GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.100

Nominal Wage Adjustment, Demand Shortage andEconomic Policy

Yoshiyasu Ono and Junichiro Ishida

November 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Nominal Wage Adjustment, Demand Shortage andEconomic Policy∗
Yoshiyasu Ono† and Junichiro Ishida‡
November 2009

Abstract
We formulate nominal wage adjustment by incorporating various concepts offairness. By applying it into a continuous-time money-in-utility model we examine macroeconomic dynamics with and without a liquidity trap and obtainthe condition for persistent unemployment, and that for temporary unemployment, to occur. These conditions turn out to be critical, since policy implications
significantly diﬀer between the two cases. A monetary expansion raises private
consumption under temporary unemployment but does not under persistent unemployment. A fiscal expansion may or may not increase short-run private consumption but crowds out long-run consumption under temporary unemployment.
Under persistent unemployment, however, it always increases private consumption.
Keywords: Wage adjustment, Fairness, Phillips curve, Demand shortage, Persistent stagnation
JEL Classification Numbers: E52, E31.

∗

The authors would like to thank S. Ikeda, T. Matsumura, R. Murota, T. Ogawa and S. Tanaka
for their valuable comments. They acknowledge the financial support from the GCOE (Global Centerof Excellence) program, Ministry of Education, Culture, Sports, Science and Technology; and Grantsin-Aid for Scientific Research, JSPS.
†
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka
567-0047, Japan. E-mail:ono@iser.osaka-u.ac.jp, Tel:+81-6-6879-8578, Fax:+81-6-6878-2766.
‡
Institute of Social and Economic Research, Osaka University

1

Introduction

We extend the fair wage model of Akerlof and Yellen (1990) to a continuous-time dynamic model of a monetary economy as portrayed by Keynes (1936, chap.17). Various
concepts of wage fairness are incorporated to this framework to provide a microeconomic foundation for sluggish nominal wage adjustment. In this setup, we find thatpersistent unemployment may arise in the presence of a liquidity trap and obtain thecondition for persistent unemployment, and that for temporary unemployment, to occur. These conditions turn out to be critical, since eﬀects of fiscal and monetary policies
on aggregate demand are shown to be quite diﬀerent between the two cases.
When analyzing unemployment, whether temporary or permanent, we must consider sluggish price/wage adjustment, viz. the Phillips curve,1 since any possibility ofdemand-supply imbalance would intrinsically be avoided without it. Ono (1994, 2001)
presents a dynamic optimization model of a monetary economy with sluggish nominal
wage adjustment and shows that unemployment persists in the steady state if there isa positive lower bound of the marginal utility of liquidity. This setting, which captures
Keynes’s notion of a monetary economy,2 has recently been used in various analyses of
persistent stagnation.3 All of these previous studies, however, simply take the original
Phillips curve as given, leaving the nominal wage adjustment process as a complete
black box.
Since the original Phillips curve sorely lacks any microeconomic foundation for the1

As an alternative to the sticky-price approach, Mankiw and Reis (2002) propose a stickyinformation model where information diﬀuses slowly through the population.
2
Keynes (1936, chap.17) defines a non-monetary economy as an economy where there is no asset
such that its liquidity premium remains strictly positive.
3
For example, Matsuzaki (2003) finds the eﬀect of a consumption tax on eﬀective demand in thepresence of poor and rich people. Hashimoto (2004) examines the intergenerational redistribution
eﬀects of the public pensions system in an overlapping generations framework with the present typeof stagnation. Ono (2006) extends the model into a two-country framework and analyzes the spillover
eﬀects of fiscal spending. Johdo (2006) considers the relationship between R&D subsidies and unemployment. Rodriguez-Arana (2007) examines the dynamic path with public deficit in the present
stagnation case and compares it with that in the neoclassical case. Johdo and Hashimoto (2008) introduce FDI into a two-country model with the present stagnation mechanism and analyze the eﬀectof the corporation tax on employment in each country. Murota and Ono (2008) find that a preference
for money holding as status is insaiable and thus generates persistent stagnation of the present type.

1

nominal wage adjustment process, several attempts have been made to augment it inthis direction: among most notable are the New Classical Phillips curve, the New Keynesian Phillips curve and the hybrid of the two.4 Models along this line include e.g.
Yun (1996), Woodford (2003), Gali (2008), Dotsey et al. (1999), Golosov and Lucas(2007) and Gertler and Leahy (2008). The first three assume Calvo’s staggered pricing
while the others adopt the menu cost approach. They examine firm and household
reactions to a policy or parameter shock that occurs in the middle of the initial period.
Since they assume that agents cannot revise prices or wages in the middle of each period, demand-supply imbalances arise in the initial period but disappear in subsequent
periods. Krugman (1998) also analyzes Japan’s stagnation using a two-period modelin which prices are rigid only in the initial period and any demand shortages disappear
in the second period.
What is common among these studies is that they preclude the possibility of demand shortages in the steady state, almost by construction.5 The logic of unemployment in this existing literature is indispensably based on the assumption that there
is a period within which prices and wages cannot be revised. According to this logic,
unemployment is necessarily a temporary phenomenon which occurs only in the adjustment process, as any demand shortages would eventually dissipate once people’s
expectations are corrected and prices are completely adjusted. Therefore, althoughcertainly insightful in understanding short-run fluctuations, these existing settings faceserious diﬃculties in explaining long and persistent recessions because, for them to occur, it must be either (i) that people continuously make false expectations about pricesor (ii) that the price/wage adjustment process is extremely slow. Neither appears to
be a likely cause for persistent recessions, such as the Great Depression of the 1930s,
Japan’s lost decade of the 1990s, and many other prolonged economic downturns which
we observe every now and then.6 In many of those instances, the recessions persisted
4

See Woodford (2003) for properties of these Phillips curves.
In these settings, the existence of demand shortages implies that the deflation rate cumulatively
expands. If this were to continue, prices would reach zero within a finite time and the feasibility
condition would eventually be violated.
6
See Kehoe and Prescott (2007) for many “great depressions” of the twentieth century.
5

2

for more than a decade, which should be more than enough for people to adjust expectations correctly. It is also implausible to think that the price/wage adjustment
process is so slow that even after all these years, prices and wages cannot be adjusted
to the equilibrium levels to clear the markets.
It is our view that, to account for long and persistent deviations from full employment, we need an alternative framework where the presence of unemployment does nothinge on unexpected events or policies. To illuminate this point, we construct a modelof perfect foresight with no unexpected shocks where everyone precisely understands
the current state of the economy.7 Moreover, we adopt a continuous-time setting which
inherently has no border of periods: at any point in time, firms are free to set any prices
and wages, so that there is no “adjustment” in this particular sense. We instead consider a nominal wage adjustment process which rests on workers’ inherent concerns
for fairness. This adjustment mechanism is introduced to a dynamic general equilibrium setting to provide our own version of the Phillips curve with its microeconomic
foundation: in the current variation, the inflation rate is governed by the liquidity premium, the subjective discount rate and the unemployment rate; in particular, aroundthe steady state, the inflation rate depends only on the unemployment rate, as in theoriginal Phillips curve.
Within this framework, we show that unemployment due to demand shortages may
persist in the presence of a liquidity trap, thereby pointing to a specific route throughwhich persistent recessions arise as an equilibrium phenomenon. Aggregate demand
falls short of its full-employment level because people hold onto liquid assets, viz.
money, in the shadow of a recession, which in turn stagnates aggregate demand. The
driving force behind persistent unemployment is hence our perpetual and insatiable
carving for money, whereas the role of nominal wage rigidity is fundamentally diﬀerent
from the previous models: in fact, while the insatiable demand for money is the direct cause of persistent unemployment, nominal wage rigidity works more as a buﬀer,
7

Obviously, this is not to say that expectations play no role in aggregate fluctuations; they mostcertainly do. What we argue here is rather that unemployment could persist even under perfectforesight where there are no surprises.

3

preventing the economy from falling into a catastrophic disaster. Given this construction, we identify conditions that delineate persistent and temporary unemployment andshow that policy implications diﬀer substantially between the two cases: that is, whatworks under temporary stagnation may not work under persistent stagnation. In lightof these findings, we argue that diﬀerent sets of remedies may be needed to restore full
employment once the economy gets stuck in the state of persistent stagnation.
Despite all the advancements made over the past years, it still remains to be adaunting task to fully understand the underlying mechanism of prolonged recessions
which we face from time to time. The main aim of this paper is to contribute to thislong-standing issue, especially focusing on why recessions, once started, often persist
for so long. On a grand scale, this paper shares a lot with the New Keynesian paradigm
in its motivation and fundamental approach: we turn to Keynes as the source of insighton malfunctioning economies.8 The problem is that his book “The General Theory of
Employment, Interest, and Money” is painfully convoluted, and critical insights arescattered throughout the book in a rather unstructured way. Still, many attempts
have been made to pick up pieces of his insight and reduce them down to their essence.
An emerging consensus seems to be that the crux of Keynesianism lies in various sorts
of market imperfection, especially nominal rigidity, whose eﬀects are amplified throughthe eﬀect of aggregate demand.
We have no intention of arguing with this general consensus in this paper, as there
is little doubt that nominal rigidity constitutes a critical component of Keynes’s theory.
It is our stance, however, that nominal rigidity alone does not exhaust all of Keynes’s
insight. We argue that there is an overlooked aspect of his theory that is of independent
importance, i.e., the role of insatiable liquidity preferences. This paper is an attempt
to recast and revitalize this insight in a modern macroeconomic framework to derive its
dynamic policy implications. Although this insight has somehow escaped economists’8

Several prominent economists make this point rather emphatically. Greg Mankiw notes “If you
were going to turn to only one economist to understand the problems facing the economy, there
is little doubt that the economist would be John Maynard Keynes.... His insights go a long way
toward explaining the challenges we now confront (New York Times, November 28). “The Keynesiansbasically got it right,” says George Akerlof (2007).

4

attention, especially in a rigorous context, we believe that the current framework sheds
light on a diﬀerent side of Keynesianism and provides a useful prescription of prolonged
recessions.

2

The wage dynamics

Before we set up a dynamic optimization problem, we first illustrate the wage adjustment process which describes how nominal wages are determined in this economy. As
stated, the driving force behind this whole process is workers’ inherent concerns forfairness. Having derived the equilibrium wage dynamics we then incorporate it into a
dynamic general equilibrium framework to close the model in the next section.

2.1

The setup

There are continua of identical workers and firms, both with unit measure. At any
instance, each worker is either employed or unemployed. Since firms are all identical,
unemployed workers are randomly assigned to firms whenever vacancies exist: as aconsequence, each firm hires workers of equal size in any equilibrium. Let x(t) denotethe number of workers newly hired at time t which is constant across firms. Moreover,define X(t) as the aggregate rate of employment X(t) ∈ [0, 1], where X(t) = 1 meansthat the economy is in the state of full employment.
Although employed workers can in principal quit and leave their respective firms at
will, workers must incur some flow cost while they are unemployed.9 We assume thatthe cost of unemployment is prohibitively large, so that workers would not choose tovoluntarily leave their firms when there is a positive probability of being unemployed.
This means that their mobility is heavily influenced by the aggregate rate of employment: workers are perfectly mobile when X(t) = 1, but they are virtually held up by
their respective firms when X(t) < 1. These assumptions are made for clarity andtractability rather than for realism.10
9The cost of unemployment is meant to capture physical costs of job search as well as more
psychological costs of anxiety or social stigma inherently attached to unemployment.
10
All we need is the fact that workers’ mobility is more limited when there are more unemployed5In this model, since both firms and workers are identical and there is no information asymmetry, job separations occur only for exogenous reasons (no voluntary job
separations on any equilibrium path) as long as unemployment exists. We assume thateach employed worker randomly separates from the current firm at Poisson rate α. The
aggregate rate of employment is then obtained as
Z t
X(t) =
x(s)eα(s−t) ds.

(1)

−∞

The time diﬀerentiation of (1) yields
Ẋ(t) = −αX(t) + x(t).

(2)

Note that when X(t) = 1 and x(t) = α, full employment continues. Throughout theanalysis, we restrict attention to the case where α is relatively small so that ρ > α.11

2.2Wage setting

At each instance t, each firm i ∈ [0, 1] oﬀers a (take-it-or-leave-it) wage w(i, t) > 0 thatapplies equally for all of its employed workers. Define
R1
σ(i)w(i, t)di
W (t) ≡ 0 R 1,
σ(i)di0(3)

as the economy-wide average wage, where σ(i) is the weight given to firm i. When
unemployed, each worker receives unemployment benefits b(t), which we later normalize

to zero for all t.12
We suppose that workers in this economy have a strong preference for being treatedfairly, and their motivation and productivity depend heavily on this fairness concern.
To be more precise, let ω(t) denote the nominal wage level that is perceived as fairby employed workers at time t: in what follows, we simply refer to ω(t) as the fairworkers waiting for job oﬀers and the expected duration of unemployment is longer. Our assumptions
should thus be viewed as a way to capture this aspect of reality in an analytically tractable manner.
11
If the job-separation rate α is regarded as the rate of death as a worker, it is naturally less than
the subjective discount rate ρ.
12
The presence of unemployment benefits plays no role in the analysis, but it helps clarifying thedefinition of the fair wage which we discuss next.

6

wage. When w(i, t) ≥ ω(t), each worker perceives that the current wage is “fair” andproduces θ (per unit of time); when ω(t) > w(i, t), each worker perceives that thecurrent wage is “unfair” and produces nothing by withholding work eﬀort. Note that
this specification is a variant of the fair wage-eﬀort hypothesis, put forth by Akerlof
and Yellen (1990), which posits that a worker’s productivity goes down when the wage
level dips below what is perceived as fair.

2.3

The fair wage

Under this setup, what becomes critical is how workers develop the perception of
fairness. At a general level, since fairness is inherently a distributional concern, it
should be subject to various kinds of social comparisons. While it is undoubtedly true,
what social comparisons can imply is potentially very broad and somewhat vague.
More restrictions are thus needed to pin down a tightly specified process of fairness
formation. To do this, we build on the following four popular views in social psychology
and behavioral economics:
1. The entitlement eﬀect: Once a high wage is oﬀered, people develop a sense ofentitlement which persists over time (Falk et al., 2006).
2. The anchoring eﬀect: One’s perceptions, preferences and valuations are initiallymalleable but, once imprinted, become fairly persistent over time (Ariely et al., 2003).
3. Opinion-based transmission of perceptions: Others’ perception of fairness influences
one’s own perception of fairness (Folger and Kass, 2000; Umphress et al., 2003).
4. The belief in the just world: People are motivated to help others who have beentreated unfairly, to make the world fair and just again (Lerner, 1980).
For illustrative purposes, we momentarily consider a discrete-time version of themodel where each firm hires new workers and revises its wage contract at interval of∆t, i.e., at time t ∈ T ≡ {..., t0 − ∆t, t0 , t0 + ∆t, ...}. We posit that the fair wage inthis economy is computed through the following two-stage process.1313Note that the current specification is a way, possibly among some others, to capture the four
aforementioned views in a unified form.

7

First Stage:

At any t ∈ T , workers can be classified into either one of the threeclasses: remaining workers who continue to be employed, incoming workers who are
newly hired, and unemployed. The fair wage is computed mainly from the viewpointof the remaining workers. First, those remaining workers have in mind a (common)
wage level, denoted by ν(t), that they are rightfully entitled to. Taking this as thebasis, they also take into account the well-beings of unemployed workers (the justworld hypothesis).14 The fair wage at time t0 is obtained as the average, weighted by
the number of each class of workers, of these concerns:
ω(t0 ) =ν(t0 − ∆t)X(t0 − ∆t)(1 − α∆t) + b(t0 )(1 − X(t0 )),1 − x(t0 )∆t(4)where the denominator equals the total number of the remaining workers and theunemployed, as is clear from (2). At this point, the incoming workers, with no priorwork experiences, basically have no idea of what is supposed to be fair and simply accept
and internalize their predecessors’ view (opinion-based transmission of perceptions).
Second Stage: Immediately after time t0 , the incoming workers are quickly assimilated into the remaining workers. With new wage contracts in eﬀect, all the employedworkers, both remaining and incoming, then readjust their perception ν(t). Two factors enter into this readjustment process: on one hand, their perception is influenced
to some extent by the current average wage (the entitlement eﬀect); on the other hand,
it is also influenced by their past perceptions (the anchoring eﬀect). Given this, theiradjusted perception is obtained as
δW (t0 ) + (1 − δ)ω(t0 ) = ν(t0 )X(t0 ) + b(t0 )(1 − X(t0 )),(5)where δ ∈ [0, 1] measures the relative salience of the entitlement eﬀect.
The fair wage is shaped by going through this two-stage process repeatedly overtime. Since we normalize b(t) = 0 for all t, combining (4) and (5) yields
ω(t0 ) =(δW (t0 − ∆t) + (1 − δ)ω(t0 − ∆t))(1 − α∆t),
1 − x(t0 )∆t

14

(6)

Since workers are all homogeneous and unemployed workers are simply unlucky to be in thatstate, workers are to some extent willing to take a wage cut, by lowering their fair wage, in order tohelp them out.

8

which characterizes how the fair wage in this economy evolves over time.

2.4

The equilibrium wage adjustment

We assume perfect market competition among firms, each of which acts as a pricetaker in the goods market. Let P (t) denote the aggregate price of the good. At timet ∈ T , each firm unilaterally oﬀers a wage w(i, t) to its employed workers, taking thesequence of all the aggregate variables {W (s), P (s), x(s)}ts=−∞ as given. This meansthat each firm also takes the fair wage ω(t) as given when it oﬀers its own wage. Since
the cost of unemployment is prohibitively large, there is a fine line between X(t) < 1
and X(t) = 1, and the nature of wage setting diﬀers completely depending on whether
the economy achieves full employment or not.
We first characterize the equilibrium wage adjustment process in the presence of
unemployment, i.e., X(t) < 1. In this case, workers are completely immobile and haveno choice but to accept wages oﬀered by their respective firms. While bargaining poweris entirely in the hands of firms, workers can withhold work eﬀort whenever they feelthey are slighted. With this fairness concern as a credible threat, firms can lower thewages only down to the fair wage level. The following is a formal representation of thisfact.
Proposition 1 When X(t) < 1, w(i, t) = ω(t) for all i ∈ [0, 1].
Proof: First, if firm i chooses to hire a worker, it must be that w(i, t) ≥ ω(t). To see
this, note that if ω(t) > w(i, t) > 0, the worker who never quits produces nothing andconsequently yields a negative profit. It is then strictly better not to hire the worker
in the first place. Second, it is also straightforward to see that firms have no incentive
ex post to oﬀer a wage that is strictly larger than the fair wage, because that would
only decrease their profits. It follows from these observations that firms simply oﬀer
the fair wage at every instance when X(t) < 1.
Q.E.D.
9

The proposition means that the evolution of the fair wage totally dictates theequilibrium wage dynamics in the presence of unemployment. First, since w(i, t) = ω(t)for all i, we have W (t) = ω(t) for any given weight {σ(i)}1i=0 . It then follows from (3)and (6) thatW (t0 ) = W (t0 − ∆t)(1 − α∆t) + W (t0 )x(t0 )∆t.

(7)

The equilibrium wage dynamics is governed by this adjustment process. Letting ∆t →0, we obtain
W (t0 ) − W (t0 − ∆t)x(t0 ) − αẆ (t0 )
= lim
= lim
= x(t0 ) − α.
∆t→0 1 − x(t0 )∆tW (t0 ) ∆t→0
W (t0 − ∆t)∆t(8)

The nature of wage setting drastically changes once the economy achieves full employment, i.e., X(t) = 1. Workers are now fully mobile in search of the best wageoﬀer available to them without facing any risk of being unemployed. Consequently,
the market price P (t) dictates the equilibrium dynamics, and the fair wage may play
no role. Since workers are mobile, competition among firms for workers drives up thewage oﬀers up to θP (t). We can thus establish the following result.
Proposition 2 When X(t) = 1, w(i, t) = θP (t) for all i ∈ [0, 1].
Proof:First, it is evident that no firm oﬀers w(i, t) > θP (t) because it yields astrictly negative profit. Suppose that θP (t) > wmax ≡ maxi w(i, t). Then, a firm canoﬀer a wage that is slightly larger than wmax and attract all workers away from otherfirms. This means that the only equilibrium with perfect worker mobility is to oﬀer
w(i, t) = θP (t).
Q.E.D.
In either case, whether there is unemployment or not, w(i, t) = W (t) for all i ∈ [0, 1].
Perfect market competition then forces all firms to break even and earn zero profit inequilibrium. The following zero-profit condition thus always holds:
θP (t) = W (t).
10

(9)

3

General equilibrium

In the previous section, we have derived the nominal wage adjustment process whichstems from workers’ fairness concerns. We now incorporate this process into a dynamic
general equilibrium framework which admits the possibility of equilibrium unemployment.
The problem we consider is a standard dynamic money-in-utility optimization problem (in what follows, we abbreviate time notation to simplify exposition). The lifetime
utility of a representative household is given byZ ∞U=[u(c) + v(m)] exp(−ρt)dt,

(10)

0

where ρ is the subjective discount rate, c is real consumption and m ≡ M/P is realmoney balances. The representative household maximizes U subject to the flow budget
equation:ṁ = wX − πm − c − z,(11)where π is the inflation rate, z is the lump-sum tax-cum-subsidy and w is the real
wage W/P. Since all firms earn zero profit under perfect market competition, theonly storable asset in this economy is the real balances m.15 The first-order optimal
condition of this problem isċv 0 (m)u00 (c)cη +ρ+π = 0
where η ≡ − 0
,
c
u (c)
u (c)

(12)

and the transversality condition islim λ(t)m(t) exp(−ρt) = 0,

t→∞

(13)

where λ(t) is the costate variable of m which equals u0 (c).
The government’s budget constraint is
z + μm = g,
15

(14)

Even if there are government bonds, the present analysis is unchanged since the Ricardian equivalence holds.

11where g represents government purchases and μ is the monetary expansion rate:
Ṁ
= μ.
M

(15)

Given the definition of m, this can also be written asṁ= μ − π.
m

(16)

The general equilibrium properties of the model diﬀer sharply, depending on whether
there is unemployment or not. In the presence of unemployment, (8) and (9) giveπ=Ẇ= x(t) − α.
W(17)

On any equilibrium path, all employed workers are motivated enough to exert eﬀort
and hence the total production is always θX, as discussed in the previous section.
Therefore, under perfect commodity price adjustment, we must havec + g = θX.
From (2), (12) and the time derivative of (18),µ¶´v 0 (m)c+g ³x
−
π
−
ρ
=
η
−
α
.
u0 (c)
c
X
Combined with (17) and (18), (19) yields
µ 0
¶
Ṗ
c
v (m)
αηθ c + g=π=−ρ+(− 1) ,
Pc + θη u0 (c)
c
θ

(18)

(19)

(20)

which is our version of the Phillips curve. Substituting (20) into (12) and (16) andrearranging the results produces
µ 0
¶
ṁ
c
v (m)
αηθ c + g
= μ−
−ρ+
(
− 1) ,
mc + θη u0 (c)
c
θ
µ 0
¶
ċ
θ
v (m)
c+g
=
− ρ − α(− 1) .
0cc + θη u (c)θ(21)

These two equations constitute an autonomous dynamic system with respect to m andc in the presence of unemployment.
12Remark: Under the current Phillips curve (20), the inflation rate is governed by theliquidity premium v 0 (m)/u0 (c), the subjective discount rate ρ and the unemployment
rate 1 − (c + g)/θ. More importantly, substituting (12) into this, we obtain
π=

c+g
ċ
+ α(
− 1).
θ
θ

(22)

This indicates that as the economy approaches the steady state (ċ → 0), the inflation
rate depends solely on the unemployment rate. The current framework hence provides
a microeconomic foundation for the relationship between the unemployment rate andthe inflation rate, as the original Phillips curve posits.
If full employment is realized (X = 1), on the other hand, we havec = θ − g.

(23)

From (12), (16) and (23), we obtainṁv 0 (m)=ρ+μ− 0,mu (θ − g)which is the same as the standard dynamics of the money-in-utility model (see e.g.
Blanchard and Fischer, 1989). From the beginning, therefore, P takes the level thatsatisfies
ρ+μ=v0 (M/P ),u0 (θ − g)(24)

and thereafter rises at the same pace as μ, so that M/P remains at the constant levelthat satisfies (24). From (9), θP = W and hence W also rises at the rate of μ. Thus,in this case the Phillips curve forms a vertical line as μ changes.

4

Temporary Unemployment

Using the dynamic equations obtained in the previous section we draw the phase diagram and analyze the properties of the present dynamics. This section first considers
the case with no liquidity trap, where the marginal utility of liquidity has no positive13

lowerbound, i.e., limm→0 v 0 (m) = 0, so that the demand for liquidity would eventually dissipate. Under this condition, any equilibrium path reaches the full-employment
steady state and unemployment occurs only during the adjustment process.
From (21), the boundary curve of m dynamics and that of c dynamics are given byµ¶θηα c + gθη0ṁ = 0: v (m) = ρ −(− 1) + μ(1 + ) u0 (c).
c
θ
c
¶
µ
c+g
− 1) u0 (c).
(25)
ċ = 0: v 0 (m) = ρ + α(θThe right-hand side of the ṁ = 0 curve is obviously a decreasing function with respect
to c whereas the right-hand side of the ċ = 0 curve can be sloped either positively or
negatively. If the aggregate demand c + g is less than the full-employment supply θ,and μ is non-negative, the right-hand side of the ṁ = 0 curve is larger than that of theċ = 0 curve. Thus, since v00 (m) < 0, the ṁ = 0 curve is located on the left-hand sideof the ċ = 0 curve. Figure 1 illustrates the two curves in the case where the right-handside of the ċ = 0 curve is negatively sloped and hence the ċ = 0 curve is positively
sloped.16 If μ = 0, the intersection point of the two curves is given by (23) and (24),which is A in the figure. Therefore, there is a unique saddle path that converges toA, which we refer to as the full-employment path. Along the path deflation gradually
declines and eventually becomes zero.
If μ > 0, on the other hand, the steady state given by (21) does not exist within therange where c + g ≤ θ. This is illustrated in figure 2. Then, along the full-employment
path, the economy reaches E, at which (23) and (24) are valid, within a finite time
and thereafter stays there. As the figure shows, an increase in μ lowers m but leaves
c unaﬀected in the new steady state. Therefore, if μ unexpectedly increases when theeconomy passes through B, it jumps up to D and thereafter follows DE. Eventually,
it reaches E, where the inflation rate is μ: an increase in μ is hence translated into
an one-for-one increase in the inflation rate. If the monetary authority reduces themonetary expansion rate to zero in the new steady state so as to stabilize prices,
16

When g = 0, it is valid if ηρ > α and ρ > α. Note that a positively sloped unique saddle pathobtains even if the boundary curve is negatively sloped.

14

private consumption stagnates in the short run, as the economy jumps down to F andmoves along FA. The monetary authority thus faces the short-run tradeoﬀ between
inflation and consumption. It should gradually and intermittently decrease μ so as to
avoid a sudden downward spike in private consumption.
The eﬀect of an increase in the government purchases g is illustrated in figure 3.
From (21), around the full-employment steady state where c = θ − g, the dynamics ofm and c are given byṁηθμ=m θ − g + θη

and

ċ
θμ
=
,
c θ − g + θηand hence on the equilibrium path c and m move so that they satisfy¯
ċ/c1
m dc ¯¯== .
¯c dm path ṁ/m η(26)

increase in g must satisfy¯
ηm dc ¯¯= m¯c dm steady stateη(27)

From (23) and (24), on the other hand, the change in the steady state induced by an

v 00 (m)mwhere ηm = − 0(> 0),v (m)where ηm is the elasticity of money utility. By comparing (26) and (27), we find thatan increase in g shifts the steady state from E to B if η m < 1 and to D if ηm > 1, as
illustrated in figure 3. In either case, an increase in g leads to a one-for-one decreasein c in the long run (complete crowding out). In the short run, however, it may or
may not stimulate private consumption, depending on the elasticity of money utility: cincreases if the elasticity is smaller than one and decreases if the elasticity is larger thanone, since B and D are respectively located above and below the previous equilibrium
path. If a fiscal expansion occurs at A, the path jumps to either F or H and thereafter
traces the new saddle path.
To sum up, along the full-employment path, any government interventions, either
monetary or fiscal, aﬀect aggregate demand in the short run, one way or the other.
Since the economy eventually reaches full employment, however, any policies have only
a transitory eﬀect as market forces bring the economy back on track in the long run.
We summarize our findings as follows.
15

Proposition 3 If the marginal utility of liquidity has no positive lowerbound, i.e.,
limm→∞ v 0 (m) = 0, there is a unique equilibrium path that reaches the full-employment
steady state. Along the full-employment path,• a monetary expansion raises private consumption in the short run, while it does not
aﬀect private consumption in the long run,
• a fiscal expansion totally crowds out private consumption in the long run, while it
may or may not increase private consumption in the short run.

5

Persistent unemployment

We have so far examined the case where the full-employment steady state, given by(23) and (24), exists and is eventually reached. Under certain conditions, however,
our model also admits the possibility of persistent unemployment where the fullemployment steady state fails to exist. In this case, market forces alone are not
suﬃcient to bring the economy back on track, and the government may have somerole to play, even in the long run. It should be noted that the presence of persistent
unemployment in this economy is not caused by real wage rigidity which tips the balance on the supply side of the market: although wages are sluggish due to fairness
concerns, prices are fully flexible so that the real wage rate is always equal to themarginal productivity, i.e., θ = w(= W/P ) from (9). It rather arises as a monetaryphenomenon driven purely from the demand side.

5.1

Cases for insatiable liquidity preferences

The money demand curve is given by the relationship between m and the liquidity
premium v0 (m)/u0 (c), which represents the nominal interest rate. It is negatively slopedsince the liquidity premium decreases as m increases. We now introduce a liquidity
trap to this, which arises if v 0 (m) has a positive lowerbound:
lim v 0 (m) = β > 0.

m→∞

16

(28)

There is an indication that this property captures what Keynes envisioned as an essenceof a monetary economy, i.e., an economy in which there is an asset whose marginalutility stays positive (Keynes, 1936, chap.17).17 As demonstrated in Ono (1994, 2001)
and also in this model, this constant and perpetual craving for money can be a driving
force for persistent demand shortages. Here, we oﬀer two interpretations to justify thisassumption.
Money as a secondary reward: One interpretation is to take the presence of thelowerbound literally as a formal representation of our perpetual craving for liquid assets, viz. money. Although the assumption of this kind is somewhat non-standard inmodern macroeconomics, we argue that it is not as provocative as some may think:after all, there is no firm scientific ground to believe that the marginal utility of money
should converge to zero either. To make a case for this interpretation, we focus on thenature of money as a reward as opposed to other rewards, called “primary rewards”
in neuroscience, that are indispensable for the survival of the species. This distinction, which is clearly made in neuroscience, is potentially important in economics aswell because that should have deep connection to the way we specify preferences for“secondary rewards” such as money, power and fame.18
The satiable nature of the demand for primary rewards such as food is intuitively
clear. For instance, we all know from our experiences that if we eat enough, the marginal value of food eventually diminishes to zero and even negative. The assumptionof the form limc→∞ u0 (c) = 0, which is fairly conventional in economics, is presumably
meant to capture this gut intuition. This does not necessarily mean, though, that we
can apply this same logic to secondary rewards which derive their value from theirassociated primary rewards. The diﬀerence is clear at a superficial level: money is not
something we directly consume but something we learn to appreciate though the asso17

Using aggregate quarterly data in Japan and the Japanese survey data called NIKKEI RADAR,
Ono, Ogawa and Yoshida (2004) empirically find this property to be well supported using both parametric and non-parametric methods.
18
In neuroscience, Izuma et al. (2008) show that the brain areas activated by social rewards (acquisition of good reputation) overlap with those activated by monetary rewards.

17

ciation between money and consumption. There is no convincing reason to believe thatour gut intuition about our appetite, or any other more instinctive needs for primary
rewards, should directly be applied to our craving for money as a secondary reward:
at the very least, we never get tired of accumulating wealth in the same way as we gettired of eating food. In fact, evidence in neuroscience suggests that the diminishingnature of marginal utility, called “devaluation” in the field, for primary and secondary
rewards is represented in diﬀerent brain areas: the orbitofrontal cortex (limbic area)for primary rewards and the dorsal striatum (cognitive area) for secondary rewards.19
Although research along this line is still at its infant stage and there remains a lot tobe seen, these findings seem to confirm a belief that the valuation of secondary rewards
is related more to human cognition whereas that of primary rewards is related more
to sheer instinct, indicating that simplistic analogy between primary and secondary
rewards is not necessarily warranted.
Other-regarding preferences: An alternative interpretation, which is more widely
accepted in economics, is to see this as a consequence of other-regarding preferences,such as the preference for “status” or the fear of lagging behind.20 Suppose, for instance,that agents are “status-seeking” in that they gain utility from the diﬀerence in wealthholdings. In this case, we can replace v(m) by v(m − m) where m denotes the social
average of m. Since agents are all homogeneous and m always equals m in the present
setting, v 0 (m − m) stays to be fixed at v0 (0) for any level of m. Our assumption (28)then follows if we let this value be β.
19

Evidence, based on both animal studies (Ballein and Dickinson, 1998, 2000; Ballein and Ostlund,2007) and human studies (Valentin et al., 2007), suggests that the orbitofrontal cortex is the region
responsible for the devaluation of primary rewards. Research on the devaluation of secondary rewards
is relatively scarce, but Pine et al. (2009) recently finds that the devaluation of secondary rewards isrepresented in the dorsal striatum. We thank S. Tanaka for enlightening us on the subject.
20
See, for instance, Clark and Oswald (1998) for an extensive survey on the “status” literature.

18

5.2

Dynamics under insatiable liquidity preferences

We now characterize equilibrium dynamics under (28). If the full-employment supplyθ is suﬃciently large and satisfiesβ
u0 (θ − g)> ρ + μ,(29)

the liquidity premium exceeds the time preference for any m whenever c attains its
full-employment level. This means that there exists no m that can satisfy (24) andhence the full-employment steady state fails to exist. In this case, along the boundary
curve of c given in (25), c gradually approaches cu defined by
Ru ≡cu + g
β
=
ρ
+
α(
− 1).
u0 (cu )θ(30)

If μ is suﬃciently small, there exists an equilibrium path that leads to this stagnation
steady state (the stagnation path, for short) which satisfies the transversality condition.
Proposition 4 Suppose that the marginal utility of liquidity has a positive lowerbound,i.e., limm→∞ v0 (m) = β > 0. If (29) holds, the full-employment steady state does notexist. Moreover, if μ is small enough to satisfyβu0 (cu )> μ,there arises a unique equilibrium path that leads to the stagnation steady state. Alongthe stagnation path, the steady-state consumption level converges to cu which falls shortof the full-employment level θ − g and deflation occurs.
Proof: We can show that there exists a well-defined cu such that0 < cu < θ − g.
To see this, define the diﬀerence of the two sides of (30) as
Φ(c) ≡βu0 (c)− ρ − α(
19

c+g
− 1),
θ

which is continuous in c. The intermediate-value theorem then guarantees the existenceof a well-defined cu ifgΦ(θ − g) > 0 > lim Φ(c) = −ρ − α( − 1).
c→0θThe first inequality directly follows from (29) whereas the second inequality holds byassumption. This property also implies Φ0 (c) > 0 around the stagnation steady state(where Φ(c) = 0) and hence
ηβθ
ηRu θ
(= u ) > α.
cu u0 (cu )c(31)

Since Φ(c) is the inside value of the parenthesis of ċ/c in (21) when the stagnation
steady state is reached, the boundary curve of the c dynamics is positively sloped withrespect to c around the stagnation steady state, as illustrated in figure 4. From (20)and (30) the steady-state inflation rate is then obtained as
π u = α(cu + g− 1) < 0.
θ

(32)

From (21), (30) and (32), the transversality condition (13) is valid if and only if0>βṁ−ρ=μ− 0 u .
mu (c )(33)

Evidently, this condition holds when μ is suﬃciently small.
Q.E.D.
This situation is depicted in figure 4. In sum, on the equilibrium path, the economy
never reaches the full-employment steady state. Along the stagnation path, the pricelevel continuously falls, resulting in persistent deflation which makes the real balances
m keep expanding while satisfying the transversality condition. The consumption level
falls short of the full-employment level θ − g and, due to this insuﬃcient level of aggregate demand, unemployment persists in the steady state. Note that cu given by (30)
is decreasing in α, meaning that a recession becomes severer as the wage adjustment
becomes less sluggish, since an improvement in the wage adjustment worsens deflation
20

and makes holding money less costly. In the current setup, therefore, the role of nominal rigidity diﬀers fundamentally from the conventional New Keynesian models where
nominal rigidity is the direct cause of demand shortages.21

5.3

Fiscal and monetary policies

We are now in a position to draw policy implications in the stagnation phase with persistent unemployment. We start with the eﬀect of a monetary expansion on aggregate
demand. An important observation here is that cu is independent of μ, so that a change
in μ does not aﬀect either the upper bound of c or the steady-state inflation rate aslong as μ satisfies (29) and (33). This fact leads to a serious implication regarding theeﬀectiveness of monetary policies in the stagnation phase.
There are two cases we need to consider, but the implications are roughly the same.
Suppose first that cu is not much smaller than the full-employment level θ − g suchthatβu0 (θ − g)−ρ<βu0 (cu ),(34)

It then follows from (29) and (33) thatμ<βu0 (θ − g)− ρ ⇒ stagnation,

β
β
−
ρ
<
μ
<
⇒ both stagnation and full employment,u0 (θ − g)u0 (cu )β

u0 (cu )< μ ⇒ full employment.

Only the stagnation path exists when μ is relatively small (less than β/u0 (θ − g) − ρ).
Since cu is independent of μ, this means that a small change in μ is totally ineﬀective when the economy is in this state. This situation is illustrated in figure 5. Oncethe monetary authority raises μ enough to violate (29) and hence enable m to validate (24), the full-employment steady state, represented by E in figure 6, is restored.
Consequently, there emerges a new equilibrium path given by BE which leads to the21

As mentioned at the outset of this section, the real wage always equals the marginal productivityof labor θ due to perfect flexibility of commodity prices, so that there is no real rigidity in our model.

21

full-employment steady state. Note, however, that the stagnation path AU also exists
in this range. In the presence of multiple paths, there is no guarantee that the economy jumps to the new full-employment path, because it requires coordination among
all agents. This means that the monetary authority needs an extra push, if it is to put
an end to persistent stagnation. This can in fact be done by raising μ even higher, inwhich case only the full-employment path survives. A policy intervention of this kindapparently comes with a cost, however, as the economy necessarily suﬀers from highinflation. The classic dilemma between inflation and unemployment surfaces, now witha long-run implication: if it reduces μ in order to stabilize prices, the full-employment
steady state disappears and the economy returns to the stagnation path AU which
results in persistent unemployment.
On the other hand, if cu is much smaller than θ − g andβu0 (θ − g)−ρ>βu0 (cu ),we findμ<β
u0 (cu )<μ<β
u0 (cu )β

u0 (θ − g)⇒ stagnation,− ρ ⇒ no equilibrium path,β− ρ < μ ⇒ full employment.
u0 (θ − g)

The situation is the same in the case above when μ is either relatively small (lessthan β/u0 (cu )) or relatively large (β/u0 (θ − g) − ρ): only the stagnation path exists inthe former case while only the full-employment path exists in the latter. There is adiﬀerence in the case in-between: no equilibrium path exists in this intermediate range.
Even in this case, though, we can draw the same conclusion: a modest increase in μ is
totally ineﬀective and of no help to get the economy out of the stagnation phase.
We now turn to the eﬀect of a fiscal expansion along the stagnation path which isalso drastically diﬀerent from that along the full-employment path. This is illustrated

22

in figure 7. Since (30) and (31) yield
dcu
=³
dg

α

´ > 0,

ηβθ−α
cu u0 (cu )the equilibrium path shifts upward from AF to DH and thus c rises in both the shortrun (from B to D) and the long run (from cu to cu0 ). If the government expands g
suﬃciently to satisfyρ>βu0 (θ − g),as in the case of g2 , the full-employment steady state, K in the figure, is eventuallyreached. It should be noted that if the government reduces g to the previous level, theconsumption c returns to the previous level as well. In order to maintain a high levelof consumption, the government must continuously hold g at the same level.
Since market forces cannot be relied upon to restore full employment automatically,
the role of the government is inherently diﬀerent in the stagnation phase. Government
interventions are not only eﬀective, even in the long run, but also indispensable torealize full employment. We summarize our findings as follows.
Proposition 5 Along the stagnation path,• a modest monetary expansion is totally ineﬀective and has no impact on neither private consumption nor the inflation rate,• a substantial monetary expansion may restore full employment, although it necessarily entails high inflation,• a fiscal expansion raises private consumption, in both the short run and the long run.

6

Conclusion

We incorporate various concepts of fairness to the fair wage model of Akerlof and Yellen(1990) and extend it to a dynamic general equilibrium model of a monetary economy
as portrayed by Keynes (1936, chap.17). In this “Akerlof-Yellen meets Keynes” framework, we derive a version of the Phillips curve with its microeconomic foundation.
23

We show that the inflation rate is governed by the liquidity premium, the subjective
discount rate and the unemployment rate; around the steady state, in particular, it
depends on the unemployment rate only, as in the original Phillips curve. More importantly, we also show that unemployment due to demand shortages may arise in
the steady state. Eﬀects of monetary and fiscal expansions are very diﬀerent between
under temporary unemployment and under persistent unemployment.
An increase in the monetary expansion rate raises short-run aggregate demandwhen the economy is on the full-employment path, although the steady-state aggregate
demand is unaﬀected. If demand shortages occur under persistent stagnation, on theother hand, a modest increase in the monetary expansion rate is totally ineﬀectivein stimulating aggregate demand. The monetary expansion rate must be significantly
high to get the economy out of persistent stagnation. This obviously comes with a cost,
however, since the economy inevitably suﬀers from high inflation once full employment
is realized.
The eﬀect of an increase in government purchases is also quite diﬀerent between
under temporary unemployment and under persistent stagnation. An increase in government purchases can either increase or decrease private consumption in the shortrun, depending on the elasticity of money utility, when the economy is on the fullemployment path: if the elasticity is higher (resp. lower) than one, it increases (resp.
decreases) private consumption in the short run. Since the economy eventually restores
full employment, though, an increase in government purchases totally crowds out private consumption in the long run. In contrast, under persistent stagnation, an increasein government purchases never fails to increase private consumption in both the short
run and the long run.

24

Appendix: Stability under persistent stagnation
This appendix shows the saddle-path stability of the dynamics. In the case where thefull-employment steady state exists, there is a unique equilibrium path since the present
model has essentially the same structure as the standard money-in-utility model. Thus,
we focus on the stability when the economy is on the path that leads to the stagnation
steady state.
For simplicity, we consider the case where μ = 0 and g = 0. Since m diverges
to infinity, we consider h(= 1/m) instead of m and examine the stability of the two
dynamic equations derived from (21):µ 0¶cv (1/h)θḣ =− ρ + αη(1 − ) h,c + θη
u0 (c)cµ 0¶
θcv (1/h)cċ =− ρ − α( − 1) ,c + θη
u0 (c)
θaround the stagnation steady state where c = cu and h = 0. The partial derivatives of
the above two equations around the stagnation steady state arecu ηm Ru,
cu + θη
∙
µ
¶¸
c
β
θ
= h×d
− ρ + αη(1 − ) /dc
c + θη u0 (c)cηRu θ − αcu=,
cu + θηθcu η m Ru=.
(cu + θη)h

∂ ḣ
∂h
∂ ḣ
∂c
∂ ċ
∂c
∂ ċ
∂h

= πu +

Since ηm = 0 because of the liquidity trap, from (31) and (32), we find(∂ ḣ ∂ ċ∂ ḣ ∂ ċ(ηRu θ − αcu )π u)( ) − ( )( ) =
< 0,
∂h ∂c
∂c ∂h
cu + θηi.e., one of the characteristic roots is positive and the other is negative. Note that c
is jumpable whereas h(= 1/m = P/M) is not since W is not jumpable and P = W/θ
from (9). We can thus conclude that the path is saddle-path stable.

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29

cc = 0Agm = 0

0

0
0
O

v (m) = u(  g )Figure 1: Full-employment Steady State

m

c

E

g

c = 0Am = 0D
F
B

Ov(m) = +u(  g )v(m) = u(  g )mFigure 2: Monetary Expansion
When Approaching Full Employment

c

E



g

B

g
F
A

D
H

O

v(m) = +u(  g )v(m) = +
u()

m

Figure 3: Fiscal Expansion
When Approaching Full Employment

c

g

m = 0cuc = 0m
O

Figure 4: Stagnation Steady State

c
g

m = 0cuc = 0m
O

Figure 5: Monetary Expansion under Persistent Stagnation(when  is small.)

c

g

E

m = 0cu
BUc = 0AOv(m) = +u(  g )mFigure 6: Monetary Expansion under Persistent Stagnation(when  is large.)

c

K

g2

c = 0 (g = g2)cuHc = 0 (g = g1)
Fc = 0 (g = 0)

E
cu
D

A

O

B

v (m) = +u(  g2 )mFigure 7: Fiscal Expansion under Persistent Stagnation