Discussion Paper No. 946

GROWTH, SECULAR STAGNATION
AND WEALTH PREFERENCE

Yoshiyasu Ono

September 2015

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

September 2015

Growth, Secular Stagnation and Wealth Preference*by
Yoshiyasu Ono†

Abstract
In 1960s-1980s Japan enjoyed high economic growth. In the early 1990s, however, the growthrate drastically declined and thereafter Japan has been suffering secular stagnation. This paper
proposes a dynamic macroeconomic model that can consistently explain such a drastic change
in economic performance. Wealth preference plays an important role. In the early stage
consumption grows at the same pace as productivity increases. Once consumption reaches acertain level, however, it deviates from the full-employment level and aggregate demand
deficiency appears. After that the economic growth rate asymptotically approaches zero even if
productivity keeps on increasing, and secular stagnation arises.

JEL: E32, E13
Keywords: Long-run stagnation, aggregate demand deficiency, wealth preference, fair wages,
economic growth.

*This research is financially supported by the Grants-in-Aid for Scientific Research (#15H05728), the Japan
Society for the Promotion of Science (JSPS). The author thanks seminar participants at Fifth Graz Schumpeter
Summer School held in the University of Graz (July 2015) and Norges Bank (August 2015) for their valuablecomments.
†

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

1. Introduction
Japan has been suffering secular stagnation for more than two decades since stock-pricebubbles burst in the early 1990s. Before and after the bubble burst Japan’s growth rate
drastically decreased, the unemployment rate jumped up, and price and wage indices stopped
increasing but rather declined although the Bank of Japan vastly expanded money supply, as
shown in Figure 1 (Japan). Surprisingly, Japan’s nominal GDP in 2014 is the same as that in1992 (488 trillion yen). Other developed countries, such as the USA and the EU, have also
faced serious business situations since the Lehman shock of 2008. They are now worrying if
they would fall into such long-run stagnation as Japan’s. Therefore, the central banks
significantly expand money supply so that business activity should recover. However, theirGDPs have not so much responded to the monetary expansions as expected. See Figure 2 forcase of the USA.
In the literature such macroeconomic downturns are regarded as either declines in
productivity or short-run recessions that arise during the process of price adjustment. For
example, Kehoe and Prescott (2002), Hayashi and Prescott (2002) and McGratten and Prescott(2012) mention that recent stagnations are mainly due to decreases in productivity. DSGE
models (e.g., Yun, 1996; Erceg et al., 2000; Smets and Wouters 2003, 2005, 2007; Christiano
et al., 2005) apply the Calvo pricing (1983) to analyze macroeconomic fluctuations. Therefore,
the fluctuations are due to neither aggregate demand deficiency nor involuntary unemployment
but to monopolistic pricing of firms. Krugman (1989) adopts a period model and assumes pricerigidity within a period.1 Aggregate demand deficiency appears only in the period in which an
unexpected shock arises; it disappears in the long run. There are also researchers who consider

1

Eggertsson and Krugman (2012) consider a period model with sluggishness of some firms’ price setting inwhich there are borrowers that face credit constraints and savers that do not. A sudden decline in the credit
constraint makes borrowers reduce consumption and leads to aggregate demand deficiency.

1

a credit constraint that leads to misallocation of capital and decreases in future productivity(Kiyotaki and Moore, 1997; Hall, 2011). In this setting stagnation is interpreted to arise due toa decline in productivity.
While those approaches are useful to analyze each relevant situation, however, they may
not fit the case of secular stagnation such as Japan’s. In fact, it may be difficult to accept theview that the price adjustment process takes more than two decades or that in the early 1990s
Japan’s productivity suddenly stopped increasing and since then has never improved for morethan two decades. In the IMF annual conference of 2013 Summers (2013) expressed a similar
concern and criticized too much reliance on the DSGE approach in solving economic crises. He
emphasized the need for researchers to work on secular stagnation rather than temporary
business fluctuations. This paper shares this view and presents a model that can treat atransition from the phase of growth with full employment to that of secular stagnation withaggregate demand deficiency.
A model of stagnation that occurs in the steady state has firstly been proposed by Ono(1994, 2001) and extended by Ono and Ishida (2014). The crux of these models is thesustainment of liquidity (or wealth) preference. 2 Without it, if there is aggregate demand
deficiency, wages and prices decline, real wealth increases, and hence the relative preference
for consumption over wealth holding increases, stimulating consumption enough to recover2This type of model has been extended in various directions. For example, Matsuzaki (2003) and Hashimoto(2004) examine the effect of redistribution policy on aggregate demand in the present type of stagnation. Johdo(2006) analyzes the effect of R&D subsidies on aggregate demand and employment. Rodriguez-Arana (2007)
examines the dynamic path with public deficit in the present stagnation case and compares it with that in the casewhere full employment prevails. Ono (2006, 2014) applies the stagnation model into a two-country framework
and analyzes the international spillover effects of various policies. Johdo and Hashimoto (2009) consider the roleof FDI in the two-country model with the present stagnation mechanism. Using the implications of this model Ono(2010) analyzes Japan's long-run stagnation and economic policies. Murota and Ono (2011) applies the concept ofstatus preference to explain the insatiability of wealth preference and show that aggregate demand deficiency
arises in the steady state. Hashimoto and Ono (2011) examine the effects of child allowances on the population
growth and aggregate demand in the stagnation steady state. Murota and Ono (2012) consider a preference fordeposit holdings in this setting and find zero nominal interest rates and excess reserves held by commercial banks
to appear in the persistent stagnation. Apart from this line of research, Michaillat and Saez (2014) show that
aggregate demand deficiency arises in the steady state if the marginal utility of wealth holding is constant.

2

full employment. However, if the marginal utility of wealth holding stays positive whereas thatof consumption continues to decline as it increases, consumption may not reach a level large
enough to fill the deficiency of aggregate demand and then secular stagnation occurs. In thosemodels productivity is assumed to be constant and aggregate demand deficiency always
appears along the dynamic equilibrium path. In practice, however, Japan enjoyed high growthwith full employment in the past and then has fallen into long-run stagnation. Moreover, thissequence is reverse to the Keynesian implication that aggregate demand deficiency occurs inthe short run but eventually disappears in the long run.
This paper considers wealth preference in an economy with persistent productivity
growth. Along the dynamic equilibrium path the economy initially achieves full employment
and grows at the same pace as the productivity growth, but eventually falls into secularstagnation with little (or even zero) growth, steady deflation and underemployment, although
the productivity keeps on increasing and the money stock continues to expand. Thisimplication may be more in conformity with Japan’s experience, shown in Figure 1, than theimplications of the conventional models.

2. The modelA representative household has the following utility functional:∞∫0 [𝑢(𝑐) + 𝑣(𝑚) + 𝜑(𝑎)] exp(−𝜌𝑡) 𝑑𝑡,
where 𝑢(𝑐) is utility of consumption 𝑐, 𝑣(𝑚) is preference for money 𝑚 due to transaction
motive and 𝜑(𝑎) is preference for holding wealth 𝑎. The household maximizes it subject to theflow budget equation and the stock budget constraint:𝑎̇ = 𝑟𝑎 + 𝑤𝑥 − 𝑐 − 𝑅𝑚 − 𝑧, 𝑎 = 𝑚 + 𝑏,(1)

where 𝑎 is real total assets, 𝑏 is interest-bearing asset whose real and nominal interest rates are3

𝑟 and 𝑅 respectively, 𝑤 is the real wage, 𝑥 is the realized labor supply and 𝑧 is the lump-sum
tax-cum-subsidy. The labor endowment is 1 but 𝑥 may be less than 1 because there may be
unemployment.
𝑥  1.
The Hamiltonian function of this problem is
𝐻 = 𝑢(𝑐) + 𝑣(𝑚) + 𝜑(𝑎) + 𝜆(𝑟𝑎 + 𝑤𝑥 − 𝑐 − 𝑅𝑚 − 𝑧),
and the first-order optimal conditions are
𝜆 = 𝑢′(𝑐),𝜆𝑅 = 𝑣′(𝑚),𝜆̇ = (𝜌 − 𝑟)𝜆 − 𝜑′(𝑎).
They are summarized as𝜑′ (𝑎)

𝑐̇𝑣 ′ (𝑚)𝜑 ′ (𝑎)𝜌 + 𝜋 + 𝜂 𝑐 = 𝑅 + 𝑢′ (𝑐) = 𝑢′ (𝑐) + 𝑢′ (𝑐) ,(2)where 𝜋(= 𝑅 − 𝑟) is the inflation rate and 𝜂 = −𝑢′′𝑐/𝑢′. The transversality condition is
lim𝑡→∞ 𝑢′ (𝑐)𝑎 exp(−𝜌𝑡) = 0.

(3)

The monetary authority expands money supply 𝑀 at an exogenous rate 𝜇,
𝑀̇
𝑀

= 𝜇,

and hence 𝑚(= 𝑀/𝑃) satisfies𝑚̇𝑚= 𝜇 − 𝜋.

(4)

The government budget equation is
𝜇𝑚 + 𝑧 + 𝑏̇ = 𝑟𝑏.
Note that the stock of government bond 𝑏 equals the household’s interest-bearing asset
because the firm value is zero under linear technology and competitive behavior, which will bementioned later in this section. The fiscal authority controls the bond stock to be eventuallyfinite and thus
4

lim𝑡→∞ 𝑏𝑡 = 𝑏̅ ≪ ∞.

(5)

The representative firm is competitive and utilizes labor to produce commodities withproductivity 𝑦 that grows at an exogenous rate 𝛿:𝑦 = 𝑦0 exp(𝛿𝑡),(6)where 𝑦0 is the initial level of the productivity. Because the labor endowment is normalized to1, productivity 𝑦 also represents the full-employment output while realized output 𝑦𝑥 may beless than 𝑦. Commodity price adjustment is perfect and thus always𝑐 = 𝑦𝑥.

(7)

In order to accommodate aggregate demand shortages, sluggish wage adjustment isassumed. In the recent new Keynesian literature the standard setting of price and wage
adjustment is that of Calvo (1983). However, it does not suit the present analysis because it
never allows aggregate demand shortages to arise. Aggregate demand shortages are explicitly
analyzed by Krugman (1998) in a discrete-time setting. However, it cannot treat long-run
stagnation because the length of each period is exogenously given and a demand shortage
appears only in the period in which an exogenous shock occurs. Therefore, in the following
analysis a dynamic version of the Akerlof-type fair wage model (Akerlof, 1982; Akerlof andYellen, 1990), proposed by Ono and Ishida (2014), is adopted because it can deal with
unemployment as well as full employment during the adjustment process and in the steadystate.
There are three kinds of workers, employed, unemployed and newly hired ones. The
employed randomly separate from the current job at an exogenously given Poison rate 𝛼.3
Therefore, employment 𝑥 follows
𝑥̇ = −𝛼𝑥 + 𝜒,
3

(8)

The Calvo model (1983) also assumes an exogenous Poison rate with which a firm (or a labor union) has theopportunity of revising prices (or wages) at each time point. The present 𝛼 instead represents the opportunity ofrevising the fair wage that appears when some of the incumbent workers leave the job, as mentioned later.

5where 𝜒 is the number of the newly hired per unit time. While workers are employed, they
conceive fair wages in mind on the basis of their past wages, their fellow workers’ fair wages
and the unemployment level of the society. Formally, at the end of the previous period (𝑡 − ∆𝑡)each employed worker conceives the rightful wage 𝜈(𝑡 − ∆𝑡), which is the wage that he or she
believes fair if full employment prevails. Next, he or she calculates the average of the rightful
wages of the employed, which are the same to each other, and the income of the unemployed,
which is assumed to be zero. The average is taken as the fair wage 𝑊𝐹 (𝑡) at time 𝑡.
Because the number of the employed and that of the unemployed are respectively𝑥(𝑡 − ∆𝑡) and 1 − 𝑥(𝑡 − ∆𝑡), the employed calculate the rightful wage 𝜈(𝑡 − ∆𝑡) so that it
satisfies𝜈(𝑡 − ∆𝑡)𝑥(𝑡 − ∆𝑡) + 0 × [1 − 𝑥(𝑡 − ∆𝑡)] = 𝑊𝐹 (𝑡 − ∆𝑡),

(9)

at the timing denoted by (A) in figure 3. The newly hired have no preconception about the fairwage or the rightful wage and simply follow the fair wage of the employed. When theemployed calculate the fair wage 𝑊𝐹 (𝑡) at time 𝑡, namely at the timing (B) in figure 3, thenumber of the employed is 𝑥(𝑡 − ∆𝑡)(1 − 𝛼∆𝑡) and the total number of people that they care is
1 − 𝜒(𝑡)∆𝑡 because the number of the newly hired is 𝜒(𝑡)∆𝑡. Therefore, given that the
unemployed earn no income, the fair wage 𝑊𝐹 (𝑡) is
𝑊𝐹 (𝑡) =𝜈(𝑡−∆𝑡)𝑥(𝑡−∆𝑡)(1−𝛼∆𝑡)
1−𝜒(𝑡)∆𝑡.

From (9) and the above equation, one obtains
𝑊𝐹 (𝑡)−𝑊𝐹 (𝑡−∆𝑡)∆𝑡= 𝜒(𝑡)𝑊𝐹 (𝑡) − 𝛼𝑊𝐹 (𝑡 − ∆𝑡).

Reducing ∆𝑡 to zero yields the dynamics of 𝑊𝐹 :𝑊̇𝐹
𝑊𝐹= 𝜒 − 𝛼,6(10)implying that the fair wage is updated at the replacement pace of employment.4 As 𝛼 is larger,
the declining speed of 𝑊𝐹 is higher. Thus, the Poison rate 𝛼 is a proxy of flexibility of the fairwage.
If there is unemployment, the firm will set wage 𝑊 equal to the fair wage 𝑊𝐹 because 𝑊𝐹
is the lowest wage under which the employees properly work. The commodity price 𝑃 isalways equalized to 𝑊𝐹 ⁄𝑦 because there is no commodity supply if 𝑃 < 𝑊𝐹 ⁄𝑦 and excess
commodity supply if 𝑃 > 𝑊𝐹 ⁄𝑦. If full employment prevails, in contrast, the firm tries to pickout workers from rival firms to expand the market share by increasing 𝑊 from 𝑊𝐹 so long as
the marginal profits are positive (i.e., 𝑃𝑦 > 𝑊). Therefore, 𝑊 is set higher than the fair wage
𝑊𝐹 that follows (10), and is instantaneously equalized to 𝑃𝑦. Once 𝑊 is thus determined, 𝑊𝐹
is ex post updated to equal 𝑊.
In sum, 𝑃 follows the movement of the fair wage 𝑊𝐹 when there is unemployment
whereas 𝑊 follows the movement of 𝑃 regardless of 𝑊𝐹 when full employment maintains.
Anyway one has𝑊

𝑤 = 𝑃 = 𝑦,(11)

under linear technology and competitive behavior. Consequently, the firm earns no profit andthus the interest-bearing assets indeed consist of only the government bond. From (6) and (11),one immediately obtains
𝑊̇

𝜋 + δ = 𝑊.

(12)

Given the firm behavior mentioned above, the dynamic equations of 𝑐 and 𝑚 are obtained.
If full employment prevails (𝑥 = 1), from (7),𝑦 = 𝑐.

This equation must be 𝑊̇𝐹 /𝑊𝐹 = 𝜒/𝐿0 − 𝛼 , where 𝐿0 is the total population. However, because 𝐿0 isnormalized to unity, (10) is valid.
47Then, from (2) and (6) 𝑃 moves so that it satisfies𝜋=𝑣 ′ (𝑚)+𝜑′ (𝑎)𝑢′ (𝑐)− (𝜌 + 𝜂𝛿),

(13)

and 𝑊 follows the movement of 𝑃 so as to satisfy (11). Therefore, from (12)𝑊̇=𝑊𝑣 ′ (𝑚)+𝜑 ′ (𝑎)𝑢′ (𝑐)− 𝜌 + (1 − 𝜂)𝛿.

(14)

When 𝑥 = 1, from (8) 𝜒 = 𝛼 (i.e., the number of job separation equals that of the newly hired)
and thus from (10) 𝑊̇𝐹 = 0. If 𝑊 that follows (14) is lower than 𝑊𝐹 , 𝑊 is set equal to 𝑊𝐹
because otherwise the employees do not properly work, and aggregate demand deficiency
appears. Therefore, variables 𝑚 , 𝑐 and 𝑎 that satisfy (14) are valid only when (14) is
non-negative. Substituting 𝜋 given by (13) into (4) yields 𝑚̇. These results are summarized as
𝑊̇

≥0↔
𝑊
𝑐̇

𝑣 ′ (𝑚)+𝜑′ (𝑎)𝑢′ (𝑐)
𝑦̇= 𝑦 = δ,𝑐≥ 𝜌 − (1 − 𝜂)𝛿:𝑚̇

= 𝜇 + 𝜌 + 𝜂𝛿 −𝑚𝑣 ′ (𝑚)+𝜑 ′ (𝑎)𝑣 ′ (𝑚)+𝜑 ′ (𝑎)𝑢′ (𝑐)𝑢′ (𝑐), 𝜋=− (𝜌 + 𝜂𝛿).

(15)

When 𝑊̇ /𝑊 given by (14) is negative, there is aggregate demand deficiency (i.e., 𝑥 < 1).
In this case 𝑊 is set equal to 𝑊𝐹 of which the dynamics is (10) because 𝑊 cannot be lower
than 𝑊𝐹 . Substituting 𝑥 and 𝑥̇ obtained from (7), which are
𝑐̇

𝑐

𝑐

𝑥̇ = (𝑐 − 𝛿) 𝑦 , 𝑥 = 𝑦 ,into (8) gives 𝜒, the number of the newly hired. Applying this 𝜒 to (10) and utilizing (12) inwhich 𝑊 = 𝑊𝐹 yields 𝜋:
𝑊̇

𝑐̇

𝑐

𝑐

𝜋 = 𝑊 − 𝛿 = 𝑥̇ + 𝛼(𝑥 − 1) − 𝛿 = 𝑦 + 𝛼 (𝑦 − 1) − 𝛿 (𝑦 + 1) < 0.

(16)

𝜋 is negative because 𝑊̇ /𝑊 < 0 in the present case. Substituting this 𝜋 into (2) andrearranging the result gives the dynamics of 𝑐. Because 𝜋 is negative, the dynamics of 𝑚
represented by (4) is always positive. The results are
𝑊̇

<0↔
𝑊

𝑣 ′ (𝑚)+𝜑′ (𝑎)𝑢′ (𝑐)< 𝜌 − (1 − 𝜂)𝛿:

8

𝑦̇

𝑊̇

𝜋 = 𝑊 − 𝛿 < 0,𝑣 ′ (𝑚)+𝜑′ (𝑎)
𝑐
𝑐
−
𝜌
−
𝛼
(
−
1)
+
𝛿
(
+ 1),
′
𝑢 (𝑐)
𝑦
𝑦

𝑐

𝑐̇

𝑚̇

= 𝜇 − 𝜋 > 0.

= δ > 0, (𝜂 + 𝑦) 𝑐 =
𝑦
𝑚

(17)

Note that when 𝑊̇ /𝑊 equals zero in (14), the values of the equations in (15) and (17) are
𝑐̇
𝑐

= δ,

𝑚̇
𝑚

= 𝜇 + 𝛿,𝜋 = −𝛿,i.e., the dynamics in the two cases smoothly connect to each other.

3. Transition from growth to secular stagnation
In order that the dynamics is tractable the following functions are assumed:𝑢(𝑐) = ln 𝑐, 𝑣(𝑚) = 𝛾 ln(𝑚), 𝜑(𝑎) = 𝛽𝑎.

(18)With these functions the marginal utility of 𝑚 (= 𝛾/𝑚) reaches zero while that of wealth (= 𝛽)remains positive.5 The property that the wealth preference remains positive plays the crucialrole in making the economy switches from full-employment growth to secular stagnation. Thissection obtains the phase diagram of the dynamics of 𝑐 and 𝑚 when 𝛽 is positive. Along the
dynamic equilibrium path the economy grows at the same pace as the productivity growth
before falling into secular stagnation. In the next section the case where there is no wealthpreference (𝛽 = 0) is considered and persistent growth with full employment is shown to berealized.
With the functions given in (18) the 𝑊̇ = 0 curve obtained from (14) isγ
𝑊̇ = 0: 𝑐 = 𝜌/ (𝑚 + 𝛽),(19)

and the dynamic equations in (15) and (17) are5

Murota and Ono (2011) show that status preference with respect to money holding makes the marginal utilityof mony holding constant. The present assumption applies this property to wealth holding. Michaillat and Saez(2014) consider a model in which net household wealth is always zero and hence the marginal utility of wealthholding is positive and constant. There is also an empirical research that supports the existence of such wealthpreference. Using aggregate quarterly data in Japan and the Japanese survey data called NIKKEI RADAR, Ono,
Ogawa and Yoshida (2004) empirically find that the preference for financial asset holding does not disappear as
the money holding increases using both parametric and non-parametric methods.

9

𝑊̇

γ

≥ 0 ↔ 𝑐 ≥ 𝜌/ (𝑚 + 𝛽):
𝑊
𝑐̇
𝑐
𝑊̇
𝑊

𝑦̇

= 𝑦 = δ,

𝑚̇
𝑚

γ

γ

= 𝜇 + 𝜌 + 𝛿 − (𝑚 + 𝛽) 𝑐, 𝜋 = (𝑚 + 𝛽) 𝑐 − (𝜌 + 𝛿);

(20)

γ

< 0 ↔ 𝑐 < 𝜌/ (𝑚 + 𝛽):
𝑦̇

𝑐

𝑐̇

γ

= δ > 0, (1 + 𝑦) 𝑐 = (𝑚 + 𝛽 +
𝑦
𝑚̇
𝑚

𝛿−𝛼
𝑦

) 𝑐 − (𝜌 − 𝛼 − 𝛿),= 𝜇 − 𝜋 > 0, 𝜋 < 0.

(21)

Figure 4 illustrates the 𝑊̇ = 0 curve given by (19) and the phase diagram of 𝑚 and 𝑐. In
the area above the 𝑊̇ = 0 curve full employment prevails and thus (20) is valid. From (20), theboundary curve of 𝑚 satisfiesγγ𝑚̇ = 0: 𝑐 = (𝜇 + 𝜌 + 𝛿)/ (𝑚 + 𝛽) > 𝜌/ (𝑚 + 𝛽),i.e., this curve is located above the 𝑊̇ = 0 curve given by (19). The dashed dynamic path
represents the one that is tangent to the 𝑊̇ = 0 curve. Along any path located above it or above
the right-hand part of the 𝑊̇ = 0 curve from point D, 𝑚 will eventually reach zero within afinite time span. Therefore, any feasible path must start in the area between the dashed line andthe left-hand part of the 𝑊̇ = 0 curve from D. Any path starting from that area eventuallyreaches the 𝑊̇ = 0 curve and enters the area with aggregate demand deficiency. In the figure
point B denotes the point at which the path reaches 𝑊̇ = 0 curve. Obviously, there is acontinuum of such paths and one of them must be chosen, as mentioned below.
Once the path enters the area with aggregate demand deficiency, it follows the dynamic
equations in (21) and the 𝑐̇ = 0 curve isγ

𝑐̇ = 0: 𝑐 = (𝜌 − 𝛼 − 𝛿)/ (𝑚 + 𝛽 +

10

𝛿−𝛼
𝑦

).

(22)

This curve in the cases where 𝛿 > 𝛼 and 𝛿 < 𝛼 is respectively illustrated in figures 4 and 5. Inorder for this curve to be located within the area where 𝑐 and 𝑚 are both positive, it must be
satisfied that6
𝜌 − 𝛼 − 𝛿 > 0.
Otherwise, all paths in this area will eventually enter the area with full employment and 𝑚 willreach zero within a finite time span –i.e., there is no feasible path.
In figure 4 (the case where 𝛿 > 𝛼) the 𝑐̇ = 0 curve shifts upward as 𝑦 increases while infigure 5 (the case where 𝛿 < 𝛼) it shifts downward as 𝑦 increases. In either case, when a path
reaches the 𝑊̇ = 0 curve represented by (19), 𝑐, 𝑦 and 𝑚 satisfyγ

𝑐 = 𝑦 = 𝜌/ (𝑚 + 𝛽).
By applying this 𝑦 as a function of 𝑚 to the 𝑐̇ = 0 curve given by (22), one finds that the levelof 𝑐 for the same 𝑚 along the 𝑐̇ = 0 curve satisfies
𝜌−𝛼−𝛿

γ

γ

𝑐 = (𝜌−𝛼+𝛿) {𝜌/ (𝑚 + 𝛽)} < 𝜌/ (𝑚 + 𝛽),where the right-hand side is the 𝑊̇ = 0 curve given by (19). Therefore, the 𝑊̇ = 0 curve islocated above the 𝑐̇ = 0 curve when the path reaches the area with aggregate demand
deficiency. It is also located above the 𝑐̇ = 0 curve for the case where 𝑦 = ∞ because from(22) 𝑐 on this curve satisfiesγγ

𝑐 = (𝜌 − 𝛼 − 𝛿)/ (𝑚 + 𝛽) < 𝜌/ (𝑚 + 𝛽).
In the area below the 𝑊̇ = 0 curve, there is only one path along which 𝑐 is finite andpositive, which is BE in figures 4 and 5. The mathematical proof of the saddle-path stability is

6

Lawrence (1991) estimates the subjective discount rate to be from 4% to 21% and Andersen et al. (2008)
estimates it to be 10.1%. The average of the annual GDP growth rates of OECD member countries in 2005-2013
was 1.34% (World Bank national accounts data, and OECD National Accounts data files, 2015).

11

given in the appendix. From the dynamic equation of 𝑐 in (21), the steady-state level of 𝑐 on
this path is
𝑐=

𝜌−𝛼−𝛿
𝛽

,

(23)

While productivity 𝑦 keeps on expanding. Thus, from (16) and (21) 𝑚̇/𝑚 satisfies𝑚̇

lim 𝑚 = 𝜇 + 𝛿 + 𝛼.

𝑡→∞Using (5), (18), (23) and the above property one finds that the transversality condition (3) in thepresent case,lim𝑡→∞𝑚+𝑏̅𝑐𝛽

exp(−𝜌𝑡) = 𝜌−𝛼−𝛿 lim𝑡→∞ 𝑚 exp(−𝜌𝑡) = 0,

is valid if𝜌 − 𝛼 − 𝛿 > 𝜇.

(24)

Along any other path than BE, 𝑐 either approaches zero, violating the transversality condition,or continues to expand and then 𝑚 reaches zero within a finite time span, as illustrated in figure
4. Therefore, BE is the unique equilibrium path and the starting point B is determined. Once B
is determined, the path that reaches B in the area where 𝑊̇ > 0 is uniquely chosen and thewhole shape of the dynamic equilibrium path is determined. Note that if 𝜇 is so high as toviolate (24), the saddle path does not satisfy the transversality condition and then no dynamic
equilibrium path exists.
As shown in the two figures, along the equilibrium path consumption 𝑐 grows at the samepace as productivity 𝑦 does when 𝑦 is small. The economy enjoys economic growth with full
employment in the developing stage with low productivity. As 𝑦 reaches the level that satisfiesγ

𝑦 = 𝜌/ (𝑚 + 𝛽),which is point B in figures 4 and 5, consumption 𝑐 starts to deviate from full-employment
output 𝑦 and the economy faces aggregate demand deficiency. Thereafter, 𝑐 stays lower than

12the value given by (23) even though productivity 𝑦 continues to increase. Thus, once theeconomy reaches the matured stage with high productivity, it suffers secular stagnation. Wageand price deflations continue, making 𝑚 expand and 𝑣 ′ (𝑚) equal zero eventually, and thenfrom (2) the nominal interest rate 𝑅 is𝑣 ′ (𝑚)𝑅 = 𝑢′ (𝑐) = 0,i.e., zero interest rates appear.
The steady-state level of consumption given by (23) satisfies𝛿 ↑, 𝛽 ↑, 𝛼 ↑ ⇒ 𝑐 ↓.
Higher productivity growth, greater wealth preference and more frequent job separation
worsen stagnation, representing the paradoxes of toil, thrift and flexibility, respectively. In thepresence of aggregate demand deficiency higher productivity enlarges the deflationary gap andmakes deflation more serious, which harms the incentive to consume (toil). Greater wealthpreference directly reduces the incentive to consume (thrift). More frequent job separation
raises the wage adjustment speed and thus worsens deflation, which also harms the incentive toconsume (flexibility). Therefore, they decrease aggregate demand and make the stagnation
more serious.

4. Wealth preference and secular stagnation
Secular stagnation with aggregate demand deficiency arises due to the wealth preference
𝛽𝑎 . While the marginal utility of wealth preference is 𝛽(> 0) , the marginal utility of
consumption 𝑐 decreases if 𝑐 increases at the same pace as productivity 𝑦 does. Eventually,
people want to accumulate wealth rather than to consume so much as to maintain full
employment, and aggregate demand deficiency arises. Because the wealth preference remains

13

to be 𝛽 , this situation is unchanged and the economy falls into secular stagnation withaggregate demand deficiency.
Without the wealth preference, full employment prevails along the dynamic equilibrium
path. To show this, let us obtain the dynamic path in the case where 𝛽 = 0. From (20) in which
𝛽 = 0, 𝑠(= 𝑐/𝑚) satisfies
𝑠̇
𝑠

𝑐̇

𝑚̇

𝜌+𝜇

(> 𝛾 ),(= 𝑐 − 𝑚) = 𝛾𝑠 − (𝜌 + 𝜇).

Therefore, the equilibrium path is𝑐𝑠 (= 𝑚) =𝛾

𝜌𝑐 = 𝑦 = 𝑦0 exp(𝛿𝑡).

(25)

From (19) in the case where 𝛽 = 0, one finds that𝑐𝑊̇ > 0 if

𝜌

𝑚

>𝛾,

and hence the equilibrium path represented by (25) is indeed located in the area with full
employment. (25) implies that 𝑐 and 𝑚 expand at the same pace as 𝑦 does and full
employment continues to hold –i.e., aggregate demand deficiency does not arise. Applying
𝑐/𝑚 given by (25) and 𝛽 = 0 to 𝜋 in (20) yields
𝜋 = 𝜇 − 𝛿.
From (6), (18) and (25), the transversality condition (3) is valid because
lim𝑡→∞

𝑚+𝑏̅
𝑐

𝛾

𝑏̅

exp(−𝜌𝑡) = lim𝑡→∞ (𝜌+𝜇 + 𝑦) exp(−𝜌𝑡) = 0.

When 𝛽 = 0, the 𝑚̇ = 0 curve obtained from (20) is a straight line that starts from O in
figures 4 and 5:
𝑐=(

𝜇+𝜌+𝛿
𝛾

) 𝑚,

implying that there is no upper bound of 𝑐 on the curve. Therefore, there is a dynamic path that
does not intersect the 𝑚̇ = 0 curve, which is given by (25). However, if 𝛽 > 0, no matter how
small it is, there is an upper bound of 𝑐 on the 𝑚̇ = 0 curve, as shown in figure 4. A dynamic

14

path along which full employment prevails intersects the 𝑚̇ = 0 curve, and thereafter 𝑚
declines and reaches zero within a finite time span; that is, the path is infeasible. Therefore, thedynamic equilibrium path must intersect the 𝑊̇ = 0 curve and enter the area with aggregate
demand deficiency, resulting in secular stagnation.

5. Conclusions
Many developed countries, such as Japan, the EU and the USA, have recently experienced
drastic downturns of economic growth and lapsed into secular stagnation. Conventional
macroeconomic approaches successfully deal with short-run recessions but cannot necessarily
treat such a transition from economic growth to secular stagnation consistently in one unified
framework. This paper considers an economy with productivity growth and wealth preference,and finds that there is a unique dynamic equilibrium path along which the economy initiallyenjoys steady growth but eventually falls into secular stagnation with aggregate demand
deficiency.
Along the path consumption expands at the same speed as productivity does in the earlystage. As productivity increases and consumption follows it, the marginal utility of
consumption gradually declines and eventually the desire for consumption up to thefull-employment level is dominated by the desire for wealth holding. Then, aggregated demand
deficiency arises. As long as the wealth preference exists, this situation is unchanged eventhough productivity keeps on growing and the money stock continues to expand, and theeconomy suffers secular stagnation. The nominal interest rate approaches zero. This process
may be in conformity with the experiences of developed countries.

15Appendix: local stability
The local stability of the dynamics in the case where 𝑊̇ < 0, given by (21), around thesteady state E is examined. Along the dynamics 𝑚 and 𝑦 continue to expand while 𝑐 stays to
be finite at the level given by (23). Replacing 𝑚 and 𝑦 by 𝑞(= 1/𝑚) and ℎ(= 1/𝑦)
respectively changes the dynamic equations in (21) to𝑐𝑐̇ = ((γ𝑞 + 𝛽 + (𝛿 − 𝛼)ℎ)𝑐 − (𝜌 − 𝛼 − 𝛿)) 1+𝑐ℎ,𝑞̇ = (𝑐ℎ((𝛾𝑞+𝛽)𝑐−𝜌)+𝛼(𝑐ℎ−1)(1+𝑐ℎ)− (𝜇 + 𝛿)) 𝑞,ℎ̇ = −δℎ.
Around the steady state E:(𝑐, 𝑞, ℎ) = (

𝜌−𝛼−𝛿
𝛽

, 0, 0),the characteristic function is𝛽𝑐 − 𝜎| 00𝑐2𝛾−(𝛼 + 𝛿 + 𝜇) − 𝜎
0(𝛿 − 𝛼)𝑐 2| = 0.
0−𝛿 − 𝜎This yields one positive and two negative roots for 𝜎 (i.e., eigenvalues). 𝑐 is jumpable, ℎ (orequivalently 𝑦) is historically given and 𝑞 (or equivalently 𝑚(= 𝑀/𝑃)) cannot jump because
𝑊 is set equal to 𝑊𝐹 and 𝑃 is chosen to satisfy (11) in the area where 𝑊̇ < 0. Therefore, in thisarea the dynamic path is uniquely determined, which is BE in figures 4 and 5.
Among a continuum of paths in the area where 𝑊̇ > 0 the path that reaches theintersection point of the saddle path obtained above and the 𝑊̇ = 0 curve, which is B, ischosen. Because 𝑃 is jumpable when 𝑊̇ > 0 while 𝑦 is historically given, the initial point of
the path in this area is determined, and the whole trajectory uniquely obtains.

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Real GDP (trillion yen, 2005 price)
600.0

2000-2014

500.0

90’s

400.0

80’s

300.0

1970’s

200.0

100.0

0.0

0.0

50.0

100.0

150.0

200.0

Real monetary base (trillion yen, 2005 price)

Figure 1: Money supply and GDP in Japan(Cabinet Office, the Japanese Government; Bank of Japan)

20

250.0

Real GDP (billion dollars, 2010 price)
18000.0

2000~2014

2008

16000.014000.012000.0
10000.090’s

8000.080’s
6000.01970’s4000.02000.0
0.0
0

500

1000

1500

2000

2500

3000

Real monetary base (billion dollars, 2010 price)

Figure 2: Money supply and GDP in the USA(Bureau of Economic Analysis; Fed, St. Louis)

21

3500

𝑡

𝑡 − ∆𝑡
𝑊𝐹 (𝑡 − ∆𝑡)(A)

𝑊𝐹 (𝑡)

𝜈(𝑡 − ∆𝑡)

𝜈(𝑡)

(B)

Figure 3: Determination timing of the fair and rightful wages

22

𝑐
𝜇+𝜌+𝛿

𝑚̇ = 0𝛽

full employment (𝑐 = 𝑦)𝜌𝑊̇ = 0𝛽stagnation (𝑐 < 𝑦)
D
𝜌−𝛼−𝛿

E

𝛽

𝑦→∞

𝜌

𝑦↑

B

𝛾/𝑚1 +𝛽𝑦=𝑦0O𝑐̇ = 0𝜌
𝛾/𝑚1 +𝛽

A

𝑚0

𝑚

𝑚1
Figure 4: Dynamic equilibrium path (𝛿 > 𝛼)

23

𝑐
𝜌

𝑊̇ = 0𝛽

full employment(𝑐 = 𝑦)𝜌−𝛼−𝛿stagnation(𝑐 < 𝑦)
𝑦↑

𝛽
𝜌

B

A

O

𝑚0

𝜌
𝛾/𝑚1 +𝛽𝑐̇ = 0E𝑦→∞𝛾/𝑚1 +𝛽

𝑦0

𝑦=

𝑚

𝑚1

Figure 5: Dynamic equilibrium path (𝛿 < 𝛼)24𝑐𝑚̇ = 0𝑊̇ = 0

𝜇+𝜌+𝛿
γ
𝜇+𝜌
γ
𝜌
γ

𝑚

O
Figure 6: Persistent growth (𝛽 = 0)25