Discussion Paper No. 856

DIFFERENCE OR RATIO:IMPLICATION OF STATUS PREFERENCE
ON STAGNATION

Yoshiyasu Ono
Katsunori Yamada

October 2012
Revised March 2013
Secondly Revised January 2014

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Difference or Ratio:
Implications of Status Preference on Stagnation*
by
Yoshiyasu Ono† and Katsunori Yamada‡

Abstract
We consider a dynamic macroeconomic model of households that regard relative
affluence as social status. The measure of relative affluence can be the ratio to, or thedifference from, the social average. The two specifications lead to quite different equilibrium
consequences: under the ratio specification full employment is necessarily realized, whereas
under the difference specification there is a case where a persistent shortage of aggregate
demand arises. Furthermore, using data of an affluence comparison experiment we
empirically find that the difference specification fits the data better than the ratio
specification. Therefore, affluence comparison can be a cause of persistent stagnation.

JEL classification: C91, E12, E24
Keywords: status, stagnation, unemployment.

*

Permission by Masayuki Sato for usage of the data set originally used in Yamada and Sato (2013) is greatly
appreciated. We are indebted to helpful comments by Gerhard Illing and seminar participants in
Ludwig-Maximillian’s University, Munich, and in the Economic and Social Research Institute, Cabinet Office,the Governemnt of Japan. This research is financially supported by a Grant-in-Aid for Scientific Research fromthe Japan Society for the Promotion of Science (JSPS).
†

Institute of Social and Economic Research, Osaka University. Adress: 6-1 Mihogaoka, Ibaraki, Osaka
567-0047, Japan. E-mail: ono@iser.osaka-u.ac.jp‡

Institute of Social and Economic Research, Osaka University. Adress: 6-1 Mihogaoka, Ibaraki, Osaka
567-0047, Japan. E-mail: kyamada@iser.osaka-u.ac.jp

1. Introduction
Recently many countries suffer from serious long-run recessions. One is the GreatRecession that spreads worldwide in the wake of the 2008 international financial crisis.
Another is Japan’s ‘lost decades’ that started when its asset bubble burst in 1990. Facing suchserious economic circumstances, economists now more than ever need an analyticalframework that can treat inefficient macroeconomic outcomes and valid policy options forrecovery from chronic stagnation.
The currently dominant research agenda for dealing with stagnation is the New
Keynesian approach promoted by researchers such as Christiano et al. (2005) and Blanchard
and Galí (2007). They considered microeconomic foundations of price sluggishness andanalyzed macroeconomic fluctuations. This type of analysis is quite successful in examining
short-run recessions that fade out as prices adjust. However, because it treats perturbations
around the full-employment steady state, in order to analyze chronic and serious stagnation
with unemployment we need a different theoretical framework. Along this line, in the recent
IMF annual conference Summers (2013) criticized too much reliance on the DSGE approachin solving economic crises and emphasized the need for researchers to work on long-run
recessions rather than short-run business fluctuations. This paper shares the same view andadopts a long-run stagnation model.
A long-run stagnation model in a dynamic optimization framework was first explored by
Ono (1994, 2001), following the spirit of Chapter 17 of Keynes’s General Theory.
Households in this model have an insatiable preference for money, which causes a liquidity
trap to appear. Prices continue to adjust, but nevertheless shortages of aggregate demand andemployment persist in the steady state. Murota and Ono (2011) also presented a model of
persistent stagnation in which status preference plays the same role in creating persistent1

stagnation as does the insatiable preference for money. They considered three objects ofstatus preference—consumption, physical capital holding, and money holding—and foundthat an economy grows or stagnates depending on which is the primary measure of status. If it
is money (an unproducible asset), persistent stagnation with unemployment occurs.
The above-mentioned insatiable desires for absolute and relative money holdings werediscussed by Keynes (1972, p. 326). He wrote: “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 situation of our fellow human beings may be, andthose which are relative in the sense that we feel them only if their satisfaction lifts us above,
makes us feel superior to, our fellows.” It may, however, be ambiguous whether the target ofpeople’s desire is to hold money or wealth. In the literature of status preferences, such as
Corneo and Jeanne (1997) and Futagami and Shibata (1998), status concerns are often
defined with respect to wealth holdings.
Following this convention, we present a model with status preference for wealth, insteadof money holdings, and explore the possibility of persistent stagnation. In this analysis there
are two specifications of relative affluence. One is that people care about the difference of
their wealth holdings from the social average. The other is that people care about the ratio of
those to the social average.1 Murota and Ono assumed that people care about the differenceof money holdings because this specification was necessary for persistent stagnation to
occur. Corneo and Jeanne (1995) and Futagami and Shibata (1998) took the ratio as themeasure of status because that specification was required for endogenous growth to occur in

1

Clark and Oswald (1998) considered both the difference and ratio specifications of social status andexplored tax policy implications for both cases in a static setting.

2their models.2 We examine both cases and find that persistent stagnation with unemployment
occurs under the difference specification but not under the ratio specification. Thus, if thedifference specification reflects the real world, our disequilibrium model offers a potential toprovide adequate policy implications for long-run stagnation.
We then empirically examine the two specifications to see which is more plausible. Data
are borrowed from the hypothetical discrete choice experiment of Yamada and Sato (2013).
They conducted a large-scale socially representative survey and estimated the effects ofincome comparisons by applying the data to the random utility model framework of Train(2009). While the framework is mostly used for parameter estimations of utility functions, as
in Viscusi et al. (2008), we instead conduct a horse race between the two specifications ofstatus preference and apply the Akaike Information Criteria (AIC) and R-squared to thecomparison. We find that the difference specification fits the data much better than the ratio
specification does, and hence the model that accommodates persistent stagnation is
supported.
Previous macroeconomic studies of the status preference, such as Cole et al. (1992),
Konrad (1992), Zou (1994), Corneo and Jeanne (1997) and Futagami and Shibata (1998),
investigated the effects of status preference on the economic growth rate. There are also
quantitative approaches that tested the validity of status preference, e.g. Abel (1990), Gali
(1994) and Bakshi and Chen (1996). They argued that observed asset price volatility can be
explained by the motivation to keep up with the Joneses. Our purpose is to relate statuspreference to the possibility of persistent demand deficiency and long-run stagnation.

2

Under the difference specification and decreasing returns to real capital, Murota and Ono (2011) showedthat endogenous growth occurs when households regard real capital as status.

32. Two Specifications of Status Concern
We consider a representative household that cares about relative affluence, whose utility
is
,
where
,

exp

is the utility of consumption c,
represents status preference,. Functions

and

,

(1)

is the utility of money m for transactions,is total asset holdings, andis the social average ofsatisfy
0,
0,

0,

0

∞;

0

∞,

∞

0,
,

Two types of status preference

0.

(2)

are considered. One is that the household cares about

the difference (case D), and the other is that it cares about the ratio (case R).3
Case D:,,

Case R:

0;

,
0.

,

(3)

The flow budget equation and the asset budget constraint are respectively
,
,
where

is the real interest rate,interest-bearing assets,

is the real wage,(4)is the amount of employment,isis the nominal interest rate, and is a lump-sum tax. Obviously

satisfies,

3

Quite different theoretical implications obtained from the ratio and difference specifications documented
below have nothing to do with so called the Keeping Up with the Joneses (KUJ) and the Running Away fromthe Joneses (RAJ) effects. While the utility function with the difference specification is always a KUJ type, theutility function with the ratio specification can be KUJ and RAJ, depending on the marginal rate of substitution.

4

whereis the inflation rate. The number of representative households is normalized to unityand each representative household owns one unit of labor endowment. Therefore, the amountof employment x straightforwardly represents the employment rate.
Maximizing (1) subject to (4) gives a Ramsey equation and portfolio choice summarized
as
,

,

,

(5)

where
/ ′

=

,

,

,

.

The transversality condition is
lim →

0.

(6)

The firm sector is competitive and uses only labor with linear technology, whereisthe labor productivity and is assumed to be constant. In this case, the firm sector infinitely
expands production if nominal commodity priceifis lower thanis higher than

/ . Thus, with perfect flexibility of ,,/ , but produces nothingtakes the value that satisfies(7)

as long as a finite and positive amount of the commodity is traded. Since the profits in thiscase are zero, the firm value equals zero. Therefore, interest-bearing assets b consist of only
government bonds.
In the money market,,
where M is the nominal money stock. The monetary authority is assumed to keep M constant,for simplicity, and thus

5

.

(8)

The fiscal authority finances interest paymentsby collecting lump-sum tax and issuing

new bonds.4 Formally,.
The fiscal authority adjusts

and so that the no-Ponzi-game condition is satisfied. In theneighborhood of the steady state, in particular, it adjusts tax so that,
where

equal :
(9)is the long-run debt level that the government targets.

Due to the perfect flexibility ofin the commodity market,
.

(10)

1.

(11)

If W is also perfectly flexible,

Substituting (8), (10) and (11) into (5) yields
,

Since

, from (3)

,

0 ,
,

Case R:(12)

satisfies,

Case D:

.

.

(13)

Equation (12) has the same structure as the dynamics of the standard money-in-the-utilityfunction model (Blanchard and Fischer, 1989, pp. 239-243), and thus

initially jumps to thelevel that makes m satisfy
Case D:,,4

We ignore government purchases, for simplicity, but even when government purchases are considered thearguments presented are essentially the same.

6

Case R:

,

,

(14)

and the steady state is immediately reached. However, the steady state, and hence, theequilibrium path, may not exist in case D as shown below.
In case R, the value of m that satisfies (14) definitely exists, since from (2)
lim →

∞

lim →

0 .

Thus, the full-employment steady state is indeed realized. In case D, however, there is no mthat satisfies (14) if.

(15)

This happens because for any m,which is inconsistent with the first equation in (14). Note that bothand0 can beindependently set and hence we can innocuously consider the case where the condition of(15) is satisfied. In particular, if productivity
strong, (15) is likely to hold. Thenis high or if the status preference

given by (12) is negative for any positive

0 is. Moreover,

if
lim →
it is strictly negative even when0,0, implying that

becomes negative within a finite

time.5 Thus, there is no feasible path for case D if the condition (15) is satisfied.

5This condition is required to avoid hyperinflationary paths in the standard money-in-the-utility-function
model. See Obstfeld and Rogoff (1983) for this property.

7

3. Persistent Stagnation under the Difference Specification
In the previous section, we find that under the difference specification (case D) andflexible adjustments of

andthere is no dynamic equilibrium path if (15) holds. This isbecause preference for money holding always dominates preference for consumption. This
would naturally suggest a shortage of aggregate demand although it is not allowed to exist
due to the assumption of perfect flexibilities of prices and wages. Therefore, we introduce
sluggish wage adjustments to the model so as to allow for a shortage of aggregate demand.
Consequently, we find that the dynamic equilibrium path exists and that shortages ofaggregate demand and employment remain even in the steady state.6
Recent dominant settings of wage adjustments are the New Classical, the New
Keynesian, and the hybrid Phillips curves. They well fit to analyze short-run fluctuations
around the full-employment steady state, but not to examine persistent stagnation because
they are set up so that the inflation-deflation rate cumulatively expands as long as market
disequilibrium exists.7 Thus, the possibility of unemployment in a steady state, which we
focus on, is intrinsically eliminated. In order for the unemployment steady state to bepossible, we adopt the conventional Walrasian wage adjustment process:
1 ,
where

(16)

is the adjustment speed. Obviously, this is a simplified form of wage adjustment

without a microeconomic foundation but we can provide a microeconomic foundation forthis type of adjustment. In the appendix we indeed apply the wage adjustment mechanismwith a microeconomic foundation proposed by Ono and Ishida (2013) to the present model

6

Obviously, nominal wage sluggishness does not exclude the case where full employment is reached in thesteady state.
7

See Woodford (2003) for properties of those Phillips curves.

8and show that the same steady state with the same stability property as presented below
obtains.
Becausealways moves in proportion to, as shown by (7), the wage adjustment

mechanism (16) leads to1 .
From (4), (5), (8), (9), (10) and the above equation, we obtain an autonomous dynamic
system:
,

1 ,
1 ,
.

where

,

(17)

is given by (13). The full-employment steady state given by (14) is

eventually reached as long as it exists.8 It always is the case in case R, and also in case D with(15) being invalid.
However, if (15) holds in case D, a steady state with full employment given by (14) does
not exist. Then, the first and second equations of (17) form a two-dimensional autonomous
dynamic system with respect to and

:
1 ,
1 .

Becauseis sluggish,initially takes the value that satisfies (7) for the initialdetermines the initial level of(18). Then, while jumps to the amount that is on the saddle path.9

Along this path, a demand shortage remains and deflation continues.
8

The uniqueness and the stability of the present dynamics are proved in the same way as in Ono (1994,
2001), who treats the case where b = 0.

9

In the steady state of the present dynamics, from (18) satisfies
Φ

≡

1

Φ

0.

0.

(19)

From (15), one has(20)

Therefore, for (19) to have a positive solution, it must be valid that
Φ 0

0,

(21)

and then satisfies
0,which implies a persistent demand shortage. Deflation continues, making m diverge toinfinity. Nevertheless, the transversality condition (6) is valid since
lim →
and

. Note that in this state,
0 and thus the second equation of (5) yields
0,i.e., the zero interest rate holds.
Let us mention the economic implication of the difference between the two relative
affluence specifications. In case R, in which households care about the ratio of their asset
holdings to the social average, the marginal utility of real money balances, represented by
1 ⁄

′
of

, converges to zero as

approaches infinity. Thus, there is a levelthat equalizes the desire to accumulate real money balances to the desire to consume

sufficient commodities to realize full employment, and then the steady state with full9

The dynamic equations given in (18) are mathematically the same as those in the case where there is astrictly positive upper bound on the marginal utility of money, as analyzed by Ono (2001). He showed that there
is a unique dynamic path and that it converges to the stagnation steady state if (15) is valid.

10employment obtains. In case D, in which households care about the difference, the desire toaccumulate assets

0 remains strictly positive. Thus, if (15) holds, no matter how muchassets the households accumulate, the desire to accumulate money as an asset stays to behigher than the desire to consume sufficient commodities to realize full employment. A
demand shortage remains despite continued declining prices and expanding real balances.

4. Experimental Evidence of the Two Specifications of Status
In the previous section we showed that persistent stagnation arises as an equilibriumoutcome when households care about not the ratio of their asset holdings to, but thedifference from, the social average. To see relevance to the real world, we investigate whichof the two specifications of relative affluence is more plausible.
We use the data set created by the hypothetical discrete choice experiment of Yamada
and Sato (2013), which includes 48,172 observations from 10,203 respondents. They
conducted an original Internet-based survey in February 2010 with Japanese subjects, andinvestigated the intensity and sign of income comparisons against the social average.10 By
applying the data to a random utility model framework, they estimated the following utility
function with relative affluence:11,
where y is the subject’s income and

,

(22)

is the social average of income. We replace (22) by thetwo utility specifications with a single composite variable given in (3), so that we can focus

10

By setting up an experiment such that parameters were fully randomized and choice situations were
orthogonal, they exploited full potential of the discrete choice experiment framework to find if the subjects had
altruism or jealousy. See section 3 of Yamada and Sato (2013) for details of the experimental setting.
Experimental details are provided in a supplemental material of this paper.
11This setting was first presented by Dupor and Liu (2003) and Liu and Turnovsky (2005).

11

on a comparison of the two. We then apply the conditional logit model framework andcompare their AICs to see which specification better fits the data.
Note that there is a gap between the theoretical structure in the previous sections and theexperimental setting given below. In the choice experiment of Yamada and Sato (2013), therelative affluence is associated with income, whereas in our model it is with asset holdings.
That said, income is a predictor of asset holdings under the permanent income hypothesis.
Moreover, Headey and Wooden (2004) found that income and asset holdings are both
important determinants of subjective well-being, and that the positive effect of asset holdings
on subjective well-being is taken away when adding an income term as an additional control.
This evidence suggests that income is a good proxy for asset holdings in the happinessanalyses. Therefore, we take income y as a proxy for assets a and replace
by

,

,

given in (3).

To facilitate the experimental data of Yamada and Sato (2013) to conduct a horse race ofthe ratio and difference specifications, let us reformulate the model to the following random
utility model:,,

where represents each income scenario,

is,Case D:
Case R:is the marginal utility from the status,,
is the constant term, and ϵ is the error term that

follows an independent and identical distribution of extreme value type 1 (IIDEV1). The
probability

that respondents prefer income situation to income situation is given by
Prob

,

,

12

, for all.

By assuming IIDEV1 for the error term we consider a conditional logit model (McFadden,1974) and estimate the parameterof the random utility function using the maximized

likelihood method. We also assume that irrelevant alternatives are independent (IIA), andthat the random components of each alternative and those within each subject are
respectively uncorrelated.

Table 1: Estimation results from the conditional logit model

Dep var: Utility

Relative income

Pseudo R2

AIC

N

Sign

robust s.e.

Ratio

0.4430***(0.1059)

0.005363261.28

48172

Difference

0.0337***(0.0004)

0.225551721.87

48172

Robust standard errors clustered by subjects, *** p < 0.01.

Table 1 reports the results of the conditional logit model estimations. The first row is forthe ratio specification, and the second for the difference specification. In both cases, therelative income terms,

and/ , have a positive and significant effect. The striking

finding here is a significant difference in the AIC between the two specifications. The AIC
under the difference specification is smaller than that under the ratio specification. Also, thepseudo R-squared for the difference specification (which is 0.22) is much higher than that for
the ratio specification (which is 0.0053). Therefore, the difference specification fits the data
better than the ratio specification.12

12

After we finished this project, we have learned a study by Mujcic and Frijters (2013). Using their own
experimental data on ranking comparisons, they also found that the difference specification fitted the data better
than the ratio specification.

13

In section 3 we have found that with the difference specification persistent stagnation
can occur while with the ratio specification full employment is always reached in the steadystate. Because the present experimental result supports the former, we may conclude that ourmodel accommodates persistent stagnation and unemployment.

5. Conclusion
When relative affluence compared to the social average is taken as status, the measurecan be the ratio to, or the difference from, the social average. The two specifications lead tomutually quite different scenarios of business activity. If it is the ratio, full employment isnecessarily reached in the steady state. If it is the difference, there is a case where
unemployment and stagnation due to shortage of aggregate demand appear in the steadystate. This case arises particularly if the output capacity is high or if the desire for the relative
affluence is strong.
Using the experimental data on income comparison carried out by Yamada and Sato(2013), we find that the difference specification fits the data better than the ratio specification
does. Therefore, relative affluence can be a cause of persistent stagnation, and our model canbe a good platform to analyze persistent stagnation. Furthermore, since the mathematicalstructure of the present model is essentially the same as that of Ono (1994, 2001), the same
policy implications as those of Ono hold. They are quite different from those under theconventional models and are more in conformity with classical wisdom of Keynes (1936): an
increase in government purchases expands private consumption, while improvements in
productivity and wage adjustments reduce private consumption and worsen stagnation.

14

Appendix: Stability with a microeconomic foundation of sluggish wage adjustment
In the text we assume the conventional Walrasian wage adjustment process that lacks amicroeconomic foundation, represented by (16). Ono and Ishida (2013) extended thefair-wage hypothesis a la Akerlof (1982) and Akerlof and Yellen (1990) to a dynamic setting
and proposed a microeconomic foundation of wage adjustment that converges to theconventional Walrasian one. Furthermore, they applied it to a money-in-the-utility-function
model and obtained the condition that makes the stability and uniqueness of the steady state
hold. This appendix introduces that wage adjustment mechanism, instead of (16), to thepresent model and shows that under conditions (20) and (21) the unemployment steady state
given by (19) is reached.
Let us start the analysis by summarizing the dynamics of fair wages presented by Ono
and Ishida (2013). Employed workers randomly separate from the current job at the Poison
rate , and therefore total employment , which also represents the employment rate because
the population is normalized to unity, changes in the following way:
,
where

(A1)

is the number of workers that are newly hired. While workers are employed, they

form fair wage

in mind by referring to their past wages, their fellow workers’ fair wages(which equal their own) and the unemployment situation of the society. More precisely, theyfirst consider the rightful wage , which is the wage that they believe fair if everybody is
employed. Therefore,∆ , implying the rightful wage that is ex post conceived at time

∆ , is calculated so that the current fair wage equals the average ofzero income of the unemployed. Because the number of the employed is∆and the∆ , itsatisfies
∆

∆

∆ .

15

(A2)

Newly hired workers, in contrast, do not have any preconception about the fair wage andsimply follow the incumbent workers’ conceptions.
∆ . Therefore, when the incumbent workers

At time the number of new comers is, the total number of workers that they care is 1

calculate the fair wage

∆ .

Because the rightful wage that they have in mind is the one that was ex post conceived at time
∆ , which is
∆

1

∆ ,

∆

in (A2), and the number of the incumbent workers is

is formed as follows:
∆

∆

∆
∆

.

Substituting (A2) into the above equation and rearranging the result leads to
∆

∆ .

∆

Therefore, by reducing ∆ to zero we obtain.

(A3)

The representative firm is competitive and takes commodity priceas given. In thepresence of unemployment, it will set the wage equal to the fair wage because the fair wage isthe lowest wage under which the employees properly work. The commodity price adjusts to⁄ since there is no commodity supply if⁄ and excess commodity supply if⁄ . Under full employment the firm tries to pick out workers from rival firms to
expand the market share by raising the wage so long as the marginal profits are positive,
makingequal toNote thatthat.

follows the movement of the fair wage in the presence of unemployment andfollows the movement ofin the absence of unemployment. Thus, anyway we have,

which yields

16

.

(A4)

From (5), the time differentiation of (10), (A1), (A3) and (A4), in the presence of
unemployment we obtain an autonomous dynamic system of and .
,

≡
,

≡

/

,

/

1 ,
,

1 ,

instead of (17). Note that in the neighborhood of the steady state, where

(A5)
0,

in (A5)equals1 ,as is the case in the dynamics given by (17). This is the Walrasian adjustment in which
adjustment speed

is the Poison rate of job separation or equivalently 1/ is the average

duration of employment. The steady state condition of the first dynamic equation of (A5) isequivalent to (19) and then a shortage of aggregate demand persists. In this steady state, from
(5) and (8) we find,0.

Therefore,
lim →

exp

0,

i.e., deflation continues and nevertheless the transversality condition is valid.
Let us next examine the dynamic stability. The characteristic equation of the dynamics
given by (A5) is0.
The partial derivatives ofandare obtained from (A5) as follows:
/

17

,

(A6)

/

,
/

,

/

in the neighborhood of the state in which.

0. Therefore,
/

/

,

(A7)

where
0 in case D,in case R.
If full employment is achieved in the steady state,

(A8)
and

0 and then from (A7)and (A8),
/

1

0.

(A9)

If unemployment continues in the steady state, which occurs only in case D, from (13) we
find0.
From (20) and (21) Φ(A10)satisfies
Φ

0.

lim →

0,

(A11)

Therefore, if

which holds e.g. when the elasticity ofis constant, from (A7), the first equation of(A8), (A10) and (A11) we obtain
0.

18

(A12)

From (A9) and (A12), in either case one of the two solutions of (A6) is positive and the otheris negative. Note that in the presence of unemployment

follows the movement ofcannot jump, as mentioned below equation (A3). Because is jumpable while, whichis not, thedynamics is saddle-path stable.
Having shown the validity of the saddle stability in the case where the steady state isreached from the region with unemployment, we next turn to the case where full employment
has already been realized. In this caseflexibly follows the movement of , as mentioned

below equation (A3). Then the firm produces

and always equals it.
(A13)

Substituting (A13) into (5) and (8) and rearranging the result gives,.

Therefore, it has the same property as the standard money-in-the-utility-function model, as
discussed in Blanchard and Fischer (1989).

jumps to the steady-state level given by (14)and the steady state is immediately reached.

19

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