Endogenous Keynesian Business Cycles∗
Hans Jørgen Whitta-Jacobsen†
This version: March 2004
Abstract
Nominal rigidity is shown to work as a natural mechanism for rational
expectations endogenous fluctuations. A nominal price (the nominal wage
rate) is assumed to adjust sluggishly, but competitively between any two successive periods in response to the excess supply or demand of labor in the
first of the periods. The standard case of instantaneous, within the period,
competitive wage adjustment is kept within the analysis as a special case.
It turns out that under realistic circumstances that rule out the possibility
of endogenous fluctuations in the standard case, a certain and plausible degree of sluggishness in the nominal wage adjustment implies existence of such
fluctuations. Along endogenous business cycles caused by the type of nominal rigidity considered, the rate of unemployment varies countercyclically,
substantial fluctuations can occur for positive and reasonably small values of
the elasticity of labor supply, and welfare concerns give a relatively strong
case for stabilization policies aiming at keeping the economy at a steady state
with unemployment at the natural rate all the time.
JEL Classification: E32 and E63.
Keywords: Endogenous business cycles, nominal rigidity.
∗
This paper has benefited from comments and suggestions by Jean-Pascal Bénassy, Birgit Grodal and Christian Groth. I would like to thank Departament d’Economia i Empresa, Universitat
Pompeu Fabra, Barcelona for hospitality during the period where this paper was started.
†
Institute of Economics and EPRU, University of Copenhagen, Studiestraede 6, DK-1455
Copenhagen K., DENMARK. E-mail:[email protected]
1
Introduction
This paper assumes a nominal rigidity of traditional Keynesian type: A certain
money price, the nominal wage rate, is assumed not to be free to adjust within each
short period to equate price taking supply and demand, but to adjust sluggishly
between any two successive periods in reaction to the excess supply or demand
of labor in the first of the periods. It is well-known, and a long-lived Keynesian
insight, that nominal rigidities may be of importance for an economy’s business cycle
properties, since such rigidities influence the way the economy reacts to nominal and
real exogenous shocks. We bring a new insight: Sluggish, competitive adjustment
of nominal wages may itself be the factor that causes business cycles to occur.
We study a simple and standard, dynamic monetary model with no intrinsic
uncertainty. Our interest is in the occurrence and properties of endogenous fluctuations in the form of rational expectations “stationary sunspot equilibria”. Prices
are assumed to adjust competitively in reaction to price taking supplies and demands, but the nominal wage rate possibly adjusts non-instantaneously. The standard assumption of instantaneous wage adjustment is kept within the analysis as
a special case. In this special case endogenous fluctuations can, according to our
model, only occur if labor supply is sufficiently decreasing in the real wage rate, a
result well-known from the literature. We show that under realistic assumptions
on economic fundamentals, possibly implying an increasing labor supply curve, a
sufficiently sluggish adjustment of the nominal wage rate may imply indeterminacy
of rational expectations equilibrium and the existence of endogenous fluctuations.
In this precise sense nominal rigidity, or sluggish nominal wage adjustment, can be
a mechanism for endogenous fluctuations.
In the literature several mechanisms for endogenous fluctuations have been provided in various types of models, e.g., strong complementarities in intertemporal
choice, Azariadis (1981), Woodford (1984), Grandmont (1985), Azariadis and Guesnerie (1986), strong complementarity between labor and capital in production, Reichlin (1986), Woodford (1986), Grandmont, Pintus, and de Vilder (1998), strong
productive externalities, Benhabib and Farmer (1994), Farmer and Guo (1994),
Benhabib and Farmer (1996), countercyclical markups, Gali (1994). For a comparison, see Schmitt-Grohe (1997).
1
The new mechanism for endogenous fluctuations suggested here may be of interest for mainly two reasons: First, the mechanism is plausible in the sense that
the degree of sluggishness in wage adjustment required for indeterminacy under
reasonable other circumstances is not unrealistic empirically. Furthermore, endogenous fluctuations caused by this mechanism have some properties of interest from
a macroeconomic perspective.
First, because the fluctuations we study are rooted in sluggish adjustment of
nominal wages, output variations will be accompanied by variations in unemployment rates according to a countercyclical pattern, and substantial fluctuations in
output and employment can occur for positive and realistically low values of the
real wage elasticity of labor supply.
Second, our analysis gives a relatively strong welfare based motivation for stabilization policy. The monetary steady state of the model we consider, at which
output and unemployment are at their “natural” rates, is better as regards welfare than endogenous fluctuations according to which output and unemployment
fluctuate around the natural rates, and the welfare loss of fluctuations is probably
not negligible. This may motivate policies aiming at affecting rational expectations
dynamics such as to stabilize output and unemployment at the natural rates.
Finally, an often heard objection to Keynesian economics is that although nominal rigidity may imply that shocks cause unemployment to fluctuate around the
natural rate, this gives no strong case for stabilization policy. Over a sequence of
short periods, the competitive reactions of the wage rate should themselves bring
unemployment back towards the natural rate. We incorporate explicitly into the
analysis the type of wage adjustments between the short periods of rigid wages,
that this objection points to. We show that this may well be exactly the factor
that makes the full rational expectations dynamics of the economic system (without shocks) become volatile, thus causing unemployment to fluctuate permanently
around the natural rate rather than drawing it towards the natural rate all the time.
The intention here is to study the implications for endogenous fluctuations of
perfectly competitive price adjustment, adding only the feature that it may take
time for competitive market forces to bring the nominal wage rate to its equilibrium value. The underlying idea is that prices are determined by the anonymous
market forces of supply and demand, but an excess demand can only give rise to a
2
pressure on a price after a short period of “manifestation”. We measure the degree
of excess demand in the labor market by the rate of employment, for which reason
our wage adjustment rule will look like an “old” (expectations augmented) Phillips
curve. This will allow us an empirical evaluation of the rule’s basic elasticities, and
thereby an assessment of the empirical plausibility of the sluggish wage adjustment
mechanism for indeterminacy and endogenous fluctuations.
Our intention is not to study the implications of nominal rigidity in the form
where imperfect competitors determine nominal wages or prices and have to preset
these for a period, possibly with some form of staggering as well. This leads to other
types of Phillips curves, such as a Lucas “surprise” supply function or a so-called
new (Keynesian) Phillips curve. A main difference between old and new Phillips
curves lies in the timing of events: According to the old, a measure of excess demand,
which can be the employment rate or the output gap, in the current period has an
impact on (wage) inflation from the current period to the next one, while according
to new Phillips curves excess demand in the current period influences inflation from
the previous to the current period. In other words, according to old Phillips curves,
excess demand leads inflation, while according to new, inflation leads the output
gap. There is overwhelming empirical evidence that the first of these possibilities
is the more realistic, e.g. Fuhrer and More (1995) and Gali and Gertler (1999),
in particular if the period considered is short, such as a quarter. This is the main
reason why we consider a nominal rigidity of the “old” type.1
In Section 2 we set up the elements of the monetary economy under consideration. Section 3 is concerned with the standard case of instantaneous price adjustment and summarizes some known results, while Section 4 presents the model
with sluggish competitive wage adjustment. Section 5 reports the positive results
concerning Keynesian business cycles and gives the intuition. Section 6 offers some
concluding remarks.
1
Gali and Gertler (1999) write, for instance: “... the benchmark new Phillips curve implies
that inflation should lead the output gap ... Yet, exactly the opposite pattern can be found in the
data ... This lead of the output gap ... (is) ... consistent with the old Phillips curve theory but
in direct contradiction to the new.” Although the contribution of Gali and Gertler can be seen as
an attempt to rescue the new Phillips curve, we find that for the present purposes the empirical
evidence is in favor of an assumption with respect to timing where excess demand leads inflation.
3
2
A Simple Monetary Model
We consider a monetary dynamic, general equilibrium model, which is a simplification to “no capital” of the cash-in-advance economy suggested by Woodford (1986).2
The exclusion of capital in the production function will imply that the dynamic
equation describing rational expectations equilibrium will be one-dimensional (univariate). This may be desirable for two reasons. First, in the one-dimensional case
a better intuitive understanding of why and how the nominal rigidity considered
promotes the existence of endogenous rational expectations fluctuations can be obtained. Second, at present the learning stability or instability of sunspot equilibria
in the neighborhood of a steady state is better understood for the one-dimensional
case, see Evans and Honkapohja (2003).
Time runs discretely in periods indexed by t. In each period the commodities
are labor, output, and money. The nominal wage rate and the nominal output price
in period t are wt and pt respectively. The money stock stays constantly at M > 0.
A representative firm produces output, yt , from labor input, lt , in period t
1/β
according to the production function, yt = lt , where β ≥ 1.
An infinitely lived representative worker obtains utility in period t according to
the von Neumann-Morgenstern utility function
P∞
s=t
δ s−t [u(cs )/δ − v(ns )], where δ
is the discount factor, 0 < δ < 1, and cs and ns are consumption and work time
in period s, respectively. On u and v we impose differentiability assumptions and
u0 > 0, u00 < 0, u0 (0) = ∞, v 0 > 0, v 00 > 0, v 0 (0) = 0, and v 0 (1) = ∞ (maximal work
time in a period is one). In case the firm earns positive profits (β > 1) these go
to a separate representative capitalist who does not work, but obtain utility from
consumption according to the utility function
P∞
s=t
δ s−t u(cs ).
The consumers are assumed to be subject to cash-in-advance constraints. In
any period t, the value of a consumer’s consumption cannot exceed the amount of
money, mt , held at the beginning of the period, pt ct ≤ mt . This gives rise to a
“transactions demand” for money.3
2
The equilibrium conditions derived below are equivalent to the equilibrium conditions of an
appropriately defined overlapping generations model, as explained by Woodford (1986). For the
present analysis the period length should be thought of as the amount of time during which a
nominal price (the wage rate ) is insensitive to current circumstances (excess demand for labor),
which should be relatively short. The model interpretation with an infinitely lived agent and
cash-in-advance constraints is therefore more appropriate.
3
We make the assumption that profits go to a separate capitalist class because it has been
4
The consumers are also subject to traditional budget constraints stating that
in any period the increase in money holding equals what is earned minus what is
spent, and money holdings must always be positive. However, if the cash-in-advance
constraints are binding in all periods the optimal behavior will be as follows. The
capitalist will in each period spend on consumption exactly the profits received in
the preceding period, while the worker will over the succession of any two periods
(t, t + 1), decide on nt , mt+1 , and ct+1 to maximize Et u(ct+1 ) − v(nt ), subject
to mt+1 = wt nt , and ct+1 = mt+1 /pt+1 . Hence, labor supply, nt , in period t is
the solution to maximizing Et u([wt /pt+1 ] nt ) − v(nt ) with respect to nt , where the
expectation, Et , is taken with respect to pt+1 at time t. We only consider rational
expectations equilibria, so the subjectively expected distribution over pt+1 at time
t will be equal to the equilibrium distribution over pt+1 conditional on current
realizations pt , yt etc. The above problem has for any wt > 0 and any distribution
over pt+1 > 0, a unique solution, nt , strictly between zero and one given by the first
order condition,
v 0 (nt ) = Et u0 (
wt
wt
nt )
.
pt+1 pt+1
(1)
Let n(ω t ) be the labor supply curve under deterministic expectations, that is, the
solution in nt to v 0 (nt ) = u0 (ω t nt )ω t , where ω t := wt /pt+1 . Log-differentiation gives
for the elasticity ε(ω t ) := n0 (ω t )ω t /n(ω t ),
ε(ω t ) =
1 − R(ω t nt )
> −1,
R(ω t nt ) + N (nt )
(2)
where nt = n(ω t ), R(c) := −u00 (c)c/u0 (c), and N(n) := v00 (n)n/v 0 (n).
The behavior described above is optimal conditional on the cash-in-advance
constraints being binding in all periods. Along an equilibrium that stays sufficiently
close to a steady state the cash-in-advance constraints will indeed be binding in all
periods simply because one must have u0 (ct ) > δ (pt /pt+1 ) u0 (ct+1 ) for all t, whenever
ct and ct+1 are always sufficiently close to a common value, c, and nominal prices are
close to being constant (as they will be close to a steady state). In the continuation
it will therefore be of interest to identify equilibria that stay close to a steady state.
analyzed elsewhere how the occurrence of endogenous fluctuations is affected if the decision on
labor supply is influenced by profit income, Tvede (1991) and Jacobsen (2000). The assumption
that the capitalist is also subject to a cash-in-advance requirement is made for simplicity and
preserves the equivalence to an overlapping generations economy where nominal income earned as
young is saved and spent in the old age.
5
It will everywhere be assumed that the output and money markets clear instantaneously and competitively. The labor market possibly has a sluggish (competitive)
wage adjustment implying that supply and demand of labor are not necessarily equal
in all periods. Plausible assumptions below on the fundamentals of the considered
economy will imply that along the equilibria studied labor supply will exceed labor
demand in all periods, just to varying degrees. It is therefore natural to assume
that actual employment is given by labor demand in all periods, implying that the
firm is never rationed in demand for labor.
The worker may be rationed in supply of labor and experience that not all of
nt , but only the smaller demanded quantity, lt , can be sold. This will have an
implication for how much the worker can actually consume next period, but will
not affect labor supply, nt , the amount of labor the worker would like to sell in
period t.
The fact that the cash-in-advance constraints are binding in all periods implies
that the money holdings of the worker and the capitalist at the beginning of period t are last period’s incomes, wt−1 lt−1 and (pt−1 yt−1 − wt−1 lt−1 ), respectively,
so total money demand is pt−1 yt−1 , while the output demands in period t are the
output values of these money holdings, giving a total demand of pt−1 yt−1 /pt . Since
equilibrium in the money market implies M = pt−1 yt−1 , the AD-curve is simply,
yt =
M
.
pt
(3)
(1−β)/β
Profit maximization by the firm (the marginal product lt
equal to wt /pt )
implies a supply of output in period t, or an AS-curve,
yt =
Ã
pt
βwt
!
1
β−1
.
(4)
Instantaneous clearing of the output market, AD = AS, then implies that in any
period t, for any given wt , the nominal output price adjusts to ensure,
1
wt =
β
µ
1
M
¶β−1
pβt ,
(5)
implying, by also using pt = M/yt , that,
ω t :=
wt
yt+1
=
.
pt+1
βytβ
6
(6)
3
Instantaneous Adjustment of Nominal Wages
We first give a brief account (meant for comparison) of some known results for
the standard case of overall instantaneous price adjustment where in each period,
in addition to the adjustment of pt just described, wt adjusts to ensure nt = lt .
Inserting nt = lt = ytβ and (6) into the first order condition, (1), gives,
βytβ v0 (ytβ ) = Et yt+1 u0 (
yt+1
).
β
(7)
This temporary equilibrium condition and the rational expectations and/or learning dynamics associated with it, have been studied in numerous contributions, e.g.
Azariadis (1981), Woodford (1984), Grandmont (1985, 1986), Azariadis and Guesnerie (1986), Woodford (1990), Evans and Honkapohja (1994, 2001).
The perfect foresight dynamics associated with (7) is βytβ v 0 (ytβ ) = yt+1 u0 (yt+1 /β).
For any yt+1 > 0, this equation has a unique solution yt < 1 defining (globally) a
function yt = f (yt+1 ). A monetary steady state is a y > 0, such that y = f (y), or
βy β−1 v 0 (y β ) = u0 (y/β). Under our assumptions there is a unique such y, and y < 1.
At this steady state the intertemporal real wage is, from (6), ω = 1/(βy β−1 ) and thus
constant. The slope of f at steady state is f 0 (y) = ε(ω)/ [β(1 + ε(ω))].4 If f 0 (y) <
−1 (> 1 is not possible since ε > −1), perfect foresight equilibrium is indeterminate.
Indeterminacy, which is equivalent to ε(ω) < −β/(1 + β), implies the existence of
truly stochastic stationary sunspot equilibria arbitrarily close to the steady state.
(We provide some definitions and general results later on in connection with the
more general model). Some of these stationary sunspot equilibria are learning
stable under an appropriately defined adaptive learning scheme, an application of
the recent result in Evans and Honkapohja (2003).
The condition ε(ω) < −β/(1 + β) (≤ − 12 ) is not empirically plausible, but
it is not strictly necessary for the existence of rational expectations endogenous
fluctuations either. However, an assumption of ε(ω 0 ) ≥ 0 for all ω 0 can be shown to
imply determinacy and non-existence of stationary sunspot equilibria (as actually
shown in footnote 6 below).
4
One can find f 0 (yt+1 ) by implicit differentiation of (7) and then measure at yt+1 = yt = y to
find f 0 (y). This is actually done for the more general model studied later.
7
4
Sluggish Adjustment of Nominal Wages
We now generalize the model with instantaneous competitive wage adjustment to
a situation where the nominal wage rate possibly reacts sluggishly, but still competitively, to an excess supply or demand.
In any short period t, in which the wage rate is wt , a short run equilibrium given
by AS = AD is established, exactly as described in Section 2. Hence, given wt , the
nominal price pt adjusts in accordance with (5), implying a level of production,
yt = M/pt , and a certain level of employment, lt = ytβ . In period t there will also be
a labor supply, nt depending, as given by (1), on wt and the expectations concerning
pt+1 . To distinguish logically between individual and aggregate magnitudes we will
let total employment in period t be denoted by Lt , and total labor supply by Nt . In
equilibrium these will equal lt and nt respectively, but the individual worker takes
Lt and Nt , and hence the rate of employment, et := Lt /Nt , as given.
We assume that market forces imply that the expected value of any period’s
nominal wage rate is predetermined from the previous period’s realizations and
expectations, among them the degree of labor shortage measured by et , in a way
that obeys,
Ã
Et wt+1
Et pt+1
= P (et )
wt
pt
!α
, P 0 (e) > 0 and 0 ≤ α ≤ 1.
(8)
In this Phillips-curve type wage adjustment rule, α measures the degree to which expected inflation builds itself into wage inflation with (8) allowing anything between
“fully” (α = 1, the most natural assumption) and “not at all” (α = 0). Alternatively, α measures the degree to which it is the real, as opposed to the nominal,
wage rate that reacts to a given demand pressure on the labor market.
It should be noted that (8) generalizes the model with instantaneous market
clearing. A special case is that P is vertical at et = 1, as illustrated by the vertical lines in Figure 1, a) and b). (Formally this particular case is included below
by proceeding in terms of the inverse of P ). With a vertical P , the economy’s
equilibrium condition, as derived below, will specialize to exactly (7), arising from
instantaneous wage adjustment. However, in the wide range of functions P allowed
by (8), the vertical one seems special.
8
P
1
e
e*
1
Figure 1, a)
P
1
e*
e
1
Figure 1, b)
It is important that (8) imposes a restriction on the expected value of the nominal wage rate, not on the wage rate itself. This allows the wage rate of any period to
be a random variable, given past realizations and expectations. (The actual process
for the wage rate will be determined in a full rational expectations equilibrium as
defined and described below). If it had been wt+1 rather than Et wt+1 that entered
into (8), then wt+1 would be determined deterministically given current variables
and the equilibrium distribution of pt+1 , but then, due to (5), pt+1 would also have to
9
be be determined deterministically ruling out a stochastic process for pt+1 in rational expectations equilibrium. Hence, the possibility of stochastic sunspot equilibria
would be ruled out a priori.5
The natural rate of employment is defined by P (e∗ ) = 1, that is, as a degree
of employment that does not imply any increase or decrease in wage inflation over
what is caused by expected price inflation. Such an e∗ is assumed to exist, see also
Figure 1, and it is natural to assume that e∗ < 1, except in the case where P is
vertical at one, where e∗ = 1. An assumption of e∗ < 1 (or P vertical at one) will
imply that all equilibria sufficiently close to steady state will have et ≤ 1 in all
periods since the steady state rate of employment turns out to be exactly e∗ . This
justifies our earlier claim that labor supply always exceeds labor demand (weakly)
along the studied equilibria.
From (5), Et wt+1 /wt = Et pβt+1 /pβt . Inserting this into (8) gives,
Et pβt+1 (Et pt+1 )α
/
= P (et ) .
pαt
pβt
(9)
To include formally as a special case the vertical P , we proceed in terms of the
inverse of P , that is, the function H := P −1 from (0, ∞) into (0, ∞), that gives the
rate of employment in period t needed to create a certain (excess) wage inflation
from t to t + 1, and we allow H to be “flat at one”. Taking the inverse, P −1 , on
both sides of (9) gives,
Ã
!
Et pβt+1 (Et pt+1 )α
et = H
/
, H 0 (z) ≥ 0 for all z ≥ 0.
pαt
pβt
(10)
The vertical P corresponds to H(z) = 1 for all z. The natural rate of employment
is e∗ = H(1). The relation between the elasticity of P , π(e) := P 0 (e)e/P (e) > 0,
and that of H, η(z) := H 0 (z)z/H(z) > 0, is η(z) := 1/π(H −1 (z)) = 1/π(e).
To derive the equilibrium condition under sluggish wage adjustment first note
5
The only exception would be the case where (8) did not determine wt+1 , because P were
vertical, that is, the standard case of immediate wage adjustment. However, letting wt+1 rather
than Et wt+1 enter into (8) would still, interestingly, make the kind of stochastic sunspot equilibria
often studied in standard models of immediate price adjustment “shaky”. In the space of all
possible P -functions, the phenomenon would only pertain to the special case of a vertical P . The
slightest departure from wage and price adjustment being literally instantaneous would rule out the
phenomenon. Expressed differently, if one insists on continuity in the equilibrium correspondence
as the slope of P goes to infinity, it cannot be wt+1 that enters in (8), but it can be Et wt+1 .
10
that since et = Lt /Nt , it follows from (10) that,
Nt =
H
µ
Lt
Et pβ
t+1
pβ
t
)α
/ (Et ppt+1
α
t
¶.
(11)
Inserting that in equilibrium Nt = nt , Lt = lt = ytβ , and pt = M/yt gives,
ytβ
⎛
nt =
⎜
H ⎝ytβ−α
´ ⎞.
³
β
1
yt+1
(Et y 1 )α
t+1
Et
(12)
⎟
⎠
Finally, inserting into the first order condition (1), the expression for nt of (12), as
well the expression for wt /pt+1 of (6), gives,
⎛
⎛
⎞
⎞
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
β
⎜
⎜
⎟
⎟
y
y
t+1
t
β 0⎜
0
⎜
⎟
⎛
βyt v ⎜ ⎛
³
´β ⎞ ⎟ = Et yt+1 u ⎜
³
´β ⎞ ⎟
⎟.
1
1
⎜
⎜
⎟
⎟
E
E
t
t
⎜ ⎜ β−α
⎜
yt+1
yt+1
⎟⎟
⎜ β−α
⎟⎟
⎝ H ⎝yt
⎝
⎠
⎠
βH ⎝yt
1
1
α ⎠
α ⎠
(Et y
t+1
)
(Et y
t+1
(13)
)
This is the temporary equilibrium condition under a possibly sluggish wage adjustment. It states for any given distribution over yt+1 , what yt must fulfill in order to
be an equilibrium output of period t. A rational expectations dynamic equilibrium
is a stochastic (in particular a deterministic) process for yt , such that (13) is fulfilled
for all t.
If H(z) = 1 for all z, then (13) reduces to exactly (7), verifying the claim that
overall instantaneous price adjustment is a special case.
5
Keynesian Business Cycles
5.1
Perfect Foresight Dynamics
The perfect foresight dynamics associated with (13) appears by deleting the expectations operators. We write this as,
G(yt , yt+1 ) :=
βytβ v0
Ã
!
Ã
ytβ
yt+1
/u0
β−α
H ((yt /yt+1 ) )
βH ((yt /yt+1 )β−α )
!
= yt+1 .
(14)
A monetary steady state is a constant solution yt = y > 0,
βy
β−1 0
v
Ã
yβ
e∗
!
=u
11
0
Ã
!
y
,
βe∗
(15)
where it is used that H(1) = e∗ . It follows from our assumptions on u and v
that there is a unique monetary steady state. At this steady state, since output is
constantly y, it follows from yt = M/pt , that also the nominal price level is constant,
and then from (5), that the nominal wage rate is constant. The rate of employment
must then be at the natural rate all the time, et = e∗ for all t. (Formally this
follows from (10) with pt+1 deterministic and pt+1 /pt = 1). Also the intertemporal
real wage is constant and given by (6), ω t = ω = 1/(βy β−1 ).
It is a direct consequence of the Implicit Function Theorem that if the derivative
of the function G with respect to yt measured at the steady state (yt = yt+1 = y) is
not zero, then (14) defines a differentiable function, yt = f (yt+1 ), in a neighborhood
around the steady state. To prove Lemma 1 below is a matter of using this fact
and conducting the appropriate differentiation (proofs are given in Appendix).
Lemma 1 If β [1 + ε(ω)] − (β − α)η(1) 6= 0, then there are two neighborhoods
around y, such that for yt+1 chosen in the first of these neighborhoods there is a
unique yt in the second neighborhood solving (14), and the function yt = f (yt+1 )
thus (locally) defined is continuously differentiable.
We will everywhere assume that the condition in Lemma 1 is fulfilled. This
amounts to eliminating a knife edge case. Hence the perfect foresight dynamic
yt = f (yt+1 ) will be locally well-defined around the steady state y. We will also
need the slope of f at steady state.
Lemma 2 The derivative of f measured at the monetary steady state, i.e. at yt =
yt+1 = y, is
f 0 (y) =
5.2
ε(ω) − (β − α)η(1)
.
β [1 + ε(ω)] − (β − α)η(1)
(16)
Endogenous Fluctuations
This subsection serves to provide some relevant definitions and to demonstrate
that our model falls within the generality of Chiappori, Geoffard, and Guesnerie
(1992), so that the fundamental results of that article concerning the existence of
endogenous fluctuations can be used.
e ,µ ) =
Write the temporary equilibrium condition (13) in the general form Z(y
t
t+1
0, where Ze is a function defined on pairs of a current output level, yt , and a probabil-
ity distribution, µt+1 , over future output, yt+1 . Chiappori, Geoffard, and Guesnerie
12
e , µ ) = 0, so
(1992) study general (multi-dimensional) dynamics of the form Z(y
t
t+1
(13) is a special (one-dimensional) case. In the general notation of Chiappori, Geof-
e ,δ
fard, and Guesnerie, the perfect foresight dynamic is Z(yt , yt+1 ) := Z(y
t yt+1 ) = 0,
where δ yt+1 is the distribution assigning probability one to yt+1 , and a (monetary)
steady state is a y > 0, such that Z(y, y) = 0. If Z10 (y, y) 6= 0 (corresponding to the
condition of our Lemma 1), then Z(yt , yt+1 ) = 0 implicitly defines a differentiable
function, yt = f (yt+1 ), around the steady state y, and f 0 (y) = −Z20 (y, y)/Z10 (y, y)
(corresponding to (16)).
Still in the general notation of Chiappori, Geoffard, and Guesnerie, a rational
expectations stationary sunspot equilibrium of order k consists of k different output
levels, yi , and distributions µ(yi ), i = 1, ..., k, where each distribution only assigns
e , µ(y )) = 0 for i = 1, ...k, and the
positive probability to y1 , ..., yk , such that Z(y
i
i
Markov matrix of transition probabilities, M := (µ(yi ))ki=1 , is irreducible. (If people
believe that a Markov chain over states s1 , ..., sk , with transition probabilities as in
M , governs the economic system such that state si will imply output yi , then this
theory will become correct or self-fulfilling, although the Markov chain is completely
extrinsic to the economic system). If one distribution µ(yi ) assigns strictly positive
probability to at least two different output levels, the sunspot equilibrium is truly
stochastic. Otherwise it is a perfect foresight k-period cycle (defined as k different
output levels yi , i = 1, ..., k, such that Z(y1 , y2 ) = 0, Z(y2 , y3 ) = 0, ..., Z(yk , y1 ) = 0).
5.3
Rational Expectations Stationary Sunspot Equilibria
It follows from Chiappori, Geoffard, and Guesnerie (1992), Theorem 3, for our
model, that if perfect foresight dynamics is indeterminate in the sense f 0 (y) > 1
or f 0 (y) < −1, then there are truly stochastic stationary sunspot equilibria of any
order arbitrarily close to the monetary steady state.
The denominator in (16) may be negative. In that case f 0 (y) > 1 (implying
indeterminacy). The denominator may be positive, in which case f 0 (y) < 1. Still if
the denominator is positive one has f 0 (y) < −1 if and only if,
ε(ω) < 2
β −α
β
η(1) −
.
1+β
1+β
(17)
Hence, whenever this condition is fulfilled, either f 0 (y) > 1 or f 0 (y) < −1. This
suffices for,
13
Theorem 1 Assume (17). Then for any neighborhood around y, and for any k ≥ 2,
there is a truly stochastic stationary sunspot equilibrium of order k, for which all
output levels, y1 , ..., yk , belong to the neighborhood.
Note that the processes for pt and wt associated with the equilibrium process
for yt are given by pt = M/yt and (5).
The condition for the denominator in (16) being positive is,
ε(ω) >
β −α
η(1) − 1.
β
(18)
If both (17) and (18) are fulfilled, then f 0 (y) < −1. The result of the recent
contribution, Evans and Honkapohja (2003), is that exactly in the case f 0 (y) < −1,
there will be stationary sunspot equilibria of ordre 2 arbitrarily close to the steady
state, which are learning stable under a natural adaptive learning rule.6
Theorem 2 Assume (17) and (18). Then for any neighborhood around y, there
is a learning stable, truly stochastic stationary sunspot equilibrium of order 2, for
which both output levels, y1 and y2 , belong to the neighborhood.
5.4
The Plausibility of Endogenous Keynesian Business Cycles
For β = α the basic condition (17) for endogenous fluctuations specializes to the
standard one, ε(ω) < −β/(1+β). Our assumptions on α and β just allow α = β = 1.
However, while α = 1 is a natural restriction, 1/β should be labor’s share, so β
should be strictly larger than one with a plausible value around 3/2.
For η(1) = 0 (or π(e∗ ) = ∞), the case of instantaneous wage adjustment, the
condition (17) specializes again to the standard one. Sluggish wage adjustment
means η(1) > 0, so whenever β > α, condition (17) implies a relaxation of the
requirement on ε(ω) as compared to instantaneous wage adjustment.
In particular, and importantly, there are realistic restrictions on the remaining
parameters, such that with η(z) = 0 for all z, the condition (17) cannot be fulfilled,
6
For the details of the learning rule, the reader is referred to Evans and Honkapohja (2003).
Our equilibrium condition (13) does not fall completely within the generality of the equilibrium
condition studied by Evans and Honkapohja, which is yt = Et F (yt+1 ), where F may be nonlinear. However, the full analysis done in Evans and Honkapohja (2003) bears over to a generality
including (13).
14
and indeed stationary sunspot equilibria do not exist, while for η(1) sufficiently
large, or π(e∗ ) sufficiently small, (17) is fulfilled and stationary sunspot equilibria therefore exist. One such restriction is ε(ω 0 ) ≥ 0 for all ω 0 .7 In this precise
sense the considered type of nominal rigidity serves as a mechanism for endogenous
fluctuations.
The feature that sluggish and competitive wage adjustment can give rise to
endogenous fluctuations is not only a logical possibility. We now argue that for
plausible calibrations of ε, α, and β, the requirement on η, or π, for the existence
and learning stability of endogenous fluctuations is not unrealistic.
As said, it is natural to let β be around 3/2, and to consider for α the most usual
assumption of expected price inflation building itself perfectly into wage inflation,
that is, α = 1. Note that lower values for α only tend to make the basic condition
(17) for fluctuations more easily fulfilled. Given α = 1 and β = 3/2, Figure 2 shows
the combinations of ε(ω) and η(1), which are compatible with the conditions (17)
and (18), with [17] and [18] referring to these, respectively.
Everywhere below the line [17], endogenous fluctuations exist arbitrarily close
to the monetary steady state. One should compare with the indicated little piece
where −1 < ε < −β/(1 + β) and η(1) = 0, giving fluctuations under instantaneous
market clearing. Realistically, ε(ω) is positive, but relatively close to zero, so a
plausible sufficient (approximate) condition on the speed of wage adjustment for
endogenous fluctuations is η(1) > 3/2, or π(e∗ ) < 2/3. Everywhere below [17] and
above [18], indeterminacy takes the form f 0 (y) < −1, so this is the area in which
there are stationary sunspot equilibria close to the steady state known to be learning
stable. For ε close to zero, the condition for being in this area is approximately
3/2 < η(1) < 3, or 1/3 < π(e∗ ) < 2/3.
7
It follows directly that with ε(ω) ≥ 0, (17) is not fulfilled for η(1) = 0, but will be for η(1)
sufficiently large. Since (17) is just a sufficient condition for endogenous fluctuations it does
not follow that endogenous fluctuations do not exist when η(z) = 0 and ε(ω 0 ) ≥ 0 everywhere.
However, it follows from (23) of the Appendix that η(z) = 0 and ε(ω 0 ) ≥ 0 everywhere imply
that f 0 (yt+1 ) > 0 everywhere. Since also f 0 (y) < 1 under these circumstances, the function f
is globally well-defined, everywhere increasing, and has exactly one positive intersection with the
45◦ -line, which is from above. This implies global determinacy (global stability of y according to
the (backward) perfect foresight dynamic f), which excludes that f can have cycles and hence,
due to results in, e.g., Grandmont (1986), excludes the existence of stationary sunspot equilibria.
15
ε(ω)
1.5
[17]
1
[18]
0.5
η(1)
0
0
1
2
3
4
5
-0.5
-1
Figure 2
The period length in the present analysis should be thought of as the amount of
time during which the nominal wage rate is insensitive to the current conditions of
supply and demand. Such a period could reasonably be considered to be up to one
year. A rule of thumb in connection with Phillips-curves is that each percentage
point that the rate of unemployment is below the natural rate implies annual inflation to be one half of a percentage point above “core inflation”.8 This points to an
annual Phillips curve like, log Et wt+1 − log wt = 12 (et − e∗ ) + (log Et pt+1 − log pt ),
or Et wt+1 /wt = exp( 12 (et − e∗ ))Et pt+1 /pt ). For the model studied in this paper one
would then have P (e) = exp( 1 (e − e∗ )), and hence π(e) = 1 e, and π(e∗ ) ∼
= 1 , since
2
2
2
e∗ should be thought of as being close to one. This places π(e∗ ) right in the middle
of the interval yielding both existence and learning stability of stationary sunspot
equilibria close to the steady state. This empirical exercise suggests that the nomi-
nal rigidity mechanism for endogenous fluctuations is not in sharp contradiction to
empirically based parameter values as is the case for several other mechanisms for
endogenous fluctuations.
8
See for instance footnote 20 in Chapter 5 of Romer (2001).
16
5.5
The Intuition for Endogenous Keynesian Business Cycles
To provide an intuition for how and why the assumed nominal rigidity promotes
endogenous fluctuations it is clarifying to consider for a while a particular case of a
stationary sunspot equilibrium, namely a two-period deterministic cycle. Consider
a sequence of outputs where yt goes from high to low back to high etc. over the
periods. How can this be sustained as an equilibrium outcome under rational expectations (perfect foresight)? Due to the decreasing AD-curve, (3), the price level
pt must be low when output is high and vise versa. Due to the AS-curve, (4), wt
must also be low when output is high, so the intertemporal real wage ω t = wt /pt+1
must, under correct expectations, be low exactly when output is high. With instantaneous wage adjustment, output is in each period given directly by labor supply,
so labor supply needs to be high exactly when the intertemporal real wage is low.
In other words, labor supply must be decreasing in the real wage to a sufficient
degree. This explains the standard condition ε(ω) < −β/(1 + β).
Sluggish wage adjustment, or the short run non-clearing of the labor market
associated with it, implies a separation between labor supply on the one side and
employment and output on the other. For the considered volatility in output it is
still required that yt is high in periods where ω t is low, but this is possible without
the low ω t implying a high labor supply. It can suffice for having a high level
of employment when output is high that the rate of unemployment is low, and
a low rate of unemployment in periods where output is high will exactly tend to
create high wages and prices in the (next) periods where output is low. This is as
required and expected along the considered path. Thus sluggish wage adjustment
can substitute for a perversely sloped labor supply curve as a mechanism for perfect
foresight equilibrium cycles. This also explains why a certain degree of sluggishness
may be required. From Figure 1, a), e.g., it is clear that if the P -function is close to
the vertical one then only little separation between employment and labor supply
is possible. This separation is, however, essential in the above argument. To have
a sufficient amount of separation the P -curve must be sufficiently flat, that is, π
must be sufficiently low.
17
6
Concluding Remarks
Shortly stated our conclusion is that sluggishness in the competitive adjustment
of nominal wages may make the phenomenon of rational expectations endogenous
fluctuations (more) plausible: Under realistic assumptions on other fundamentals,
which exclude the existence of rational expectations fluctuations under instantaneous wage adjustment, a certain and not unrealistic degree of sluggishness in the
adjustment of nominal wages will imply the existence and learning stability of stationary sunspot equilibria close to a monetary steady state. The fact that such
fluctuations are rooted in a sluggish nominal wage adjustment has several implications that may be of macroeconomic interest:
1. From (9), using again pt = M/yt , one has that in any period,
E
β−α t
yt
³
1
yt+1
´β
1 α
(Et yt+1
)
= P (et ) ,
or, approximately, (yt Et (1/yt+1 ))β−α = P (et ). Hence there will (for β > α) be
a tendency that unemployment is high in periods where output is low. Over the
Keynesian cycle the rate of unemployment tends to vary countercyclically.
2. The endogenous fluctuations studied here appear around steady state and
therefore unemployment fluctuates around the natural rate. The rigidity of the
nominal wage rate makes unemployment a permanent problem in the sense that it
continues to make unemployment go back to higher levels (above the natural rate)
all the time.
3. Because of the separation between labor supply and employment possible
with a sluggish wage adjustment, there can be substantial fluctuations in output
and employment even for (numerically) low values of the elasticity of labor supply.
It is well known that in models where labor supply and employment are tied together
it is difficult to create realistic fluctuations in output for realistically low values of
the elasticity of labor supply.
4. The variations over the Keynesian business cycle in activity and unemployment are bad for welfare in the sense that the households would be better off if
the economy were permanently at the monetary steady state with unemployment
at the natural rate all the time. A welfare loss from fluctuations follows from the
“concavity” of utility functions, but it is sometimes argued that this loss must
18
be small since it is of second order. In our model, the nominal rigidity assumed
implies a wedge between the marginal product of labor and the marginal rate of
substitution between consumption and leisure, a wedge referred to as “the gap” by
Gali, Gertler, and López-Salido (2002). The authors show that the gap implies that
plausibly the welfare costs of business fluctuations are considerable. In the present
context there is no reason to believe that the welfare loss of fluctuating around,
rather than staying at, the steady state is negligible. Hence, nominal rigidity as
assumed here may both be the facor causing endogenous business cycles and be
the factor that causes business cycles to involve considerable welfare losses. In this
sense the nominal rigidity mechanism for endogenous fluctuations gives a relatively
strong case for stabilization policy aiming at keeping the rate of unemployment at
the natural rate all the time.
19
A
Proofs
Proof of Lemma 1. As noted in the text this is just a matter of showing that
the condition expressing that the derivative of the function G defined in (14) with
respect to yt measured at yt = yt+1 = y is not zero. For convenience we restate (14)
here,
G(yt , yt+1 ) :=
βytβ v0
Ã
!
Ã
ytβ
yt+1
/u0
β−α
H ((yt /yt+1 ) )
βH ((yt /yt+1 )β−α )
!
= yt+1 .
(19)
We first find the derivative G01 (yt , yt+1 ) at an arbitrary point and then measure
at steady state. For later use we will express the derivative in terms of the elasticity G0 (yt , yt+1 )yt /G(yt , yt+1 ). By log-differentiation of G one finds, not writing
everywhere what is inside v 0 (·), u(·), H(·) etc.,
G01 (yt , yt+1 )yt
G(yt , yt+1 )
β−1
β 0
v Hβyt − yt H (β − α)
= β+ 0
v
H2
00
³
yt
0
u yt+1 H (β − α) yt+1
+ 0
u β
H2
00
= β+N
Ã
Ã
³
´β−α−1
!⎡
yt
yt+1
1
yt+1
´β−α−1
1
yt+1
yt
yt
⎛Ã
ytβ
yt
⎣β − (β − α)η ⎝
H ((yt /yt+1 )β−α )
yt+1
!
⎛Ã
yt+1
yt
⎝
−R
(β
−
α)η
βH ((yt /yt+1 )β−α )
yt+1
!β−α ⎞⎤
⎠⎦
!β−α ⎞
⎠.
What is inside N (·) must be nt (formally, ytβ /H = lt /(lt /nt ) = nt ), and what is
h
inside R(·) must be ω t nt (formally, yt+1 /(βH) = yt+1 /(βytβ
ih
i
ytβ /H = ω t nt where
we used (6) on the way). Hence, not writing everywhere what is inside η(·),
G01 (yt , yt+1 )yt
= β + N(nt ) [β − (β − α)η] − R(ω t nt )(β − α)η
G(yt , yt+1 )
= β [1 + N (nt )] − (β − α)η [R(ω t nt ) + N (nt )]
Ã
!
1 + N (nt )
= [R(ω t nt ) + N(nt )] β
− (β − α)η
(20)
R(ω t nt ) + N (nt )
Ã
!
1 + N (nt )
= [R(ω t nt ) + N(nt )] β
− β + β − (β − α)η
R(ω t nt ) + N (nt )
"
#
1 − R(ω t nt )
= [R(ω t nt ) + N(nt )] β
+ β − (β − α)η)
R(ω t nt ) + N(nt )
20
⎡
⎛Ã
!β−α ⎞⎤
y
t
⎠⎦ ,
= [R(ω t nt ) + N(nt )] ⎣β(1 + ε(ω t )) − (β − α)η ⎝
yt+1
where (2) was used in the last line. Measuring where yt = yt+1 = y and ω t = ω
gives that G01 (y, y) 6= 0 ⇔ β [1 + ε(ω)] − (β − α)η(1) 6= 0.
Proof of Lemma 2. From total log-differentiation of the equation implicitly
defining f, namely G(yt , yt+1 ) = yt+1 , one gets,
G0 (y ,y
)y
t+1
1 − 2G0 t(ytt+1
dyt yt+1
,yt+1 )
2
=
.
G01 (yt ,yt+1 )yt
dyt+1 yt
(21)
G(yt ,yt+1 )
Here the last line of (20) gives an expression for the denominator. A similar expression for the numerator is found by log-differentiation of G with respect to yt+1 ,
G02 (yt , yt+1 )yt+1
G(yt , yt+1 )
00
v β
=
y
v0 t
H 0 (β − α)
³
yt
yt+1
H2
´β−α−1
0
u0 1 H + yt+1 H (β − α)
− 00
u β
H2
= N
Ã
!
³
yt
2
yt+1
yt+1
´β−α−1
yt
yt
2
yt+1
yt+1
⎛Ã
ytβ
yt
(β − α)η ⎝
β−α
H ((yt /yt+1 ) )
yt+1
Ã
!⎡
yt+1
!β−α ⎞
⎠
⎛Ã
yt+1
yt
⎣1 + (β − α)η ⎝
+R
β−α
βH ((yt /yt+1 ) )
yt+1
!β−α ⎞⎤
⎠⎦ .
Here again, what is inside N (·) must be nt , and what is inside R(·) must be ω t nt .
Then,
1−
G02 (yt , yt+1 )yt+1
= 1 − N(nt )(β − α)η − R(wt nt ) [1 + (β − α)η]
G(yt , yt+1 )
= 1 − R(wt nt ) − (β − α)η [R(wt nt ) + N (nt )]
(22)
"
#
1 − R(wt nt )
= [R(wt nt ) + N(nt )]
− (β − α)η
R(wt nt ) + N (nt )
⎡
⎛Ã
!β−α ⎞⎤
yt
⎠⎦ ,
= [R(wt nt ) + N(nt )] ⎣ε(ω t ) − (β − α)η ⎝
yt+1
where (2) was used again in the last line. Inserting the last lines of (20) and (22)
21
into (21) gives the slope of f at an arbitrary point where f is well-defined,
f 0 (yt+1 ) =
ε(ω t ) − (β − α)η
µ³
yt
´β−α ¶
yt+1
yt
µ³
´β−α ¶ .
yt+1 β(1 + ε(ω )) − (β − α)η
yt
t
yt+1
(23)
Measuring at the monetary steady state, yt = yt+1 = y and ω t = ω, gives the
expression for f 0 (y) stated in Lemma 2.
22
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24