Discussion Paper No. 4

Growth, Stagnation and Status Preference

Ryu-ichiro Murota and Yoshiyasu Ono

September 2008

ＧCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Growth, Stagnation and Status Preference
Ryu-ichiro Murota∗ and Yoshiyasu Ono†

Abstract

We consider three objects of people’s status preference, consumption, physical capital holding and money holding, and showthat an economy grows or stagnates depending on which object
people most seriously take as status. If the main object of statuspreference is consumption, a steady state with full employment
is reached. If it is physical capital (viz. a producible asset), permanent growth with full employment occurs. However, if it ismoney (viz. an unproducible asset), stagnation with persistent
unemployment arises.

JEL Classiﬁcation: E12; E24; O41
Keywords: Status Preference; Persistent Unemployment; LongRun Growth∗

School of Economics, Kinki University, 3-4-1 Kowakae, Higashi-Osaka, Osaka 5778502, JAPAN. E-mail: murota@eco.kindai.ac.jp.
†
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki,
Osaka 567-0047, JAPAN. E-mail: ono@iser.osaka-u.ac.jp. Ono’s research is financially
supported by the Grants-in-Aid for Scientific Research, JSPS.

1

1

Introduction

Many economists (e.g., J. M. Keynes, A. Smith, and T. Veblen) have longpointed out that people care about their relative positions in society. For
example, Keynes (1972, p. 326) states
Now it is true that the needs of human beings may seem to be
insatiable. But they fall into two classes—those needs which are
absolute in the sense that we feel them whatever the situationof our fellow human beings may be, and those which are relativein the sense that we feel them only if their satisfaction lifts us
above, makes us feel superior to, our fellows. Needs of the secondclass, those which satisfy the desire for superiority, may indeed
be insatiable; for the higher the general level, the higher still theyare.
Recently, the literature on social status has been expanding and various
factors that generate social status are analyzed.1 For instance, Corneo andJeanne (1997a) and Rauscher (1997) model Veblen’s (1922) conspicuous consumption, which signals high income and thus high status. Bakshi and Chen
(1996), Clemens (2004), Cole, Mailath and Postlewaite (1992), Corneo andJeanne (1997b), Fisher (2004, 2005), Futagami and Shibata (1998), Gong
and Zou (2001), Long and Shimomura (2004), and Zou (1994, 1995, 1998)
present models where social status comes from the relative or absolute levelof physical capital or wealth holding. Fershtman, Murphy and Weiss (1996)
and Kawamoto (2007) adopt the idea that the relative amount of human capital generates social status. Clark and Oswald (1998) more generally analyze
the case where people’s actions aﬀect their positions in society.
1See, e.g., Weiss and Fershtman (1998) for a survey.

2

We consider three objects of status preference together, viz. consumption(conspicuous consumption), physical capital (capitalist spirit) and money(miser Scrooge), and ﬁnd that depending on which object people value mostan economy exhibits quite diﬀerent performances: converging to the full employment steady state without growth, persistent growth with full employment, stagnation with persistent unemployment, and unemployment growth.
Demand for producible objects creates employment. Thus, if people takeconsumption or physical capital holding to be more important than money
holding as exhibiting social status, full employment is reached. In particular,
if physical capital holding is the object, people continue to accumulate it
and hence an economy perpetually grows, as in the capitalist-spirit economy
discussed by Weber (1958).2 However, if money is the object, people do not
suﬃciently demand producibles, causing persistent unemployment to arise.
It is in conformity with the following statement by Keynes (1936, p. 235):
Unemployment develops, that is to say, because people want themoon;– men cannot be employed when the object of desire (i.e.
money) is something which cannot be produced and the demand
for which cannot be readily choked oﬀ.
Is frugality beneﬁcial to society? This is an important issue that has long
been discussed. For example, Smith (1910, p. 304) states
Whatever, therefore, we may imagine the real wealth and revenueof a country to consist in, whether in the value of the annual
produce of its land and labour, as plain reason seems to dictate;or in the quantity of the precious metals which circulate within it,
2

Zou (1994) proposes a model of such capitalist spirit and shows that perpetual capital
growth occurs.

3

as vulgar prejudices suppose; in either view of the matter, every
prodigal appears to be a public enemy, and every frugal man a
public benefactor.
Keynes (1936, chapter 23) criticizes it by quoting from B. Laﬀemas, W. Petty,
N. Barbon, J. Cary, B. Mandeville, J. A. Hobson, A. F. Mummery and T.
R. Malthus, and insists that frugality causes unemployment to occur. In thepresent paper we show that in the presence of status preference frugality canbe virtue or vice, depending on which asset people want to hold as exhibiting
status.
The remainder of the present paper is organized as follows. Section 2
presents a model that considers three objects of people’s status preference
together. Section 3 shows how people’s status preference inﬂuences the dynamic behavior of the economy. Section 4 concludes. Proofs of dynamic
stabilities under various growth and stagnation patterns are set out in appendices.

2

The Model

2.1

Households

We consider an economy populated by identical households whose size isnormalized to unity. The representative household’s utility depends on not
only consumption ct and real money holding mt but also social status st :
 ∞[u(ct ) + v(mt ) + χ(st )] exp(−ρt)dt,
U=0

where ρ (> 0) is the subjective discount rate and the felicity function satisﬁes4u (·) > 0, u (·) < 0, u (0) = ∞, u (∞) = 0;v  (·) > 0, v  (·) < 0, v  (0) = ∞, v  (∞) = 0,
lim v  (mt )mt > 0,

mt →0


lim v  (mt )mt = 0;

(1)

mt →∞



χ (·) > 0, χ (·) < 0.
Social status st consists of three factors, viz. consumption, physical capital
holding and real money holding, each in excess of the social average:
st = θc (ct − ct ) + θk (kt − k t ) + θm (mt − mt ),(2)

where ct , k t and mt represent respective social averages. θc , θk and θm indicate
the magnitude of each factor’s contribution to social status, representing
conspicuous consumption, capitalist spirit and miser Scrooge respectively.
Real total asset at consists of physical capital and money:
at = kt + mt .

(3)ȧt = rt kt − πt mt + wt lt + ξt − ct ,(4)

The ﬂow budget equation iswhere rt is the capital rent, wt is the real wage, πt (≡ Ṗt /Pt ) is the inﬂation
rate of commodity price Pt , and lt is the realized amount of labor supply.
Each household’s labor endowment is assumed to be unity. However, since
there may be unemployment, under the short-side assumption lt is given bylt = min{ltd , 1},(5)where ltd represents labor demand. ξt is the real proﬁt income that the individual receives from ﬁrms. As is shown later, in the presence of excess labor5demand real proﬁt ξt is positive even under perfect competition and a linear
homogeneous production function.
The individual maximizes U subject to (3) and (4). The ﬁrst-order optimal conditions areu(ct ) + θc χ (st ) = λt ,(6)v  (mt ) + θm χ (st ) − πt λt = ηt ,(7)

θk χ (st ) + rt λt = ηt ,(8)

λ̇t − ρλt = −ηt ,(9)and the transversality condition islim λt at exp(−ρt) = 0,

t→∞

(10)

where λt and ηt are the co-state variable associated with (4) and the Lagrangian multiplier associated with (3) respectively, and st is given by (2).

2.2

Firms

The ﬁrm sector has the following linear homogeneous production function:yt = F (kt , lt ) = f (kt /lt )lt ,where f (·) satisﬁes
f (0) = 0, f (∞) = ∞, f  (·) > 0, f  (·) < 0, f  (0) = ∞, f  (∞) = 0.
Under perfect competition it determines ltd so as to satisfy
wt = f (kt /ltd ) − f  (kt /ltd )(kt /ltd ).

(11)

Thus, kt /ltd is expressed as a function of only wt :1
kt /ltd = h(wt ), h(∞) = ∞, h (wt ) = − 
≥ 0.
f (h(wt ))h(wt )6(12)Substituting (12) into (5) gives
lt = min{kt /h(wt ), 1}.

(13)

From (5) and (11), we ﬁnd
wt = f (kt /lt ) − f  (kt /lt )(kt /lt ) if 1 ≥ ltd (= lt ),

wt < f (kt ) − f (kt )kt if

ltd > 1 (= lt ).

(14)

Whereas the labor market adjustment is sluggish, the capital market adjustment is assumed to be perfect and thus the capital rent always equals themarginal productivity of capital for any lt :rt = f  (kt /lt ).

(15)

Since real proﬁt ξt is
ξt = f (kt /lt )lt − rt kt − wt lt ,using (14) and (15) we obtainξt = 0 if 1 ≥ ltd (= lt ),ξt > 0 if ltd > 1 (= lt ).
When ξt is strictly positive, it is transferred to households, as shown by (4).

2.3

Dynamics

Since households are identical, ct , kt and mt equal each social average:
ct = ct , kt = k t , mt = mt .
Therefore, social status st given by (2) is always
st = 0,7(16)

i.e., a desire for social status is never satiated.
From (6)-(8), (15) and (16), we obtainf  (kt /lt ) +θk χ (0)v  (mt )θm χ (0)
+
=
−π
+
,
t
u (ct ) + θc χ (0)u (ct ) + θc χ (0) u (ct ) + θc χ (0)(17)

where the left-hand side represents the total beneﬁt of holding physical capital, consisting of the capital rent and the utility of social status, and theright-hand side represents the net beneﬁt of holding money, consisting of theutility of liquidity and that of social status minus the inﬂation cost. From(6), (8), (9), (15) and (16), the dynamics of consumption is given byθk χ (0)
ċt

−ρ ,
= σ(ct ) f (kt /lt ) + ctu (ct ) + θc χ (0)(18)

where σ(ct ) ≡ −[u (ct ) + θc χ (0)]/[u (ct )ct ] > 0.
Nominal money supply Mt is assumed to be constant at M and henceṁt /mt = −πt . Substituting πt given by (17) into this equation yieldsχ (0)(θk − θm )v  (mt )ṁt++ f  (kt /lt ).
=− mtu (ct ) + θc χ (0) u(ct ) + θc χ (0)(19)

Nominal wage adjustment is assumed to be sluggish in the following manner:Ẇt= γ ltd − 1 ,Wtwhere the initial level of nominal wage W0 is historically given.3

(20)
Since

ẇt /wt = Ẇt /Wt − πt , (12), (17) and (20) lead tov  (mt )χ (0)(θk − θm )ktẇt
−1 − 
++ f  (kt /lt ). (21)
=γ



wt
h(wt )u (ct ) + θc χ (0) u (ct ) + θc χ (0)3Many economists have explored how the real or the nominal wage is adjusted. Although this issue is interesting and important, the present paper does not address the issue
and assumes the simple adjustment of the nominal wage, in order to focus on influencesof status preference on the dynamic behavior of the economy.

8Commodity price Pt perfectly adjusts so that the commodity market is always
in equilibrium:k̇t = f (kt /lt )lt − ct .

(22)(18), (19), (21) and (22), where lt is given by (13), formulate an autonomous
dynamic system with respect to ct , mt , wt and kt .

3

Economic Consequences of Status PreferenceThis section demonstrates the relationship between people’s status preference represented by θc , θk and θm and the dynamic behavior of an economy,
viz. convergence to the full employment steady state, persistent growth withfull employment, persistent growth with unemployment, and stagnation withpersistent unemployment.

3.1

Full Employment Steady StateLet us ﬁrst examine the condition for the full employment steady state to bereached. From (18), (19), (21) and (22) in whichl = ld = 1, ċ = 0, k̇ = 0, ṁ = 0, ẇ = 0,
the full employment steady state is given byc∗ = f (k ∗ ),θk χ (0)(≡ ϕ(k ∗ )),u (f (k ∗ )) + θc χ (0)θm χ (0)v  (m∗ )+(≡ φ(m∗ , k ∗ )),ρ= 
u (f (k ∗ )) + θc χ (0) u (f (k ∗ )) + θc χ (0)ρ = f  (k ∗ ) +(23)

where ∗ represents each value in the full employment steady state. The
second and third equations imply equality among the time preference rate,9the marginal beneﬁt of holding physical capital k and that of holding money
m. The marginal beneﬁt of k is the sum of the marginal productivity and themarginal contribution to status while that of m is the sum of the marginalutility of liquidity and the marginal contribution to status.
Since the marginal beneﬁt of k is naturally assumed to decrease as kincreases, ϕ(·) and φ(·, ·) given in (23) satisfyθk χ (0)u f θk, ϕ = f  − < 0;θc[u + θc χ (0)]2
θm χ (0),φ(0, k ∗ ) = ∞, φ(∞, k ∗ ) = u (f (k ∗ )) + θc χ (0)
∂φ
∂φ
< 0,
> 0,
∂m
∂k

ϕ(0) = ∞, ϕ(∞) =(24)

and hence k ∗ and m∗ uniquely exist if θc is so large as to satisfy
θk
≤ ρ,
θc

θm χ (0)≤ ρ.
u (f (k ∗ )) + θc χ (0)(25)

Furthermore, in appendix A it is proven that the dynamic path along which
the full employment steady state is reached is uniquely determined. Thus,
Proposition 1. The full employment steady state represented by (23) isreached if the contribution of consumption to social status, θc , is so largethat (25) is valid.
The area of θk and θm satisfying (25) is given by region A in ﬁgure 1. Notethat the border of the second inequality of (25) forms a negative relationshipbetween θk and θm . It is because from the second equation of (23) and theproperty that ϕ < 0 given in (24) a rise in θk increases k ∗ and hence reducesu (f (k ∗ )), which decreases θm that equalizes both sides of the second propertyof (25).

10

3.2

Permanent Growth

Let us next treat the case where the full employment steady state does notexist. In this subsection it will be shown that if
θk
> ρ,
θc

θk ≥ θm ,(26)

steady growth occurs although the marginal productivity of capital decreasingly converges to zero as k expands. The ﬁrst property of (26) implies thatthe marginal beneﬁt of holding capital dominates the time preference rateeven when capital holding expands to inﬁnity whereas the second property
implies that people take capital more seriously than money as a measure ofsocial status.
Under (26) capital k continues to accumulate and eventually f  (·) declines
to zero while c and m approach respective constant levels that make ċ givenby (18) and ṁ given by (19) zero:θk χ (0)= ρ,u (c) + θc χ (0)v  (m)θm χ (0)θk χ (0)=+.
u (c) + θc χ (0)u (c) + θc χ (0) u (c) + θc χ (0)

(27)
(28)

Since ẇ/w = 0, as shown in appendix B, from (21) and (28) the labor market
reaches full employment:(ld =)

k
= 1.
h(w)

(29)

In appendix C it is proven that there is a unique equilibrium path thatreaches the steady state given above.
Note that k continues to accumulate and yet transversality condition (10)is valid, as proven below. Since m is ﬁnite and a = k + m,
λ̇t ȧtλ̇t k̇t+ − ρ ≤ lim+ − ρ.
t→∞ λtt→∞ λt
at
kt
lim

11

Since c is ﬁnite and k continues to expand, from (22) we obtain
f (k/l) c
c
k̇t
lim
−
= lim
= lim f  (k/l) − lim = 0,t→∞ kt
k→∞
k→∞ k
k/l→∞
k/l
k

(30)

where l = min{ld , 1} and the second equality results from de l’Hôpital’s rule.
Thus, using (6), (8), (9), (15) and (30) one ﬁnds
λ̇t ȧtθk χ (0)+ −ρ≤− < 0,t→∞ λtatu (c) + θc χ (0)limwhich implies the validity of (10).
Such full employment growth occurs only if initial price level P0 (andhence initial real money balances m0 ) jumps to the value leading to it since
the dynamic path is saddle point stable as shown by appendix C. If P0 jumps
to a lower level than the value, P continues to fall and m expands to inﬁnity.4
We next show that a continuum of such paths is equilibrium paths under acertain condition. In this case not only full employment growth but also
unemployment growth is possible.
The above case in fact occurs if θk and θm satisfy (26) and
ρ−γ
θm
≥
,
θk
ρ

(31)

which are shown by region B in ﬁgure 1. Since both v  (m) and f  (k/l)
converge to zero along the deﬂationary paths, from (19) and (21) in the statein which ẇ/w = 0,5 we ﬁndṁ= −π = −γ(l − 1) ≥ 0,m
(θk − θm )χ (0)= −γ(l − 1).
u (c) + θc χ (0)4(32)

If P0 jumps to a higher level than the value, m continues to decline. From (19), suchinflationary paths are ruled out since from (1) limmt →0 v  (mt )mt > 0. See, e.g., Blanchard
and Fischer (1989, chapter 5) for the infeasibility of such hyper-inflationary paths.
5
See appendix B for the property that ẇ/w = 0.

12

Since m continues to expand, as shown by the ﬁrst property of (32), (28) is
replaced by the second equation of (32) while (27) is still valid. Substituting(27) into the second property of (32) yieldsθm ρ(≤ 1 under (26))
l = 1− 1−θk γθm ρ − γ ρ=(≥ 0 under (31)),
−
θk
ρ
γ

(33)

implying that unemployment persists. Although both k and m expand to
inﬁnity, the transversality condition is valid since from (6), (7), (9), (19) and(30) we obtain
λ̇t ȧtλ̇t ṁtθm χ (0)+ − ρ ≤ lim+−ρ=− 
< 0.
t→∞ λtt→∞ λtat
mt
u (c) + θc χ (0)lim

Along these paths people continue to accumulate capital and money sothat they raise their social status. Since money is unproducible, a desire forit does not create any labor demand and causes persistent unemployment
and deﬂation to occur. Furthermore, the stability type of unemployment
growth is sink, as shown in appendix D, and hence there is a continuum ofequilibrium paths with permanent growth and unemployment.
We summarize the preceding discussion below.
Proposition 2. Under (26) and (31) there are two types of dynamic paths:
a unique path that leads to permanent growth with full employment and acontinuum of paths that lead to permanent growth with unemployment.
This proposition suggests that unemployment growth will most probably
occur.
If (26) is valid and (31) is violated, the value of l given by (33) is lessthan zero and hence the deﬂationary paths are infeasible. This is the casewhere the relative preference for money over physical capital (θm /θk ) is small
13

enough. People desire to accumulate physical capital rather than money
and hence full employment is realized. This case is represented by region
C in ﬁgure 1. Note that it appears only if ρ > γ, that is, preference for
consumption (ρ) dominates the beneﬁt of holding money due to deﬂation(γ).
Proposition 3. Permanent growth with full employment occurs if the contribution of capital holding to social status, θk , is so high as to satisfy (26)and violate (31).

3.3

Stagnation with Unemployment

Let us ﬁnally consider the case illustrated by region D in ﬁgure 1. In thiscaseθm χ (0)> ρ (if θk ≤ ρθc ),u (f (k ∗ )) + θc χ (0)(34)θm > θk ,where k ∗ is given by the second equation of (23). Note that k ∗ exists if
θk ≤ ρθc , as shown by (24) and (25). The ﬁrst property of (34) implies thatthe marginal beneﬁt of money is always higher than the time preference rateand hence money is preferred to consumption whereas the second one implies
that money is more important than capital as a source of social status. Thus,c and k stay ﬁnite while m continues to expand and l = ld < 1. From (18),(19), (21) and (22), the steady state in this case is given byc = f (k/l)l,f  (k/l) +θm χ (0)θk χ (0)
=
−γ(l
−
1)
+
,
u(c) + θc χ (0)u (c) + θc χ (0)θm χ (0)
ρ = −γ(l − 1) + ,
u (c) + θc χ (0)ṁ= −π = −γ(l − 1) > 0.
m
14

(35)

Substituting the ﬁrst equation of (35) into the second equation yields k/l
as a function of l, denoted by g(l):f  (k/l) +θk χ (0)θm χ (0)
=
−γ(l
−
1)
+
u (f (k/l)l) + θc χ (0)u(f (k/l)l) + θc χ (0)⇒ k/l = g(l).

From g(l) given above and the ﬁrst and third equations of (35), l is determined so that it satisﬁes
G(l) ≡ −γ(l − 1) +θm χ (0)= ρ,u (f (g(l))l) + θc χ (0)(36)

where from the deﬁnitions of ϕ(·) and φ(·, ·) given in (23) G(·) satisﬁes
G(0) = γ,G(1) = φ(∞, g(1)) = ϕ(g(1)).

(37)

Once l is determined, therefore, c and k are determined and the steady-state
deﬂation rate takes the value given by the last equation of (35).
Let us ﬁnd the condition under which such l exists within (0, 1). From(24) and the second property of (34),ϕ(0) > φ(∞, 0),ϕ(∞) (= θk /θc ) < φ(∞, ∞) (= θm /θc ),∂φ(∞, k)> 0.
ϕ (k) < 0,
∂k

(38)

Therefore, there is a unique g(1) that satisﬁes the second property of (37).
If ρ < θk /θc , where the right-hand side is the lower bound of ϕ(·), as isclear from (38), it is obvious that ρ < ϕ(g(1)) and thus from (36) and (37)G(l) < G(1). If θk /θc ≤ ρ, from the second and third equations of (23) andthe ﬁrst property of (34),ρ = ϕ(k ∗ ) < φ(∞, k ∗).
15

Since this property, the second property of (37) and the ﬁrst property of (38)yield
0 < g(1) < k ∗ ,and ϕ (k) < 0, as shown in (38), we ﬁnd(ϕ(k ∗ ) =) ρ < ϕ(g(1)).
Thus, from (36) and (37) G(l) < G(1).
Having established that always G(l) < G(1) and assuming the steadystate to be unique, we ﬁnd that in order for l given by (36) to exist within(0, 1) G(0) must be lower than G(l). From (36) and the ﬁrst property of (37)this condition is equivalent toρ > γ.

(39)

Since G(0) < G(l) < G(1) and (36) gives a unique solution for l, we ﬁndG (l) > 0.

(40)

Since k̇ = 0 and v  (m) = 0 in the present steady state, from (6), (7), (9)and (19) we obtainθm χ (0)λ̇t ȧtλ̇t ṁt< 0,+ − ρ ≤ lim+−ρ= − t→∞ λtt→∞ λtat
mt
u (c) + θc χ (0)limwhich represents the validity of the transversality condition. In appendix E
it is shown that there is a unique equilibrium path and hence
Proposition 4. A steady state with persistent unemployment without growth
is reached if (39) is valid and the contribution of money holding to social
status, θm , is so large as to satisfy (34).
16

If money is the main object of status preference, households permanently
accumulate it whereas consumption and capital converge to respective ﬁnite
levels. Since money demand creates no employment, unemployment persists.6
Let us ﬁnally note that, if (39) does not hold, the case mentioned inproposition 2 is always valid under (26), the case mentioned in proposition
3 disappears, and l given by (36) does not exist. Thus, region C in ﬁgure
1 disappears and in region D there is no equilibrium path. This case isillustrated by ﬁgure 2.

4

Concluding Remarks

We consider an economy in which there are three objects of people’s statuspreference, consumption, capital and money, and nominal wages adjust ina sluggish manner. If the main object of status preference is producible,
viz. consumption or physical capital, labor demand is created and hence
full employment is reached. In particular, if physical capital is most desired,endogenous and permanent capital growth occurs even under decreasing returns to capital. This is in conformity with Max Weber’s capitalist-spirit
economy. However, if people take money as the most important object ofstatus, no demand for labor is created and hence persistent unemployment
and deﬂation occur. This is consistent with what Keynes (1936, p. 235)
states.
Adam Smith considers frugality to be beneﬁcial since it urges people
to accumulate physical capital and generate permanent growth. In contrast,
John M. Keynes regards it as harmful since it keeps people from consumption
6

This result is similar to Ono (1994, 2001), who shows that unemployment arises as asteady-state phenomenon if utility of money holding is insatiable.

17and leads to unemployment. We ﬁnd that it can be virtue or vice, dependingon which object people most seriously take as a measure of social status.
If the object is physical capital, permanent growth arises. If it is money,
persistent stagnation occurs and people are left unemployed.

18

Appendix A Stability of the Full Employment
Steady StateWe ﬁrst examine the case where excess labor demand arises along the paththat converges to the steady state represented by (23). Linearizing (18), (19),(21) and (22), where lt = 1 < ltd , in the neighborhood where ct = c∗ , kt = k ∗ ,wt = w ∗ (= f (k ∗ ) −f  (k ∗ )k ∗ ) and mt = m∗ and applying (12) and (23) to theresults yields the characteristic equation of the dynamics of (ct , kt , wt , mt ):−θk χ (0)σc∗ u−z[u + θc χ (0)]2σc∗ f −1f − z[χ (0)(θk − θm ) − v  ]w ∗u
γw ∗
∗ 
−
w
f
+
[u + θc χ (0)]2
k∗


∗ 
[χ (0)(θk − θm ) − v ]m u−m∗ f 
[u + θc χ (0)]2

0

0

0

0
= 0,
γw
w ∗ v −z−u + θc χ (0)f  k ∗ 2
m∗ v −z0− u + θc χ (0)∗where z denotes a characteristic root. It reduces to γw ∗
 2m∗ v ∗ − z = 0,−z− z − κz + σc ϕu + θc χ (0)f  k ∗ 2
whereκ ≡ f −θk χ (0)σc∗ u
.
[u + θc χ (0)]2

Thus, besides the following two roots:z=γw ∗m∗ v 
> 0,
<
0,
z
=
−
u + θc χ (0)f  k ∗ 2

there are two roots, z1 and z2 , which using (24) one ﬁnds to satisfy
z1 z2 = σc∗ ϕ < 0,
implying that the characteristic equation has two positive and two negativeroots. Note that c0 is jumpable, k0 is unjumpable, and the degree of freedom19

of choosing the initial level of mt and wt is one since once P0 is chosen thenboth of m0 (= M/P0 ) and w0 (= W0 /P0 ) are determined. Therefore, thedynamic path is uniquely determined.
Next, we treat the case where excess labor supply arises (lt = kt /h(wt ) <1) in the interim state. By linearizing (18), (19), (21) and (22) in the neighborhood where ct = c∗ , kt = k ∗ , wt = w ∗ and mt = m∗ , and applying (12)and (23) to the results, we obtain the following characteristic equation of the
dynamics of (ct , kt , wt , mt ):−θk χ (0)σc∗ u−z[u + θc χ (0)]2−1[χ (0)(θk − θm ) − v  ]w ∗u[u + θc χ (0)]2[χ (0)(θk − θm ) − v  ]m∗ u
−[u + θc χ (0)]2
−

σc∗
0
k∗
f
f − f k∗−z
0
k∗f  k ∗ 2
= 0.
γw ∗
w∗
γw ∗
w ∗v − ∗ +  ∗ 2 − z− k∗ku + θc χ (0)f k∗mm∗ v 
−z
0
− ∗
− 
k
u + θc χ (0)−0

It reduces toz 4 + B3 z 3 + B2 z 2 + B1 z + B0 = 0,
where using (24) we ﬁnd B0 to satisfy
γσc∗ w ∗ m∗ v  ϕ> 0,
B0 = −  ∗ 2 
f k [u + θc χ (0)](A1)

and using (14) we ﬁnd B3 to beB3 =θk χ (0)σc∗ uγw ∗m∗ v 
.
−f−+[u + θc χ (0)]2f  k ∗ 2 u + θc χ (0)(A2)

From (A1), there are three possibilities: (a) two positive and two negative
characteristic roots, (b) four positive ones, and (c) four negative ones. As γchanges from zero to inﬁnity, the third term of (A2) increases from zero to
inﬁnity whereas the other three terms remain negative and unchanged since20c∗ , k ∗ and m∗ that satisfy (23) do not depend on γ. Consequently B3 changes
from negative to positive. However, since B0 > 0 for any positive γ, any root
cannot be zero, implying that any root cannot change its sign as γ changes.
Since B3 must be negative in case (b) and positive in case (c), cases (b) and(c) are ruled out and only case (a) is valid, implying the saddle path stability
to hold and the path to be unique.7

Appendix B The Dynamics of the Real Wage
This appendix shows that ẇ/w converges to zero if capital k permanently
accumulates.
First we treat the case where m converges to a ﬁnite value. Since π = 0
in this steady state, (21) impliesẇ= γ(ld − 1),w(B1)

where from (12) ld is
ld =

k
.
h(w)

In the steady state, from (30) and (B2) we obtain




l˙d
h (w)w ẇ
h (w)w ẇk̇= − lim,= − limw→∞ h(w)w→∞ h(w)
ld
k
w
w

(B2)

(B3)

where from (12) h (w) ≥ 0. We here assumeh (w)w> 0,
w→∞ h(w)lim(B4)

which is indeed the case under a production function of the Cobb-Douglas or
CES type whose elasticity of substitution is greater than 1, as derived from(11) and (12). From (B1), (B3) and (B4) we obtainl˙d 0 ⇐⇒ ld  1.
ld
7

See, e.g., Gandolfo (1997, chapter 25) for this logic.

21

Therefore, the labor market is stable and eventually ld = 1 and ẇ/w = 0.
Next, we consider the case where m as well as k continues to expand.
Since π < 0 in the steady state, from (21)ẇ
= γ(ld − 1) − π.
w

(B5)Since from (B3)-(B5) we ﬁndπl˙d 0 ⇐⇒ ld  1 + ,dlγld converges to 1 + π/γ (< 1) and then ẇ/w = 0.

Appendix C Stability of Full Employment Growth
Under full employment growth w rises to inﬁnity while P stays ﬁnite. Therefore, W increases to inﬁnity and hence ltd > 1 for t < ∞, as is clear from(20). Furthermore, ltd converges to 1, as shown in appendix B. Thus, from(5),lt = 1 ≤ ltd .
Replacing kt and wt by bt (≡ kt −1 ) and nt (≡ wt −1 ) respectively and applying
the above property to (18), (19), (21) and (22) yieldsθk χ (0)
ċt

−ρ ,
= σ(ct ) f (1/bt ) + ctu (ct ) + θc χ (0)ḃt = −f (1/bt )b2t + ct b2t ,


1
ṅt
χ (0)(θk − θm )v  (mt )−− f  (1/bt ),= −γ−1 + nth(1/nt )bt
u (ct ) + θc χ (0) u (ct ) + θc χ (0)ṁtχ (0)(θk − θm )v (mt )+
+ f  (1/bt ).
=− 
mtu (ct ) + θc χ (0) u (ct ) + θc χ (0)
(C1)

22

By linearizing (C1) around the steady state in which (29) holds and c andm are given by (27) and (28), we obtain the characteristic equation:
ζc − z−σcf  /b2

0

0

b2

f  − 2(f − c)b − z

0

0

ψn

n(f  + γ/h)/b2−ψm−f  m/b2

ζn − z −ζm n/m0= 0,ζm − zwhereθk χ (0)σcu
v  m
> 0,
>
0,
ζ
≡
−
m
[u + θc χ (0)]2u + θc χ (0)(C2)
h (1/n)[v  − χ (0)(θk − θm )]uζn ≡ −γ.
< 0 (from (B4)), ψ ≡ −h(1/n)n[u + θc χ (0)]2

ζc ≡ −

Therefore, the characteristic roots are ζn (< 0), ζm (> 0), and the twosolutions ofz 2 − [ζc + f  − 2(f − c)b]z + ζc [f  /b − 2(f − c)]b + σcf  = 0.

(C3)

Since (w =) f − f  k > 0 and k (= 1/b) is large enough while c is given by(27), we ﬁndf  k − 2(f (k) − c) < 0,
implying that the constant term of (C3) is negative. Thus, there are two
positive and two negative characteristic roots. Since the degree of freedom
in choosing the initial values of ct , bt , nt and mt is two, the present dynamics
is saddle path stable.

23Appendix D Stability of Unemployment Growth
By replacing kt , wt and mt by bt (≡ kt −1 ), nt (≡ wt −1 ) and xt (≡ mt −1 )respectively in (18), (19), (21) and (22), where lt = kt /h(wt ) < 1, we obtain



χ
(0)
ċt
θ
k
−ρ ,
= σ(ct ) f  (h(1/nt )) + ctu (ct ) + θc χ (0)ḃt = −f (h(1/nt ))bt /h(1/nt ) + ct b2t ,1v  (1/xt )ṅtχ (0)(θk − θm )= −γ
−1 + 
−− f  (h(1/nt )),nth(1/nt )bt
u (ct ) + θc χ (0) u (ct ) + θc χ (0)χ (0)(θk − θm )ẋtv  (1/xt )−− f  (h(1/nt )) (= π < 0).
= xt
u (ct ) + θc χ (0) u (ct ) + θc χ (0)(D1)
Linearizing (D1) in the neighborhood of the steady state where c is givenby (27) and applying (12) to the result yields the following characteristic
function:ζc − z
b2
ψn
ψx

σc
0
hn2
f
b
− + 2bc − z(f − f  h)0
hf h3 n2
= 0,
1
γ
n
γn
−
−z
ζm
hb2
f  h3 bn hn
x
x
0
− 2
π + ζm − z
hn
0

(D2)

where ζc , ψ and ζm are the same as those in (C2).
In the present steady state ζc and ψ are ﬁnite. Since l < 1 and k (= 1/b)
and m (= 1/x) continue to expand, using (1), (11) and (12) we ﬁndf (h)/h = f (k/l)/(k/l) = 0, 1/n = f (h) − f  (h)h,1/(hn) = f (h)/h − f  (h) = 0, bc = 0, ζm = 0, hb = 1/l.
Calculating (D2) and applying the above properties to the result leads to



γ
1 2c
γ
2
f+ −
= 0.
(ζc − z) (π − z) z −  3 z −  4
f h bnf h bnnl24Therefore, two of the four roots are ζc (> 0) and π (< 0) and the other tworoots, z1 and z2 , satisfyγγ
z1 + z2 =  3 < 0, z1 z2 = −  4
f h bnf h bn



1 2c
f+ −
n
l


> 0,

where the second inequality holds since f and 1/n (= w) continue to expand
while c/l is ﬁnite. These properties show that z1 and z2 are negative. Since
there are one positive and three negative roots whereas the degree of freedom
in choosing the initial values of ct , bt , nt and xt is two, the stability type of
the present dynamics is sink.

Appendix E Stability of Stagnation with Unemployment
We replace mt by xt (= 1/mt ) and linearize (18), (19), (21) and (22), wherelt = kt /h(wt ) < 1, in the neighborhood of the steady state where c, k andl are given by (35) and x = 0. By applying (1) and (12) to the results weobtain the characteristic function of the dynamics of (ct , kt , wt , xt ):−θk χ (0)σcu−z[u + θc χ (0)]2−1

χ (0)(θk − θm )wu−[u + θc χ (0)]2
0

σcl
0
−
k


l3
fl
k
−z
f
−
f
0
k
f  k 2
l
= 0.
v 
wl γwl3w
γwl− +  2 − zkkf ku + θc χ (0) x2000γ(l − 1) − z

It reduces to(γ(l − 1) − z)(−z 3 + 2 z 2 + 1 z + 0 ) = 0,

25

(E1)

whereθk χ (0)σcu
f l wl γwl3+−
+  2 ,[u + θc χ (0)]2
k
k
f k




3
f l γwlθk χ (0)σcu
θm χ (0)σcu wl f wl2 γf  wl3++ 2 −  2 ,−1 ≡ 
[u + θc χ (0)]2 k
f  k 2[u + θc χ (0)]2 k
k
f k



2

 


σcwl θk χ (0)γu f lθm χ (0)u f− f  γ + 
0 ≡ −  2
< 0.


2
f k
[u + θc χ (0)][u + θc χ (0)]22 ≡ −(E2)
Note that 0 < 0 since from (36) and (40) the square bracket of the right-handside of 0 is negative.
One of the four roots of (E1) is obviously γ(l − 1), which is negative.
Let z1 , z2 and z3 , where z1 ≤ z2 ≤ z3 , denote the other three roots. Since0 = z1 z2 z3 < 0, there are two possibilities:
z1 < 0, z2 > 0, z3 > 0;z1 < 0, z2 < 0, z3 < 0.

(E3)

From the second property of (34) and the second equation of (E2),
f l wl γwl3
θk χ (0)σcuf wl2 γf  wl3−+
1 > 
−  2 ,+[u + θc χ (0)]2 kkf  k 2
k2
f k
and hence1 = −z1 z2 − z2 z3 − z3 z1 > 0 if

f l wl γwl3−
+  2 ≤ 0.
kk
f k

From this property we ﬁnd that only the ﬁrst combination in (E3) is possible.
Furthermore, from the ﬁrst equation of (E2)
2 = z1 + z2 + z3 > 0 if

f l wl γwl3−
+  2 > 0,kk
f k

implying that only the ﬁrst combination in (E3) is possible. Thus, there
are always two negative and two positive characteristic roots and hence aunique dynamic path obtains since the degree of freedom in choosing theinitial values of ct , kt , wt and xt is two.
26

References[1] Bakshi, G. S. and Z. Chen (1996), “The Spirit of Capitalism and StockMarket Prices,” American Economic Review, Vol. 86, pp. 133-157.
[2] Blanchard, O. J. and S. Fischer (1989), Lectures on Macroeconomics,
Cambridge, Massachusetts: The MIT Press.
[3] Clark, A. E. and A. J. Oswald (1998), “Comparison-Concave Utility
and Following Behaviour in Social and Economic Settings,” Journal of
Public Economics, Vol. 70, pp. 133-155.
[4] Clemens, C. (2004), “Status, Risk-Taking and Intertemporal Substitution in an Endogenous Growth Model,” Journal of Economics, Vol. 83,
pp. 103-123.
[5] Cole, H. L., G. J. Mailath, and A. Postlewaite (1992), “Social Norms,
Savings Behavior, and Growth,” Journal of Political Economy, Vol. 100,
pp. 1092-1125.
[6] Corneo, G. and O. Jeanne (1997a), “Conspicuous Consumption, Snobbism and Conformism,” Journal of Public Economics, Vol. 66, pp. 55-71.
[7] ———– (1997b), “On Relative Wealth Eﬀects and the Optimality of
Growth,” Economics Letters, Vol. 54, pp. 87-92.
[8] Fershtman, C., K. M. Murphy, and Y. Weiss (1996), “Social Status,
Education, and Growth,” Journal of Political Economy, Vol. 104, pp.
108-132.
[9] Fisher, W. H. (2004), “Status Preference, Wealth and Dynamics in theOpen Economy,” German Economic Review, Vol. 5, pp. 335-355.
27

[10] ———– (2005), “Current Account Dynamics in a Small Open-Economy
Model of Status Seeking,” Review of International Economic, Vol. 13,
pp. 262-282.
[11] Futagami, K. and A. Shibata (1998), “Keeping One Step Ahead of theJoneses: Status, the Distribution of Wealth, and Long Run Growth,”
Journal of Economic Behavior and Organization, Vol. 36, pp. 109-126.
[12] Gandolfo, G. (1997), Economic Dynamics: Study Edition, Berlin:
Springer-Verlag.
[13] Gong, L. and H. Zou (2001), “Money, Social Status, and Capital Accumulation in a Cash-in-Advance Model,” Journal of Money, Credit, andBanking, Vol. 33, pp. 284-293.
[14] Kawamoto, K. (2007), “Preferences for Educational Status, Human Capital Accumulation, and Growth,” Journal of Economics, Vol. 91, pp.
41-67.
[15] Keynes, J. M. (1936), The General Theory of Employment, Interest andMoney, London: Macmillan.
[16] ———– (1972), “Economic Possibilities for Our Grandchildren”, in Essays in Persuasion, The Collected Writings of John Maynard Keynes,
Vol. IX, London: Macmillan, pp. 321-332. Originally Published in 1930.
[17] Long, N. V. and K. Shimomura (2004), “Relative Wealth, StatusSeeking, and Catching-Up,” Journal of Economic Behavior and Organization, Vol. 53, pp. 529-542.
[18] Ono, Y. (1994), Money, Interest and Stagnation, New York: Oxford
University Press.
28[19] ———– (2001), “A Reinterpretation of Chapter 17 of Keynes’s General
Theory: Eﬀective Demand Shortage under Dynamic Optimization,” International Economic Review, Vol. 42, pp. 207-236.
[20] Rauscher, M. (1997), “Conspicuous Consumption, Economic Growth,and Taxation,” Journal of Economics, Vol. 66, pp. 35-42.
[21] Smith, A. (1910), The Wealth of Nations; introduction by Edwin R. A.
Seligman, Vol. 1, New York: Dutton. Originally Published in 1776.
[22] Veblen, T. (1922), The Theory of the Leisure Class. An Economic Studyof Institutions, London: George Allen Unwin. Originally Published in
1899.
[23] Weber, M. (1958), The Protestant Ethic and the Spirit of Capitalism,
New York: Charles Scribner’s Sons. Originally Published in 1904.
[24] Weiss, Y. and C. Fershtman (1998), “Social Status and Economic Performance: A Survey,” European Economic Review, Vol. 42, pp. 801-820.
[25] Zou, H. (1994), “‘The Spirit of Capitalism’ and Long-Run Growth,”
European Journal of Political Economy, Vol. 10, pp. 279-293.
[26] ———– (1995), “The Spirit of Capitalism and Saving Behavior,” Journal of Economic Behavior and Organization, Vol. 28, pp. 131-143.
[27] ———– (1998), “The Spirit of Capitalism, Social Status, Money, andAccumulation,” Journal of Economics, Vol. 68, pp. 219-233.

29θkθ k = [ ρ ( ρ − γ )]θ m

θk = θm

(B)

(C)

ρθ c

(D)

(A)

θm

o
(A) Full Employment Steady State(B) Full Employment Growth or Unemployment Growth(C) Full Employment Growth(D) Stagnation with Persistent Unemployment

Figure 1: Growth and Stagnation Patterns (ρ > γ)30θkθk = θm
(B)

ρθ c

(D’)

(A)

θm

o
(A) Full Employment Steady State(B) Full Employment Growth or Unemployment Growth(D’) No Equilibrium Path

Figure 2: Growth and Stagnation Patterns (ρ ≤ γ)31