Discussion Paper No. 1017

COUNTRY SIZE,
SPECIALIZATION PATTERNSANDSECULAR DEMAND STAGNATION*
Yoshiyasu Ono

November 2017

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

November 2017

Country size, specialization patterns
and secular demand stagnation *byYoshiyasu Ono †Abstract
Using a dynamic two-country two-commodity Ricardian model where preference for money(or wealth) leads to aggregate demand deficiency, this paper examines the relationship between
the two countries’ relative population size and their specialization patterns, employment andconsumption. When the countries have similar population sizes, they specialize in respective
commodities with comparative advantage. In this case a larger foreign, or a smaller home,
population raises the relative price of the home commodity. It raises home real income andconsumption per capita if full employment prevails in the home country. If unemployment
appears, however, home employment and consumption per capita decrease.

JEL classification: F41, E24, E32
Keywords: secular demand stagnation, liquidity trap, unemployment, population,
specialization pattern.

*

The author thanks seminar participants at Osaka and Kobe Universities for their valuable comments. A partof this research was carried out while the author visited CES, Munich University. Financial supports by JSPS
KAKENHI Grant (JP15H05728 and JP22530178) and the Joint Usage/Research Center Program by the Ministryof Education, Culture, Sports, Science and Technology, Japan, are gratefully acknowledged.
†

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
This paper considers a dynamic two-country two-commodity Ricardian model withendogenous specialization patterns where secular aggregate demand stagnation can occur in
one or both countries. Using this model, it analyzes the relationship between the two countries’
relative population size and their specialization patterns, employment and consumption.
Nowadays persistent stagnation of aggregate demand and depopulation may be the two
major threats that many developed countries face. They welcome expansions of foreign
demand in hope of creating new home employment but worry about home depopulation in fearof decreasing production capacity. This is indeed the case for Japan. Japan regards
depopulation as a threat of being dominated by Asian emerging economies but favorably viewsthe emergence because it is expected to expand foreign demand for Japanese products andstimulate the Japanese economy. This paper shows that a change in the international relative
population size yields quite different effects on income and consumption per capita, dependingon whether full employment prevails or not.
In a Ricardian trade model with two commodities, if the home population is much larger
than the foreign one, the home country produces both commodities while the foreign country
specializes in the commodity with comparative advantage. 1 In this case, the international
relative price of the two commodities equals the home autarky one, which is not affected by achange in the relative population. Thus, population changes vary neither per-capita income nor
consumption in the two countries. If the two countries have similar population sizes, theyspecialize in respective commodities with comparative advantage. In this case, as the relative
foreign population is larger, the relative price approaches the foreign autarky level from the1

Ono and Shibata (2006) shows that this Ricardian property holds also in a dynamic Heckscher-Ohlin modelwith endogenous capital accumulation.

1

home autarky level, improving the home terms of trade. It expands home, and reduces foreign,
per-capita income and consumption as long as full employment prevails in both countries.
If households have insatiable preference for money (or wealth) holding, however, there isan upper bound of consumption, as proved by Ono (1994, 2001) and Ono and Ishida (2014). In
this case, an improvement in the terms of trade may increase the real value of full-employment
production so much as to exceed the upper bound of home consumption and yield aggregate
demand deficiency. Then, a further rise in the relative price of the home commodity, due to anincrease in the relative foreign population, decreases world demand for the home commodity,
reduces home employment and consumption per capita, and worsens aggregate demandstagnation.
The crux of the present analysis is aggregate demand stagnation due to preference formoney (or wealth). Analyses on stagnation have recently extended in many ways. Someeconomists attribute it to productivity declines and others to market distortions, but there are
not many that consider aggregate demand shortages. Typical productivity approaches are
Kehoe and Prescott (2002), Hayashi and Prescott (2002) and McGratten and Prescott (2012).
DSGE models take business fluctuations to be generated by monopolistic competition with the
Calvo pricing (1983) in the commodity or labor market, where no aggregate demand shortages
appear. They are Yun (1996), Erceg et al. (2000), Smets and Wouters (2003, 2005, 2007) andChristiano et al. (2005) in the case of a closed economy and Adolfson et al. (2007, 2008) in thecase of an open economy. Kiyotaki and Moore (1997), Aghion et al. (1999), Matsuyama(2007), Hall (2011), Liu and Wang (2014) and Azariadis and Kaas (2016) consider a credit
constraint that firms or investors face, and mention that the constraint distorts capital allocationand deteriorates productivity. A mechanism of unemployment that persists in a steady state ispresented by the search theory (e.g. Pissarides, 2000) but it is of the frictional type.

2

While those researches do not treat aggregate demand shortages, Krugman (1998)
proposes a period model in which aggregate demand shortages appear due to a zero lower
bound of the nominal interest rate and nominal price rigidities. 2 In this setting aggregate
demand shortages appear only in the period in which an unexpected shock arises, and disappear
in the long run.
However, the present stagnation of Japan lasts for more than two decades. It may not beplausible to consider that negative productivity shocks last more than two decades in Japan orthat Japan’s market adjustment process takes such a long time. Similar consideration may
apply to the other developed countries that have been suffering from aggregate demandstagnation since the financial crisis of 2008, such as EU and the USA.
This paper adopts yet another approach to stagnation. It considers secular stagnation of
aggregate demand due to preference of money (or wealth) holding, which arises in richcountries even without market distortions. This property was firstly presented by Ono (1994,
2001) and has recently rediscovered by Michaillat and Saez (2014) and Michau (2015).
Moreover, Ono (2006, 2014, 2017) has extended the model to an open-economy context where
specialization patterns are exogenously given. He has analyzed own and international spillover
effects of productivity improvements, fiscal expansions and trade policies on employment,
income and consumption. This paper introduces endogenous specialization patterns to thatframework and examines the relationship between the relative population size and the twocountries’ specialization patterns, employment and consumption.

2

Eggertsson and Krugman (2012) consider a period model where some firms’ price settings are sluggish andthere are borrowers who face credit constraints and savers who do not. A credit crunch makes borrowers reduce
consumption and leads to aggregate demand deficiency.

3

2. Household and Firm Behavior
Suppose that there are two countries, home and foreign, and two commodities, 1 and 2.
Their populations are respectively 𝐿𝐿 and 𝐿𝐿∗ . Variables with asterisk * imply foreign ones.
Households: Households in the two countries have the same utility functional:∞(∗)

where 𝑐𝑐𝑖𝑖

(∗)

(∗)

∫0 [𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) + 𝑣𝑣(𝑚𝑚(∗) )] exp(−𝜌𝜌𝜌𝜌) 𝑑𝑑𝑑𝑑,(1)

and 𝑚𝑚(∗) are respectively each household’s consumption of commodity 𝑖𝑖 (𝑖𝑖 =1, 2) and real money holding.

Function 𝑢𝑢�(. , . ) is homothetic and hence, once real aggregate consumption 𝑐𝑐 (∗) is given at
(∗)

(∗)

each point in time, the optimal levels of 𝑐𝑐1 and 𝑐𝑐2 satisfy
(∗)

(∗)

(∗)

(∗)

𝑢𝑢�2 (𝑐𝑐1 , 𝑐𝑐2 )/𝑢𝑢�1 (𝑐𝑐1 , 𝑐𝑐2 ) = 𝑝𝑝2 (𝜔𝜔)/𝑝𝑝1 (𝜔𝜔) = 𝜔𝜔,
(∗)

(∗)

𝑝𝑝1 (𝜔𝜔)𝑐𝑐1 = 𝛿𝛿(𝜔𝜔)𝑐𝑐 (∗) , 𝑝𝑝2 (𝜔𝜔)𝑐𝑐1 = [1 − 𝛿𝛿(𝜔𝜔)]𝑐𝑐 (∗) ,1 > 𝛿𝛿(𝜔𝜔) > 0, 𝛿𝛿′(𝜔𝜔) > 0,(2)where 𝜔𝜔 is the relative price of commodity 2 to commodity 1, 𝑝𝑝𝑖𝑖 (𝜔𝜔) is the real price ofcommodity 𝑖𝑖 (𝑖𝑖 = 1, 2), and 𝛿𝛿(𝜔𝜔) is the share of consumption expenditure on commodity 1.
The real and nominal prices satisfy
𝑃𝑃𝑃𝑃∗

𝑝𝑝1 (𝜔𝜔) = 1 = 1∗,
𝑃𝑃

𝑃𝑃

𝑃𝑃

𝑃𝑃∗

𝑝𝑝2 (𝜔𝜔) = 2 = 2∗,(∗)

where 𝑃𝑃(∗) is the general price index, 𝑃𝑃𝑖𝑖

𝑃𝑃

𝑃𝑃

𝑃𝑃

𝑃𝑃

𝜖𝜖 = 1∗ = 2∗,
𝑃𝑃1

𝑃𝑃2

(3)

is the nominal price of commodity 𝑖𝑖 measured ineach currency, and 𝜖𝜖 is the nominal exchange rate. The general price index is chosen so that achange in the relative price 𝜔𝜔 will not affect the level of 𝑢𝑢�(. , . ). Therefore, 𝑢𝑢�(. , . ) into which
(∗)

(∗)

𝑐𝑐1 and 𝑐𝑐2 given in (2) are substituted satisfies𝑢𝑢� �𝛿𝛿(𝜔𝜔)

𝑝𝑝1 (𝜔𝜔)𝑐𝑐 (∗) ,1−𝛿𝛿(𝜔𝜔) (∗)
𝛿𝛿(𝜔𝜔) 1−𝛿𝛿(𝜔𝜔)𝑐𝑐 � = 𝜙𝜙�𝑐𝑐 (∗) �𝑢𝑢� � (𝜔𝜔) , (𝜔𝜔) � ≡ 𝑢𝑢(𝑐𝑐 (∗) ),𝑝𝑝2 (𝜔𝜔)
𝑝𝑝1
𝑝𝑝2

4

𝛿𝛿(𝜔𝜔) 1−𝛿𝛿(𝜔𝜔)�� (𝜔𝜔), (𝜔𝜔) �
𝑑𝑑𝑢𝑢
𝑝𝑝
𝑝𝑝
1

𝑑𝑑𝑑𝑑

2

= 𝑢𝑢�1𝛿𝛿(𝜔𝜔)

𝑑𝑑� 𝑝𝑝 (𝜔𝜔)�
1

𝑑𝑑𝑑𝑑+ 𝑢𝑢�21−𝛿𝛿(𝜔𝜔)𝑑𝑑� 𝑝𝑝 (𝜔𝜔) �

= 0.

(4)

= 𝛿𝛿𝑝𝑝1 > 0.

(5)

2

𝑑𝑑𝑑𝑑

Because 𝑢𝑢�2 /𝑢𝑢�1 = 𝜔𝜔 = 𝑝𝑝2 (𝜔𝜔)/𝑝𝑝1 (𝜔𝜔) from (2), the second equation in (4) yields
𝑝𝑝1′ (𝜔𝜔) = −From (3) one obtains(1−𝛿𝛿)𝑝𝑝1
𝜔𝜔

of which the time derivative yields

𝑝𝑝2′ (𝜔𝜔) =

< 0,

𝛿𝛿𝑝𝑝2
𝜔𝜔

𝑃𝑃 = 𝜖𝜖𝑃𝑃∗ ,𝜖𝜖̇𝜋𝜋 = + 𝜋𝜋 ∗ ,𝜖𝜖where 𝜋𝜋 and 𝜋𝜋 ∗ are respectively the home and foreign inflation rates. From this equation andthe non-arbitrage condition between home and foreign assets:𝜖𝜖̇𝑅𝑅 = + 𝑅𝑅∗ ,𝜖𝜖where 𝑅𝑅 and 𝑅𝑅∗ respectively represent the home and foreign nominal interest rates, one finds
𝑅𝑅 − 𝜋𝜋 = 𝑟𝑟 = 𝑅𝑅∗ − 𝜋𝜋 ∗ ,(6)i.e., the real interest rate 𝑟𝑟 is internationally the same.

From the first equation in (4), the utility functional given by (1) reduces to∞∫0 �𝑢𝑢(𝑐𝑐 (∗) ) + 𝑣𝑣(𝑚𝑚(∗) )� exp(−𝜌𝜌𝜌𝜌) 𝑑𝑑𝑑𝑑,
𝑢𝑢′ > 0, 𝑢𝑢′′ < 0, 𝑣𝑣′ > 0, 𝑣𝑣′′ < 0,which is maximized subject to each flow budget equation and asset constraint:𝑎𝑎̇ (∗) = 𝑟𝑟𝑎𝑎(∗) + 𝑤𝑤 (∗) 𝑥𝑥 (∗) − 𝑐𝑐 (∗) − 𝑅𝑅 (∗) 𝑚𝑚(∗) ,𝑎𝑎(∗) = 𝑚𝑚(∗) + 𝑏𝑏 (∗) ,(7)

where 𝑤𝑤 (∗) is real wage, 𝑥𝑥 (∗) is realized employment, 𝑎𝑎(∗) is real total asset, 𝑏𝑏 (∗) is realinternational lending and borrowing, and the real interest rate 𝑟𝑟 is internationally the same, as

shown in (6). Because all firms are assumed to be competitive and utilize only labor as input

with constant-returns-to-scale technology, the firm values are zero. Therefore, 𝑎𝑎(∗) consists of5money 𝑚𝑚(∗) and international net asset 𝑏𝑏 (∗) . Each household’s labor endowment is normalized

to 1 and may be underemployed –i.e., 𝑥𝑥 (∗) can be regarded as the employment rate.
From the Hamiltonian function of the utility maximization:

𝐻𝐻 (∗) = 𝑢𝑢(𝑐𝑐 (∗) ) + 𝑣𝑣(𝑚𝑚(∗) ) + 𝜆𝜆(∗) �𝑟𝑟𝑎𝑎(∗) + 𝑤𝑤 (∗) 𝑥𝑥 (∗) − 𝑐𝑐 (∗) − 𝑅𝑅(∗) 𝑚𝑚(∗) �,

one obtains the first-order optimal conditions: 3𝜆𝜆(∗) = 𝑢𝑢′ �𝑐𝑐 (∗) �, 𝜆𝜆(∗) 𝑅𝑅(∗) = 𝑣𝑣 ′ �𝑚𝑚(∗) �,

Equations (6) and (8) yield
𝑐𝑐̇ ∗

𝑐𝑐̇

𝜂𝜂 ∗ ∗ = 𝜂𝜂 = 𝑟𝑟 − 𝜌𝜌,𝑐𝑐𝑐𝑐

where 𝜂𝜂 (∗) = −𝑢𝑢′′ �𝑐𝑐 (∗) �𝑐𝑐 (∗) /𝑢𝑢′ �𝑐𝑐 (∗) �.

𝜆𝜆̇(∗)𝜆𝜆(∗)

𝑅𝑅(∗) �= 𝑟𝑟 + 𝜋𝜋 (∗) � == 𝜌𝜌 − 𝑟𝑟.
𝑣𝑣 ′ �𝑚𝑚(∗) �𝑢𝑢′ �𝑐𝑐 (∗) �

,

(8)

(9)

Firms: The home and foreign labor productivities of commodity 𝑖𝑖 (𝑖𝑖 = 1,2) are respectively 𝜃𝜃𝑖𝑖

and 𝜃𝜃𝑖𝑖∗ , both of which are constant. Without loss of generality, they satisfy
𝜃𝜃1
𝜃𝜃2

𝜃𝜃∗

> 1∗ ,
𝜃𝜃2(10)i.e., the home (or foreign) country has comparative advantage in production of commodity 1(or commodity 2). Because all firms are competitive, if sector 𝑖𝑖 (𝑖𝑖 = 1,2) operates,
𝑝𝑝𝑖𝑖 (𝜔𝜔)𝜃𝜃𝑖𝑖 = 𝑤𝑤,

𝑝𝑝𝑖𝑖 (𝜔𝜔)𝜃𝜃𝑖𝑖∗ = 𝑤𝑤 ∗ .

(11)

3. Three specialization patterns
Because the labor productivities of the two commodities are constant, at least one of thetwo countries specializes in the sector with the comparative advantage represented by (12). The
home (or foreign) country specializes in commodity 1 (or 2) if it specializes. This section finds

Apparently, by replacing 𝑢𝑢(𝑐𝑐) with 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) one obtains the intratemporal and intertemporal optimal
conditions given by (2) and (8) all at once.
3

6

that three specialization patterns appear depending on the relative population size 𝛾𝛾(= 𝐿𝐿∗ /𝐿𝐿).
Apparently, 𝛾𝛾 increases with a larger foreign population and a smaller home population.

From the home and foreign demand functions of the two commodities given in (2), world

demand for commodity 1 and that for commodity 2 are respectively 𝛿𝛿(𝜔𝜔)(𝐿𝐿𝐿𝐿 + 𝐿𝐿∗ 𝑐𝑐 ∗ )/𝑝𝑝1 (𝜔𝜔)and [1 − 𝛿𝛿(𝜔𝜔)](𝐿𝐿𝐿𝐿 + 𝐿𝐿∗ 𝑐𝑐 ∗ )/𝑝𝑝2 (𝜔𝜔). Therefore, the market clearing of the commodities yields
𝛿𝛿(𝜔𝜔)(𝑐𝑐 + 𝛾𝛾𝑐𝑐 ∗ ) = 𝜃𝜃1 𝑥𝑥1 + 𝜃𝜃1∗ 𝛾𝛾𝑥𝑥1∗ ,𝑝𝑝1 (𝜔𝜔)𝜃𝜃2 𝑥𝑥2 +𝜃𝜃2∗ 𝛾𝛾𝑥𝑥2∗
𝜃𝜃1 𝑥𝑥1 +𝜃𝜃1∗ 𝛾𝛾𝑥𝑥1∗= 𝜑𝜑(𝜔𝜔) ≡1−𝛿𝛿(𝜔𝜔)𝜔𝜔𝜔𝜔(𝜔𝜔),𝛾𝛾 = 𝐿𝐿∗ /𝐿𝐿,

𝜑𝜑′ (𝜔𝜔) < 0,(12)where 𝜑𝜑(𝜔𝜔) is the demand ratio of commodity 2 over commodity 1, and 𝑥𝑥𝑖𝑖 and 𝑥𝑥𝑖𝑖∗ arerespectively home and foreign employment per capita in sector 𝑖𝑖 (𝑖𝑖 = 1, 2), satisfying
𝑥𝑥1 + 𝑥𝑥2 = 𝑥𝑥,𝑥𝑥1∗ + 𝑥𝑥2∗ = 𝑥𝑥 ∗ .

(13)Under the comparative advantage assumed in (10) one obtains the following specialization
patterns, where the numbers given after h or f imply the commodities produced in the home (h)or foreign (f) country:
Pattern (h12-f2): 0 < 𝑥𝑥𝑖𝑖 < 𝑥𝑥 (for 𝑖𝑖 = 1, 2), (𝑥𝑥1∗ , 𝑥𝑥2∗ ) = (0, 𝑥𝑥 ∗ ),Pattern (h1-f2): (𝑥𝑥1 , 𝑥𝑥2 ) = (𝑥𝑥, 0), (𝑥𝑥1∗ , 𝑥𝑥2∗ ) = (0, 𝑥𝑥 ∗ ),Pattern (h1-f12): (𝑥𝑥1 , 𝑥𝑥2 ) = (𝑥𝑥, 0), 0 < 𝑥𝑥𝑖𝑖∗ < 𝑥𝑥 (for 𝑖𝑖 = 1, 2).

From the third equation in (12) to which (13) and the above properties are applied, one obtains
𝛾𝛾𝑥𝑥 ∗𝑥𝑥𝜃𝜃𝜃𝜃< 𝜑𝜑( 1) 1∗ in pattern (h12-f2),𝜃𝜃2 𝜃𝜃2𝛾𝛾𝑥𝑥 ∗𝑥𝑥𝜃𝜃∗ 𝜃𝜃

> 𝜑𝜑( 1∗ ) 1∗ in pattern (h1-f12).
𝜃𝜃2 𝜃𝜃2

Therefore, with (11) the following lemma obtains:

Lemma 1: As the size of the foreign employed relative to the home employed, 𝛾𝛾𝑥𝑥 ∗ /𝑥𝑥,
increases, the three specialization patterns consecutively appear:𝛾𝛾𝑥𝑥 ∗
𝑥𝑥

𝜃𝜃

𝜃𝜃

𝜃𝜃

𝜃𝜃

𝜃𝜃

< 𝜑𝜑( 1) 1∗ ⇒ Pattern (h12-f2): 𝜔𝜔 = 1, 𝑤𝑤 = 𝑝𝑝1 ( 1)𝜃𝜃1 , 𝑤𝑤 ∗ = 𝑝𝑝2 ( 1)𝜃𝜃2∗ .
𝜃𝜃2 𝜃𝜃2

𝜃𝜃2

7

𝜃𝜃2

𝜃𝜃2

𝜃𝜃

𝜃𝜃

𝜑𝜑( 1) 1∗ <𝜃𝜃2 𝜃𝜃2𝛾𝛾𝑥𝑥 ∗
𝑥𝑥

𝛾𝛾𝑥𝑥 ∗𝑥𝑥𝜃𝜃∗ 𝜃𝜃

𝜃𝜃∗ 𝜃𝜃

< 𝜑𝜑( 1∗ ) 1∗ ⇒ Pattern (h1-f2):𝜃𝜃2 𝜃𝜃2𝛾𝛾𝑥𝑥 ∗𝑥𝑥

𝑤𝑤 ∗ = 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ ,𝑤𝑤 = 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 ,𝜃𝜃

= 𝜑𝜑(𝜔𝜔) 1∗ ,

𝜃𝜃∗

𝜃𝜃2

𝜃𝜃1
𝜃𝜃2

> 𝜔𝜔 >

𝜃𝜃∗

𝜃𝜃1∗
𝜃𝜃2∗

,

𝜃𝜃∗

> 𝜑𝜑( 1∗ ) 1∗ ⇒ Pattern (h1-f12): 𝜔𝜔 = 1∗ , 𝑤𝑤 = 𝑝𝑝1 ( 1∗ )𝜃𝜃1 , 𝑤𝑤 ∗ = 𝑝𝑝2 ( 1∗ )𝜃𝜃2∗ .
𝜃𝜃2 𝜃𝜃2

𝜃𝜃2

𝜃𝜃2

𝜃𝜃2

The implication of lemma 1 is straightforward. The country with much more people
working produces both commodities while the other specializes in the commodity withcomparative advantage. Therefore, the relative price equals the autarky level of the country that
produces both commodities. If the number of the employed is similar between the twocountries, both countries specialize in respective commodities with comparative advantage.
In the home and foreign money markets real money balances 𝑚𝑚(∗) satisfy
𝑚𝑚(∗) =

𝑀𝑀(∗)
𝑃𝑃(∗)

,

where 𝑀𝑀(∗) is each country’s nominal money supply per capita, which is assumed to beconstant over time, for simplicity. Time differentiation of the equation yields
𝑚𝑚̇(∗) = −𝜋𝜋 (∗) 𝑚𝑚(∗) .

(14)

The flow budget equation in (7) to which 𝑤𝑤 (∗) in lemma 1 and 𝑚𝑚̇(∗) in (14) are applied turns to𝑏𝑏̇ = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥 − 𝑐𝑐,𝑏𝑏̇ ∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ 𝑥𝑥 ∗ − 𝑐𝑐 ∗ ,(15)which are the two countries’ current account equations, respectively. International lending andborrowing 𝑏𝑏 and 𝑏𝑏 ∗ always satisfy

𝑏𝑏 + 𝛾𝛾𝑏𝑏 ∗ = 0,(16)where 𝛾𝛾 = 𝐿𝐿∗ /𝐿𝐿. The absolute values of 𝑏𝑏 and 𝑏𝑏 ∗ are assumed to be negligibly small so that

one can ignore the redistribution effect of a change in population on foreign assets or debts

among people within a country. The case where 𝑏𝑏 and 𝑏𝑏 ∗ are non-negligible is discussed in
section 6.

8

Because the two countries have the same subjective discount rate 𝜌𝜌, one find the following

property, of which the proof is set out in Appendix A.

Lemma 2: The economy immediately jumps to the steady state where 𝑟𝑟 equals 𝜌𝜌, the current
account 𝑏𝑏̇(= −𝑏𝑏̇ ∗ ) equals zero, and real wages and consumption are constant, satisfying
𝑐𝑐 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥,

𝑐𝑐 ∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ 𝑥𝑥 ∗ .

4. The relative population size, employment and consumption
Having set up the model, let us find the relationship between the relative population size
and the two countries’ employment and consumption. This section first treats the benchmark
case where full employment prevails in both countries, and shows the condition under whichfull employment cannot be achieved and secular stagnation of aggregate demand arises in eachcountry. Then, the cases where one of the two countries or both of them suffer from seculardemand stagnation are examined.

4-1. Full employment in both countries
If both countries achieve full employment:𝑥𝑥 = 1, 𝑥𝑥 ∗ = 1,

from lemmas 1 and 2 one finds

𝑐𝑐 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 , 𝑐𝑐 ∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ ;
𝜃𝜃

𝜃𝜃

𝜃𝜃

𝜃𝜃

𝜃𝜃

𝛾𝛾 < 𝜑𝜑 � 1 � 1∗ → pattern (h12-f2): 𝜔𝜔 = 1,𝜃𝜃2 𝜃𝜃2𝜃𝜃∗ 𝜃𝜃

𝜑𝜑( 1) 1∗ < 𝛾𝛾 < 𝜑𝜑( 1∗ ) 1∗ → pattern (h1-f2):𝜃𝜃2 𝜃𝜃2𝜃𝜃∗ 𝜃𝜃

𝜃𝜃2 𝜃𝜃2𝜃𝜃2

𝜃𝜃∗𝜑𝜑( 1∗ ) 1∗ < 𝛾𝛾 → pattern (h1-f12): 𝜔𝜔 = 1∗ .
𝜃𝜃2 𝜃𝜃2

𝜃𝜃2

9

𝜃𝜃

𝛾𝛾 = 𝜑𝜑(𝜔𝜔) 1∗ ,
𝜃𝜃2

(17)

𝛽𝛽

𝑢𝑢−1 ( )

𝑐𝑐

𝜌𝜌

𝜃𝜃∗

𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1
𝜃𝜃2

𝜃𝜃𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1 )𝜃𝜃1
𝜃𝜃2

𝛾𝛾̅

O

𝛾𝛾

𝛾𝛾̿

Figure 1: Full employment in the home country

Thus, if the relative foreign population 𝛾𝛾 is low enough, specialization pattern (h12-f2) appears

and 𝛾𝛾 affects none of 𝜔𝜔 , 𝑐𝑐 and 𝑐𝑐 ∗ . If 𝛾𝛾 is between 𝜑𝜑(𝜃𝜃1 /𝜃𝜃2 )𝜃𝜃1 /𝜃𝜃2∗ and 𝜑𝜑(𝜃𝜃1∗ /𝜃𝜃2∗ )𝜃𝜃1 /𝜃𝜃2∗ ,pattern (f1-h2) appears. Because 𝑝𝑝1′ (𝜔𝜔) < 0 and 𝜑𝜑′(𝜔𝜔) < 0 , as shown in (5) and (12)respectively, from (17),𝑑𝑑𝑑𝑑

and hence

𝑑𝑑𝑑𝑑

=

𝜃𝜃2∗

𝜑𝜑′(𝜔𝜔)𝜃𝜃1

< 0,

𝑑𝑑𝑑𝑑

𝑑𝑑𝛾𝛾

= 𝑝𝑝1′ (𝜔𝜔)𝜃𝜃1

𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

> 0,

𝑑𝑑𝑐𝑐 ∗𝛾𝛾 ↑ → 𝜔𝜔 ↓ , 𝑥𝑥 = 1, 𝑐𝑐 ↑ , 𝑥𝑥 ∗ = 1, 𝑐𝑐 ∗ ↓𝑑𝑑𝑑𝑑= 𝑝𝑝2′ (𝜔𝜔)𝜃𝜃2∗

𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

< 0,

in pattern (f1-h2).

(18)

A larger 𝛾𝛾 increases the ratio of foreign products to home products and lowers the relative price𝜔𝜔 of the foreign commodity. Thus, it improves the home terms of trade and increases 𝑐𝑐 but
worsens the foreign terms of trade and decreases 𝑐𝑐 ∗ . If 𝛾𝛾 is higher than 𝜑𝜑(𝜃𝜃1∗ /𝜃𝜃2∗ )𝜃𝜃1 /𝜃𝜃2∗ ,pattern (h1-f12) appears and 𝜔𝜔, 𝑐𝑐 and 𝑐𝑐 ∗ stay constant for any 𝛾𝛾.

This relationship between 𝛾𝛾 and 𝑐𝑐 is illustrated in figure 1, where 𝛾𝛾̅ and 𝛾𝛾̿ are the criticalvalues of 𝛾𝛾 at which the specialization pattern changes. From (17), they are10𝜃𝜃𝜃𝜃∗ 𝜃𝜃

𝜃𝜃

𝛾𝛾̅ = 𝜑𝜑 � 1 � 1∗ ,𝛾𝛾̿ = 𝜑𝜑( 1∗ ) 1∗ .

𝜃𝜃2 𝜃𝜃2𝜃𝜃2 𝜃𝜃2

From (9) in which 𝜋𝜋 = 0, and lemma 2, 𝑃𝑃 and 𝑃𝑃∗ must satisfy𝜌𝜌 =𝑣𝑣 ′ (𝑀𝑀/𝑃𝑃),
𝑢𝑢′ (𝑐𝑐)𝜌𝜌 =𝑣𝑣 ′ (𝑀𝑀∗ /𝑃𝑃∗ ),𝑢𝑢′ (𝑐𝑐 ∗ )(19)where 𝑐𝑐 and 𝑐𝑐 ∗ are given in (17). In the presence of a liquidity trap, however, there may be no

solutions of 𝑃𝑃 and 𝑃𝑃∗ . In the present setting a liquidity trap emerges if the marginal utility ofmoney 𝑣𝑣 ′ (𝑚𝑚) has a positive lower bound 𝛽𝛽, 4𝑣𝑣 ′ (∞) = 𝛽𝛽 > 0,(20)

because under this property the home money demand function given in (9) satisfies
𝑅𝑅 =𝑣𝑣 ′ (𝑚𝑚)𝛽𝛽(> ′ (𝑐𝑐)𝑢𝑢′ (𝑐𝑐)𝑢𝑢

for any 𝑚𝑚).

𝑅𝑅 remains to be strictly positive as 𝑚𝑚 increases, implying a liquidity trap. 5

In this case, if and only if the home (or foreign) consumption 𝑐𝑐 (or 𝑐𝑐 ∗ ) given in (17) is solarge as to satisfy

𝜌𝜌 ≤

𝜌𝜌 ≤

𝛽𝛽
𝑣𝑣 ′ (𝑀𝑀/𝑃𝑃)�<�,𝑢𝑢′ (𝑐𝑐)𝑢𝑢′ (𝑐𝑐)𝛽𝛽𝑢𝑢′ (𝑐𝑐 ∗ )�<𝑣𝑣 ′ (𝑀𝑀∗ /𝑃𝑃∗ )�,𝑢𝑢′ (𝑐𝑐 ∗ )where 𝑐𝑐 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 ,where 𝑐𝑐 ∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ ,(21)

there is no 𝑃𝑃 (or 𝑃𝑃∗ ) that satisfies the first (or second) equation in (19), implying that there is no

steady state with full employment in the home (or foreign) country. In order for the
full-employment steady state to be feasible, the two inequalities in (21) must be invalid.
Given that the left- and right-hand sides of each inequality in (21) respectively represent
the time preference rate and the liquidity premium, (21) shows that the marginal desire for

4

The validity of this property is empirically shown by Ono et al. (2004) using both a parametric and a
non-parametric approach.
If 𝑣𝑣 ′ (𝑚𝑚) continues to be positive as 𝑚𝑚 expands, 𝑣𝑣(𝑚𝑚) does not imply the transaction motive. It should be
wealth preference. In Appendix B the wealth preference and the narrowly-defined liquidity preference due to thetransaction motive are separately introduced to the present model. It is shown that in this setting the following
analysis is still valid while the home nominal interest rate is zero in the steady state.
5

11

holding money dominates that for consumption if the household consumes enough to realize
full employment. Thus, aggregated demand deficiency and unemployment will appear in thesteady state, which will be shown in the following sections. Note that a country with high value
productivity, 𝑝𝑝1 𝜃𝜃1 or 𝑝𝑝2 𝜃𝜃2∗ , is likely to satisfy (21) and suffer from secular stagnation of

aggregate demand.

𝑐𝑐

𝛽𝛽

𝑢𝑢−1 � �𝜌𝜌𝜃𝜃1∗ ↑, 𝜃𝜃2∗ ↑𝜃𝜃2 ↑𝜃𝜃1 ↑, 𝜃𝜃2∗ ↑
A

F-F

F-UF

F-U

𝜃𝜃1 ↑, 𝜃𝜃2∗ ↑𝜃𝜃1∗𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( ∗ )𝜃𝜃1
𝜃𝜃2

FU-U

FU-UF

FU-F

U-U

U-UF

U-F

𝜃𝜃1∗ ↑

𝜃𝜃1 ↑, 𝜃𝜃2 ↑𝜃𝜃𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1 )𝜃𝜃1
𝜃𝜃2

O

𝜃𝜃1∗
𝜃𝜃2∗

𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( )𝜃𝜃2∗

𝜃𝜃

𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1)𝜃𝜃2∗
𝜃𝜃2

𝛽𝛽

𝑢𝑢−1 � �𝜌𝜌𝑐𝑐 ∗Figure 2: Patterns of full- and underemployment

From (5), (10) and (17), the maximum and minimum values of 𝑐𝑐 and 𝑐𝑐 ∗ under full

employment, 𝑐𝑐 𝑀𝑀(∗) and 𝑐𝑐 𝑚𝑚(∗) respectively, are𝜃𝜃∗

𝑐𝑐 𝑀𝑀 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1 ,𝜃𝜃2𝜃𝜃𝑐𝑐 𝑀𝑀∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1 )𝜃𝜃2∗ ,𝜃𝜃2

𝜃𝜃

𝑐𝑐 𝑚𝑚 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1)𝜃𝜃1 ,
𝜃𝜃2

𝜃𝜃∗

𝑐𝑐 𝑚𝑚∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1∗ )𝜃𝜃2∗ .
𝜃𝜃2

12

As shown in figure 2, they partition the positive quadrant of the (𝑐𝑐, 𝑐𝑐 ∗ ) plane into nine areas.
From (21), full employment consumption 𝑐𝑐 (∗) cannot exceed 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌) . Therefore, full
employment always appears and (17) holds when𝛽𝛽

𝛽𝛽

point A: (𝑐𝑐, 𝑐𝑐 ∗ ) = �𝑢𝑢′−1 ( ), 𝑢𝑢′−1 ( )�,
𝜌𝜌

𝜌𝜌

(22)

is located in F-F of figure 2. From (5), 𝑐𝑐 𝑀𝑀(∗) and 𝑐𝑐 𝑚𝑚(∗) given above satisfy𝜃𝜃1 ↑ → 𝑐𝑐 𝑀𝑀 ↑, 𝑐𝑐 𝑚𝑚 ↑, 𝑐𝑐 𝑀𝑀∗ ↑;𝜃𝜃1∗ ↑ → 𝑐𝑐 𝑀𝑀 ↓, 𝑐𝑐 𝑚𝑚∗ ↑;𝜃𝜃2 ↑ → 𝑐𝑐 𝑚𝑚 ↑, 𝑐𝑐 𝑀𝑀∗ ↓;𝜃𝜃2∗ ↑ → 𝑐𝑐 𝑀𝑀 ↑, 𝑐𝑐 𝑀𝑀∗ ↑, 𝑐𝑐 𝑚𝑚∗ ↑;
(∗)

(∗)

i.e., changes in the productivity parameters, 𝜃𝜃1 and 𝜃𝜃2 , move the boundary curves andlocate point A defined by (22) in different areas. F and U in the area names imply full
employment and underemployment in each country, respectively.
In the following the condition under which A is located in each area is found and therelationship between the relative population size and the two countries’ specialization patterns,
consumption and employment in each case is examined.

4-2. Case U-U
If 𝜃𝜃1 and 𝜃𝜃2∗ are so large that point A given by (22) is located in U-U of figure 2, where
𝛽𝛽

𝜃𝜃

𝜃𝜃∗

𝑢𝑢′−1 ( ) < min. �𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1 )𝜃𝜃1 , 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1∗ )𝜃𝜃2∗ �,
𝜌𝜌

𝜃𝜃2

𝜃𝜃2

the two inequalities in (21) hold for any 𝜔𝜔 ∈ [𝜃𝜃1∗ /𝜃𝜃2∗ , 𝜃𝜃1 /𝜃𝜃2 ]. The marginal desire for holding

money dominates that for consumption in both countries and aggregate demand deficiency
appears. In order to treat aggregate demand deficiency, a wage adjustment mechanism that canaccommodate labor demand shortages must be assumed. Moreover, it should not intrinsically
avoid the possibility of full employment because the present unemployment is caused not by
wage sluggishness but by aggregate demand deficiency.

13

To describe sluggish wage adjustment the Calvo pricing is widely adopted in DSGE
models. However, it never allows any demand-supply discrepancy to appear either during theadjustment process or in the steady state. The search theory (e.g. Pissarides, 2000) explains
unemployment but, in turn, never allows full employment to be reached. Moreover, theunemployment that it discusses is of the frictional type. Thus, neither of them can apply toanalyze involuntary unemployment due to aggregate demand deficiency.
Ono and Ishida (2014) extend the Akerlof-type fair wage model (see Akerlof, 1982, andAkerlof and Yellen, 1990) to a dynamic context and present a wage adjustment mechanism that
can apply to both the cases of full employment and underemployment. They show that it
asymptotically reaches the conventional Walrasian adjustment if unemployment appears in thesteady state and is perfect if full employment prevails. Therefore, for simplicity, it is now
assumed that home and foreign nominal wages adjust as follows:𝑊𝑊̇ (∗)𝑊𝑊 (∗)= 𝛼𝛼�𝑥𝑥 (∗) − 1�,in the presence of unemployment. 6 Because 𝜔𝜔 is constant in the steady state, from (11) real

wage 𝑤𝑤 (∗) is constant and hence the wage adjustment given above yields
𝜋𝜋 (∗) =𝑊𝑊̇ (∗)𝑊𝑊 (∗)= 𝛼𝛼�𝑥𝑥 (∗) − 1�.

(23)

By substituting (20) and (23) to (9) one finds that if (21) is valid, 𝑐𝑐 (∗) in the steady state withaggregate demand deficiency is expressed as a function of 𝑥𝑥 (∗) :𝛽𝛽𝑢𝑢′ (𝑐𝑐 (∗) )= 𝜌𝜌 + 𝛼𝛼�𝑥𝑥 (∗) − 1�

→ 𝑐𝑐 (∗) = 𝑐𝑐(𝑥𝑥 (∗) ) ≡ 𝑢𝑢′−1 �𝑐𝑐 ′ (𝑥𝑥 (∗) ) > 0, 𝑥𝑥 (∗) < 1.

𝛽𝛽𝜌𝜌+𝛼𝛼�𝑥𝑥 (∗) −1�

�,

Because 𝑐𝑐 (∗) given in lemma 2 satisfies (24) in the present case, 𝑥𝑥 (∗) satisfies
𝑐𝑐(𝑥𝑥) = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥,

6

𝑐𝑐(𝑥𝑥 ∗ ) = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ 𝑥𝑥 ∗ ,Schmitt-Grohé and Uribe (2016, 2017) assume basically the same wage adjustment mechanism.

14

(24)

(25)

𝜃𝜃

𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1 )𝜃𝜃1
𝜃𝜃2

𝑐𝑐

𝛽𝛽

𝑢𝑢−1 � �
𝜌𝜌

𝑐𝑐̅

𝑐𝑐̿

𝛾𝛾̅

O

𝛾𝛾

𝛾𝛾̿

Figure 3: Underemployment in the home country

where from (12) and lemma 2 𝜔𝜔 satisfies
𝛾𝛾𝑥𝑥 ∗
𝑥𝑥

𝜃𝜃

𝜃𝜃

𝜃𝜃

< 𝜑𝜑( 1) � 1∗ � → pattern (h12-f2): 𝜔𝜔 = 1,

𝜃𝜃

𝜃𝜃

𝜃𝜃2

𝜑𝜑( 1) � 1∗ � <𝜃𝜃2𝛾𝛾𝑥𝑥 ∗
𝑥𝑥

𝜃𝜃∗

𝜃𝜃2

𝛾𝛾𝑥𝑥 ∗
𝑥𝑥

𝜃𝜃

𝜃𝜃∗

𝜃𝜃2

𝜃𝜃

< 𝜑𝜑( 1∗ ) � 1∗ � → pattern (h1-f2):
𝜃𝜃2

𝜃𝜃2

= 𝜑𝜑(𝜔𝜔) � 1∗ �, 𝑐𝑐(𝑥𝑥) + 𝛾𝛾𝛾𝛾(𝑥𝑥 ∗ ) =𝜃𝜃

𝜑𝜑( 1∗ ) � 1∗ � <
𝜃𝜃2

𝜃𝜃2

𝜃𝜃2

𝜃𝜃2

𝛾𝛾𝑥𝑥 ∗𝑥𝑥

𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥
𝛿𝛿(𝜔𝜔)

𝜃𝜃∗→ pattern (h1-f12): 𝜔𝜔 = 1∗ .

⇒ 𝑥𝑥 (∗) = 𝑥𝑥� (∗) (𝜔𝜔, 𝛾𝛾),
(26)

𝜃𝜃2

Thus, 𝑐𝑐 (∗) , 𝑥𝑥 (∗) and 𝜔𝜔 are fully determined. A change in 𝛾𝛾 does not affect the variables inpattern (h12-f2) or (h1-f12), while in pattern (h1-f2) it varies them as follows:𝛾𝛾 ↑ → 𝜔𝜔 ↓ , 𝑥𝑥 ↓ , 𝑐𝑐 ↓ , 𝑥𝑥 ∗ ↑ , 𝑐𝑐 ∗ ↑ ,𝛾𝛾𝑥𝑥 ∗
𝑥𝑥↑in pattern (h1-f2),(27)

of which the proof is set out in appendix C. An increase in the relative foreign population 𝛾𝛾

expands the production ratio of commodity 2 to commodity 1, which reduces 𝑝𝑝2 (𝜔𝜔) and raises
15

𝑝𝑝1 (𝜔𝜔). Thus, world demand for commodity 1 declines, which reduces home employment 𝑥𝑥

and consumption 𝑐𝑐. World demand for commodity 2 expands and hence foreign employment

𝑥𝑥 ∗ and consumption 𝑐𝑐 ∗ increase. The last property in (27) and lemma 1 imply the three

specialization patterns to appear consecutively as 𝛾𝛾 increases.

From (25) and (26), 𝑐𝑐 in patterns (h12-f2) and (h1-f12), respectively denoted by 𝑐𝑐̅ and 𝑐𝑐̿,and the critical values of 𝛾𝛾 that separate the three specialization patterns, 𝛾𝛾̅ and 𝛾𝛾̿ , satisfy𝜃𝜃𝑐𝑐̅ = 𝑐𝑐(𝑥𝑥̅ ) = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1)𝜃𝜃1 𝑥𝑥̅ ,
𝜃𝜃2

𝜃𝜃∗

𝑐𝑐̿ = 𝑐𝑐(𝑥𝑥̿ ) = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1 𝑥𝑥̿ ,𝜃𝜃2𝜃𝜃𝛾𝛾̅ = 𝛾𝛾̅𝑈𝑈𝑈𝑈 where 𝑥𝑥̅ = 𝑥𝑥�( 1 , 𝛾𝛾̅𝑈𝑈𝑈𝑈 ),
𝜃𝜃2

𝜃𝜃∗

𝛾𝛾̿ = 𝛾𝛾̿𝑈𝑈𝑈𝑈 where 𝑥𝑥̿ = 𝑥𝑥�( 1∗ , 𝛾𝛾̿𝑈𝑈𝑈𝑈 ).
𝜃𝜃2

(28)

Figure 3 shows 𝑐𝑐 as a function of 𝛾𝛾, where 𝑐𝑐̅, 𝑐𝑐̿, 𝛾𝛾̅ and 𝛾𝛾̿ are given by (28).
4-3. Case U-F
In the case where point A is located in U-F of figure 2 –i.e.,
𝜃𝜃

𝛽𝛽

𝜃𝜃

𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1 )𝜃𝜃1 > 𝑢𝑢−1 ( ) > 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1)𝜃𝜃2∗ ,
𝜌𝜌

𝜃𝜃2

𝜃𝜃2

which appears when 𝜃𝜃1 is large and 𝜃𝜃2∗ is small, the foreign country always achieves full

employment because the second inequality in (21) is invalid for any 𝜔𝜔 ∈ [𝜃𝜃1∗ /𝜃𝜃2∗ , 𝜃𝜃1 /𝜃𝜃2 ]. The

home country cannot attain full employment and suffers from secular stagnation because thefirst inequality in (21) is valid for any 𝜔𝜔 ∈ [𝜃𝜃1∗ /𝜃𝜃2∗ , 𝜃𝜃1 /𝜃𝜃2 ], implying that the marginal desire
for holding money dominates that for consumption. Therefore,
𝑥𝑥 < 1, 𝑥𝑥 ∗ = 1,

and then from lemmas 1 and 2 and function 𝑐𝑐(𝑥𝑥) given by (24), 𝜔𝜔, 𝑐𝑐, 𝑐𝑐 ∗ and 𝑥𝑥 satisfy
𝑐𝑐 = 𝑐𝑐(𝑥𝑥) = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥,
𝛾𝛾

𝜃𝜃

𝜃𝜃

𝑥𝑥

𝜃𝜃

𝜃𝜃

𝑐𝑐 ∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ ;𝜃𝜃

< 𝜑𝜑( 1 ) 1∗ → pattern (h12-f2): 𝜔𝜔 = 1,𝛾𝛾𝜃𝜃2 𝜃𝜃2𝜃𝜃∗ 𝜃𝜃

𝜃𝜃2𝜃𝜃∗𝜑𝜑( 1) 1∗ < < 𝜑𝜑( 1∗ ) 1∗ → pattern (h1-f2): 𝑥𝑥 = � 2 �
𝜃𝜃2 𝜃𝜃2𝑥𝑥𝛾𝛾

𝜃𝜃1 𝜑𝜑(𝜔𝜔)𝜃𝜃2 𝜃𝜃216,𝜃𝜃∗ 𝜃𝜃𝜑𝜑( 1∗ ) 1∗ <𝜃𝜃2 𝜃𝜃2

𝛾𝛾
𝑥𝑥

𝜃𝜃∗

→ pattern (h1-f12): 𝜔𝜔 = 1∗ .

(29)

𝜃𝜃2

From (29), neither 𝑥𝑥 nor 𝑐𝑐(𝑥𝑥) depends on 𝛾𝛾 in patterns (h12-f2) and (h1-f12) while both ofthem do in pattern (h1-f2).

From (15), (24), lemma 2 and 𝑥𝑥 in pattern (h1-f2) of (29),𝜃𝜃∗𝑏𝑏̇ = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥 − 𝑐𝑐(𝑥𝑥) = 0,𝛾𝛾

𝑥𝑥 = � 2�𝜃𝜃1 𝜑𝜑(𝜔𝜔)

.

(30)

By assuming 𝑏𝑏̇ to satisfy the Marshall-Lerner condition 𝑏𝑏̇𝜔𝜔 > 0 and using 𝑝𝑝1′ (𝜔𝜔) < 0 in (5)and 𝜑𝜑′ < 0 in (12), one obtains

′

∂𝑏𝑏̇
𝑥𝑥𝜑𝜑
𝑏𝑏̇𝜔𝜔 �≡ � = 𝑝𝑝1′ (𝜔𝜔)𝜃𝜃1 𝑥𝑥 − [𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 − 𝑐𝑐 ′ (𝑥𝑥)]
> 0,
𝜕𝜕𝜕𝜕

𝜑𝜑

𝑥𝑥
𝑏𝑏̇𝛾𝛾 = [𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 − 𝑐𝑐′(𝑥𝑥)] > 0,𝛾𝛾where the latter property implies

𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 − 𝑐𝑐′(𝑥𝑥) > 0.

(31)

In appendix C it is shown that the property of (31) must hold if 𝑥𝑥 exists within (0,1). Totallydifferentiating the two equations in (30) and applying the above properties of 𝑏𝑏̇𝜔𝜔 and 𝑏𝑏̇𝛾𝛾 to theresults yields

𝑑𝑑𝑑𝑑

in pattern (h1-f2).

𝑑𝑑𝑑𝑑

𝑏𝑏̇

= − ̇ 𝛾𝛾 < 0,
𝑏𝑏𝜔𝜔

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑

=

𝑝𝑝1′ (𝜔𝜔)𝜃𝜃1 𝑥𝑥 2
< 0,𝛾𝛾𝑏𝑏̇𝜔𝜔

𝛾𝛾

𝑑𝑑(𝑥𝑥)
𝑑𝑑𝑑𝑑> 0,

The implication is the following. Because a larger 𝛾𝛾 increases foreign supply ofcommodity 2, lowers 𝜔𝜔 and raises 𝑝𝑝1 (𝜔𝜔), home current account 𝑏𝑏̇ in (30) improves if 𝑥𝑥 isinvariant. Therefore, in order for the current account to deteriorate and recover the balance,from (31) 𝑥𝑥 has to be smaller than before. A smaller 𝑥𝑥 worsens deflation and urges home

households to decrease consumption 𝑐𝑐 , as shown by (24). The decrease in 𝜔𝜔 in turn

deteriorates the foreign terms of trade and reduces foreign consumption 𝑐𝑐 ∗ given in (29).
Therefore,17

𝛾𝛾 ↑ → 𝜔𝜔 ↓ , 𝑥𝑥 ↓ , 𝑐𝑐 ↓ , 𝑥𝑥 ∗ = 1, 𝑐𝑐 ∗ ↓ ,

𝛾𝛾
𝑥𝑥

↑

in pattern (f1-h2).

(32)

Because 𝛾𝛾/𝑥𝑥 increases with 𝛾𝛾, as shown above, from lemma 1 the three specialization patterns
consecutively appear as 𝛾𝛾 increases.

From (29), the maximum and minimum values of 𝑐𝑐 and the boundary levels of 𝛾𝛾 satisfy
𝜃𝜃

𝜃𝜃

𝜃𝜃

𝜃𝜃∗

𝛾𝛾̅ = 𝛾𝛾̅𝑈𝑈𝑈𝑈 ≡ � 1∗ � 𝑥𝑥̅ 𝜑𝜑( 1 ),
𝜃𝜃2

𝜃𝜃2

𝛾𝛾̿ = 𝛾𝛾̿𝑈𝑈𝑈𝑈 ≡ � 1∗ � 𝑥𝑥̿ 𝜑𝜑( 1∗ ),
𝜃𝜃2

𝜃𝜃2

𝜃𝜃

𝑐𝑐̅ = 𝑐𝑐(𝑥𝑥̅ ) = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1)𝜃𝜃1 𝑥𝑥̅ ,𝜃𝜃2𝜃𝜃∗𝑐𝑐̿ = 𝑐𝑐(𝑥𝑥̿ ) = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1 𝑥𝑥̿ .
𝜃𝜃2

(33)

Note that 𝑐𝑐̅ and 𝑐𝑐̿ are equal to, but 𝛾𝛾̅ and 𝛾𝛾̿ are different from, those in the case where bothcountries are underemployed, given in (28). Therefore, the shape of 𝑐𝑐 as a function of 𝛾𝛾 is notexactly the same as, but quite similar to, that in figure 3.

4-4. Case F-U
In the case where point A is located in F-U of figure 2:
𝜃𝜃

𝛽𝛽

𝜃𝜃∗

𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 � 1 � 𝜃𝜃2∗ > 𝑢𝑢−1 ( ) > 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1 ,
𝜌𝜌

𝜃𝜃2

𝜃𝜃2

which emerges when 𝜃𝜃1 is small and 𝜃𝜃2∗ is large, is symmetric to the previous case where point

A is located in U-F. The home country attains full employment while the foreign country

suffers from underemployment. Therefore, by interchanging the home variables and theforeign ones in (29) and (32) and rearranging the results one obtains
𝑐𝑐 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 ,
𝜃𝜃

𝜃𝜃

𝜃𝜃

𝜃𝜃

𝑐𝑐 ∗ = 𝑐𝑐(𝑥𝑥 ∗ ) = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ 𝑥𝑥 ∗ ;𝜃𝜃𝛾𝛾𝑥𝑥 ∗ < 𝜑𝜑( 1 ) 1∗ → pattern (h12-f2): 𝜔𝜔 = 1,𝜃𝜃2 𝜃𝜃2𝜃𝜃2𝜃𝜃∗ 𝜃𝜃

𝜃𝜃 𝜑𝜑(𝜔𝜔)𝜑𝜑( 1) 1∗ < 𝛾𝛾𝑥𝑥 ∗ < 𝜑𝜑( 1∗ ) 1∗ → pattern (h1-f2): 𝛾𝛾 = 1 ∗ ∗ ,𝜃𝜃2 𝜃𝜃2𝜃𝜃2 𝑥𝑥𝜃𝜃2 𝜃𝜃2𝛾𝛾 ↑ → 𝜔𝜔 ↓ , 𝑥𝑥 = 1, 𝑐𝑐 ↑ , 𝑥𝑥 ∗ ↑ , 𝑐𝑐 ∗ ↑ , 𝛾𝛾𝑥𝑥 ∗ ↑,𝜃𝜃∗ 𝜃𝜃𝜃𝜃∗𝜑𝜑( 1∗ ) 1∗ < 𝛾𝛾𝑥𝑥 ∗ → pattern (h1-f12): 𝜔𝜔 = 1∗ .
𝜃𝜃2 𝜃𝜃2

𝜃𝜃2

18

(34)

The relationship between 𝛾𝛾 and 𝑐𝑐 obtained above is similar to that in figure 1. Byinterchanging the home variables and the foreign ones in (33) one obtains 𝛾𝛾̅ and 𝛾𝛾̿ , the criticalvalues of 𝛾𝛾 that separate the three specialization patterns, in the present case. They satisfy𝛾𝛾̅ = 𝛾𝛾̅𝐹𝐹𝐹𝐹 ≡𝛾𝛾̿ = 𝛾𝛾̿𝐹𝐹𝐹𝐹 ≡𝜃𝜃𝜃𝜃1 𝜑𝜑(𝜃𝜃1 )2

𝜃𝜃2∗ 𝑥𝑥̅ ∗𝜃𝜃∗𝜃𝜃2𝜃𝜃2∗ 𝑥𝑥̿ ∗𝜃𝜃1 𝜑𝜑( 1∗ )𝜃𝜃where 𝑐𝑐(𝑥𝑥̅ ∗ ) = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1)𝜃𝜃2∗ 𝑥𝑥̅ ∗ ,𝜃𝜃2𝜃𝜃∗where 𝑐𝑐(𝑥𝑥̿ ∗ ) = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1∗ )𝜃𝜃2∗ 𝑥𝑥̿ ∗ .
𝜃𝜃2

(35)

4-5. Summary of this section
The results of this section are thus summarized by the following two propositions:

Proposition 1: A country with high value productivity suffers from secular aggregate demandstagnation and underemployment whereas a country with low value productivity attains
full employment.

Proposition 2: If a country has a much larger population than the other, it produces bothcommodities while the smaller country specializes in the sector with comparativeadvantage. The relative price, home and foreign consumption and employment are
constant with changes in the international population ratio. If the populations of the twocountries are similar, both countries perfectly specialize according to respective
comparative advantages. Then, whether the foreign country is fully employed orunderemployed, a larger foreign population or a smaller home population makes lower the
relative price of the foreign commodity. Accordingly, home consumption increases if thehome country is fully employed, whereas home employment and consumption decrease if

19

it is underemployed. Foreign consumption decreases if the foreign country is fully
employed, whereas foreign employment and consumption increase if it is underemployed.

5. Switching between full employment and underemployment
The previous section has treated the cases in which each country does not switch between
full employment and underemployment. This section examines the cases in which one or bothcountries switch between the two as the relative population 𝛾𝛾 changes.
5-1. Case FU-F
In the case where point A defined in (22) is located in FU-F of figure 2, where
𝜃𝜃∗

𝛽𝛽

𝜃𝜃

𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1 > 𝑢𝑢′−1 ( ) > 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1)𝜃𝜃1 ,
𝜌𝜌

𝜃𝜃2

𝜃𝜃2

𝛽𝛽

𝜃𝜃

𝑢𝑢′−1 ( ) > 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1)𝜃𝜃2∗ ,𝜌𝜌𝜃𝜃2the foreign country always achieves full employment, while the home country attains full
employment when 𝑐𝑐 is small and faces underemployment when 𝑐𝑐 is large.

More precisely, when 𝛾𝛾 is smaller than 𝜑𝜑(𝜃𝜃1 /𝜃𝜃2 )𝜃𝜃1 /𝜃𝜃2∗ , all properties in case F-F arevalid. Then, from (17),𝑐𝑐 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜃𝜃1 /𝜃𝜃2 )𝜃𝜃1 ,𝑐𝑐 ∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜃𝜃1 /𝜃𝜃2 )𝜃𝜃2∗ ,and hence the two inequalities in (21) are invalid, implying that the steady state with full
employment is reached in both countries.
When 𝛾𝛾 is higher than 𝜑𝜑(𝜃𝜃1 /𝜃𝜃2 )𝜃𝜃1 /𝜃𝜃2∗ , pattern (h1-f2) in (17) appears and 𝑐𝑐 increases

with 𝛾𝛾, as shown by (18), through an improvement in the home terms of trade. Once 𝑐𝑐 reaches
the upper bound 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌), which occurs when 𝛾𝛾 = 𝛾𝛾𝑐𝑐 that satisfies
𝜃𝜃

𝛽𝛽𝛾𝛾𝑐𝑐 = 𝜑𝜑(𝜔𝜔𝑐𝑐 ) � 1∗ � where 𝑐𝑐 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔𝑐𝑐 )𝜃𝜃1 = 𝑢𝑢′−1 ( ),
𝜌𝜌

𝜃𝜃2

20

(36)

𝑐𝑐
𝜃𝜃∗

𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1
𝜃𝜃2

Full employment

Underemployment𝛽𝛽𝑢𝑢−1 � �𝜃𝜃𝜌𝜌𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1 )𝜃𝜃1
𝜃𝜃2

𝑐𝑐̿

O

𝛾𝛾𝑐𝑐

𝛾𝛾̅

𝛾𝛾̿

𝛾𝛾

Figure 4: Switch from full employment to underemployment

as is clear from (17), a further increase in 𝛾𝛾 makes this 𝑐𝑐 so large as to satisfy the firstinequality in (21). Then, the home country suffers from underemployment while the foreigncountry maintains full employment, making case U-F to appear. Therefore, from (32), an
increase in 𝛾𝛾 raises the relative price of the home commodity, worsens home employment andreduces 𝑐𝑐 as far as 𝛾𝛾 is smaller than 𝛾𝛾̿𝑈𝑈𝑈𝑈 given in (33).

For any 𝛾𝛾 > 𝛾𝛾̿𝑈𝑈𝑈𝑈 , the international relative commodity price equals the foreign autarky

price 𝜃𝜃1∗ /𝜃𝜃2∗ and 𝑐𝑐 is constant at 𝑐𝑐̿ given in (33).

Figure 4 illustrates the abovementioned relationship between 𝛾𝛾 and 𝑐𝑐, where 𝛾𝛾̅ equals

𝜑𝜑(𝜃𝜃1 /𝜃𝜃2 )𝜃𝜃1 /𝜃𝜃2∗ , 𝛾𝛾̿ equals 𝛾𝛾̿𝑈𝑈𝑈𝑈 in (33) and 𝛾𝛾𝑐𝑐 is given by (36). Because 𝜔𝜔 anyway declines andthe foreign terms of trade deteriorates as 𝛾𝛾 is larger, 𝑐𝑐 ∗ given in (29) decreases. Therefore, it

21

will never make the second inequality in (21) valid and the foreign country always achieves full
employment.

5-2. Case F-UF
In the case where point A is located in F-UF, where𝜃𝜃∗

𝛽𝛽

𝑢𝑢′−1 ( ) > 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1 ,
𝜌𝜌

𝜃𝜃2

𝜃𝜃

𝜃𝜃∗

𝛽𝛽

𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 � 1� 𝜃𝜃2∗ > 𝑢𝑢′−1 � � > 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1∗ )𝜃𝜃2∗ ,
𝜃𝜃2

𝜌𝜌

𝜃𝜃2

the home country always achieves full employment. As 𝛾𝛾 increases and 𝜔𝜔 declines, the foreigncountry initially suffers from underemployment, which is the same as in case F-U, but

eventually attains full employment, which is the same as in case F-F. This is because thedecline in 𝜔𝜔 lowers foreign potential income and makes foreign consumption 𝑐𝑐 ∗ lower than theupper bound 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌) , leading the foreign country from underemployment to full

employment. Because the home country always achieves full employment, from proposition 2
a rise in the relative price of the home commodity due to an increase in 𝛾𝛾 makes consumption
𝑐𝑐 larger in pattern (h1-f2), whether the foreign country is fully employed or underemployed.

From (17) in case F-F and (34) in case F-U, the minimum and maximum values of 𝑐𝑐 arerespectively 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜃𝜃1 /𝜃𝜃2 ) and 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜃𝜃1∗ /𝜃𝜃2∗ )𝜃𝜃1 . Thus, the relationship between 𝛾𝛾 and 𝑐𝑐

is similar to that in figure 1, although 𝛾𝛾̅ equals 𝛾𝛾̅𝐹𝐹𝐹𝐹 in (35) and 𝛾𝛾̿ equals 𝜑𝜑(𝜃𝜃1∗ /𝜃𝜃2∗ )𝜃𝜃1 /𝜃𝜃2∗ , asshown in (17).

5-3. Case FU-U
In the case where point A is located in FU-U of figure 2, where
𝜃𝜃∗

𝛽𝛽

𝜃𝜃

𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1∗ )𝜃𝜃1 > 𝑢𝑢′−1 ( ) > 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1)𝜃𝜃1 ,
𝜃𝜃2

𝜌𝜌

𝜃𝜃2

22

𝜃𝜃

𝛽𝛽

𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1 )𝜃𝜃2∗ > 𝑢𝑢′−1 ( ),𝜃𝜃2𝜌𝜌the foreign country is always underemployed because the lowest full employment consumption
𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜃𝜃1 /𝜃𝜃2 )𝜃𝜃2∗ exceeds the upper bound 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌). The home country attains full

employment (case F-U) when 𝛾𝛾 is small and faces underemployment (case U-U) when 𝛾𝛾 islarge. Because (34) holds in case F-U, 𝑐𝑐 equals 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜃𝜃1 /𝜃𝜃2 )𝜃𝜃1 when 𝛾𝛾 is smaller than 𝛾𝛾̅𝐹𝐹𝐹𝐹
given in (35).

When 𝛾𝛾 > 𝛾𝛾̅𝐹𝐹𝐹𝐹 , a larger 𝛾𝛾 raises the relative price of the home commodity, and hence thefull-employment level of 𝑐𝑐 increases toward its upper bound 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌), as shown in (34).
Because the foreign country is underemployed and 𝑐𝑐 ∗ given in lemma 2 satisfies (24), when 𝑐𝑐
reaches 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌), 𝛾𝛾 equals 𝛾𝛾𝑐𝑐′ that satisfies𝜃𝜃 𝜑𝜑(𝜔𝜔 )𝛽𝛽𝛾𝛾𝑐𝑐′ = 1 ∗ ∗ 𝑐𝑐 , 𝑐𝑐 = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔𝑐𝑐 )𝜃𝜃1 = 𝑢𝑢′−1 ( ), 𝑐𝑐(𝑥𝑥 ∗ ) = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔𝑐𝑐 )𝜃𝜃2∗ 𝑥𝑥 ∗ ,𝜃𝜃2 𝑥𝑥𝜌𝜌(37)

where 𝜔𝜔𝑐𝑐 is the same as that in (36) while 𝛾𝛾𝑐𝑐′ differs from 𝛾𝛾𝑐𝑐 in (36). When 𝛾𝛾 > 𝛾𝛾𝑐𝑐′ , bothcountries suffer from underemployment, which is the same as in case U-U. Therefore, from(27) an increase in 𝛾𝛾 worsens home employment by raising the relative price of the homecommodity 1/𝜔𝜔 and lowers 𝑐𝑐. Once 𝛾𝛾 reaches 𝛾𝛾̿𝑈𝑈𝑈𝑈 given in (28), a further rise in 𝛾𝛾 does notaffect 𝜔𝜔 and 𝑐𝑐 stays equal to 𝑐𝑐(𝑥𝑥̿ ) given in (28). Thus, the 𝑐𝑐 function has a similar shape as

illustrated in figure 4 with 𝛾𝛾𝑐𝑐 replaced by 𝛾𝛾𝑐𝑐′ .
5-4. Case U-UF

In the case where point A is located in U-UF of figure 2, where
𝜃𝜃

𝛽𝛽

𝜃𝜃

𝛽𝛽

𝜃𝜃∗

𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1 )𝜃𝜃1 > 𝑢𝑢′−1 ( ), 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1 )𝜃𝜃2∗ > 𝑢𝑢′−1 ( ) > 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1∗ )𝜃𝜃2∗ ,
𝜃𝜃2

𝜌𝜌

𝜃𝜃2

𝜌𝜌

𝜃𝜃2

the home country always suffers from underemployment while the foreign country isunderemployed (case U-U) when 𝛾𝛾 is small but fully employed (case U-F) when 𝛾𝛾 is large.

From proposition 1, whether the foreign country is fully employed or underemployed, a larger
𝛾𝛾 raises the relative price of the home commodity and hence decreases home employment 𝑥𝑥
23and consumption 𝑐𝑐 if the home country is underemployed in pattern (h1-f2). Thus, the 𝑐𝑐

function has a similar shape as illustrated in figure 3, where 𝛾𝛾̅ = 𝛾𝛾̅𝑈𝑈𝑈𝑈 given in (28), which isvalid in case U-U, and 𝛾𝛾̿ = 𝛾𝛾̿𝑈𝑈𝑈𝑈 given in (33), which is valid in case U-F.
5-5. Case FU-UF
Finally, in the case where point A is located in FU-UF, where
𝜃𝜃∗

𝛽𝛽

𝜃𝜃

𝛽𝛽

𝜃𝜃

𝜌𝜌𝜌𝜌 + 𝑝𝑝1 � 1∗ � 𝜃𝜃1 > 𝑢𝑢′−1 � � > 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 ( 1)𝜃𝜃1 ,
𝜃𝜃2

𝜌𝜌

𝜃𝜃2

𝜃𝜃∗

𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 � 1 � 𝜃𝜃2∗ > 𝑢𝑢′−1 � � > 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 ( 1∗ )𝜃𝜃2∗ ,
𝜃𝜃2

𝜌𝜌

𝜃𝜃2

as 𝛾𝛾 increases, the home country switches from fully employment to underemployment while

the foreign country switches in the reverse direction. From proposition 1, whether the foreigncountry is fully employed or underemployed, an increase in 𝛾𝛾 expands (reduces) homeconsumption 𝑐𝑐 if the home country is fully employed (underemployed). Therefore, the 𝑐𝑐

function has a similar shape as illustrated in figure 4, in which 𝛾𝛾̅ = 𝛾𝛾̅𝐹𝐹𝐹𝐹 given in (35) and 𝛾𝛾̿ =𝛾𝛾̿𝑈𝑈𝑈𝑈 given in (33).

The level of 𝛾𝛾 at which the home country turns from full employment tounderemployment is either 𝛾𝛾𝑐𝑐 in (36) or 𝛾𝛾𝑐𝑐′ in (37), depending on whether the foreign country is

still fully employed or has already entered the area of underemployment at the turning point.
From proposition 1, an increase in 𝛾𝛾 always lowers 𝜔𝜔. As 𝛾𝛾 increases and makes 𝜔𝜔 reach 𝜔𝜔𝑐𝑐∗ atwhich the full-employment level of 𝑐𝑐 ∗ in pattern (h1-f2) of (17) equals its upper bound𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌):𝜃𝜃

𝛽𝛽𝛾𝛾𝑐𝑐∗ = 𝜑𝜑(𝜔𝜔𝑐𝑐∗ ) � 1∗ � where 𝑐𝑐 ∗ = 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔𝑐𝑐∗ )𝜃𝜃2∗ = 𝑢𝑢′−1 ( ),𝜌𝜌𝜃𝜃2the foreign country turns from underemployment to full employment. Similarly, the homecountry turns from full employment to underemployment when 𝜔𝜔 equals 𝜔𝜔𝑐𝑐 given in (36).
24Therefore, if 𝜔𝜔𝑐𝑐 is larger than 𝜔𝜔𝑐𝑐∗ given above, 𝛾𝛾 reaches 𝛾𝛾𝑐𝑐 before 𝛾𝛾𝑐𝑐∗ and then the homecountry turns from full employment to underemployment while the foreign country isunderemployed. If 𝜔𝜔𝑐𝑐 < 𝜔𝜔𝑐𝑐∗ , 𝛾𝛾 reaches 𝛾𝛾𝑐𝑐∗ before 𝛾𝛾𝑐𝑐 and the foreign country turns fromunderemployment to full employment before the home country turns full employment tounderemployment. Then, the critical value of 𝛾𝛾 is 𝛾𝛾𝑐𝑐′ . In either case the 𝑐𝑐 function has a similar
shape as in figure 4.

6. Non-negligible lending and borrowing
The absolute value of 𝑏𝑏, which equals −𝛾𝛾𝑏𝑏 ∗ from (16), has so far been assumed to be

negligibly small so that the redistribution effect of lending and borrowing among households

within a country due to a population change is ignored. This section considers the case of
non-negligible 𝑏𝑏(= −𝛾𝛾𝑏𝑏 ∗ ) and discusses the redistribution effect.

International assets/debts are assumed to be held in real terms and evenly distributed

among households within each country. Therefore,𝐵𝐵

𝑏𝑏 = ,
𝐿𝐿

𝑏𝑏 ∗ =

𝐵𝐵∗
𝐿𝐿∗

𝐵𝐵

= − ∗,
𝐿𝐿

where 𝐵𝐵(∗) is the total net lending of the home (or foreign) country. They satisfy
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

𝑑𝑑𝑏𝑏∗

𝐵𝐵

= − 2 ⋚ 0 and
𝐿𝐿

𝑑𝑑𝑑𝑑

𝑑𝑑𝐿𝐿∗

∗ = 0,

𝑑𝑑𝐿𝐿

𝐵𝐵= ∗2 ⋛ 0 if and only if 𝐵𝐵 ⋛ 0,
𝐿𝐿

𝑑𝑑𝑏𝑏∗
𝑑𝑑𝑑𝑑

= 0.

From the above equations, if the absolute level of 𝐵𝐵(= −𝐵𝐵∗ ) is negligibly small, as has so

far been assumed, a change in 𝐿𝐿(∗) yields no redistribution of lending and borrowing among

each population. It affects 𝑥𝑥 (∗) and 𝑐𝑐 (∗) only through a change in 𝜔𝜔. Moreover, whether 𝑏𝑏 is

negligible or not, in the case of homothetic utility the relative commodity price 𝜔𝜔 does notdepend on 𝑏𝑏, as is clear from (17) in case F-F, (26) in case U-U, (29) in case U-F and (34) in25

case F-U. Therefore, in the case where 𝑏𝑏 is non-negligible, the effects of a change in 𝐿𝐿(∗) on

𝑥𝑥 (∗) and 𝑐𝑐 (∗) consist of those through a change in 𝜔𝜔, which do not depend on 𝑏𝑏, and those

directly through a change in 𝑏𝑏.

An increase in 𝐿𝐿∗ does not affect the home net lending per capita 𝑏𝑏 so that the previous

results are valid. However, an increase in 𝐿𝐿 reduces (or expands) 𝑏𝑏 if 𝑏𝑏 is positive (ornegative). From (17), (25) and (31), the effect on 𝑐𝑐 of a change in 𝐿𝐿 through the redistributionof 𝐵𝐵 isFull employment:

Underemployment:

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

=

𝜕𝜕[𝜌𝜌𝜌𝜌+𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 ]= 𝑐𝑐 ′ (𝑥𝑥)𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

=−

= 𝜌𝜌

𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

⋚ 0 ⟺ 𝐵𝐵 ⋛ 0,𝑐𝑐 ′ (𝑥𝑥)𝜌𝜌
𝜕𝜕𝜕𝜕
⋛0
𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 −𝑐𝑐 ′ (𝑥𝑥) 𝜕𝜕𝜕𝜕⟺ 𝐵𝐵 ⋛ 0.

It is added to the effect through a change in 𝜔𝜔 explored in the previous sections.

Note that the direction of the additional effect depends on whether the home country is

fully employed or not and whether it is a creditor or a debtor. If the home country is a creditor,
for example, the additional effect of a larger home population on home consumption throughredistribution of international lending is negative (or positive) in the case where the homecountry is fully employed (or underemployed). If the home country is a debtor, the result is just
opposite.

7. Conclusion
Nowadays many developed countries face long-run aggregate demand deficiency
although they have high potential production capacity and large wealth. This situation appears
if households have insatiable preference for money (or wealth) because the relative preference
for wealth accumulation over consumption does not decline as households accumulate wealth,preventing consumption from increasing. Therefore, if a country has large potential

26

productivity, consumption cannot be large enough to achieve full employment, leading tosecular demand stagnation.
In this case, the effect of a larger foreign population on home consumption is quite
opposite to that under full employment. A larger foreign population expands supply of theforeign commodity and raises the relative price of the home commodity, which increases homereal income as far as full employment prevails in the home country. Therefore, homeconsumption increases. If the country faces secular demand stagnation, however, a higher
relative price of the home commodity due to a larger foreign population decreases demand forthe home commodity, making home employment and realized income lower. This negative
effect dominates the positive effect of foreign demand expansion due to a larger foreign
population and hence home consumption decreases.
The same result obtains in the case of a smaller home population. A smaller home
population raises the relative price of the home commodity, which improves the home terms oftrade and increases home real income and consumption per capita as far as the home country
attains full employment. If home productivity is so high that home households do not consume
enough to achieve full employment, however, aggregate demand deficiency arises in the homecountry. In this case a rise in the relative price of the home commodity due to a smaller home
population decreases home employment and consumption per capita.

Appendix A: Stability
The Euler equation in (8) gives
𝜆𝜆∗
𝜆𝜆

=

𝑢𝑢′ (𝑐𝑐 ∗ )= 𝜅𝜅 = constant over time,𝑢𝑢′ (𝑐𝑐)𝑐𝑐 ∗ = 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�.

Therefore, in the case where (𝑥𝑥, 𝑥𝑥 ∗ ) = (1, 1), from (12), (13) and lemma 1 one finds

27

(A1)

h12-f2: 𝜔𝜔 =h1-f2:

𝜃𝜃2∗ 𝛾𝛾
𝜃𝜃1

𝜃𝜃1

𝜃𝜃

𝛿𝛿�𝜃𝜃1 �2𝜃𝜃
𝑝𝑝1 �𝜃𝜃1 �
2

𝜃𝜃2

,

𝛿𝛿(𝜔𝜔)= 𝜑𝜑(𝜔𝜔),h1-f12: 𝜔𝜔 =

𝜃𝜃1∗
𝜃𝜃2∗

𝑝𝑝1 (𝜔𝜔)

𝜃𝜃∗
𝜃𝜃2
𝜃𝜃∗
𝑝𝑝1 � 1∗ �𝜃𝜃2

𝛿𝛿� 1∗ �,

�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 𝑥𝑥1 ,𝜃𝜃2 (1−𝑥𝑥1 )+𝜃𝜃2∗ 𝛾𝛾
𝜃𝜃1 𝑥𝑥1

𝜃𝜃= 𝜑𝜑 � 1�,𝜃𝜃2

�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 ,�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 + 𝜃𝜃1∗ 𝛾𝛾𝑥𝑥1∗ ,𝜃𝜃2∗ 𝛾𝛾(1−𝑥𝑥1∗ )𝜃𝜃1 +𝜃𝜃1∗ 𝛾𝛾𝑥𝑥1∗

𝜃𝜃∗= 𝜑𝜑 � 1∗ �, (A2)𝜃𝜃2which fully determines 𝑐𝑐 once 𝜅𝜅 is given. Substituting this 𝑐𝑐, which is constant over time, into(9) yields

Applying this 𝜋𝜋 into (14) gives

𝑟𝑟 = 𝜌𝜌.

𝜋𝜋 =𝑣𝑣 ′ (𝑚𝑚)− 𝜌𝜌.
𝑢𝑢′ (𝑐𝑐)𝑣𝑣 ′ (𝑚𝑚)𝑚𝑚̇ = − � ′ (𝑐𝑐) − 𝜌𝜌� 𝑚𝑚,𝑢𝑢

which immediately determines 𝑚𝑚 (or 𝑃𝑃), and then 𝜋𝜋 = 0. From (15),𝑏𝑏̇ = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 − 𝑐𝑐,

where 𝜔𝜔 and 𝑐𝑐 are fully determined in (A2) once 𝜅𝜅 is given. Because 𝑏𝑏̇ is unstable with respect
to 𝑏𝑏, 𝜅𝜅 jumps to the level that makes 𝑏𝑏̇ = 0 and the new steady state is immediately reached.
In the case where 0 < 𝑥𝑥 < 1 and 𝑥𝑥 ∗ = 1 equations (12) and (13) and lemma 1 yield𝜃𝜃

h12-f2: 𝜔𝜔 = 1,
h1-f2:

𝜃𝜃2∗ 𝛾𝛾𝜃𝜃1 𝑥𝑥𝜃𝜃2𝜃𝜃𝛿𝛿�𝜃𝜃1 �2𝜃𝜃
𝑝𝑝1 �𝜃𝜃1 �
2= 𝜑𝜑(𝜔𝜔),h1-f12: 𝜔𝜔 =

𝜃𝜃1∗

∗,

𝜃𝜃2

�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 𝑥𝑥1 ,𝛿𝛿(𝜔𝜔)

𝑝𝑝1 (𝜔𝜔)

𝜃𝜃∗
𝜃𝜃2
𝜃𝜃∗
𝑝𝑝1 � 1∗ �𝜃𝜃2

𝛿𝛿� 1∗ �𝜃𝜃2 (𝑥𝑥−𝑥𝑥1 )+𝜃𝜃2∗ 𝛾𝛾
𝜃𝜃1 𝑥𝑥1

𝜃𝜃= 𝜑𝜑 � 1�,𝜃𝜃2

�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 𝑥𝑥,

�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 𝑥𝑥 + 𝜃𝜃1∗ 𝛾𝛾𝑥𝑥1∗ ,𝜃𝜃2∗ 𝛾𝛾(1−𝑥𝑥1∗ )𝜃𝜃∗1∗ = 𝜑𝜑 � ∗ �,𝜃𝜃1 𝑥𝑥+𝜃𝜃1∗ 𝛾𝛾𝑥𝑥1

𝜃𝜃2which gives 𝑥𝑥 as a function of 𝑐𝑐 once 𝜅𝜅 is given:

𝑥𝑥 = 𝑥𝑥(𝑐𝑐, 𝜅𝜅).

Because unemployment appears and deflation continues in the home country, from (20)𝑣𝑣 ′ (𝑚𝑚) = 𝛽𝛽. Therefore, from (9) and (23), one obtains an autonomous dynamic equation of 𝑐𝑐:
28

𝑐𝑐̇

𝜂𝜂 = Φ(𝑐𝑐) ≡𝑐𝑐𝛽𝛽𝑢𝑢′ (𝑐𝑐)− 𝜌𝜌 − 𝛼𝛼(𝑥𝑥(𝑐𝑐, 𝜅𝜅) − 1).

(A3)

Because in the present case the first inequality in (21) holds and henceΦ(𝑐𝑐 𝑓𝑓 ) ≡𝛽𝛽𝑢𝑢′ �𝑐𝑐 𝑓𝑓 �− 𝜌𝜌 > 0,in order for the steady-state 𝑐𝑐 to exist it must be valid thatΦ(0) < 0,and thenΦ′(𝑐𝑐) > 0,around the steady state, where 0 < 𝑐𝑐 < 𝑐𝑐 𝑓𝑓 . Thus, the dynamics of 𝑐𝑐 given by (A3) is unstable
and 𝑐𝑐 instantaneously jumps to the level that makes (A3) equal zero. This 𝑐𝑐 is a function of 𝜅𝜅and hence 𝑥𝑥(𝑐𝑐, 𝜅𝜅) is also determined for a given 𝜅𝜅. In patterns h12-f2 and h1-f12 𝜔𝜔 is constantand in pattern h1-f3 𝜔𝜔 is a function of 𝑥𝑥. By substituting those values to (30) gives the

dynamics of 𝑏𝑏, one finds it to be unstable with respect to 𝑏𝑏. Therefore, 𝜅𝜅 is determined so that𝑏𝑏̇ = 0 and then all variables jump to respective steady-state levels and stay there. The casewhere 𝑥𝑥 = 1 and 0 < 𝑥𝑥 ∗ < 1 is analogously treated.

Finally, in the case where both inequalities in (21) are valid and hence 0 < 𝑥𝑥 < 1 and 0 <𝑥𝑥 ∗ < 1, from (12), (26) and (A1) one finds
h12-f2:h1-f2:

h1-f12:𝜃𝜃𝛿𝛿�𝜃𝜃1 �2𝜃𝜃
𝑝𝑝1 �𝜃𝜃1 �
2

�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 𝑥𝑥1 ,𝛿𝛿(𝜔𝜔)
�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 𝑥𝑥,
𝑝𝑝1 (𝜔𝜔)
𝜃𝜃∗
𝜃𝜃2
𝜃𝜃∗
𝑝𝑝1 � 1∗ �𝜃𝜃2

𝛿𝛿� 1∗ �𝜃𝜃2 (𝑥𝑥−𝑥𝑥1 )+𝜃𝜃2∗ 𝛾𝛾𝑥𝑥 ∗𝜃𝜃2∗ 𝛾𝛾𝑥𝑥 ∗𝜃𝜃1 𝑥𝑥

𝜃𝜃1 𝑥𝑥1

𝜃𝜃= 𝜑𝜑 � 1 �;𝜃𝜃2

= 𝜑𝜑(𝜔𝜔);�𝑐𝑐 + 𝛾𝛾 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 𝑥𝑥 + 𝜃𝜃1∗ 𝛾𝛾𝑥𝑥1∗ ,𝜃𝜃2∗ 𝛾𝛾(𝑥𝑥 ∗ −𝑥𝑥1∗ )𝜃𝜃1 𝑥𝑥+𝜃𝜃1∗ 𝛾𝛾𝑥𝑥1∗

From (9) and (23),𝑐𝑐̇𝛽𝛽�𝜂𝜂 − 𝜌𝜌 =� ′ (𝑐𝑐) − 𝛼𝛼(𝑥𝑥 − 1) =𝑐𝑐𝑢𝑢

and hence together with (A4) one obtains

∗
𝛽𝛽
∗
∗ 𝑐𝑐̇
−
𝛼𝛼(𝑥𝑥
−
1)
�=
𝜂𝜂
− 𝜌𝜌�,
𝜅𝜅𝑢𝑢′ (𝑐𝑐)
𝑐𝑐 ∗

29

𝜃𝜃∗

= 𝜑𝜑 � 1∗ �.
𝜃𝜃2

(A4)

𝑥𝑥 = 𝑥𝑥(𝑐𝑐, 𝜅𝜅).

Therefore, the same logic as that for the stability of the dynamics of 𝑏𝑏 in the asymmetric casementioned above applies to the present case. The steady state is immediately reached.

In sum, in all cases the steady state is instantaneously reached and the economy stays there–i.e., lemma 2 holds.

Appendix B: Wealth preference
The household behavior with wealth preference is to maximize∞∫0 [𝑢𝑢(𝑐𝑐) + 𝑣𝑣(𝑚𝑚) + 𝜙𝜙(𝑎𝑎)] exp(−𝜌𝜌𝜌𝜌) 𝑑𝑑𝑑𝑑,

where 𝑣𝑣(𝑚𝑚) now represents the transaction motive and 𝜙𝜙(𝑎𝑎) is wealth preference, subject tothe flow budget equation and asset constraint:𝑎𝑎̇ = 𝑟𝑟𝑟𝑟 + 𝑤𝑤𝑤𝑤 − 𝑐𝑐 − 𝑅𝑅𝑅𝑅, 𝑎𝑎 = 𝑚𝑚 + 𝑏𝑏.

Given the Hamiltonian function of the utility maximization:

𝐻𝐻 = 𝑢𝑢(𝑐𝑐) + 𝑣𝑣(𝑚𝑚) + 𝜙𝜙(𝑎𝑎) + 𝜆𝜆(𝑟𝑟𝑟𝑟 + 𝑤𝑤𝑤𝑤 − 𝑐𝑐 − 𝑅𝑅𝑅𝑅),

the first-order optimal conditions are

They yield

𝜆𝜆 = 𝑢𝑢′ (𝑐𝑐), 𝜆𝜆𝜆𝜆 = 𝑣𝑣 ′ (𝑚𝑚),
𝑐𝑐̇

𝜂𝜂 =
𝑐𝑐

𝜆𝜆̇
𝜆𝜆

= 𝜌𝜌 − 𝑟𝑟 − 𝜙𝜙′(𝑎𝑎).

𝑣𝑣 ′ (𝑚𝑚)+𝜙𝜙′ (𝑎𝑎)− 𝜌𝜌 − 𝜋𝜋,𝑢𝑢′ (𝑐𝑐)

instead of (9). Replacing the property of (20) by
𝑣𝑣 ′ (∞) = 0,𝑅𝑅 =𝑣𝑣 ′ (𝑚𝑚),𝑢𝑢′ (𝑐𝑐)

𝜙𝜙 ′ (∞) = 𝛽𝛽 > 0,

and applying them to the above equations yields
𝛽𝛽𝑢𝑢′ (𝑐𝑐)= 𝜌𝜌 + 𝜋𝜋 and 𝑅𝑅 = 0 for 𝑚𝑚 → ∞.

30

The first equation is the same as (24) and the second equation implies a zero nominal interest
rate under stagnation with persistent deflation.

Appendix C: Derivation of the properties in (27) and (31)Because the inequalities in (21) are valid in case U-U, using (24) one obtains
𝑐𝑐(1) = 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌) < 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 ,

𝑐𝑐(1) = 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌) < 𝜌𝜌𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ ,implying that the left-hand side of each equation in (25) is smaller than the right-hand side
when 𝑥𝑥 = 1 (or 𝑥𝑥 ∗ = 1). Therefore, in order for the solution of 𝑥𝑥 (or 𝑥𝑥 ∗ ) in (25) to exist in therange of (0,1), the left-hand side of (25) must be larger than the right-hand side when 𝑥𝑥 = 0(or 𝑥𝑥 ∗ = 0) –i.e.,

𝑐𝑐(0) > 𝜌𝜌𝑏𝑏 (∗) .

Then, in the neighborhood of the solution of 𝑥𝑥 (or 𝑥𝑥 ∗ ),𝑐𝑐′(𝑥𝑥) < 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 ,The first property is the same as (31).

𝑐𝑐′(𝑥𝑥 ∗ ) < 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ .

(A5)

Totally differentiating the two equations in pattern (h1-f2) of (26) and applying 𝜑𝜑′ < 0 in(12) and the properties in (A5) to the results yields
Ω

Ω

𝑑𝑑𝑑𝑑
𝑥𝑥

𝑑𝑑𝑥𝑥 ∗𝑥𝑥 ∗𝑑𝑑𝑑𝑑= [𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ − 𝑐𝑐 ′ (𝑥𝑥 ∗ )]𝛾𝛾𝑥𝑥 ∗ �

𝛾𝛾

𝑑𝑑𝑑𝑑

= −[𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 − 𝑐𝑐 ′ (𝑥𝑥)]𝑥𝑥 �

𝛾𝛾

−

−

𝜑𝜑′
𝜑𝜑

𝜑𝜑′
𝜑𝜑

𝑑𝑑𝑑𝑑�,

𝑑𝑑𝑑𝑑�,

Ω = [𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 − 𝑐𝑐 ′ (𝑥𝑥)]𝑥𝑥 + [𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ − 𝑐𝑐 ′ (𝑥𝑥 ∗ )]𝛾𝛾𝑥𝑥 ∗ > 0.

Because 𝜑𝜑′ < 0 in (12), these equations and (A5) give∗𝑥𝑥�𝜔𝜔 (𝜔𝜔, 𝛾𝛾) > 0, 𝑥𝑥�𝛾𝛾 (𝜔𝜔, 𝛾𝛾) > 0; 𝑥𝑥�𝜔𝜔(𝜔𝜔, 𝛾𝛾) < 0, 𝑥𝑥�𝛾𝛾∗ (𝜔𝜔, 𝛾𝛾) < 0,

(A6)

where 𝑥𝑥� (∗) (𝜔𝜔, 𝛾𝛾) is shown in (26). From lemma 2 and the home current-account equation in(15) into which 𝑥𝑥�(𝜔𝜔, 𝛾𝛾) is substituted, one finds

31𝑏𝑏̇ = 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥�(𝜔𝜔, 𝛾𝛾) − 𝑐𝑐�𝑥𝑥�(𝜔𝜔, 𝛾𝛾)� = 0,

which gives (25). By assuming that 𝑏𝑏̇ satisfies the Marshall-Lerner condition, from (5), (A5)and (A6) one obtains∂𝑏𝑏̇∗𝑏𝑏̇𝜔𝜔 �≡ � = −𝛾𝛾𝑏𝑏̇𝜔𝜔
= 𝑝𝑝1′ (𝜔𝜔)𝜃𝜃1 𝑥𝑥 + [𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 − 𝑐𝑐′(𝑥𝑥)]𝑥𝑥�𝜔𝜔 > 0,
𝜕𝜕𝜕𝜕

∂𝑏𝑏̇
𝑏𝑏̇𝛾𝛾 (≡ ) = [𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 − 𝑐𝑐′(𝑥𝑥)]𝑥𝑥�𝛾𝛾 > 0,
𝜕𝜕𝜕𝜕

𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

From (5), (A5), (A6) and (A7),
𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑

= 𝑥𝑥�𝛾𝛾 + 𝑥𝑥�𝜔𝜔

𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

=

𝜔𝜔

(A7)

𝑏𝑏𝜔𝜔

𝑝𝑝1′ (𝜔𝜔)𝜃𝜃1 𝑥𝑥𝑥𝑥�𝛾𝛾
< 0,
𝑏𝑏̇𝜔𝜔

𝑝𝑝′ (𝜔𝜔)𝜃𝜃 ∗ 𝑥𝑥 ∗ 𝑥𝑥� ∗∗ 𝑑𝑑𝑑𝑑= 𝑥𝑥�𝛾𝛾∗ + 𝑥𝑥�𝜔𝜔
= 2 ̇ ∗2 𝛾𝛾 > 0,
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑏𝑏

𝑑𝑑𝑥𝑥 ∗

𝑏𝑏̇= − ̇ 𝛾𝛾 < 0.

𝑑𝑑𝑐𝑐 ∗

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑= 𝑐𝑐 ′ (𝑥𝑥)
∗

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑

< 0,

𝑑𝑑𝑥𝑥
= 𝑐𝑐 ′ (𝑥𝑥 ∗ )
> 0,
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

These imply the properties in (27).

𝛾𝛾𝑥𝑥∗

𝑑𝑑( 𝑥𝑥 )
𝑑𝑑𝑑𝑑

> 0.

REFERENCES
Adolfson M., S. Laséen, J. Lindé and M. Villani (2007) “Bayesian Estimation of an Open
Economy DSGE Model with Incomplete Pass-through”, Journal of International
Economics, Vol. 72, pp. 481-511.
——, ——, —— and —— (2008) “Evaluating an Estimated New Keynesian Small Open
Economy Model”, Journal of Economic Dynamics and Control, Vol. 32, pp. 2690-2721.
Aghion, P., A. Banerjee and T. Piketty (1999) “Dualism and Macroeconomic Volatility”,
Quarterly Journal of Economics, Vol. 114, pp. 1359-1397.
Akerlof, G. A. (1982) “Labor Contracts as Partial Gift Exchange”, Quarterly Journal of
Economics, Vol. 97, pp. 543-569.
—— and J. L. Yellen (1990) “The Fair-Wage Effort Hypothesis and Unemployment”,
Quarterly Journal of Economics, Vol. 105, pp. 255-284.
32

Azariadis, C. and L. Kaas (2016) “Capital Misallocation and Aggregate Factor Productivity”,
Macroeconomic Dynamics, Vol. 20, pp. 525-543.
Calvo, G. (1983) “Staggered Prices in a Utility Maximizing Framework”, Journal of Monetary
Economics, Vol. 12, pp. 383-398.
Christiano, L. J., M. Eichenbaum and C. L. Evance (2005) “Nominal Rigidities and the
Dynamic Effects of a Shock to Monetary Policy”, Journal of Political Economy, Vol.
113, pp. 1-45.
Erceg, C. J., W. H. Dale and A. T. Levin (2000) “Optimal Monetary Policy with Staggered
Wage and Price Contracts”, Journal of Monetary Economics, Vol. 46, pp. 281-313.
Eggertsson, G. B. and P. Krugman (2012) “Debt, Deleveraging, and the Liquidity Trap: A
Fisher-Minsky-Koo Approach”, Quarterly Journal of Economics, Vol. 127, pp. 1469
-1513.
Hall, R. (2011) “The Long Slump”, American Economic Review, Vol. 101, pp. 431-469.
Hayashi, F. and E. C. Prescott (2002) “The 1990s in Japan: A Lost Decade”, Review of
Economic Dynamics, Vol. 5, pp. 206-235.
Kehoe T. J. and E. C. Prescott (2002) “Great Depressions of the 20th Century”, Review of
Economic Dynamics, Vol. 5, pp. 1-18.
Kiyotaki, N. and J. Moore (1997) “Credit Cycle”, Journal of Political Economy, Vol. 105, pp.
211-248.
Krugman, P. R. (1998) “It’s Baaack: Japan’s Slump and the Return of the Liquidity Trap”,
Brookings Papers on Economic Activity, Vol. 2, pp. 137-187.
Liu, Z. and P. Wang (2014) “Credit Constraints and Self-fulfilling Business Cycles”, American
Economic Journal: Macroeconomics, Vol. 6, pp. 32-69.
Matsuyama, K. (2007) “Credit Traps and Credit Cycles”, American Economic Review, Vol. 97,
pp. 503-516.

33

McGratten, E. R. and E. C. Prescott (2012) “The Great Recession and Delayed Economic
Recovery”, in L. E. Ohanian, J. B. Taylor and I. J. Wright, eds., Government Policies andthe DELAYED Economic Recovery, Hoover Institution Press: Stanford University, pp.
115-154.
Michaillat, P. and E. Saez (2014) “An Economic Business-Cycle Model”, NBER Working
Paper 19777.
Michau, J.-B. (2015) “Secular Stagnation: Theory and Remedies”, hal-01211667, Centre
National de la Recherche Scientifique, Ecole Polytechnique.
Ono, Y. (1994) Money, Interest, and Stagnation, Clarendon Press: Oxford.
—— (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.
—— (2006) “International Asymmetry in Business Activity and Appreciation of a Stagnant
Country's Currency”, Japanese Economic Review, Vol. 57, pp. 101-120.
—— (2014) “International Economic Interdependence and Exchange-rate Adjustment under
Persistent Stagnation”, Japanese Economic Review, Vol. 65, pp. 70-92.
—— (2017) “Macroeconomic Interdependence between a Stagnant and a Fully EmployedCountry”, Japanese Economic Review, forthcoming, DOI: 10.1111/jere.12156.
—— and J. Ishida (2014) “On Persistent Demand Shortages: A Behavioral Approach”,
Japanese Economic Review, Vol. 65, pp. 42-69.
——, K. Ogawa and A. Yoshida (2004) “Liquidity Preference and Persistent Unemployment
with Dynamic Optimizing Agents”, Japanese Economic Review, Vol. 55, pp. 355-371.
Ono, Y. and A. Shibata (2006) “Long-run Specialization”, Review of International Economics,
Vol. 14, pp. 1-15.
Pissarides, C. A. (2000) Equilibrum Unemployment Theory, 2nd edition, Cambridge (MA):34

MIT Press.
Schmitt-Grohé, S. and M. Uribe (2016) “Downward Nominal Wage Rigidity, Currency Pegs,
and Involuntary Unemployment”, Journal of Political Economy, Vol. 124, No. 5, pp.
1466-1514.
—— and —— (2017) “Liquidity Traps and Jobless Recoveries”, American Economic Journal:
Macroeconomics, Vol. 9, No. 1, pp. 165-204.
Smets, F. and R. Wouters (2003) “An Estimated Stochastic Dynamic General Equilibrium
Model of the Euro Area”, Journal of the European Economic Association, Vol. 1, pp.
1123-1175.
—— and —— (2005) “Bayesian New Neoclassical Synthesis (NNS) Models: Modern Tools
for Central Banks”, Journal of the European Economic Association, Vol. 3, pp. 422-433.
—— and —— (2007) “Shocks and Frictions in US Business Cycles: A Bayesian DSGE
Approach”, American Economic Review, Vol. 97, pp. 586-606.
Yun, T. (1996) “Nominal Price Rigidity, Monetary Supply Endogeneity, and Business
Cycles”, Journal of Monetary Economics, Vol. 37, pp. 345-370.

35