if
one
is
to
consider the further decline in risk premia due to increased participation in stock markets
(admittedly a somewhat circular argument), then that share rises to 80 percent or more.^^
Once such
•^See, e.g.,
rise
has taken place, the investment and output growth
boom
that followed can
Vissing-Jorgensen (1999) for a model and evidence on the relation between participation and
the equity premium.
^A
fact that
by a sequel
was not missed by economic commentators, and indeed was extrapolated to extreme
of bestsellers starting with
"Dow
36,000:
the stock market," by Glassman and Hassett (1999).
The new
limits
strategy for profiting from the coming rise in
be rationalized along standard q — theory arguments.
However, the key general equilibrium question
above story
in the
demand
tions of a low long-term cost of capital consistent with the high
How
is:
are expecta-
for savings
needed
to fund a high-investment equilibrium? Limited to rational explanations, this necessarily
means that along the expansionary path agents expect an
funding to more than
offset
the
be exogenous or endogenous.
both but highlight the
demand
increase in the availability of
such funding. The source of this
for
In our model and account of the U.S. episode
shift
we
could
include
cannot be matched by a com-
latter since the crash of 2000-2001
mensurate exogenous shock. Thus the central ingredient of
ovir
model
is
a growth-funding
feedback by which the future supply of effective funding increases as a result of the conditions created
effective
is
by a speculative expansion, and ends up lowering the cost of capital (the word
added to encompass the important decline
in the price of
new
capital relative to
consumption goods during the 1990s).
We
show that speculative growth episodes can be welfare improving but they are
susceptible to crashes.
The
latter are
more
likely if
also
bubbles develop along the expansionary
path.
Of independent
more
interest
realistic features
a "dynamically
is
than
inefficient"
the fact that in our environment rational bubbles exhibit
in the
standard setup. In the
economy, that
accumulation of capital. Their emergence
away from investment and
1.
is
bubbles can only appear in
one whose structure
is
conducive to the over-
welfare improving as they help to absorb saving
to alleviate the over-accumulation problem. However, the notion
that bubbles and investment
in figmre
is
latter,
move
in opposite directions
is
contrary to the patterns depicted
Moreover, empirical testing of "dynamic inefficiency" has been negative. Abel
et al. (1989) tested
an implication of dynamic
inefficiency,
whether the aggregate market
assets acts as a long-term "cash sink" for investors, but found no evidence of
economies.
Finally, the idea that bubbles are inherently
it
in
for
OECD
good runs counter much of the
perception of policymakers and practitioners.
In contrast, since in our model a speculative growth episode anticipates increased funding,
bubbles
may emerge even when
economy and the corporate
the current interest exceeds the rate of growth of the
sector generates a surplus.
And
while
it is still
the case that
the emergence of a bubble crowds out resources for investment, the speculative growth dy-
namics guarantees that investment
still
booms along
the path.
By
the same token,
if
the
speculative growth path crashes then no longer the conditions exist for a rational bubble to
survive (for the
same reason that a bubble cannot
exist in the
dynamically
efficient region
and
of the standard model)
in
a speculative path
is
must crash as
costly because
accumulation and because
We show that
it
it
it
well.
Finally, a
bubble that emerges early on
crowds out potentially welfare improving capital
overexposes the economy to a crash.
a growth-funding feedback arises naturally
when an expansion comes with
technological progress, especially, in the capital producing sector;
sustained
when
were
when
fiscal surpluses;
How
fiscal rules
generate
the rest of the world has lower expansion potential; and
financial constraints are relaxed
all
when
by the expansion
itself.
Arguably, these ingredients
simultaneously present in the U.S. during the 1990s.
does technological progress, especially in the capital producing sector, generate
On
feedback from growth to funding?
comes and with
it
one end, technological progress raises future
saving available for investment.
logical progress in capital
On
in-
the other, relatively fast techno-
producing sectors reduces the saving needed to finance a higher
steady-state capital-output ratio. In this context, a fundamental expansion in productivity
and stock market speculation are not necessarily competing explanations
tive
growth episode, as most observers had
might form an integral part
it
during the 1990s.
A
of
an specula-
technological revolution
— both as cause and consequence — of a speculative growth
equilibrium.
What
about
fiscal policy?
We note that
a significant share of increased available funding
in the U.S. speculative expansion of the 1990s
was attributable to public
sector's saving.
To
the extent that pro-cyclical government revenues increase public saving, they reinforce the
feedback from growth to saving. In the short run,
of the
boom and can
fiscal
surpluses can arise as a consequence
facilitate the initial rise in investment.
surpluses can play a central role in
More importantly,
making the speculative equilibrium
feasible
fiscal
by providing
the funding necessary to sustain high investment in the long run. This consideration has
a particularly stark implication
during a speculative growth episode. The
for fiscal policy
possibility of using the fiscal sinpluses that result
spending could be
illusory, as these surpluses
from such expansions to cut taxes and
raise
might be necessary to sustain the speculative
growth equilibrium that generated them.
A
similar role
is
played by external saving, the other significant source of increase in
aggregate saving during the 1990s.
on short run
interest rates
In the short run, capital flows alleviate the pressure
brought about by the investment boom. In the long run,
potential productivity in the high investment equilibrium
then the speculative growth path
itself yields
is
higher at
if
the
home than abroad,
a reallocation of global investment toward
domestic assets. Moreover, the speculative growth path
itself
may be made
feasible
by a
decline in opportunities abroad.
Finally,
if
the expansion relaxes financial constraints faced by productive entrepreneurs,
then aggregate saving
is
The expansion
reallocated toward them.
of the 1990s clearly
achieved this goal by facilitating the reallocation of capital toward the mostly small and
emerging companies of the new economy
sectors.
Some of these companies were undoubtedly
bubbles and crowded out capital from good firms, but
many
others were the pillars of the
information technology revolution that was so central to the episode.
The speculative growth episodes
to
which our theory
relates are a recurrent
phenomenon.
Earlier episodes of vigorous economic expansion under speculative asset valuations have
been documented by economic historians. In the case of the U.S.,
can be observed in the expansions of the turn of the 20th
this
phenomenon
centiury, the 1920s,
also
and the 1960s
Equally important, our theory can be brought to bear on sustained
(see, e.g., Shiller 2000).
low cost-of-capital equilibria, such as the prolonged expansions exhibited in a number of
East Asian economies in the post-war period.
In fact, the key feedback from growth to
funding in these economies has been documented.
between income growth and saving
Examining the aggregate relationship
in a cross-country panel of 64 countries over the period
1958-1987, Caroll and Weil (1994) find that growth Granger causes saving with a positive
but that saving does not Granger cause growth.
sign,
growth followed by strong increases
in saving rates
is
The pattern
of an acceleration in
particularly clear in the high-growth,
high-saving East Asian economies of Japan, South Korea, Singapore, and
(1997) elaborate on this evidence
et al.
saving
In
is
its
and show that the estimated impact of growth on
not only statistically significant but also very large in economic terms.
economic theme,
growth episodes
literature
Hong Kong. Gavin
is
(e.g.,
this
paper
is
part of a long tradition of studies of speculative
Kindleberger 1989).
divided between those
2000), and those who, based
on the
who
In terms of the recent U.S. experience, this
see a case of speculative behavior (e.g., Shiller
significant acceleration in underlying U.S. productivity
growth, conclude that the expansion was driven by a technological revolution that affected
real
fundamentals
Prom
(e.g..
Greenwood and Jovanovic,
this perspective, oiu: contribution
two views need not be mutually
is
1999; Hobijn
and Jovanovic, 2000).
to provide a unified perspective under
exclusive.
opportunities and stock market speculation
On
which these
the contrary, an expansion of technological
may come hand on
hand.
This perspective finds support in the evidence of Beaudry and Portier (2003), who device
a novel semi-structural
VAR
procedure to conclude that the
US
business cycle
driven by a shock that does not affect productivity in the short run but that
is
is
largely
strongly
They argue
related to future productivity growth.
that such shock could well represent a
coordination device. Similarly, Christiano and Fisher (2003) conclude that the explanatory
power of RBC
style
models
shocks. In our model, the
and the
single shock,
is
greatly enhanced by the introduction of procyclicaJ investment
comovements emphasized by Christiano and Fisher
latter
may be mostly
of the coordination sort hinted
arise
from a
by Beaudry and
Portier's findings.
On
the methodological side, our paper belongs to the literature on multiple equilibria.
This literature
is
too extensive to review here
some of the
(see, e.g.,
Benhabib and Farmer (1999)), but
and
Krugman
it is
interesting to point out
who
develops a trade model with external economies and adjustment costs. In that model,
if
similarities
differences with
(1991),
adjustment costs are too small, then multiple equilibria arise and expectations dominate.
On
the other hand,
when adjustment
costs are large enough, the multiple equilibria region
disappears and only history matters. In contrast, in our model multiple equilibria arise only
for intermediate
adjustment
costs.
As
in
Krugman, we need adjustment
costs to be small in
order to be able to afford a transition to a high capital equilibrium. Unlike Krugman's,
also
need adjustment costs to be large enough to generate
boom
the investment
sufficient capital gains to justify
along the transition.
We also relate on the methodological and substantive side to
the literature on bubbles in
The seminal work by
Tirole (1985) set
up the foundation
of bubbles in general equilibrium, but
when embedded
in the basic unique-equilibrimn
general equilibrium.
sis
we
OLG
model
it
In particular,
for the analy-
generates implications that are at odds with speculative growth episodes.
it
implies that investment
and bubbles experience negative comovement and
that the latter only can arise in a dynamically inefficient economy. Neither of these elements
is
observed dmring these episodes. Partly for this reason, a large literature has developed to
modify the basic structure. For example, several papers have demonstrated that, in the presence of externalities that create a wedge between private and social returns on investment,
bubbles can arise even
1992;
if
the bubbleless economy
Grossman and Yanagawa,
logic a step further
when only
1993;
is
dynamically
King and Ferguson,
and shows that with segmented
efficient (e.g., Saint-Paul,
1993). Ventura (2003) takes this
financial
markets bubbles
may emerge
the marginal savers face interest rates below the rate of growth of the economy.
Olivier (2000) addresses the negative-comovement
problem and shows how bubbles on firm
creation can lead to more rather than less investment. Aside from the difference in the specific
mechanism we emphasize,
in our
comovement with investment and
model conventional rational bubbles exhibit positive
arise
even when
private) interest rates.
6
all
investors
and savers
face high (and
The
rest of this
paper
organized as follows. In section
is
model of speculative growth. Our analysis
generations model, to which
is
we present a prototypical
2,
based on the Diamond (1965) overlapping-
we add adjustment
costs
and a
assumption about the
special
saving function. Adjustment costs allow us to characterize stock market
role in facilitating the transition
from low- to high-investment
booms and their key
Our
equilibria.
function generically captiures the growth-funding feedback that
is
special saving
central to the existence of
a high-valuation equilibrium. Section 3 discusses the possibility of bubbles and the
fragility
they introduce to a speculative growth episode. Section 4 presents specific mechanisms that
may
have facilitated an speculative growth episode in the
fiscal surpluses, capital flows,
5 shows
how
and technological progress
financial constraints a la Kiyotaki
to generate the type of growth-funding
Speculative
In this section
we
during the 1990s. Namely,
in the
equipment
and Moore (1997)
Section
sector.
also have the potential
mechanisms that we emphasize. Section 6 discusses
and Section 7 concludes. Several appendices
welfare issues,
2
US
follow.
Growth
present a prototypical model of speculative growth.
Our
analysis
on a linearized version of the Diamond (1965) overlapping-generations model.
is
We
based
capture
the growth-funding feedback mechanism through a generic funding function that relaxes
Diamond's
stability condition.^
Our
focus in this paper
the violation of the stability condition (although
on) as
,
is
is
not so
we do provide
much on what
specific
is
behind
mechanisms
later
on the other features of the economy that make such modelling change a potentially
important source of speculative growth episodes. In Section 4 we discuss specific features
that
may have
other hand,
we
this structure
is
played a role in the
little
episode during the 1990s.
focus on building the basic structure of the economy.
In this section, on the
One key
ingredient in
the presence of the "right" amount of adjustment costs to capital.
much adjustment
too
US
costs, transitions are too costly
adjustment
With too
and equilibrium becomes unique. With
costs, there are neither stock
market booms nor multiple
equilibria,
which are central features of the speculative growth episodes we wish to model.
2.1
The Growth-funding feedback
Consider a standard Diamond (1965) overlapping-generations structure with no population
growth and a unit mass of young and old agents who coexist
^See
Diamond
(1965), page 1134.
at
any date
t.
Each generation
is
born with a unit of labor, L
=
to be used
1,
when young,
for
which
wage Wt determined
in a competitive, full-employment labor market.
constunption good
used as a numeraire.
is
Consumption goods
function at any time
exogenous rate
t
is
determined by the
is
The economy's
labor, Lt-
level of technology. At,
The production
which grows at an
=
+ 7)At.
(l
given by a constant returns technology, F{Kt,AtL). Letting kt
denote capital per effective worker, we write the marginal product of capital
n
Mkt),
The
labor market
be written
of generation
=
t
is
sj
=
chooses the level of saving
per unit of effective labor can
>
0.
(1)
member
St,
that maximizes lifetime
utility.
in the level of saving per effective unit
given by
is
=
s{wt,rt),
denotes the interest rate between periods
Diamond
terized
t
(2)
and
i
+
1, s^,
>
0,
and
(1965) model, there are no adjustment costs to capital so that
capital next period
is
<
Sr
<
rt
=
fk{kt+\)
equal to today's saving. Equilibrium then
by a nonlinear difference equation
^
^
the stability condition in
panel
0.
when young,
summarized
St/AtL, which we assume
and the stock of
If
as:
00.
The long run feedback
2.1.1
In the
Kt/AtLt
a function of consumption during youth and old age. Each
St
rt
w'
w{kt)
relevant features of preferences are
where
=
as:
Lifetime utility
of labor,
<
= Wt/AtL
competitive, and the wage wt
is
Wt
The
single
7:
Ai+i
Production
and
are produced with capital, Kt,
receives a total
it
(a) of
Figure 3 can
The
for capital accumulation:
siw{kt),r{kt+i))
(1965)
is
°Note that even
if
,g-
relaxed, then multiple steady-states as in
essential feature of this saving function
region where saving increases rapidly as capital
of this feature but simply assume
it.^
A
rises.
is
that there
utility
is
particularly simple formulation of such situation
would violate the
stability conditions.
8
a
For now, we do not explain the source
one stays within purely neoclassical features, these need not be exotic.
technology we use here a log
charac-
1+7
Diamond
arise.
is
With
is
the
a:
I-;.
MulUplo Btea^ aleles
\r>
modcf
the Dlarrond
b:
MulUple steady stales
Ir
the linear modet
A
/
iu'ii-t ',r i.'.-,''
1
/
/
/
/
/
/
/
/
/
/
1
Figure
3:
to start with a saving function that
is
"
A
"
I
L-
Multiple Steady States
and
linear in k
r,
with a step at some level of capital
k":
=
s{w{kt),rt)
So
So
with
So, 5
capital,
on
TTo
and
Sj-O strictly positive.
— tti/c,
with
ttq
and
k<k°;
+ Skkt + Srn,
+ + Skh + Srn-, k > k°,
Now
the marginal product of capital be linear in
let
strictly positive
tti
(4)
(5
and k <
(henceforth,
tto/tti
we
will focus
equilibria that satisfy this constraint), so that:
n=
=
r{kt+i)
TTo
-
TTikt+i
Replacing this interest rate expression into the saving function
capital accumulation expression (3), solves the model.
that
we captm-e the
Assumption
0:
Assumption
1
Assumption
for a
The
(4),
and the
result into the
following assumptions ensure
scenario that concerns us.
A=
Sriri
(Minimum
1 states
+
(1
+7-
s^)
and ^si+^
>
grovi^th-saving feedback):
jump
that the
<
5
A;°.
5=
>
in the saving function at
fc°
[k^
—
aid^iiim) /^
must be high enough
steady state to exist to the right of k°.
Proposition 1 (Multiple Steady States)
If
Assumption
and
1
are satisfied, the
So
fc"
+
SrTTo
A
where the superscripts "n" and "s"
economy has two steady
^
o
<k°
<
,
stanrf /or
Sq
+ 5 + S^TTq
A
=
states, A;"
,
and k^
,
with:
g
fc^
normal and speculative,
(5)
respectively.
Proof: See appendix A.
The
proposition simply states that
parameter
S,
if
sufficiently strong, the
is
the growth-saving feedback, captured here by the
economy
exhibits multiple steady states.
This
is
illustrated in panel (b) of Figure 3.
Importantly, since the marginal product of capital
is
decreasing in the stock of capital,
the speculative steady state exhibits a lower cost of capital than the normal steady state:
r*
This
the reason at times
is
we
=
-TTi— <
r"
r".
A
refer to the "speculative" equilibrium as the "low cost of
capital" equilibrium. Similarly, since the valuation of
and
earnings,
in steady state the former
that valuation
is
is
an asset
is
just earnings divided
just one over the cost of capital.
Thus we
simply
its
by cost of
price divided
capital,
by
we have
also refer to the "speculative"
equilibrium as the "high valuation equilibrium."
How
requires
to
can the economy have a low cost of capital in the speculative equilibrium when
more saving
more than
to be sustained? Precisely because higher capital raises saving
offset the
at the core of the growth-funding
is
state.
To
That
is,
for
any given
see this, suppose the
level of kt the
economy
increasing saving, which can only
happen
at
is
if
economy converges
to
has a unique
to a specific steady
Then moving toward
fc".
it
the interest rate rises (since
/c*
would require
But
kt is given).
cannot be an equilibrium since a decreasing marginal product of capital implies that an
increase in the interest rate reduces investment. Conversely, suppose that the
fc*.
for funding).
in this paper.
easy to see that while the model above has multiple steady states,
equilibriuTTi.
this
mechanisms we discuss
demand
Multiple Equilibria
2.1.2
It is
enough
reduction in saving due to the decline in interest rate resulting from
lower marginal product of capital (supply of funding shifts more than
This
it
Then moving toward
fall.
But
this
A;"
equilibrium capital:
If
More
generally, the
fall
same
at
interest rate
in the interest rate leads to
logic applies to
is
any other
more
level of
a level of investment constitutes an equilibrium in the capital market
for a given level of capital,
capital)
would require lowering saving, which requires the
cannot be an equilibrium either since a
rather than less investment.
economy
then no other
can be an equilibrium. The reason
level of
is
investment (and hence of tomorrow's
simply that for any given
saving are, respectively, decreasing and increasing with respect to
10
r^.
kt,
investment and
Let us modify slightly the previous setup and introduce adjustment costs.
only creates the possibility of a stock market boom, which
is
This not
one of the features we aim
and
to explain, but also breaks the short run connection between the return on investment
the marginal product of capital (since there are capital gains), which was at the heart of
the uniqueness result above.
Let investment to maintain the effective capital stock be
from
this level of investment
is
As
investment.
all
deviations
be subject to a convex adjustment cost:^
ciIt,Kt)
where I
but
frictionless,
= \e-'Kt(J^^-j^
(6)
,
usual, the investment equation in this context takes the simple
qi-theory form:
x{qt)
where
xt
=
It/Kt and
qt is
=
,
is
responsive to such deviations.
The
(7)
When
adjustment costs
relatively unresponsive to deviations of the value of installed
When
capital firom that of uninstalled capital.
infinity.
1)
the price of a unit of installed capital.
are large {6 low) investment
goes to
+ S{qt -
7
they are low [9 large), investment
The simple model
in the previous section obtains
capital accumulation equation can
now be written
is
very
when
as:
h,r^^^h=(l^-^i,,-l))k,.
1+7
1+7
V
Importantly,
it
is
product of capital.
no longer the case that the
A
6
(8)
/
interest rate
unit of installed capital costs
qt
is
equal to the marginal
and saves adjustment costs
for
CK{It,Kt) today. This unit of capital yields qt+i and fk{kt+i) tomorrow. Thus, the return
from investing
in capital
is:
/,
^
N
^
qt
After replacing /^ in
it,
this expression
qt+i
=
(1
+ fk{h+i)
+ CK{IuKt)-
qt+i
''
can be rearranged
+ rt){qt +
CKih, Kt)) -
(ttq
as:
-
7rifct+i),
(9)
with
CKiiuKt)
^ir^(i-7) -^-^(^-7)^
= \e-\xt-^f-e-\xt--i)xt
= -(7 + i(9*-i))fe-i)-
^Subtracting effective-maintenance investment from the adjustment cost function facilitates our analysis
later
on without any substantive
cost.
11
The
a function of
qt
and
For
kt-
the old
young
capital to the
sell their
at price
qt-
allocate their saving to the purchase of the existing stock of capital, to invest in
and
capital,
to
pay
Replacing
(4), (6)
=
r{qu h)
>
l{fcf
and
—
qt
The
new
adjustment costs of installing this new capital:
for the
s{kt,
with
t,
to solve out the interest rate as
is
us find the capital market equilibrium condition.
this, let
Following production in period
young
dynamic system,
last step to fully specify the
n)
=
+ xt{qt))kt + c{x{qt),kt).
{qt
(7) into (10),
and solving out
+
1)
+ 0{qt -
-t
+
-
^{qt
1)^
(10)
for the interest rate, yields:
-
h-so-
Sk
k°} an indicator function that takes value one
when
kt
l{kt
>
> k°}s\
(11)
,
k° and zero otherwise.
Putting things together, the system governing equilibrium dynamics can be written as
a two-dimensional system in
qt+i
=
+ r{qu h))
{l
It is
-
(qt
{kt, 5t)-space:
f7
h+i=
U + Y^(5t-l)jfci,
+
-
:^{qt
1)
{qt
j
-
1)
j
-
Uo
-
tti
(l
+ ^^(gt -
1)
j
hj
.
straightforward to verify that this dynamic system has the same steady states
described in Proposition
1.
For
simply note that in steady state g
this,
=
1,
in
which case
the steady state equations of this system collapse to those of the model without adjustment
costs.
But the question that concerns us here
under which circumstances the presence of
is
adjustment costs can break the uniqueness result of the simpler model.
whether
it
possible for an
is
economy near
toward the "speculative" one. That
is,
its
In particular,
"natural" steady state to initiate a path
whether the economy can embark
in
a "speculative
growth" path.
Let us assume that 6
q
—
I
and k
appendix
=
k"'
small so that we can
for the transition
for a precise
from
A:"
be small).
to k^,
/c"
Let
dynamic system can be written
qt+i
make a
if
first
order approximation around
such transition
is
possible (see the
statement about this approximation, which also requires that the
distance between k° and
linearized
is
= :i+o(i-7) +
qt
=
{qt
-
1)
and
kt
=
{kt
—
A;"),
then the
as:
(j^ + ^^
12
qt
+ -kt-l{kt>k-}-,
Of
O-p
(12)
kt+i
In the appendix
=
-7-^qt
1+7
+ h.
(13)
we provide the exact statements that make
and
of the nonhnear one as 5
(fc"
—
k°) go to zero. In the
the dynamic system described by (12) and (13).
We
this
main
shall
hnear system the limit
om: analysis refers to
text,
add a technical condition that
ensures that the interest rate and marginal product, fk, rather than the adjustment cost
saving feature of investment, ck, dominates local dynamics for
all
values of
+ r")(l -
7)
-
Assumption
It is
0'
^+
(Technical condition):
(1
6:
>
1
useful to define the following function:
= -1 + 2-
A(^)
-,
^^^^
1+7
which corresponds to the
,
ratio of the slope of the unstable
path stemming from the normal
steady state to (minus) the slope of the saddle path stemming from the speculative steady
state (see the Appendix).
decreasing on
[0,^]
A
Observe that
and increasing on
is
non-monotonic on
strictly positive,
+oo[, with \im.g^Q K{9)
[0,
—
[0,
+oo[,
—
+cxd,
lim0^+ooA(0)
where
^+
Assumption
+
('a(^)
1'
(Minimum growth
+ r")(l-7)-l
saving feedback for a trcinsition):
5
> ^ =
i) A(fcO-fc").
>
Note that since A(^)
Assumption
m=
(l
ini{6
0,
Assumption
2 (Speculative
>
0,
A{e)
= -1 +
1'
implies
Assumption
adjustment costs region):
^(^^}
and
e{6)
=
sup{0
>
0,
1.
0_{5)
A(0)
<
9
<
9{5),
with
= -1 + ^^^^}.
Proposition 2 (Multiple Equilibria and Speculative Growth)
If
Assumptions
from
fc" to
k^
.
1
0, 0',
'
and 2
hold, there is a speculative growth path that takes the
Along that path,
qt
>
economy
0.
Proof: See appendix A.
Let us discuss here the structure of the proof with the help of the phase diagrams in
Figure
4.
The
central ingredients in each of these panels are the unstable
13
arm
of the normal
a: Intermediate
Adjustment Costs
b:
c:
Low Adjustment Costs
High Adjustment Costs
Figure
4:
Multiple Equilibria
steady state, the vertical line at k°, and the stable arm (saddle path) of the speculative
steady state. Naturally, for a speculative growth path to exist,
it
must be the case that the
unstable path of the low capital equilibrium intersects the k° line below the intersection of
the latter and the saddle path of the high capital equilibrium. This
depicted in panel
g"*"
>
1,
such that
(a).
it
By
continuity, in this case there
hits k° at the saddle
is
is
precisely the scenario
a path starting at
it
is
instructive to discuss
scenarios where the conditions for such path do not exist. Panels (b) and
6
>
The former
is
responses).
is
very
fiat
normal steady state
is
< 9)7
2
is
is
(it
is
main
if
and only
if,
is
that since our linear system
cannot state whether a speculative growth path
no speculative growth path
of the Propositions in the
the slope of the unstable path
takes enormous capital gains to justify costly
not stated as an
exactly at the boundaries of the range in
reversed, there
for the
(small deviations of q from 1 lead to large investment
very steep
approximation to the non-linear one, we
9
extreme (hence 9
On the other, when adjustment costs are too large,
^The only reason Proposition
when
Figure 4
one end, when adjustment costs are too low, the slope of the saddle path
speculative steady state
for the
(c) in
one in which adjustment costs are too low (hence
6) while the latter represents the other
On
with
path of the speculative equilibrium.
Before further characterizing the speculative growth path,
provide these examples.
(/c",g"'")
Assumption
(see the
2.
If
is
is
an
feasible or not
the inequalities in Assumption 2 are
Appendix). The same consideration applies to most
text.
14
That
investment beyond maintenance investment).
exist,
is,
adjustment costs must be large enough to generate
growth path to
for a speculative
decouple
sufficient capital gains to
the return on investment from the marginal product of capital in the short run, but not so
large that
simply too costly to finance an investment boom.
it is
Retm-ning to panel
is
it
are investment
we
These are precisely
highlighted in the introduction for the U.S. episode during the 1990s.
Fragility
Our use
of the
investment
and crashes
word "speculative"
boom and
is
primarily meant to capture the feedback between the
capital gains along the path toward the high capital equilibrium.
our framework also captures the meaning of
"fragility" in that
Assumption 3 (Non-crash adjustment costs
e^{5)
and growth. And since marginal prod-
declining, the price- earnings ratios (valuations) are also rising.
the features
2.2
see that along the speculative growth path the value of
booming and with
installed capital is
uct
we can
(a),
=
>
inf{^
0,
- -1 + ^(fc^}
K{e)
Note that Assumption 3
where the superscript
C
is
and
word.
region):
o"^ {5)
=
automatically satisfied
6 ^
sup{^
if
5
But
>
0,
0F{5),9
= -1 +
A{e)
< 5^ =
{5)
(A{9)
+
1]
,
where
^^J_^6)
A{k^
—
}.
A;°),
stands for crash.
Proposition 3 (Consolidation region)
If
Assumptions
economy from
fc"
capital stock, k°
equilibria for
0,
k^
0',
2 and 3
1',
hold,
there is a speculative growth path that takes the
to k^, but the converse is not true.
<k^ <k^
<kt
,
such that there
is
Moreover, there
is
a unique equilibrium for kt
a value
>
k'^
k'^
of the
and multiple
<k'^.
Proof: See appendix A.
The
A;*,
is
first
there
is
part of the proposition
is literally
Proposition
no path that can take the economy back to
similar to that used in Proposition
hold, the unstable path
now
2,
to
The second
2.
fc".
The
part
is
that at
structure of the proof
show that when Assumption 3 does not
from the high capital equilibrium overshoots the saddle path to the
low capital equilibrium. The
last part of
economy has a consolidation
region.
such that once the economy reaches
the proposition raises an interesting feature:
That
it,
is,
there
a region around the high capital steady state
is
no chance of a crash that takes the economy
back to the low capital equilibrium. The other side of the coin
along a speculative path
is
fragile
The
and susceptible to
15
crashes.
is
that, early on,
an economy
A
2.3
Detour: Sunspots
Of course the concepts
of crashes
paths are imprecise at best. But
as
and
it is
fragility in
a model with multiple but deterministic
conceptually straightforward to introduce a sunspot
an equilibrium-selection device, and formalize these statements.
Suppose, for example, that in the multiple equilibria region the sunspot variable
Markov chain with two
states, {u,d},
and
initial
=
value eo
u.
When
agents coordinate their expectations on the speculative growth path.
the state
When
is
is
a
u, all
the state
is d,
expectations coordinate on the path toward the low capital steady state. Define the crash
time,
Td'.
Td
and
=
inf{t
>
0,et
=
d},
the transition probabilities be such that once a crash takes place, the
let
economy
converges to the low capital equilibrium with probability one:
Puu
Pud
Pdu
Pdd
_
IJ-
1- fl
1
suppose that preferences are such that only the expected return of a project
Finally,
matters to agents (we can use Epstein-Zin preferences for this purpose). With this structure,
sunspots do not alter the expression defining the equilibrium interest rate.
modified
is
All that
is
the expression for expected future price of capital while in the speculative
growth path (and not yet
in the consolidation region),
which
is
now
[fJ-qt^i
+
(1
—
ii)qf_^_i)
instead of simply qt+i-
For expositional simplicity, we henceforth suppress sunspots from our formal analysis
but
still
use the language of crashes associated to them.^ Alternatively, our analysis can be
thought of as being conducted in the neighborhood of
/x
=
1.
Bubbles, Assets in Fixed Supply, and Speculative Growth
3
Assume momentarily that the
It is
7.
well
known from
Essentially, since a
speculative steady state
is
dynamically
inefficient,
with r*
<
Tirole (1985) that rational bubbles can emerge in such scenario.
bubble must grow at the interest
rate,
and the
latter
is
below the rate
of growth of the economy, a bubble in this environment never outgrows the economy. Thus,
if it is
*
feasible at the outset,
Similarly,
it
remains so at later dates. Moreover, rational bubbles in this
we do not model the jump onto a
speculative path.
sunspot arguments.
16
We
could do so with entirely symmetric
context are welfare enhancing since they solve an "excessive" investment problem driven by
a high demand
for a store of value rather
than by productivity considerations.
It
follows
that the emergence of a rational bubble in this context lowers aggregate investment, and
that a crash of the bubble boosts aggregate investment.
Now assume
that the low capital long run equilibrium,
steady state, this means r"
>
7. It is also well
dynamically
A;", is
known from
Tirole (1985) that rational bub-
bles cannot exist in such environment. If the interest rate exceeds the rate of
economy, a bubble grows faster than the economy until
In
efficient.
growth of the
eventually becomes inconsistent
it
with the aggregate endowment constraint. Backward induction then shows that a rational
bubble can simply not emerge in
The
most evidence point
fact that
(Abel et
this region.
al (1989))
in the direction of
dynamic
and that episodes of speculative growth as
efficiency in the U.S.
illustrated in Figure 1 exhibit
strong positive comovement between valuations and investment, would seem to be conclusive
evidence against the presence of rational bubbles during these episodes.
It
turns out that
neither property needs to hold in our speculative growth environment. Along a speculative
growth path bubbles may emerge even
where the
in the region
rate of growth of the economy. Moreover, aggregate investment
positive
comovement. The speculative growth path
is
needed
interest rate exceeds the
and the bubble experience
for a
bubble to emerge, and
if
the economy crashes into the recessionary path toward the low capital equilibrium, so will
the bubble.
Importantly, the likelihood of such crash
is
enhanced by the emergence of a
bubble.
3.1
Bubbles with
r
>7
Assumption 4 (Bubble
region):
ttq
—
tti/c"
>7>
ttq
—
tti/c"
Proposition 4 (Bubbles in high interest region)
Let Assumptions
economy
0,
0',
1',
2 and 4
initiates a speculative
(normalized by A{t)),
bt,
hold,
and
Then,
initial capital be equal to fc".
if
the
growth path, there will a range of feasible rational bubbles
such that bo
>
u
and
1
+ ^U
1+7
Proof: See appendix B.
This result
is
in a region
is
intuitive.
where the
toward a dynamically
While
at the outset of the speculative
interest rate exceeds the rate of
inefficient region
growth path the economy
growth of the economy,
where bubbles can be sustained.
17
it is
headed
But then, by
backward induction, bubbles can
required
it
is
start before the
economy reaches that
that the initial bubble be sufficiently small so that despite
region. All that
its fast
is
early growth,
reaches the dynamically inefficient region with a size consistent with the bounds imposed
by the
economy, and that
size of the
a:
it
allows capital to grow and reach that region.
Paths of q and k with and without bubbles
b:
Figure
Figure 5 shows one such example.
5:
Path of the bubble
Bubbles
Panel
(a) illustrates
the paths of q and k for two
economies along a speculative path, one with and the other without a rational bubble. Panel
(b) illustrates the
3.2
path of the bubble.
Bubbles and crashes
Proposition 5 (Bubbles and fragility)
Let Assumptions
1
0, 0',
',
2,
3 and 4 hold, and initial capital be equal to
A;".
Then, the length
of time an economy along a speculative growth path remains outside the consolidation region
(and hence
is
susceptible to a crash),
is
increasing with respect to the size of the bubble,
bt-
Proof: See appendix B.
The reason
competes
for this result is intuitive as well.
for savings
takes longer for the
bubble
still
in the standard model, a
bubble
with physical investment. This means that in the presence of a bubble,
the economy transits more slowly from
it
As
economy
fc"
to
fc*
all
along the path, which also means that
to reach the consolidation region,
crowds out investment,
it
k>
k'^.
Note that while a
now exhibits positive rather than negative comovement
with aggregate investment. In particular, the bubble can only emerge when the investment
is
booming along a
speculative path.
And
if
the latter crashes, so will the bubble.
18
,
t:
30
Change
in
RelaUve Prico o( Equipment
Govcrnmcni
25
Changs
ui
Raiaina pnca o(
Equipmanl
Pri
itn
|
aiivriga
5
-5
1990
1992
1991
199J
1994
199S
I99fi
1997
Figure
Mechanisms
4
6:
in the
1998
1999
Mechanisms
US
1670
ises
2000
in the
igso
lees
2000
is
conducive to speculative growth
At the core of
1.
this result
is
optimism
funding of large capital accumulation. In this section we discuss a
few prominent features of the
Fiscal sm-pluses, capital flows,
US economy
during the 1990s that supported this optimism:
and technological progress
in the
equipment
sector.
Fiscal surpluses
4.1
The
fiscal
surpluses generated during the U.S. speculative growth experience were the com-
bined result of
fiscal
consolidation measures and the automatic effect of procyclical tax
revenues in a booming environment.
on
less
Episode
episodes that resemble those portrayed in Figmre
effective
isao
US Epidode
Sections 2 and 3 have described an environment that
about future
1BTS
interest rates in the short
investment boom.
supported the
sustainability,
critical
It is
boom
The reduction
run and substituted
in public
for increased private saving to
in this sense that a tight policy of paying
down
effect
fund the
the national debt
Turning to the more novel issue of longer-term
in the short term.
we argue that a continuing
policy rule of generating fiscal surpluses provides
support to the speculative equilibrium.
To develop our argument, we add a government
ment taxes wage income
utility function.
at a rate r*
dt
= Dt/AtL
to the model.
We
assume the govern-
and spends gtAtL on goods that do not enter agents'
Thus the government's budget
(1
where
debt had a moderating
+ '))dt+i =
(1
constraint
+ rt)[dt -
[Ttwt
is
-
(14)
gt)],
denotes public debt per unit of effective labor.
Adding the public
sector alters two equations in the model.
19
The saving
of the
young
St
is
now a
function of after-tax wages:
st
=
s{{l-Tt)w{kt),rt)
while the capital-maxket equihbrium condition
{Ttwt
The saving
-
+ st=dt +
gt)
of the government
pubhc debt and
new
now
given by
+ x{qt)) h + c{x{qt),kt).
(16)
and of the young must fund the purchase of the existing
capital stock from the old,
(16) implicitly define a
{qt
is
(15)
,
and new investment. Combined, equations
interest rate function rt
dynamics are described by system
(8)-(9)
=
r{kt,qt,dt,Tt,gt)-
(15)-
The economy's
with the new interest rate function together with
the government budget constraint, (14), and a
fiscal
policy rule.
Let us consider a benchmark fixed-parameters policy rule under which detrended government
spending and the tax rate are fixed at g
>
and r
>
0.
To
facilitate
comparisons, we assume
that this and other policy rules result in a balanced budget in the low-valuation steady state
— which requires g — tw" — and, therefore, result
The important
point to notice
during expansions beyond
fc"
is
in the
same low valuation steady
state.
that this fixed-parameters policy creates primary siurpluses
and primary
deficits diu'ing contractions.
that the fiscal rule implies that the primary goverrmient surplus, {rivt
—
To
see this, note
5), increases
with
Wt in an expansion. Thus the response of the combined "gross" saving of government and
the young to an increase in wages
The
is
given by (t
+
Su)(l
—
r)) dwt-
increase in saving generated by the fixed-parameters rule not only facilitates the
funding of investment in the short run but, more importantly, plays a central role in
emergence of a speculative growth scenario.
cilitating the
most
clearly
path.
by focusing on the speculative steady
One can show
that
if,
state,
indexation of
other words,
also
expansion experiment
fiscal
fiscal
is
falls
surpluses are not only a
symptom
made
rather than on the entire
is
raised with the
and so does the
spending to wages, the speculative equilibrium
level of
is
no longer
fc*.
In
enough
feasible. In
of the speculative growth episode, but
it.
that the fiscal sm-pluses generated by the speculative equilibrium can be
partly spent or rebated to the taxpayer
may be an
illusion.
of the speculative equilibrium, and might swiftly disappear
giving rise to
is
similar to reducing 6 in Section 2. For a high
can be a central element of the factors that support
The notion
{k'^, 1),
instead of being fixed, government spending
endogenous increase in the wage, then aggregate saving
fact this fiscal
This feasibility point
fa-
what one might describe
as a surplus illusion.
20
The
if
surpluses could be a pillar
this equilibrium umravels
—
The
following proposition summarizes our results:^
Proposition 6 (Surplus Illusion)
Let Assumptions
0',
0,
1',
2 and 4
— 9 + oif{wt — w"")
hold. Let gt
saddle-path steady state equilibria for
a=
0, there exists
An
a'", the speculative equilibrium disappears.
an
If the
.
a™ >
upper bound for
economy has two
such that for any
a™
max{:;p^ fcO-fc"
is
a >
(*
~
A
See appendix C.
Proof:
Importantly, the parameter a"*
is less
than one
for a
wide configuration of parameters.
Capitcd flows
4.2
The
other major source of funding for the investment
account.
As a short-term funding mechanism,
boom
in the U.S.
was the current
international capital flows can moderate the
needed to fund the investment boom. Over time, the whole world may
rise in interest rates
be dragged into a speculative path.^^
We
capture the key role played role by capital flows in facilitating a speculative growth
episode with a stark contrast.
One
in
which no such episode
presence of external saving. For this purpose,
let
be
will
feasible
without the
us assume that the closed economy saving
=
function does not depend on the interest rate, Sr
In this case,
0.
if
the
home country
were in autarky, the transition from the normal steady state to the speculative one would
be impossible. This
/;",
with no role
Let us
now
endowment
Cj
is
because saving, and therefore investment, are
for interest rates in facilitating
6(1
+ 7)*
In
it,
which they save
in each period,
determined by
an investment boom.
introduce a simple foreign country.
=
fully
the young receive an aggregate
in full.
The economy
also has a
technology that uses only capital to produce consumption goods:
FiZt)
where
(foreign) capital Zt
= ^tf(j-) =
is
Ajizt);
f
>
0,7"
accumulated without adjustment
<
costs.
0,
Because
this
produc-
tion function exhibits decreasing returns to scale, there are quasi-rents which accrue to an
'in the proposition,
we
also
add the condition that the speculative equilibrium be dynamically
in order to avoid explosive assets or debt.
This
is
necessary only because
we
inefficient
are analyzing simple linear
fiscal rules.
'"Ventura (2001) emphasizes an alternative portfolio channel connecting the current account and a domestic bubble. In his model, the m.ain effect of the bubble
attempt
to rebalance their portfolios
to finance the investment that
is
by investing
in
is
to raise domestic wealth.
As domestic agents
non-bubbly domestic equity, external borrowing
required to build domestic equity.
21
rises
unmodeled
which does not save
factor of production,
(this is
not important). Let the mar-
ginal product of capital from this technology be linear in capital,
TTi
strictly positive
and z
<
tto/tti
(henceforth,
we
constraint). Since capital markets are integrated,
n=
r{zt+i)
Equilibrium in global capital market
s{kt)
+e=
{qt
is
will focus
must be
it
= n^-
on
it
in (18) to obtain
available for investment in
It is
now
can
fully
an
that:
(17)
now:
(l
+ 7)2;t+i,
=
still
(18)
given by (9) and
(8).
^=;
-
which here we mean saving
fc):
straightforward to see that with Sq
—
sq
+e—
'^°y^V
^
sf,
—
Sk
and
s^
— ^^ we
,
reproduce the analysis of Section 2 with s^{kt,rt) replacing s{kt,rt) in Section's
would have been
economy now can
economy while
foreign output declines.
strength, the domestic
Interestingly, as
it
economy
a speculative path which
if
rises,
and foreign saving flows
Over time,
pulls the foreign
into the
home
as the growth-saving feedback gains in
economy by reversing the
capital flows.
have played an important role during the 1990s, the mechanism
may
above also implies that
there
is
an exogenous decline
a transition into a speculative episode in the
transition
initiate
infeasible in autarky.
Early on in this path, the interest rate
rise in
— —
effective saving function (by
2 formulae. In particular, the domestic
facilitate
and
ttq
(17):
Zt+l
and replace
niz, with
Trizt+i
+ xt{qt))kt + c{x{qt), kt) +
can now solve zt+i out from
—
equilibria that satisfy this
while the arbitrage and capital accumulation equations for k are
We
ttq
comes together with the decline
in foreign opportunities, this
home economy. Furthermore,
in opportunities abroad, interest rates
may
if
the
need not
the short run either.
Let us suppose that with s^{kt,rt) replacing s{kt,rt), Assumptions
Assumption
2 does not hold, so that no transition
speculative steady state
is
and
1
hold but
from the normal steady state to the
possible. Consider a situation
22
0, 0'
where the economy
is initially
in
the low steady state. Let us
<
TTo'
TTo,
now imagine
SO that Sq shifts to Sq
Assumption
A(A;°
5:
-
A;")
that e and
=
Sq
+
(e'
—
e)
>
sg'
-
sg
>
A(fc°
—
(ttq'
-
ttq
unexpectedly
— ttq)^^
-
A;")
>
shift to e*
and
e
>Sg.
^x^^+i)
>^
This assumption implies that there are two steady states under the new parameters.
Denote them by
fc^
=
fc„
+ ^^^and
k'^
=
ks
+^^^.
Proposition 7 (Change in conditions in the rest of the world)
If
Assumption 5
By
tTq.
holds, there is
there are multiple equilibria
contrast,
with capital stock
no transition from
there
fc",
is
result
is
to
under the parameters
fc*
under the parameters
e
and
a saddle path that takes the economy from
speculative growth path that takes the
The
fc"
economy from
intuitive. If saving in the foreign
A;"
to k^
.
fc"
it
k^
Along these paths,
country increase or
if
qt
and a
>
0.
the rate of return
in the foreign country decreases, causing a reallocation of saving to the domestic
a speculative growth scenario can emerge where
starting
tt'q:
to
and
e
economy,
would have been impossible before the
shift.
Technologiccd progress in the equipment sector
4.3
The
productivity growth that
came with
technological progress was exceptional.
two disappointing decades beginning with the
recovered more than half of
growth
in industrial
and
its lost
oil
price shocks of the 1970s, the
economy
productivity growth. Most prominently, productivity
electronic machinery accelerated
1973—1990 to more than 6%
After
for 1995-2000.
from an annual rate of
2%
for
This acceleration reduced the price of machinery
boom. This mechanism
and
electronic devices,
fits
within the speculative growth perspective we have highlighted up to now, once one
which
in turn contributed to the investment
broadens the definition of funding to that of
of saving, a decline in the price of
economy's stock of
new
effective funding.
capital raises the
That
is,
for
any given
level
power of that saving to build the
capital.
While some of the technological progress
exogenous to the capital deepening process
in
equipment producing sectors may have been
itself,
we continue here with our endogenous
perspective (which in any event would act as a multiplier to the exogenous shocks).
We
capture this endogeneity through a positive spillover from the capital accumulation process
to the production of equipment goods. ^^ This simple channel allows us to preserve
^^See
Murphy
et al.
and Jaimovich (2003)
much
of
(1989) for a model of this sort based on increasing returns in the equipment sector,
for
a countecyclical markups version.
23
=
our previous formulae. Moreover, here we set 5
as the technological spillover ends
up
playing a similar role in the current model.
We assume that the technology in the equipment goods sectors can transform one unit of
consumption goods into X{k) units of equipment goods (with
in this sector ensures that the price of
Since
corresponds to Tobin's q
qt
the price on a
new
equipment goods
— that
is
A'
>
0).
Perfect competition
is:
as the value of a unit of installed capital over
— we preserve the formulae of Section
(uninstalled) unit of capital
2,
with two exceptions: the capital market equilibrium condition and the arbitrage equation.
The former
is
now:
s{kt,rt)
But
^ p(kt){qt + xt{qt))kt + pikt)c{x{qt),kt).
(19)
apparent that by dividing both side of this condition by p{kt) and defining the
is
funding function
effective
as:
\
P(,
sP[kt,rt)
_
=
s{kt,rt)
'
P{h)
we can
rewrite (19) as:
s^ikun)
which
=
+ xt{qt))kt +
{qt
entirely analogous to the equilibrium capital
is
c{x{qt),kt),
(20)
market condition
in Section 2.
Moreover, suppose that we model the aggregate increasing returns aspect of the equip-
ment
sector as a simple step function:
P*) =
1
then
(1
— p) >
The only
is
''"'°^
^'
P <
1,
k>
(21)
k°.
isomorphic to 5 in terms of the effective funding or saving function.
difference with the
model
in Section 2
is
that (1
- p)
not only effective
affects
funding, but also the arbitrage equation, since now:
...1
=
,,
w
,r
W PJ^t) + n){qt + ck{Iu ^^))^(^^
,
,
(1
T.-
However, nothing of our fundamental message
is
changed by
(TTQ-TTlfct+l)
Appendix
C.3.
24
,
(22)
this modification. In particular,
the pre-conditions and features of the speculative path remain valid.
in
.
•
p(fc,^,)
We
prove these claims
Financial Constraints and Growth-funding feedbacks
5
Another key ingredient
firms in
in the
new technology
US
episode
We
sectors.
is
the reallocation of funding toward small growth
develop this model here with two purposes. First, to
show that a growth-funding mechanism need not operate through aggregate
reallocation of funds can play a similar role. Second, to
show a more
explicit
saving; the
model of
6;
here the growth- funding feedback stems from the relaxation of financial constraints brought
about by the expansion.
Let us simplify interest rate determination by assuming that there
to capital, h, sector with (low) productivity
interest rate
More
is
pinned down at
substantively,
r,
which
is
is
a constant returns
always active so that the "riskless"
r.
assume that only a fraction
yu
<
1
of the
new born can
work
in the conventional sector described in Section 2, while the rest of the
work
in a sector that has
no
capital.
Yt
Young
=
Thus, aggregate output
At{^^Lf{kt)
invest
young
and
(lenders)
is:
+ {l-^l)L + rht).
lenders have undiscounted log-utility preferences, so they save half of their wages,
which
in aggregate
saving
is
amounts to
(1
—
m)/2.
We
assume that
(1
—
n)
is
large
enough so
never binding in equilibrium. Entreprenem:s, on the other hand, satisfy a
(detrended) consumption
c,
after
-c).
In equilibrium, entrepreneurs borrow from lenders,
=
{xt
minimum
which they save and invest as much as they can:
fj,{w{kt)
dt
their
df.
+ qt + l^'H^t - lf)h -
-
KHh)
c).
(23)
Against this loans, entrepreneurs post as collateral the capital they acquire:
qt+ih/Ci-
Neither
+ r).
new investment nor output can be pledged
lateralized loans are
made
(this is
our financial constraint). Col-
at the "riskless" interest rate r. Uncollateralized loans
(per unit) S/sr resources in monitoring
and therefore are made
at interest rate r
consume
-I-
6/sr-
For a given interest rate, neither the accumulation equation nor the arbitrage condition
described in Section 2 change in this model.
The main change
the two regions of the phase diagram. Rather than k°,
25
now
is
in the criterion separating
there
is
a condition that splits
.
the region according to whether the marginal loan
not (and hence at rate r
+
For this purpose,
6).
let
By
—
replacing (23), the arbitrage condition, and the accumulation equation into this expres-
we
sion,
nt
us define
qt+ih
_
= dt- -—
1 + r
,
nt
collateralized (and hence at rate r) or
is
-
obtain:
(7+g(gt-l)+^(gt-l)^)fc.-M(^(fct)-c)+(^^(g,-l)^ + ^7(g^-l)+
2
When
We
are
n{qt, kt)
now
1
<
the interest rate
is r,
in the setting of Section 2
+r
while
2
when
n^
>
and simplifying, the separating region
=
[7
+
(^
+ 7)
{qt
-
1)
+ 0{qt -
1
the interest rate
is
r
+ 5/sr.
with only a slightly more complicated function
separating the two regions of the phase diagram. Note that
this
/t"
+r
+r
I
is
if
nj
—
then
0,
rt
=
r.
Using
defined by the following equation
If] kt-fi {w{kt)
/+^
-c)+
1
+r
-I
Financial constraints with intermediate adjustment costs
Figure
7:
Financial Constraints
Figure 7 illustrates the analogue of panel
(a) in
Figure
4.
steady states and the possibility of a speculative growth path.
feedback
is
now
arises
sufficiently large,
from the
which
fact that the financial constraint
significantly drops the interest rate
the main claims in this section; in particular,
it
the context of this model.
26
is
on
As
before, there are
two
The key funding-growth
no longer binding when k
loans.
Appendix
D
proves
reproduces the main result of the paper in
^)^^
Welfare
6
Do speculative
in it?
growth episodes represent a Pareto improvement across generations involved
The answer
to this question
is
not generic, as
it
depends on the
specific
channel behind
the growth funding feedback. For example, in the financial constraints model of the previous
section the answer
is
yes.
But
it
needs not be the case
saving function as some savers after
if
time at which
t° (the
is
(5
a property of the aggregate
reaches k°) are hurt by
kt first
the sharp decline in interest rates.
Generic results
6.1
But what
t°.
generic in our models
is
generic that
It is also
capital close
results in
if
is
the welfare improvement of
the economy
is
dynamically
enough to the high capital equilibrium are
all
efficient
generations born before
then
also better
off.
all
generations with
We summarize
these
two propositions, and explain them through their short proofs.
Proposition 8 (Welfare for early generations)
Let the conditions for the multiple equilibria outcome in the cannonical model in Section
2 hold.
Then
all
generations
<
t
t° are better off
along a speculative path than in the low
capital steady state.
Proof:
The basket consumed by
individuals of generation
affordable along the speculative growth path
u;^
>
u;"
_
rt
>
r" since
qt
Therefore, (24)
>
is
t
in the
5"
1 + r"
+ i"^-!—
1 + rt
>
fc",
r^
>
and
rq
is
in the
(24)
low capital equilibrium.
<tQ, which are the ones that concern us
1, kt
low capital steady state
if:
where s" denotes detrended savings by an individual
For generations
t
>
0,
in this proposition,
and wt > Wn
since kt
>
fe"
we have
and Wk >
0.
always verified for these generations.
Q.E.D.
Proposition 9 (Welfare for late generations)
Let the conditions for the multiple equilibria outcome in the cannonical model in Section 2
hold.
Then
if
the speculative steady state
is
dynamically efficient
(r*
>
near the speculative steady state are better off than in the normal steady
if
i~^
<
(1
+7)|^ ~
1'
then
than in the normal steady
all
-y),
all
state.
generations
Conversely,
generations near the speculative steady state are worse off
state.
27
,
Observe that
(1
+ 7)|^ -
<
1
7.
Proof:
For generations
is
We
ambiguous.
The
t
>
still
whether the key inequality in the previous proposition
t°,
> w^
have wt
but now the sign of
—
rt
r"
is
s"
=
(g"
+ x" + \d-\x^ -
7)2)A;"
=
(1
satisfied
ambiguous.
market equilibrium condition in the low equilibrium
capital
is
is
+ 7)A;".
In the speculative steady state, we can rewrite (24) as
/(F) - F/'(F) Since /
is
strictly concave,
{}e
-
-
/(fc")
>
fc"/'(fc")
we have
Y^T^fc" (/'(^")
/'(fc^))
(25)
that
f{k')
-
/(fc")
<
{k'
k^f'{k')
-
/(fc")
-
k^f'ik")
<
k^)f'{k')
-
-
k^)f'{k^).
Therefore,
-
k'^ifik^)
Assume
first
/'(F))
<
f{k')
-
that r*
=
fik")
>
Y^^^"
so that (25)
is
verified
with a
<
k'{f'{k^)
-
f'{kn).
Then
7.
(/'(fc")
-
f'ikn)
<
k-
strict inequality. Since
(/'(fc-)
-
limt^+00 wt
/'(F))
=
w^ and limt_>+oo
n—
r^,
this proves the first claim in the proposition.
Assume now
that r*
=
/'(fc*)
<
(1
+ 7)|^ -
1.
Then
3^i^/c" (/'(F) - /'(F)) > F (/'(F) - /'(F))
so that (25)
is
,
violated. This proves the second claim in the proposition.
Q.E.D.
6.2
A
comment on Bubbles
cind
Welfare
In the classic model of rational equilibrium bubbles, the latter are good because they reduce
the overaccumulation problem of a dynamically efficient economy. This welfare enhancing
features of bubbles runs counter
most observers intuition during speculative growth episodes.
28
Our model
shares that positive side of bubbles for generations very late in the speculative
growth episode but they have a negative side as
by delaying the
arrival of the consolidation phase.
In other words,
if
arise early in the speculative
Final
growth path
more
likely to
be
But bubbles that
beneficial.
risk derailing this potentially prosperous process.
Remarks
This paper builds a theoretical framework
We
they overexpose the economy to crashes
^^
bubbles emerge late in the process once the economy has already
arrived to the consolidation region, they are
7
well:
characterize this
phenomenon
optimism about the future
for thinking
about episodes of speculative growth.
as a low effective cost-of-capital equilibrium based
availability of funds for investment.
Our framework
on
highlights
the key short-term and long-term funding mechanisms necessary to sustain a speculative
growth equilibrium.
Ours
is
not a framework of "irrational exuberance," although
pretation of such episodes. These occur
when a
it
offers
speculative growth path
is
a natural internot backed by a
long-run funding mechanism strong enough to eventually validate the capital gains dynamics
that are needed in the short run to reallocate resources toward the sectors that drive the
boom.
A
sector, as
prototypical example
it
may
is
when investment and
speculation focus on the "wrong"
have been the case in Japan during the 1980s with
its
large real estate
boom.
The
ulative
and
U.S. experience during the 1990s probably had elements of both:
growth component
built
on the information technology
foreigners' confidence, as well as
late 1990s.
markets
Prom
boom
this perspective,
some elements of
it
is
not at
if
bubbly, was
during the 1990s, even
regard, an interesting question that
we
all
sector,
A
rational spec-
sound
fiscal policy,
irrationality especially during the
clear
whether much of the U.S. asset
condemned
leave for future research
is
to be reversed.
On
this
whether a central bank
may
should attempt to raise interest rates upon the emergence of bubbles.
It
case that doing so not only crashes the bubbles but also the underling,
and mostly welfare
well be the
enhancing, speculative growth episode.
'"Note that this postponement has a distributional
before
t°,
which are unambiguously better
off
effect as well as
it
extends the number of generations
with the speculative growth episode.
29
—
A
Proof of Propositions
Proof of Proposition
The
(1
1:
=
conditions for a steady state are q
+ 7)fc.
<
/c"
A:° is
by Assumption
This
in Section 2
0.
a steady state
Similarly k^
last inequality holds
>
if
k^
and only
is
=
r
1,
TVo
if A:"
— nik,
+ s^k + Sr'r + l{k >
= ^ai^ <
a steady state
by Assumption
sq
if
if
=
This inequality holds
k°.
and only
k°}6
k^
—
'^
—— >
fc°.
1.
Q.E.D.
Proof of Proposition
Let
qt
=
{qt
—
1)
2:
and
kt
=
{h —
normal steady state can be written
qt
1+7
1+7
Qt
characterized by the following
n=
(l+r")(l-7) +
TTi
1
+
+7-
Sfc
kt
^
0j^n
is
+
Sr
h+1 = 7—
This system
dynamic system around the
as:
(l+0(l-7)
Qt+i
k"), then the linearized
^
+ kf
2x2
matrix:
+ ^A;"
tti
+
^
Sk-
1+7
The two
eigenvalues
A"*"
and A
of
fi
are real since the discriminant
D
of det(f)
— xl)
positive:
D
(l
+ r")(l-7) +
TTl^fc"
+
f^
+7
+
1
+4
fc"
7ri
1
A"*"
and A
A+
=l+
are given
=1+
It is
^
1
Sr
1
+7
>0.
by
(l+r")(l-7)+^+^fc"-l+/ d+rn)(l-7)+^ + ^fc"-l V4(.,+i±2^)|^
2
and
A-
+
(l+r-)(l-7)+^^+^fc"-l-y
(l+.")(l-7) +
straightforward to check that
A" <
1<
30
A+.
^
+ ^fe"-i; %4(.,+i±2^)|^
is
Let us denote by (x+,
and {x
1)
the corresponding eigenvectors:
1)',
,
A+ -
+
„
1
1+7
X' -
=
X
1
<0.
gfc"
1+7
This shows that the normal steady state
=
Similarly, let qf
—
{qt
1)
=
and kf
{kt
—
(1
+ r'){l - 7) +
+
3^
+7
(
Then the
A;*).
around the speculative steady state can be written
9t+i
a saddle point.
(fc", 1) is
linearized
dynamic system
as:
1
fe'
q!
1+7-Sfc
+
TTi
fc?
s.
ek'
fc:
1+7*
t+1
This system
is
2x2
characterized by the following
(l
Q=
+ r^)(l-7) +
^
*
matrix:
+ ^fc«
7ri
+
i±^
1+7
The two
is
eigenvalues
A®"*"
and A*
of fi* are real since the discriminant
D^
of det{Q.^
— xl)
positive:
D
(1
+ r')(l - 7 +
A*"*"
and A*~ are given by
A*+
=
(l+r-)(l-7) +
1
+
+
-^
1+7
^
fc*
-
+4
1
Sr
+ ^fc^-l +
^
TTi
+
1
+ 7-Sfc,
'-
Sr
ek'
-)r^
1+7
>
0-
a+r^)(i-7)+^+^fc^-i; V4(.i + i±I^)f^
2
and
A"It is
=1+
(l+r^)(l-7)+^+^fe--l-y a+'-^)(l-7)+^+^fe=-l]'+4(7ri+i±^)f^
straightforward to check that
A"Let us denote by
(x**"*",
1)
and (x*~,
< 1<
1)',
A"+.
the corresponding eigenvectors:
1+7
A^-
-
1+7
31
1
<0.
This shows that the speculative steady state
Let us
and
1
now
are verified and
As we do
functions of
=x+ + 0{5),
We
will
>
/i;"|
x*~,
and
6,
and A*~
A*"*"
fc°
- X- + 0(6),
prove the following
small enough, and that
if
^_. j
<
k^ and k
of {k^{S),
:
1) is
manifold of
[fc°,
A;'g
[1,
+00 and
[
parameterized by
-there exists 5
\'+{6)
(A;,
5
>
and
m
5
all
5^"+
It is clear
1 are verified.
that k'{5)
=
A;"
Let us
+ 0(<5),
= A+ +
0(6) and X'-{d)
=
A"
~^„
then a transition
is
possible for 6
<
^0
,
+ 0{S).
is
possible.
twice continuously differentiable in the
is
:
<
there exist continuously differentiable
5,
-^
[A;", fc°]
[1,
+00 such
[
6
g|~(fc)) in the region k
parameterized by
>
and
continuous
all
^aZ^n then for 5 small enough, no transition
such that for
k^] —>
(A;", 1) is
These variables are
k^, the following results hold:
>
-there exists S
functions q^~
>
0.
and X'-{5).
Because the dynamic system we consider
regions k
us vary k^ and 5 such that assumptions
also vary.
^_~i
result. If
>
>
a saddle point.
as long as assumptions
x*+(<5), x'-{S), X'+iS)
x'-{S)
let
e5 for some e
and do not depend on
denote them by k'{6),
x'+{5)
—
|fc°
this, k^, x*"*",
S,
7 and
fix A;", sq, Sr, ttq, tti,
(fc*, 1) is
(A;,
>
[k^,k^(d)],
q^~^{k)) in the region k
such that
for all 5
that the stable manifold
<
5
e
[A:",
and
A;
and the unstable
k^\.
e
[k^,k^{5)],
and
[A;",A;°],
|g|-(fc)
-
1
-
x'~{5){k
-
k'{5))\
<
m5'^
and
|g"+(A;')-l-x+(A;-fc")|
Using the
A^+(5)
=
^ k^ + 0{5), x'+{5) = x+ + 0{5), x'-{5) = x' + 0(<5),
A^~(5) = A" + 0{5), we see that there exists ^>l>OandM>0
fact that k'{5)
A+ + 0{5) and
such that for
< m^^
alU <1 and
A;
e
[A;°, A:^((5)],
|9|-(A;)
-
1
and
k'
- X- (A; -
G
[A;", A;°],
k\5))\
< M6^
and
|9"+(A:')
The condition
for
state to be possible
is
-
1
-
x+(A;
-
A:")|
<
McJ^.
a transition from the normal steady state to the speculative steady
that the intersection of the unstable
32
arm
of the
normal steady state
=
with the vertical k
ko line
below the intersection of stable arm (the saddle path) of
lies
the speculative steady state with the vertical k
Theq=
+a;+(A;-
1
Similarly the q
intersects the k
fc") line
= l+x~{k-k^{S))
k^{S))). Therefore,
5
if
<5,&
line.
line at the point (fc°, l
—
k
line intersects the
k^ line at the point
=
ko line to
lie
+ x-{k^ -
We
+ x-(fc° -
+ M5^ >
k'{5))
Similarly, a siifRcient condition
1
(A;°,
l+x~(fc° —
+ a;+(A; -
arm
below the intersection of stable
(the saddle path) of the speculative steady state with the vertical k
1
+ a;+(fc° - fc")).
necessary condition for the intersection of the unstable
of the normal steady state with the vertical k
arm
= ko
= k^
-
=
ko line
is
MS"^.
(26)
-M5^>1 + x+{k - fc") + M6^.
(27)
1
A;")
is
k'{5))
can rewrite (26) and (27) as
A+ 1
1
- A" <
k^-k°
TT +
A;0 - A;"
2M5'^
T7^
k^
-
fc"
and
A+ 1
Using the
fact that
when
transition
5
<
(5
1
- A"
we have kept
1
sufficient condition for
and that
if
reexamine
We
^_~i
>
k^
—
if
1
<
-
k'^
2M5^
-
fcO
A;"
>
^_~i
<
k'-
fco
/fcO
eS,
we
k"--
see that a necessary condition for a
k°
2M5
fc"
£
a transition
k'
1
l-\- ^
when 6^0.
^^
This shows that
fcO
- A"
A+ -
Notice that
k'
is
A+ and that a
<
^oZfcn
>
k°
-k°
- fc"
is
2M5
e~'
then a transition
is
possible for 6 small enough,
I^Zpr then for 5 small enough, no transition
this condition.
have that
33
is
possible.
Let us
now
—
= -1 + 2-
A"
1-
]
4(^i
1+-;
+ ^±i^)f|^
Wl + rn)(l-r) +
^^+^k'^-l\
= AWA
a decreasing function of
is
T
non-monotonic in
is
interval (^, +oo),
6. It is
increasing in the interval
Sr
In addition lim^^o
decreasing on
addition,
and decreasing
(0, 9)
in the
where
Ml
It is
+ l±I^)i^
4(vr,
T=
T =
0,
+
+ r")(l-7)-l
(l
and lime^+oo
and increasing on
(0, 9)
we have limg^o A
—
-l-oo
and
T =
O.As a result,
A
+oo), and reaches a
{9,
lime_>_)_oo
A=
is
non-monotonic
minimum
for 9
in 6.
=
9.
In
-|-oo.
Q.E.D.
This shows that
that
if
^_~i
>
if
^_~i
<
^o~^„
,
then a transition
is
possible for S small enough,
^o~^„ then for S small enough, no transition
is
appendix, we will often have to go through a similar argument.
same
details
and instead
and
possible. In the rest of the
We
will
state directly the equivalent of the condition
not go through the
^Zl >
^oZfcn
-
The
only exception concerns the discussion of bubbles where more complex issues arise.
Proof of Proposition
The
3:
condition for a transition from the speculative steady state to the normal steady
state to be possible
is
that the intersection of the unstable
state with the vertical k
=
ko line
lies
Going through the same arguments
enough and
fc*
, ...
of the speculative steady
above the intersection of stable arm (the saddle path)
of the normal steady state with the vertical k
transition for 5 small
arm
—
—
ko line.
as in the proof of proposition 2, the condition for a
fc°
>
e'5
A+ -
1
Q.E.D.
34
can be expressed
A;0
-
fc"
as:
B
Proof of Propositions
Proof of Proposition
in Section 3
4:
In principle, there could be a third steady state once bubbles are feasible:
TTo
— TTifc'' = 7 and
see why,
6
>
0.
Assumption 4 guarantees that
assume the contrary.
It is clear
from assumption 4 that
bubble in this steady state would be negative, since
which
is
this steady state
=
b
<
fc^
does not
<
where
exist.
To
But the associated
k^.
so+s^ttq- Afc''
{k'', 1, b),
so+Sr7ro— AA;"
=
0,
a contradiction.
In the proof of this proposition,
we have
to perform two linearizations, one around
the normal steady state and one around the speculative steady state.
This
is
because
the characteristics of the dynamic system are different around the two steady states.
The
normal steady state has a two-dimensional unstable manifold and a one-dimensional stable
manifold, whereas the speculative steady state has a one-dimensional unstable manifold and
a two-dimensional stable manifold.
We
and show that
for S small
for every
bubble
<
consider the possibility of a small initial bubbles,
A+-1
^_~i
if
<
k'-k°
prEf^r
<
enough and
m^ >
then there exists
<
6o
mb6, and that
<
there exists
0,
,
6o
<
ffibS
m^ >
if
6o
< MbS
for
some Mb >
such that a transition
>
^_~i
is
0,
possible
^o~^„ then for 5 small enough,
such that no transition
is
possible with initial
6o-
Let us
terms in
gi
first linearize
=
(qt
-
1)
,
kt
the system around the normal steady state.
—
•+1
{h -
l
k"qt
-\
=
(1
bt+i
{rt
—
r"), bt
=
bt
and
5
we have that
+ r")(l - 7) +
by
—+
1-^7
A;"
-r-.
qt
+
1
TTi
+7-
Sfc
kt
-I-
-\
bt,
+h
1-hr"
=
Similarly,
h
O 7"
1+1 Qt
n=
to second order
ir
+ T-Sfcr^
H
bt,
by
Qt+i
fc"),
Up
1-^7
we
order terms in
bt.
linearize the
^ =
(qt
—
1)
,
system around the speculative steady
kt
=
(kt
—
k^),
n—
35
{n
—
r*), bt
—
bt
and
state.
6,
Up
to second
we have that
—
-
n =
kt H
k''qt H
Of
=
qt+i
(1
h,
Of
+ r")(l -
7)
+
Of
—+
+7
A;*
T-!1
qt
+
+
TTi
l
+ J-Sk
h
H
bt,
9k'
qt +
+7
1 + r* = T"; h1 + 7
kt
1
—
^t+i
The dynamic system around
the normal steady state
(l+r")(l-7) +
fi"
=
^
+ ^/c"
is
TTi
characterized by the matrix
+
to.
i
1
1+7
i±^
^
Similarly, the
1+7
dynamic system around the speculative steady
state
characterized by
is
the matrix
(l
+ r^)(l-7) +
Q.'
^
+ ^A;^
tti
+
r^
i±^
\
1
1+r
Q
Keeping the same notations as above,
A"~
<
1
and
^f^ >
1,
it is
1+7
clear that the eigenvalues of
with corresponding eigenvectors (x"+,
1,0)',
(x"^,
Q"
are
1, 0)'
A""*"
and
>
1,
(0, 0, 1)
where
A"+
(l+r")(l-7)+^ + ^fc"-l+y
=
1
+
=
1
+
d+r")(l-7) + =||^ + ^fc"-l]V4(.i +
ii^)f^
2
and
A"-
(l+r")(l-7)+^ + ^fc"-l-y (l+r'')(l-7)+^^ + ^fc"-l" V4(., + i±^)f^
Therefore, the tangent plane to the unstable manifold of the normal steady state
plane through the normal steady state with directing vectors
the tangent line to
its
stable manifold
is
(x""*",
1,0)
and
is
(0,0, 1),
the
and
the line through the normal steady state with
directing vector (x"~, 1,0)'.
Similarly, the eigenvalues of fi* are
A*"*"
>
1, A**"
<
1
and ^^~-
<
1,
with corresponding
eigenvectors (x*+,l,0)', (x*~,l,0)' and (0,0,1) where
A"+
=1+
(l+r«)(l-7) +
and
(l+r^)(l-7) +
A^
1
+
^
^
+ ^fc=-l+/ (l+r-)(l-7)+^ + ^fc--l V4(..+i±2_:£.)Ml
2
+ ^fc-'-l-/
(l+r.')(l-7) +
36
^
+ ^fc^-l %4(., + i±2^)Mi
Therefore, the tangent hne to the unstable manifold of the speculative steady state
the line through the speculative steady state with directing vectors
stable manifold
tangent plane to
its
directing vectors
(a;*'~,
now vary
Let us
s
^^
TT^+li^+^-a
>
=
As we do
ables are
>
<5
ttq
-
k^{5)
—
7
this, k^, a;"+,
A;°((5)
=
=
x+
+ 0(5),
— A;"| >
|fc°
x^-((J)
A
— 1+
A
—
^
—
=
+ 0(6),
X-
=
and lims-^ojiS)
r"
A"+(5)
-
It
is
>
e;;"
0,
—
X
<
et"
1
k^ and
functions q'f
These
vari-
and X'-jS)
^
=
fc"
x"~((5),
+
0{S),
= A-+0(5), x^+(5) -
A"
+ 0(.5),
with
,
+4(,rt+i±^)^
0.
+ r'^
fc
>
>
[k^,k^{6)] X
G
[0,
^>f >
[/c",fc°]
<
for all 5
6,
there exist continuously differentiable
x [0,Mb5] -> [l,+oo[ and g"+
(fc*((5),
Mft^],
and
1,0)
is
:
[A;",fc°]
e
[fc",A;°]
m>
x [0,Mb5]
^
[l,+oo[ such
parameterized by {k,b,q^^{k,b)) in the region
and the unstable manifold of
(A;,6,5^+(fc)) in the region {k,b)
-there exists
twice continuously differentiable in the
is
the following results are true:
such that
[fc°,fc"]
:
/c°,
that the stable manifold of
(fc',6')
Also
o
-there exists S
and
0.
ttiA;".
x'^~^{5),
clear that k'{S)
X++0{5), A"-(5)
^ A+ + 0((5)
X'+{S)
Because the dynamic system we consider
G
-
ttq
>
=
9
l+r"
{k,b)
=
A;^((5),
(l+r")(l-r")+^ + ^fc"-l-^[(l+r'^)(l-r'')+:Llff;i + ^fc"-l]
<
fc"
eS for some e
also vary.
A'*"
Let us denote them by
6.
i-r
regions k
that
fc".
= x-+0((5),
x++0((5), x"-(^)
way
s^{6) in such a
x"", A"+,A"",x*+, x^~, A*+ and
continuous functions of
all
>
A"+(5),A"-(5), x'+{6), x'-{6), X'+{5) and A"-((5).
x"+((5)
=
7 ((5) and s^
for 5
ttiA;"
and lim^^o
and the
(0,0, 1).
assumptions 0,1 and 4 are verified and
fixed,
,
the plane through the speculative steady state with
is
)
for
fc"
=
k^
5,
Note that 7(5) < r"
k^{S)
and
1,0)'
(a;^"'",l,0)
is
(fc",l,0)
is
parameterized by
x [0,MbS].
such that
for all 5
<d,
{k, b)
G
[fe°,
fc^(5)]
x
[0, Mfc(5],
x [0,Mb6],
\qf{k,b)
-
-
1
x'-{d){k
-
k'iS))\
< m5^
and
|g^+(fc', 6')
Using the
A'*- (5)
=
A"
-I-
-
1
= /c" +
A"+(5) = A+
fact that k'{6)
0(5) and
-I-
-
x"+(5)(fc
0(5),
0(6),
-
x'*-(<5)
we
37
fc")|
-
x"
<
mS^.
-t-
0(5), x"+(5)
see that there exists
= x+
-h
0(5),
I>f>OandM>0
such that
for all 5
<5,
6
{k,b)
and {k\b') e
[k^,k'{6)] x [0,Mb5],
\ql-{k,b)-l-x-{k-k'(5))\
[fc",/c°]
x [0,MbS],
<M6^
and
\q'^+(k',b')-l-x+ik-k'')\ <M5'^.
The
1
q
=
1
+
+ x~^{k^ — A;").
the line k
=
x~^{k
/c")
k^,q
=
I
—
plane intersects the k
Similarly the q
1
is
—
= l + x~{k — k^{S))
+ x~{k^ —
+ a;-(A;° -
Therefore,
k'^{5)).
k'{5))
+ M^^ >
k^ plane along the line k
plane intersects the k
5
if
<
(5
+ x+{k -
1
we have
,
A;")
-
=
—
k'^,q
=
k^ plane along
that
M5'^.
(28)
a necessary condition for the intersection of the unstable manifold of the normal steady
state with the vertical
fc
=
/eq
plane to
lie
below the intersection of stable manifold of the
=
speculative steady state with the vertical k
Similarly, a sufficient condition
1
We
+ x-{k° -
ko plane in the region b
6
[0,
MhS].
is
k'{5))
- M5^ >
+ x+(fc -
1
A;")
+ M5^.
(29)
can rewrite (28) and (29) as
A+ 1
1
- A"
<
—--
A;^
A;°
T7.
A;0
7z:
A;"
2M5'^
+
A:^
-
A;"
and
A+ 1
Using the
5
fact that k^
— k^ >
1
- A"
e5,
<
A:*
A;0
we
- A;°
- A;"
2M5'^
k^
-
k'^'
see that a necessary condition for a transition
<^is
A+-1
1
and that a
_ A" ^
sufficient condition for
A;0
-
A;"
a transition
A+-1
1- A"
2MS
A:"-A:°
k'
-
k°
k^ -k-^
38
^
e
is
2M5
~'
when
'
^
Notice that
>
rrih
^
~i
when
-»
such that a transition
>
then
^aZ^ri
that no transition
is
for 5
is
S -> O.This
shows that
possible for 5 small
small enough, for every
if
and only
-1
For
exactly the
of a transition
is
<
enough and
frib
>
0,
f^rf^, then there exists
<
6o
<
there exists
mt,6,
<
6o
and that
<
if
f^hS such
if
A+
is
j^
possible with initial bubble 6o.
This condition holds
which
if
same condition
fc"-A:°
^
fco
-
fc^
as without bubbles.
The
discussion of the possibility
therefore entirely similar.
Q.E.D.
Proof of Proposition
that the projection of the trajectory of the transition path with initial
It is also clear
bubble
6o
>
on the plane
bubble.
initial
5:
To
6
=
will lie
below the trajectory of the transition path with zero
see this, note that the projection of a trajectory with positive bubble can
only cross the projection of a trajectory with zero bubble from below. This proves the claim
since the projection of a transition path, with or without bubble ends in the speculative
steady state
By
initial
(fc*, 1, 0).
the capital accumulation equation, this in turn implies that a transition path with
bubble
6o
>
will reach every
plane k
=
h with k^
< h<
k'^
after a larger
of periods than the transition path with zero initial bubble.
Q.E.D.
C
Proof of Propositions
C.l
We
in Section 4
Fiscal
can rewrite the capital market equilibrium condition as
s{{l-T)w{kt),rt)
+ Tw{kt) -gt=
iqt
where
Let us define a net saving function
39
+ xt +
^0~'^{xt-j)'^]
h + du
number
s{w{kt),rt)
=
s((l
-
T)w{kt),rt)
+ Tw{kt) - gf
In this proposition, we perform a comparative statics exercise varying
lently
pj and keeping
as{{i~-r)Mkthrt)+rw{k,)^-g
on
the normal steady state fixed.
^^^^ ^^
^
Therefore,
as{{i-r)w{k,),n)+rMk,.)-g
rj.^^^^
_
let
a
(or equiva-
us denote by Sk
—
numbers do not depend
a.
Note that a
first
order approximation of wages around the normal steady state
w{kt)
We
therefore have the following
~f
fi.^
^
s{w{kt),rt)
=
first
y
{
=
s'^
—
Sk
—
QTTifc",
and
Sq
=
order approximation for the net saving function
+
+
s'^kt
sq
+ 7riA:"fcf
u;"
s'^kt
where
+
srn
Srn
k<k°+ s^,
+ s^ + 6, k> k°,
"^^^^
Assumption 4 guarantees that the normal steady
region, while the speculative steady state,
The
speculative steady state,
if it
>
=
d'
+
il
state,
s(i(;(A;"),r")
r"
+ j)k',
w") - Tw').
we have
=
TTq
=
(l
-
TTlfc",
40
+ 7)fc",
in a dynamically efficient
by the following equations
k°,
- ^-^(5 + a{w' -
normal steady
is
a dynamically inefficient region.
exists, is characterized
J{w{k'),r')
Similarly, in the
state
exists, is in
if it
k'
d'
is:
.
Using these equations, we can rewrite the conditions characterizing the speculative
steady state as
>
k'
-
s^{k'
+ Sr{r' -
fc")
k°,
+ =
r")
<5
d^
+
+ 7)(fc^ -
(1
fc"),
r^-r" = -7ri(F-A:"),
7—
We
can solve these equations
In particular,
\sl
We
(5
=
[(1
-
n,sr
we
-
(1
can rewrite
+ 7) -
Sfc
for
r*
/c®,
w^
r^
,
and
d^
get
+ 7)]
-kn+S= ]^Sn7n%~S]
(fc^
" ^^^^^^^^^ '
^''
^"^^
this equation as
+ TTiSr]
(/c*-fc")+a7rifc"(A;"-/c")
+ (Q; -
[1
+ r" -
r)
7ri(A;^
7—
-
A;")] 7rifc"(A;*
+ 7ri(fc*
r"
—
-
A;")
(30)
Using the
k^
>
k^,
we
fact that 1
see that for
[(1
It is clear
+
a >
r"
r,
+ 7) -
=
1
+
r"
-
7ri(A;'*
-
fc")
>
0,
(1
+
7)
the right hand side of this equation
Sfc
+ T^isr]
(fc°
-
A;")
+ Q7rifc"(A;° -
-
is
Sfc
-
niSr
>
and
greater than
A;").
that
lim
a—+00
{[(l
+ 7)-Sfc + 7riSr](fc°-A:")+a7riA;"(/c°-A;")} =
+cx3.
This proves that for a high enough (30) has no solution.
,
We
have the stronger result that a
a > max
sufficient condition for (30) not to
[a
s —-(1
—;— ,,„,
/,
<
1 TTiA;" (A;0
-
^
A;")
41
A;0-A;", _1
j
i
),r
>
J
.
have a solution
is
A;")
)
Therefore
a
>a where
fcO-fc"
T— ; <
if
1, it
can be the case that there
no solution
is
for
A
a <
1.
Q.E.D.
External
C.2
We
continue to assume that 5
—
around q
^—
(?<
-
1
and k
1)) ^t
written
as:
Qt+i
(l
=
=
k^
{h -
from
for the transition
and
k"-),
+ r")(l-7) +
small so that
is
(7ri^
zt
=
{zt
-
we can make a
/c"
to k^,
if
first
such transition
z") then the linearized
+ ?r(l + e))
order approximation
1+7
i
is
possible. Let
dynamic system can be
+7
1
+7
(31)
«t+i
The
analysis
and proofs that follow
1+7
qt
+
refer to the
kt.
(32)
dynamic system described by
and
(31)
(32).
The
speculative steady state
7ri(5
Tl (l+7)+'ri (1+7-Sfc
is
characterized by k^
=
—
,,
,
,'^J-'^/.
^
>
r
,
and
P—
>0.
For a speculative growth path to
must be the case that the unstable path of
exist, it
the normal steady state intersects the k° line below the intersection of the latter and the
saddle path of the speculative steady state.
We
can express
this condition as
A^+-l
k'-k^
<
A,g =
where the superscript
A^+
= 1+
e stands for external
-l + (l+r")(l--7)+^
(i+e)fc"
1
+7
+
1+7
(33)
and
+ \/( -H-(l+r -)(1--7) + ^
{i+e)fc"
sqefc"(i-
1+7
+
1+7
^ 1+7
+
1+7
-)'
1+7
-i_4
7r]flfc"
1+7
1+7
2
and
A"-
= 1+
We
have
-l+(l+r'^)(l--7) +
A^
-1
^ (l+S)*:"
+
1+7
-l+(l+r -)(1--7) +
1+7 ^\/(-
(i+e)fc"
+ 21,sie*"(i-^)
1+
^
42
1+7
,^^,efen
+^T+T^
)" -^-A
7ri9fc"(i1
+7
1+7 4-4
n-^Sfc"
1+7
This
+
is
+ r")(l -
(1
-l
+
we have assumed
true because
(l
7)
+
^
>
0'
which imphes that
holds,
and therefore
+ r")(l-7) + ^^^l±^ +
^>0.
a decreasing function of
A*^ is
1+7
'Y^e
-l
T^
that Assumption
is
non-monotonic
+
in 9. It
(l
is
1+7
+^1+7
efc"
+ r»)(l-7)+ "'^iy""
1+7
increasing on
-l +
(l
(0,
9
)
'
and decreasing on
+ r")(l-7) +
{9"
+00), where
,
!^
1+7
In addition
Um^^o """^ =
0,
and Hme_,+oo
=
'^'^
0.
—€
As a
result, A*^ is
{9 ,+00),
non-monotonic
and reaches a minimum
lime_+oo A^
=
in 9. It
for ^
=
is
decreasing on
9
)
and increasing on
we have lime_o
In addition,
.
(0,
A'^
=
+00, and
+00.
Q.E.D.
C.3
Technologiccd progress
Let us assume that 6p
=
1
—p
= I and k = k"' for the
— 1) and kt = (fej — fc"),
around q
qt
=
{qt
is
small so that
transition from
we can make a
A;"
to k^,
if
first
order approximation
such transition
is
possible. Let
then the hnearized dynamic system can be written
as:
kt
qt+i
?t
+ i^t +
[(1
+ r-")7 -
r^fc"
+ ^k''
(34)
h+i =
The
analysis
<
^.^t
1+7- +
kt
^,qt +
h + ^^5, k>k°,
and proofs that follow
kt
k°-
(35)
refer to the
dynamic system described by
(35).
Assumption
5
(Minimum
technological progress):
A
'^^>^-l+^n+l±2fc,
43
-(fc°-A;").
(34)
and
<
k°\
—
Proposition 10 (Multiple Steady States)
If
Assumption 3
economy has two non-degenerate steady
is satisfied, the
states, fc"
and
k",
with:
^ £L+B <
fc»
where
A=
tti
+
and speculative,
+7-
(1
,»
feo
<
>
0,
Sk)/sr
+
1
+ -" + (^+ 7)fc"/^.,^ ^
,s^
(36)
and the superscripts "n" and "s" stand for normal
respectively.
Assumption 6 (Speculative adjustment
A+-1 +
—
A
costs region):
fc"
^'•Y+\,
,0
,,
—
<
Proposition 11 (Multiple Equilibria and Speculative Growth)
If
fc"
Assumptions 5 and 6
hold, there is a speculative
growth path that takes the economy from
to k'.
Proof:
Imposing q^i
qt+i
=
=
qi
and kt^i
a-o(i-.) +
=
kt
and solving
for k^
and
q^ in
(^ + ^)r —Ap
^
kt+
qt-\
Sp
1+7
Sr
and
r- =
Kt+1
we
k'^
>
find that
k^
5„
—
k^
>5p=
Let us
—
fc*
k^,
=
-^
—
5p
—
Ok"" ^
:rqt
ek""
p
+ h + --
and
= — 5p.
1+7
q'^
,
()p,
1+7
This
is
a steady state
if
and only
if
which we can rewrite as
^TTT-
now assume
(k°
-
A;")
.
that this inequality holds.
We now
ask under what conditions a
transition from the normal steady state to the speculative steady state
speculative growth path to exist,
it
is
possible.
For a
must be the case that the unstable path of the normal
steady state intersects the k° line below the intersection of the latter and the saddle path
of the speculative steady state.
1+7
We
can express
this condition as
1+7
44
We
can rewrite (37) as
A
\+
fe"
-ir
J
1
1
+7
n
^
AJe!L
1.0
_
•
l.n
Q.E.D.
D
Proof of Claims
The curve
—
nt
in Section 5
=
intersects the g
hne
1
a point
at
-
wjkt)
c
—
if
+r
1
the assumption that this equation has at least one solution.
example that f{kt)
^-^
and only
f'ikt)
kt
We make
(1, k) if
=
,
assumption
-fk}, this
< a <
Afcf with
be
will
and
1
+ r) > j^
|u(l
and
and the equation nt
verified
<
c
=
will
If
we assume
for
maxfc>o{/i-u;(/c)
-
have exactly two
solutions.
Let k^
there
is
We
<
k^ be the
no second
two
intersections of this curve with the g
first
this
that depends on
5,
1 line (fci
— +oo
if
intersection).
assume that there are two steady
Because in
=
framework
we
will
it is
states
now
A;"
<
fc*
where
/c"
<
fc°,
and
fc°
<
fc*
<
the interest rate in the normal steady state r
fc^.
+
S
approximate the dynamics of the system around the speculative
steady state (where the interest rate
is r).
Letting S tend to
we
0,
moving the normal
are
steady state while keeping the speculative steady state fixed.
Assumption
7:
A=
s^tti
+
(1
+7-
>
Sfc)
0,
k"
=
2^ > k°.
Assumption 8 (Minimum growth- funding feedback):
Assumption
The
line
last
9:
(1
+ /i) nik' - 7 - 3^ >
assumption
is
made
to
make
with the low interest rate region to
and
(^
+ 7) -
sure that locally, the
>
S
5
=
2^ >
n
=
S7.7ri(fc*
—
fc°).
0.
curve
is
upward sloping
its right.
Let
Et=[j+i9 + 7)
{qt
-
1)
Assuming that 5 and n* are
q
—
q*
=
l
and k
=
+ e{qt -
1)2]J
h - ^iw{kt) -c+
small, let us perform a
k^.
45
first
y// i+7+g(gt-i) j^j
^+
1
-kf
r
order approximation around
Up
nt
=
to second order terms in qt
rit
—
n^, S
and
Ht
=
we
n'^,
-fkt
+
—
-
{qt
I)
kt
,
—
—
{kt
+ -y)k'qt-{l + fi)Trik'kt-
{e
niOk^'^
E*
=
7fc*
-
=
ft
/i(u;*
if
-
c)
+
<
j^fc"
KiQk s2
and
A;"
The
analysis
and proofs that follow
+ ^i)'Kik -
•gt
—
+
1
7
k^), ^t
=
{Tt
— t^),
=
-^ <
TY—
i + 7
—
qt
1+7 9t
/to
<
+r
kt
+r
> — ^*. Note
Ef
if
Sr
= 7-
kt+i
+7
l
ft
+ r^)(l-7) +
(l
St+i
< — E'* and
Et
1
qt
1+7
have
(^°
have:
1
We
=
A;*), fco
that by assumption,
0.
+ TTikt + l{'Et<-S'] —
(38)
+
(39)
refer to the
kt-
dynamic system described by
(38)
and
(39).
For a speculative growth path to exist two conditions must be
be the case that the unstable arm of the normal steady state
line.
Second
the Ej
it
= — E*
equilibrium.
is less
satisfied.
steep than the E4
must be the case that the unstable arm of the normal steady state
line
We
can express these conditions as
=
+ M)7rifc^-7-^
(l+7)(l + ^) -TTifc'*
(l
(A/^
-
>o,
A^+-l
(l+'''''i'='-^Zlfe_(A-f+-l)
Af
_
=
(l
+ 7)(l
<
i-xf-
k"
-k
kO-k'^'
(l+.W^--7-T^ ^. _^,_
(l+7)(l + ^)-"lfc'
^
where
=
(l+rnil-l}+^
1
+
-V (l+'-»)(l-7)+^-l
^^
1
+7
2
(l+r=)(l-7)+^ -.-/
We
it
must
= — E*
intersects
below the intersection of the latter and the saddle path of the speculative
^/
A^+
First,
can rewrite
d+r=)(l-7) +
A-^ as:
46
^-l
2
,.,efc»
+* 1+7
and
(l+M)7rifc--7-i^
^^f
1
I
^
We
'^
^
~
^
"
(l+7)(l + i)-lfe»
have
*^ <
lim
SO that no transition
unstable
arm
is
possible for too large adjustment costs because the slope of the
of the normal steady state goes to
+oo
while the slope of the Et
= —5*
tends
to a finite positive limit.
Similarly
lim
so that
is
no transition
is
^•'
possible because the unstable
too steep (the slope of which tends to a
Et
= — H*
line (the slope of
— — oo,
arm
of the normal steady state
finite strictly positive limit)
which tends to
and overshoots the
0).
Therefore, a transition can only occur for intermediate adjustment costs.
Let us define the Assumption 10 as requiring that the two conditions for a transition be
satisfied.
Assumption
We
10:
'^^{9)
>
and A^iO) <
^^
have proved the following propositions.
Proposition 12 (Multiple Steady States)
If
A;"
Assumptions
and
7,
8 and 9 are
satisfied, the
economy has two non- degenerate steady
states,
k^, with:
j^n
^
JL^
Sri
<
fco
<
fc^
=
JL^
2rZ.
(40)
TTl
TTi
Proposition 13 (Multiple Equilibria and Speculative Growth)
If Assumptions
from
fc"
to k^
.
7, 8,
9 and 10 hold, there
Along that path,
qt
is
a speculative growth path that takes the economy
>
47
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