Discussion Paper No. 893

MACROECONOMIC INTERDEPENDENCE
BETWEEN A STAGNANT AND
A FULLY EMPLOYED COUNTRY

Yoshiyasu Ono

January 2014
Revised November 2014
Secondly Revised June 2017

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

February 2017

Macroeconomic interdependence
between a stagnant and a fully employed country *Yoshiyasu Ono †Abstract
This paper presents a two-country two-commodity dynamic model where one country achieves
full employment and the other suffers from secular stagnation of aggregate demand. Own andspill-over effects of changes in preference, productivity and policy parameters are examined.
Parameter changes that improve the stagnant country’s current account, such as a reduction ingovernment purchases, a decrease in foreign aid and an improvement in productivity, raise therelative price of the home commodity. Consequently, home employment shrinks, deflation
worsens and consumption decreases. The terms of trade for the full-employment country
deteriorate. Thus, income and consumption decrease in both countries.

JEL classification: F32, F41, F35
Keywords: secular stagnation, unemployment, wealth preference, current account, fiscal
expansion.

* This research was started while the author visited the Center for Economic Studies (CES) of the University of

Munich (LMU). The author thanks G. Illing and the participants of the macroeconomics seminar held in LMU for
their valuable comments. Helpful comments by the participants of the INFINITI Conference on International
Finance and Hitotsubashi Economics Seminar are also gratefully acknowledged. This research is financially
supported by the Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science(15H05728 and 22530178) and the Joint Usage/Research Center Program by the Ministry of Education, Culture,
Sports, Science and Technology, Japan.
†

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
In a closed-economy setting an increase in productivity naturally expands national income
and consumption if full employment prevails. In a two-country setting it benefits not only thehome country but also the foreign country because it lowers the relative price of the homecommodity and improves the foreign terms of trade. The effect of an increase in homegovernment purchases on the home commodity is the same as that of a decrease in home
productivity: it harms both countries.
If the two countries suffer from aggregate demand stagnation, however, the result is quite
different. Ono (2006, 2014) presents a two-country dynamic model with persistent
unemployment in both countries and shows that an increase in home productivity excessively
improves the current account and leads the home currency and the relative price of the homecommodity to appreciate so much that home employment and consumption decrease. The risein the relative price of the home commodity in turn causes foreign employment andconsumption to increase. An increase in home government purchases works in the samemanner as a decrease in home productivity. It worsens the home current account and lowers therelative price of the home commodity, which increases home employment and consumption
and decreases foreign ones.
There is yet another important case: one country faces persistent deficiency of aggregate
demand while the other country realizes full employment. It may be the case of Japan and
Korea and that of Japan and China until recently. This paper focuses on this case and examines
the own and spillover effects of changes in various parameters including productivity andgovernment purchases. It shows that an improvement in the stagnant country’s productivity
reduces both countries’ consumption levels and that an increase in the stagnant country’s
government purchases expands them.

1The effect of an international transfer is also examined. It is naturally believed that foreign
aid makes the donor country worse off and the recipient country better off. Due to this belief
foreign-aid budgets are usually cut particularly when a country faces stagnation, as Japan did inthe ‘Lost Decades’. In some important policy discussions, however, benefits of foreign aid forboth recipient and donor countries in the asymmetric case have been emphasized. The MarshallPlan in 1947 was an example. George C. Marshall, the US Secretary of State at that time,
proposed an aid of $20 billion to European countries that had significantly lost supply
capacities in World War II and faced serious shortages of supply. He then stated that it would
benefit not only European consumers by enabling them to import US commodities but also US
producers and workers by expanding the market. 1 In the context of the North-South problem,
the Independent Commission on International Development Issues, a panel lead by former
German Chancellor Willy Brandt in the early 1980s (the Brandt commission, 1980, 1983),
argued that foreign aid from the North to the South would benefit the donors as well as therecipients through not only creating better political relations but also expanding imports of theSouth and increasing employment in the North. 2This paper examines the validity of such statements in a dynamic macroeconomic
framework. It is found that foreign aid expands consumption in both countries if the recipient
country achieves full employment and the donor country faces persistent deficiency ofaggregate demand. 3

1 He stated: “The Marshall Plan, it should be noted, benefited the American economy as well. The money

would be used to buy goods from the United States, and they had to be shipped across the Atlantic on American
merchant vessels.” See Congressional Record, 30 June 1947.
2 This issue was treated by Ono (2007) but he used CES utility functions. This paper extends that analysis tothe case of general homothetic utility.
3 In the literature such a controversial case has been mentioned as a transfer paradox. The paradox arises

through not an expansion of employment but a change in the terms of trade with full employment. It arises only inthe case of Walrasian instability, multiple equilibia or some distortions in a two-country case. See Bhagwati et al(1983, 1985) for a general analysis of the transfer paradox with distortions in static two-country and three-country
frameworks. Polemarchakis (1983) extended it to an n-country economy. The paradoxical case in this paper arises
through a change in employment.

2

The stagnation considered in the present analysis is not temporary but persistent and is due
to aggregate demand deficiency. In the literature, however, stagnation has mostly been taken to
be temporary. DSGE models, for example, regard macroeconomic fluctuations as caused byfirms’ and labor unions’ monopolistic pricing a la Calvo (1983) and analyze short-run
perturbations due to unanticipated policy and technology shocks around the steady state. They
are e.g. Yun (1996), Erceg et al. (2000), Smets and Wouters (2003, 2005, 2007), Christiano etal. (2005), and Adolfson et al. (2007, 2008). In these models demand shortages appear neither
during the adjustment process nor in the steady state. Stagnation is entirely generated by firms’monopolistic pricing after an exogenous economic shock. Krugman (1998) focuses on the zero
lower bound of the nominal interest rate and shows that with nominal price rigidities aggregate
demand deficiency appears. However, it appears only within the period in which an
unanticipated exogenous shock occurs after prices are determined, and disappears in all thesubsequent periods.
There are also researchers who consider a credit constraint with which capital allocation is
distorted (Kiyotaki and Moore, 1997; Aghion et al., 1999; Matsuyama, 2007; Hall, 2011; Liuand Wang, 2014; Azariadis and Kaas, 2016). They take stagnation as a supply-side
phenomenon: capital misallocation due to a credit constraint reduces productivity and yields
productivity stagnation. By empirically analyzing the Japanese and the US economies, Kehoeand Prescott (2002), Hayashi and Prescott (2002) and McGratten and Prescott (2012) also
insist that recent stagnations are due to declines in productivity. However, it may be hard toconsider that in the early 1990s Japan’s productivity suddenly stopped increasing and has neverimproved for more than two decades.
While these models may well fit short-run macroeconomic fluctuations, the present paper
considers secular stagnation such as Japan’s Lost Decades and possibly the EU’s GreatRecession triggered by the financial crisis of 2008. In the IMF conference of 2013, Summers3

(2013) warned against relying too much on the DSGE approach in solving economic crises andemphasized the need to work on long-run recessions rather than short-run businessfluctuations. This paper follows this line of thought and considers secular stagnation of
aggregate demand.
A dynamic optimization model of secular stagnation due to aggregate demand deficiency
has firstly been presented by Ono (1994, 2001). He shows that if the marginal utility of money(or wealth) holding is insatiable, aggregate demand deficiency and deflation persistently
emerge. Although real money balances (or wealth) expand through the deflation, the marginal
rate of substitution between money and consumption does not change because of theinsatiability, implying that the wealth effect on consumption disappears. While lower prices asa result of deflation will not stimulate consumption, deflation itself makes it moreadvantageous for people to save more and consume less, leading to a steady state with seculardeflation and stagnation. 4 This stagnation mechanism has recently been discussed also by
Michaillat and Saez (2014) and Michau (2015).
There is another strand of research on secular stagnation of aggregate demand. Eggertsson
and Mehrotra (2014) introduce a borrowing constraint to an OLG model and show that with thezero lower bound of the nominal interest rate the lending and borrowing market may
persistently be uncleared. The intertemporal allocation of consumption is distorted by the

4 This model has recently been developed in various ways. Matsuzaki (2003) and Hashimoto (2004) introduce

heterogeneous households to the model and examine the effect of redistribution policy on aggregate demand.
Johdo (2006) considers the effect of R&D subsidies on aggregate demand and employment. Ono (2010) applies
the model to analyze Japan's long-run stagnation and economic policies. Murota and Ono (2011) show that if
households consider money as a target of status, aggregate demand deficiency appears. Hashimoto and Ono(2011) apply this type of stagnation model to examine the effects of child allowances on the population growth
and aggregate demand. Murota and Ono (2012) consider a preference for deposit holdings and show secularstagnation, zero nominal interest rates and excess reserves of commercial banks to appear. Murota and Ono (2015)
reformulate the model and propose a Keynesian-multiplier-like analysis with quite different implications from theconventioanl ones. Illing et al. (2016) introduce to this model heterogenous households with different timepatience and a constraint on the less patient households’ borrowing and find that relaxing the borrowing constraint
may temporarily expand but eventually reduces aggregate demand.

4

borrowing constraint, causing consumption shortages to occur. 5 Such a stagnation will
disappear if the government removes the borrowing constraint and supplies enough lending topeople.
In the model of secular demand stagnation due to people’s insatiable wealth preference, incontrast, secular demand stagnation occurs without such market distortions as the borrowing
constraint. Preference for wealth holding over consumption makes consumption too small toachieve full employment especially when the potential productivity is very large, as indeveloped countries. The present paper extends this model to a two-country framework andanalyzes the case where the home country suffers from secular demand stagnation while theforeign country achieves full employment.

2. The model
There are two countries, home and foreign. The home production sector specializes incommodity 1 and the foreign one in commodity 2. Both of them are competitive and use only
labor to produce the respective commodities with constant productivities 𝜃𝜃1 and 𝜃𝜃2∗ . Variables

without (or with) ∗ denote those of the home (or foreign) country below.

All households in the two countries have the same utility function of the two commodities.

The function is homothetic. Therefore, for a given level of aggregate consumption 𝑐𝑐 (or 𝑐𝑐 ∗ ),the function is summarized as

(∗)

(∗)

where 𝑐𝑐𝑖𝑖(∗)

𝑢𝑢(𝑐𝑐 (∗) ) ≡ 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ),is each country’s consumption of commodity 𝑖𝑖 (𝑖𝑖 = 1, 2) and satisfies
(∗)

(∗)

𝑝𝑝1 (𝜔𝜔)𝑐𝑐1 = 𝛿𝛿(𝜔𝜔)𝑐𝑐 (∗) , 𝑝𝑝2 (𝜔𝜔)𝑐𝑐2 = [1 − 𝛿𝛿(𝜔𝜔)]𝑐𝑐 (∗) ,5 Eggertsson et al. (2016) have extended this model to an open-economy setting. Eggertsson (2010) shows

aggregate demand stagnation to appear in a representative agent model with the borrowing constraint. However,the stagnation is a temporary one.

5

1 > 𝛿𝛿(𝜔𝜔) > 0, 𝛿𝛿′(𝜔𝜔) ≥ 0.

(1)

𝛿𝛿(𝜔𝜔) is the ratio of expenditure on commodity 1, 𝜔𝜔 is the relative price of commodity 2 tocommodity 1, and 𝑝𝑝𝑖𝑖 (𝜔𝜔) is the real price of commodity 𝑖𝑖 (𝑖𝑖 = 1, 2). Real prices 𝑝𝑝1 (𝜔𝜔) and𝑝𝑝2 (𝜔𝜔) satisfy

𝑃𝑃

𝑃𝑃∗

𝑃𝑃∗

𝑃𝑃

𝑝𝑝1 (𝜔𝜔) = 𝑃𝑃1 = 𝑃𝑃1∗,

𝑝𝑝2 (𝜔𝜔) = 𝑃𝑃2 = 𝑃𝑃2∗ = 𝜔𝜔𝑝𝑝1 (𝜔𝜔),𝑃𝑃

𝑃𝑃1∗ = 𝜖𝜖1,𝑃𝑃2 = 𝜖𝜖𝑃𝑃2∗ ,(2)

where 𝑃𝑃 and 𝑃𝑃 ∗ are the home and foreign general price indices, 𝑃𝑃𝑖𝑖 and 𝑃𝑃𝑖𝑖∗ are the nominal

prices of commodity 𝑖𝑖 measured in the home currency and the foreign currency, respectively,and 𝜖𝜖 is the nominal exchange rate.

The first and third equations (or alternatively the second and fourth equations) in (2) yield
𝑃𝑃 = 𝜖𝜖𝑃𝑃∗ .

The time derivative of this equation gives𝜖𝜖̇𝜋𝜋 = 𝜖𝜖 + 𝜋𝜋 ∗ ,where 𝜋𝜋 and 𝜋𝜋 ∗ are respectively the two countries’ inflation rates. Therefore, from the

non-arbitrage condition between home and foreign assets:𝜖𝜖̇𝑅𝑅 = 𝜖𝜖 + 𝑅𝑅 ∗ ,where 𝑅𝑅 and 𝑅𝑅 ∗ respectively represent the home and foreign nominal interest rates, one obtains
𝑅𝑅 − 𝜋𝜋 = 𝑟𝑟 = 𝑅𝑅 ∗ − 𝜋𝜋 ∗ ,(3)

implying that the real interest rate 𝑟𝑟 is internationally the same.

Because the home and foreign general price indices 𝑃𝑃 and 𝑃𝑃∗ satisfy
𝑑𝑑𝑃𝑃 (∗)
𝑃𝑃 (∗)= 𝛿𝛿(𝜔𝜔)

(∗)

𝑑𝑑𝑃𝑃1

(∗)
𝑃𝑃1

+ (1 − 𝛿𝛿(𝜔𝜔))

(∗)

𝑑𝑑𝑃𝑃2

(∗)
𝑃𝑃2

,

as shown by Deaton and Muellbauer (1980, p.175), from (2) one finds𝑝𝑝′ (𝜔𝜔)1and hence𝑝𝑝′ (𝜔𝜔)

0 = 𝛿𝛿(𝜔𝜔) 𝑝𝑝1 (𝜔𝜔) + (1 − 𝛿𝛿(𝜔𝜔)) 𝑝𝑝2 (𝜔𝜔),261𝑝𝑝′ (𝜔𝜔)𝑝𝑝′ (𝜔𝜔)+ 𝑝𝑝1 (𝜔𝜔) = 𝑝𝑝2 (𝜔𝜔),𝜔𝜔12

𝑝𝑝1′ (𝜔𝜔) = −(1−𝛿𝛿)𝑝𝑝1
𝜔𝜔𝛿𝛿𝑝𝑝

𝑝𝑝2′ (𝜔𝜔) = 𝜔𝜔2 > 0.

< 0,

(4)

The home and foreign representative households have the same subjective discount rate 𝜌𝜌and utility functional:∞∫0 �𝑢𝑢(𝑐𝑐 (∗) ) + 𝑣𝑣(𝑚𝑚(∗) )� exp(−𝜌𝜌𝜌𝜌) 𝑑𝑑𝑑𝑑,
𝑢𝑢′ > 0, 𝑢𝑢′′ < 0, 𝑣𝑣′ > 0, 𝑣𝑣′′ < 0,where 𝑣𝑣(𝑚𝑚(∗) ) represents utility of money. They maximize it subject to the flow budget
equation and the asset constraint:𝑎𝑎̇ (∗) = 𝑟𝑟𝑎𝑎(∗) + 𝑤𝑤 (∗) 𝑥𝑥 (∗) − 𝑐𝑐 (∗) − 𝑅𝑅 (∗) 𝑚𝑚(∗) − 𝑧𝑧 (∗) ,𝑎𝑎(∗) = 𝑚𝑚(∗) + 𝑏𝑏 (∗) ,(5)

where 𝑤𝑤 (∗) is the real wage, 𝑧𝑧 (∗) is the lump-sum tax, and 𝑥𝑥 (∗) is the realized employment.

Real total assets 𝑎𝑎(∗) consist of real money balances 𝑚𝑚(∗) and real international asset 𝑏𝑏 (∗) while
the firm value is zero under the linear technology and perfect competition. The real rate ofinterest 𝑟𝑟 is internationally the same, as shown by (3). The two countries’ labor endowments

are normalized to 1 and may be underemployed. Therefore, realized employment 𝑥𝑥 (∗) also

implies the employment rate.

The optimal household behavior is summarized as𝑐𝑐̇ (∗)

𝑣𝑣 ′ �𝑚𝑚(∗) �

𝑣𝑣 ′ �𝑚𝑚(∗) �𝜌𝜌 + 𝜂𝜂(∗) 𝑐𝑐 (∗) = 𝑢𝑢′ �𝑐𝑐 (∗)� − 𝜋𝜋 (∗) , 𝑅𝑅 (∗) = 𝑢𝑢′ �𝑐𝑐 (∗)� ,(6)of which the derivation is set out in Appendix I. The first property shows the Ramsey equation
while the second represents the equality between the liquidity premium of money and thenominal interest rate, which gives the money demand function.
(∗)

The home (or foreign) government imposes lump-sum tax 𝑧𝑧 (∗) and purchases 𝑔𝑔𝑖𝑖

commodity 𝑖𝑖 (𝑖𝑖 = 1, 2). Therefore,

(∗)

(∗)

𝑧𝑧 (∗) = 𝑝𝑝1 (𝜔𝜔)𝑔𝑔1 + 𝑝𝑝2 (𝜔𝜔)𝑔𝑔2 .
7

of

(7)

From the home and foreign demand functions of the two commodities in (1), world
demand for commodity 1 and that for commodity 2 are respectively 𝛿𝛿(𝜔𝜔)(𝑐𝑐 + 𝑐𝑐 ∗ )/𝑝𝑝1 (𝜔𝜔) and[1 − 𝛿𝛿(𝜔𝜔)](𝑐𝑐 + 𝑐𝑐 ∗ )/𝑝𝑝2 (𝜔𝜔). Given the outputs of the two commodities, the international

competitive markets perfectly adjust so that𝛿𝛿(𝜔𝜔)

𝑝𝑝1 (𝜔𝜔)(𝑐𝑐 + 𝑐𝑐 ∗ ) + 𝑔𝑔1 + 𝑔𝑔1∗ = 𝜃𝜃1 𝑥𝑥,

1−𝛿𝛿(𝜔𝜔)(𝑐𝑐 + 𝑐𝑐 ∗ ) + 𝑔𝑔2 + 𝑔𝑔2∗ = 𝜃𝜃2∗ 𝑥𝑥 ∗ ,𝑝𝑝2 (𝜔𝜔)(8)

where 𝜃𝜃1 is the home productivity and 𝜃𝜃2∗ is the foreign productivity. Real balances 𝑚𝑚(∗)

satisfy𝑀𝑀(∗)

𝑚𝑚(∗) = 𝑃𝑃(∗) ,(9)where 𝑀𝑀(∗) represents nominal money supply. It is assumed that 𝑀𝑀(∗) is constant over time but
may be increased in a once-and-for-all manner. Because the above equation yields
𝑚𝑚̇(∗) = −𝜋𝜋 (∗) 𝑚𝑚(∗) ,and 𝑧𝑧 (∗) satisfies (7), the flow budget equation in (5) reduce to(10)International asset holdings 𝑏𝑏 and 𝑏𝑏 ∗ always satisfy

(11)

(∗)
(∗)
𝑏𝑏̇ (∗) = 𝑟𝑟𝑏𝑏 (∗) + 𝑤𝑤 (∗) 𝑥𝑥 (∗) − 𝑐𝑐 (∗) − [𝑝𝑝1 (𝜔𝜔)𝑔𝑔1 + 𝑝𝑝2 (𝜔𝜔)𝑔𝑔2 ].

𝑏𝑏 + 𝑏𝑏 ∗ = 0.

3. The condition for the asymmetric steady state to arise
This section presents the condition for the asymmetric case to arise, in which the homecountry faces persistent unemployment while the foreign country achieves full employment.
Before doing so, let us obtain the steady state in which both countries attain full employment asa benchmark.
In the steady state with full employment in both countries, where8𝑥𝑥 = 1, 𝑥𝑥 ∗ = 1,

prices, wages and consumption 𝑐𝑐 (∗) are constant, and hence from (3), (6) in which 𝜋𝜋 (∗) = 0,
and (9),𝜌𝜌 =𝑣𝑣 ′ (𝑀𝑀/𝑃𝑃),
𝑢𝑢′ (𝑐𝑐)𝜌𝜌 =𝑣𝑣 ′ (𝑀𝑀∗ /𝑃𝑃 ∗ ).
𝑢𝑢′ (𝑐𝑐 ∗ )(12)Current accounts 𝑏𝑏̇ and 𝑏𝑏̇ ∗ given in (10) are zero, 𝑏𝑏 ∗ = −𝑏𝑏 from (11), and
𝑤𝑤 = 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 ,𝑤𝑤 ∗ = 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ ,under perfect competition. Therefore, 𝑐𝑐 and 𝑐𝑐 ∗ are

𝑐𝑐 = 𝑐𝑐 𝑓𝑓 ≡ 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝜃𝜃1 − [𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝑔𝑔1 + 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝑔𝑔2 ],𝑐𝑐 ∗ = 𝑐𝑐 𝑓𝑓∗ ≡ −𝜌𝜌𝜌𝜌 + 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝜃𝜃2∗ − [𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝑔𝑔1∗ + 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝑔𝑔2∗ ],where 𝜔𝜔 𝑓𝑓 satisfies (8) in which 𝑥𝑥 = 1 and 𝑥𝑥 ∗ = 1 and thus𝜔𝜔 𝑓𝑓 𝛿𝛿�𝜔𝜔 𝑓𝑓 �𝜃𝜃 −𝑔𝑔 −𝑔𝑔∗= 𝜃𝜃1∗ −𝑔𝑔1 −𝑔𝑔1∗ .
1−𝛿𝛿�𝜔𝜔 𝑓𝑓 �
2

2

2

(13)

(14)

Substituting 𝑐𝑐 𝑓𝑓 and 𝑐𝑐 𝑓𝑓∗ given in (13) respectively to 𝑐𝑐 and 𝑐𝑐 ∗ in (12) gives the steady statelevels of 𝑃𝑃 and 𝑃𝑃∗ .

However, in the presence of a liquidity trap the steady state presented above may not exist.

In the present setting a liquidity trap emerges if the utility of money is insatiable and its
marginal utility 𝑣𝑣 ′ (𝑚𝑚) has a positive lower bound 𝛽𝛽, 6lim 𝑣𝑣 ′ (𝑚𝑚) = 𝛽𝛽 > 0,

𝑚𝑚→∞

(15)

because under this property the home money demand function given in (6) satisfies
𝑅𝑅 =𝑣𝑣 ′ (𝑚𝑚)𝛽𝛽(> 𝑢𝑢′ (𝑐𝑐)𝑢𝑢′ (𝑐𝑐)

for any 𝑚𝑚).

𝑅𝑅 remains to be strictly positive as 𝑚𝑚 increases, implying a liquidity trap. 7 In this case, if andonly if 𝑐𝑐 𝑓𝑓 , the home consumption under full employment given in (13), is so large as to satisfy
6 Ono(1994, 2001) assumes this property in a closed-economy setting and proves that the dynamic
equilibrium path uniquely exists and converges to a steady state with persistent unemployment. The validity ofthis property is empirically shown by Ono, Ogawa and Yoshida (2004) using parametric and non-parametricapproaches.

9𝛽𝛽𝜌𝜌 < 𝑢𝑢′ �𝑐𝑐 𝑓𝑓 � �<

𝑣𝑣 ′ (𝑀𝑀/𝑃𝑃)𝑢𝑢′ �𝑐𝑐 𝑓𝑓 �

for any 𝑃𝑃�,

(16)

there is no 𝑃𝑃 that makes the first equation in (12) valid. Therefore, there is no dynamic pathalong which full employment is reached in the home country. Given that the left- andright-hand sides of (16) respectively represent the time preference rate and the liquidity
premium, (16) shows the case where the marginal desire for holding money dominates that for
consumption if the household consumes enough to realize full employment. The same
argument applies to the foreign country.
If the home country cannot reach full employment while the foreign country can in thesteady state, (16) is valid in the home country while foreign full-employment consumption 𝑐𝑐 𝑓𝑓∗
given in (13) satisfies𝛽𝛽𝜌𝜌 > 𝑢𝑢′ �𝑐𝑐 𝑓𝑓∗�so that there is 𝑃𝑃∗ that satisfies the second equation of (12). From (13), (16) and the above

inequality, the asymmetric case emerges when 8𝛽𝛽𝜌𝜌 < 𝑢𝑢′ �𝑐𝑐 𝑓𝑓 � ,𝛽𝛽𝜌𝜌 > 𝑢𝑢′ �𝑐𝑐 𝑓𝑓∗� ,(∗)

where 𝑐𝑐 𝑓𝑓 and 𝑐𝑐 𝑓𝑓∗ are given by (13). Thus, the asymmetric case in which 𝑔𝑔𝑖𝑖

appears if𝛽𝛽𝛽𝛽𝜌𝜌𝜌𝜌 > max �𝑢𝑢′−1 �𝜌𝜌 � − 𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝜃𝜃1 , 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝜃𝜃2∗ − 𝑢𝑢′−1 �𝜌𝜌 ��.

= 0 (for 𝑖𝑖 = 1, 2)(17)This condition is valid when home productivity 𝜃𝜃1 is high while foreign productivity 𝜃𝜃2∗ is low,
and/or when the home country owns large foreign assets 𝑏𝑏. This paper focuses on this case.

7 If 𝑣𝑣 ′ (𝑚𝑚) remains positive as 𝑚𝑚 expands, 𝑣𝑣(𝑚𝑚) may not represent the transaction motive. It would rather

imply wealth preference. In appendix II this is taken into account and the wealth preference and thenarrowly-defined transaction motive are separately introduced to the present model. It is shown that the following
analysis is still valid and that the home nominal interest rate is zero in the steady state.
8 Ono (2014) deals with the case where the first inequality is valid but the second one is not; that is, the casewhere neither country has a full-employment steady state.

104. Wage adjustment and the steady state
In the asymmetric case the home country cannot reach full employment. Therefore,sluggish wage adjustments under labor demand shortages must be present. The dominantsetting of price and wage adjustments in DSGE models is that of Calvo (1983). However, it
does not suit the present analysis because it allows neither aggregate demand shortages nor
involuntary unemployment any time. Aggregate demand shortages are explicitly analyzed byKrugman (1998) in a discrete-time setting. However, it cannot treat long-run shortages ofaggregate demand because the length of each period is exogenously given and aggregate
demand shortages appear only in the period in which an exogenous shock occurs. In contrast,
Ono and Ishida (2014) propose a dynamic extension of the fair wage model by Akerlof (1982)
and Akerlof and Yellen (1990), which can deal with unemployment as well as full employment
during the adjustment process and in the steady state. Therefore, it is adopted.

4.1. Wage and price adjustments
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 𝛼𝛼, as insearch models (e.g., Pissarides, 2000). 9 Thus, in the home country employment 𝑥𝑥 follows
𝑥𝑥̇ = −𝛼𝛼𝛼𝛼 + 𝜒𝜒,(18)

where 𝜒𝜒 is the number of the newly hired. While workers are employed, they form fair wages
in mind by referring to their past wages and the unemployment level of the society. Moreprecisely, at the end of the previous period (𝑡𝑡 − ∆𝑡𝑡) each employed worker conceives the
9 The Calvo model also assumes an exogenous Poison rate at which a firm (or a labor union) has a chance ofrevising prices (or wages) at each time point. The present 𝛼𝛼 instead represents a timing of revising the fair wage
because some of the incumbent workers that conceive fair wages leave the job, as mentioned later.

11

rightful wage 𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡) so that the realized wage 𝑊𝑊(𝑡𝑡 − ∆𝑡𝑡) is the average of the rightful

wage of the employed, 𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡), and the income of the unemployed, which is zero. Because

the number of the employed and that of the unemployed are respectively 𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡) and 1 −𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡), the rightful wage 𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡) satisfies

𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡)𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡) + 0×[1 − 𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡)] = 𝑊𝑊(𝑡𝑡 − ∆𝑡𝑡).

(19)

Having the rightful wage calculated at the end of the previous period in mind, at the
beginning of period 𝑡𝑡 the employed take the average of the rightful wage 𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡) and thezero income of the unemployed as the fair wage 𝑊𝑊𝐹𝐹 (𝑡𝑡). The newly hired, of which the numberis 𝜒𝜒(𝑡𝑡)∆𝑡𝑡, have no preconception about the fair wage or the rightful wage and simply follow
the fair wage of the employed. Therefore, when the employed calculate the fair wage 𝑊𝑊𝐹𝐹 (𝑡𝑡) at

the beginning of period 𝑡𝑡, the total number of workers that they care is 1 − 𝜒𝜒(𝑡𝑡)∆𝑡𝑡 while thenumber of the employed is 𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡)(1 − 𝛼𝛼∆𝑡𝑡). 𝑊𝑊𝐹𝐹 (𝑡𝑡) is then
𝑊𝑊𝐹𝐹 (𝑡𝑡) =𝜈𝜈(𝑡𝑡−∆𝑡𝑡)𝑥𝑥(𝑡𝑡−∆𝑡𝑡)(1−𝛼𝛼∆𝑡𝑡)
1−𝜒𝜒(𝑡𝑡)∆𝑡𝑡

.

(20)

If there is unemployment in the home country, the firm will set wage 𝑊𝑊 equal to the fairwage 𝑊𝑊𝐹𝐹 because 𝑊𝑊𝐹𝐹 is the lowest wage under which the employees properly work.

Therefore, from (19) and (20) 𝑊𝑊 changes as follows:𝑊𝑊(𝑡𝑡)−𝑊𝑊(𝑡𝑡−∆𝑡𝑡)∆𝑡𝑡= 𝜒𝜒(𝑡𝑡)𝑊𝑊(𝑡𝑡) − 𝛼𝛼𝑊𝑊(𝑡𝑡 − ∆𝑡𝑡).

Reducing ∆𝑡𝑡 to zero yields the continuous dynamics of 𝑊𝑊:𝑊𝑊̇𝑊𝑊= 𝜒𝜒 − 𝛼𝛼,(21)which implies that 𝑊𝑊 is updated at the replacement pace of the workers. The commodity price𝑃𝑃1 always adjusts to equal 𝑊𝑊 ⁄𝜃𝜃1 since there is no commodity supply if 𝑃𝑃1 < 𝑊𝑊 ⁄𝜃𝜃1 and excess

commodity supply if 𝑃𝑃1 > 𝑊𝑊 ⁄𝜃𝜃1 .

12

If full employment prevails (𝑥𝑥 = 1), in contrast, the firm tries to pick out workers fromrival firms to expand the market share by raising 𝑊𝑊 from 𝑊𝑊𝐹𝐹 so long as 𝜃𝜃1 𝑃𝑃1 > 𝑊𝑊, leading 𝑊𝑊

to equal 𝜃𝜃1 𝑃𝑃1 . The same argument applies to the foreign wages.

Thus, each commodity price, 𝑃𝑃1 in the home country and 𝑃𝑃2∗ in the foreign country,

follows the movement of 𝑊𝑊 (or 𝑊𝑊 ∗ ) given by (21) when there is unemployment. When full
employment maintains, in contrast, 𝑊𝑊 (or 𝑊𝑊 ∗ ) follows the movement of 𝑃𝑃1 (or 𝑃𝑃2∗ ). Anyway,
they satisfy

𝑊𝑊

𝑊𝑊 ∗

𝑤𝑤

(= 𝑝𝑝 (𝜔𝜔)) = 𝜃𝜃1 ,
𝑃𝑃
1

𝑃𝑃2∗

1

𝑤𝑤 ∗

(= 𝑝𝑝 (𝜔𝜔)) = 𝜃𝜃2∗ .

(22)

2

4.2. The asymmetric steady stateThis subsection obtains the asymmetric steady state, while the dynamic stability is
analyzed in appendix III. In the asymmetric case, where𝑥𝑥 ∗ = 1,

𝑥𝑥 < 1,

𝑊𝑊 follows (21) in the home country while 𝑊𝑊 ∗ and 𝑃𝑃2∗ always adjust so that full employment is

attained in the foreign country. Then, the market equilibrium conditions given by (8) reduce to
𝑐𝑐 + 𝑐𝑐 ∗ =𝑝𝑝2 (𝜔𝜔)1−𝛿𝛿(𝜔𝜔)1𝛿𝛿(𝜔𝜔)(𝑐𝑐 + 𝑐𝑐 ∗ ) + 𝑔𝑔1 + 𝑔𝑔1∗ �.
1 𝑝𝑝1 (𝜔𝜔)(𝜃𝜃2∗ − 𝑔𝑔2∗ − 𝑔𝑔2 ),They provide 𝑥𝑥 as a function of 𝜔𝜔:

𝑥𝑥 = 𝜃𝜃 �
1

𝑥𝑥 = 𝑥𝑥�(𝜔𝜔; 𝜃𝜃1 , 𝑔𝑔1 , 𝑔𝑔2 , 𝑔𝑔1∗ , 𝜃𝜃2∗ − 𝑔𝑔2∗ ) ≡ 𝜃𝜃 �1𝜃𝜃∗ −𝑔𝑔∗ −𝑔𝑔2 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝑥𝑥�𝜔𝜔 = � 2 𝜃𝜃21�

(1−𝛿𝛿)2

𝜔𝜔𝜔𝜔(𝜔𝜔)1−𝛿𝛿(𝜔𝜔)(23)(𝜃𝜃2∗ − 𝑔𝑔2∗ − 𝑔𝑔2 ) + 𝑔𝑔1 + 𝑔𝑔1∗ �,

𝑐𝑐+𝑐𝑐 ∗ 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′= �𝑝𝑝 𝜃𝜃 �
1 1(1−𝛿𝛿)𝜔𝜔

> 0,

(24)

where 𝑥𝑥� is the home employment level that clears the two commodity markets for a given 𝜔𝜔.
Note that 𝑥𝑥�𝜔𝜔 > 0 because a rise in 𝜔𝜔, the relative price of the foreign commodity, naturally
increases world demand for the home commodity and expands home employment 𝑥𝑥.

13

Because the relative price 𝜔𝜔 is constant in the steady state, from the first equation of (22),the two nominal prices, general price index 𝑃𝑃 and nominal wage 𝑊𝑊 move in parallel. The

employment rate 𝑥𝑥 is also constant, and hence from (18) 𝜒𝜒 = 𝛼𝛼𝛼𝛼, implying that the number ofjob separation equals that of new employees. Therefore, from (21) one finds𝑊𝑊̇

𝜋𝜋 = 𝑊𝑊 = 𝛼𝛼(𝑥𝑥 − 1),(25)

i.e., the price adjustment is of the Walrasian type in the steady state. Because 𝑥𝑥 < 1, 𝑃𝑃 keeps

declining, which makes 𝑣𝑣 ′ (𝑚𝑚) = 𝛽𝛽. Substituting this 𝜋𝜋 into the Ramsey equation given in (6)and making 𝑐𝑐̇ = 0 yields
𝛽𝛽𝑢𝑢′ (𝑐𝑐)− 𝜌𝜌 − 𝛼𝛼(𝑥𝑥 − 1) = 0 → 𝑐𝑐 = 𝑐𝑐̂ (𝑥𝑥; 𝛼𝛼, 𝛽𝛽).

(26)

In the steady state the home current account 𝑏𝑏̇ in (10) equals zero and the two equations in(6) in which 𝑐𝑐̇ = 0 yield 𝑟𝑟(= 𝑅𝑅 − 𝜋𝜋) = 𝜌𝜌. Thus, using the first equation in (22) one obtains
𝑏𝑏̇ = 𝜓𝜓(𝜔𝜔, 𝑥𝑥; 𝑏𝑏, 𝛼𝛼, 𝛽𝛽, 𝜃𝜃1 , 𝑔𝑔1 , 𝑔𝑔2 )≡ 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥 − [𝑝𝑝1 (𝜔𝜔)𝑔𝑔1 + 𝑝𝑝2 (𝜔𝜔)𝑔𝑔2 ] − 𝑐𝑐̂ (𝑥𝑥; 𝛼𝛼, 𝛽𝛽) = 0,
𝑥𝑥 = 𝑥𝑥�(𝜔𝜔; 𝜃𝜃1 , 𝑔𝑔1 , 𝑔𝑔2 , 𝑔𝑔1∗ , 𝜃𝜃2∗ − 𝑔𝑔2∗ ),(27)

where 𝑥𝑥� and 𝑐𝑐̂ are given by (24) and (26), respectively. Equation (27) determines 𝜔𝜔 .
Consequently, from (24) 𝑥𝑥 is determined. Substituting 𝑥𝑥 thus obtained into (26) yields 𝑐𝑐.
Foreign consumption 𝑐𝑐 ∗ is obtained from the foreign current-account equation in (10) to which

𝑥𝑥 ∗ = 1, 𝑟𝑟 = 𝜌𝜌, 𝑏𝑏̇ ∗ = 0, (11) and the second equation in (22) are applied:

𝑐𝑐 ∗ = 𝑐𝑐̂ ∗ (𝜔𝜔; 𝑏𝑏, 𝑔𝑔1∗ , 𝜃𝜃2∗ − 𝑔𝑔2∗ ) ≡ −𝜌𝜌𝜌𝜌 + 𝑝𝑝2 (𝜔𝜔)(𝜃𝜃2∗ − 𝑔𝑔2∗ ) − 𝑝𝑝1 (𝜔𝜔)𝑔𝑔1∗.

(28)

All the variables are thus determined.

Furthermore, from the analysis on the existence of the asymmetric steady state and the
dynamics around it, presented in appendix III, one obtains the following lemma:

14Lemma: The asymmetric steady state exists when (17) is valid. If changes in policy,
productivity and preference parameters occur in the asymmetric steady state, a new steady statein which 𝑐𝑐̇ (∗) = 0, 𝑏𝑏̇ (∗) = 0, 𝑚𝑚̇ > 0 and 𝑚𝑚̇∗ = 0 is immediately reached without any interim
process. Thus, on the new dynamic equilibrium path 𝜔𝜔, 𝑥𝑥, 𝑐𝑐 and 𝑐𝑐 ∗ are constant while 𝑚𝑚continues to expand, keeping 𝑣𝑣 ′ (𝑚𝑚) = 𝛽𝛽.

There should be an adjustment process of 𝑐𝑐 and 𝑏𝑏 if 𝑣𝑣 ′ (𝑚𝑚) changed as 𝑚𝑚 gradually

expanded with deflation. However, in the initial steady state 𝑣𝑣 ′ (𝑚𝑚) already reached 𝛽𝛽 andstays the same in the new steady state –i.e., gradual changes in 𝑚𝑚 due to deflation do not affect𝑣𝑣 ′ (𝑚𝑚) and hence neither 𝑐𝑐 nor 𝑏𝑏 has any adjustment process. A new steady state isimmediately reached after a parameter shift occurs.

Note that full employment cannot be reached by price and wage adjustments in the homecountry, as mentioned below. As shown in (22), home firms’ pricing behavior leads to
𝑤𝑤 = 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 ,while nominal wage 𝑊𝑊 and general price index 𝑃𝑃 decline in parallel, yielding deflation.

Because 𝜔𝜔 stays constant in the asymmetric steady state, real wage 𝑤𝑤 does not change andhence full employment is not achieved by the wage adjustment. Instead, employment adjusts toaggregate demand which depends on the deflation rate.

5. Own and spillover effects of parameter changes
This section analyzes the effects of changes in preference, productivity and policy
parameters on the relative price 𝜔𝜔 , home employment rate 𝑥𝑥 , and the two countries’
consumption 𝑐𝑐 and 𝑐𝑐 ∗ when the economy is initially in the asymmetric steady state. As

declared in the lemma, a new steady state is immediately reached after a parameter shift and15thus the effects on the steady-state values of the variables also represent the effects on the
dynamic paths of the variables.
For simplicity, it is considered that those parameter shifts occur in the neighborhood where(𝑔𝑔1∗ , 𝑔𝑔2∗ ) = (0,0),(𝑔𝑔1 , 𝑔𝑔2 ) = (0,0),𝑏𝑏 = 0.

(29)Note that under (29) the condition for the asymmetric case to arise, given by (17), turns to𝛽𝛽𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝜃𝜃1 > 𝑢𝑢′−1 �𝜌𝜌 � > 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝜃𝜃2∗ ,which is the case when the value of home potential production is high enough while that offoreign potential production is low enough.
With the Marshall-Lerner condition a rise in 𝜔𝜔 improves the home current account. 10

Therefore, from (4), (24), (27) and (29), one finds𝛿𝛿 2 (1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′∂𝑏𝑏̇= 𝜓𝜓𝜔𝜔 + 𝜓𝜓𝑥𝑥 𝑥𝑥�𝜔𝜔 = 𝑥𝑥�𝜔𝜔 �� 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑝𝑝1 𝜃𝜃1 − 𝑐𝑐̂𝑥𝑥 � > 0,
𝜕𝜕𝜕𝜕

(30)

𝜓𝜓𝑥𝑥 = 𝑝𝑝1 𝜃𝜃1 − 𝑐𝑐̂𝑥𝑥 > 0.

(31)

where

which yields𝜓𝜓𝜔𝜔 = −𝜃𝜃2∗ 𝛿𝛿𝑝𝑝1 < 0,𝜓𝜓𝑥𝑥 = 𝑝𝑝1 𝜃𝜃1 − 𝑐𝑐̂𝑥𝑥 ,An increase in home employment 𝑥𝑥 expands home production 𝑝𝑝1 𝜃𝜃1 𝑥𝑥 by 𝑝𝑝1 𝜃𝜃1 and homeconsumption 𝑐𝑐 by 𝑐𝑐̂𝑥𝑥 . The former effect improves, and the latter effect worsens, the homecurrent account. The inequality in (31) implies that the current-account improvement throughthe production expansion dominates the current-account deterioration through theconsumption expansion.
From (30), a decrease in 𝜔𝜔 worsens the home current account 𝑏𝑏̇. From the lemma, the newsteady state where 𝑏𝑏̇ = 0 is immediately reached after a parameter change. Therefore, if thehome current account represented by (27) turns to be positive after a parameter change, 𝜔𝜔 must

decreases so that the home current-account balance recovers. Formally, it is restated as follows:
10 In appendix III it is proved that the Marshall-Lerner condition is valid if the asymmentric steady state exists.

16

Proposition 1: A parameter change that improves the home current account raises the relative
price of the home commodity 1/𝜔𝜔 (or equivalently, decreases the relative price of the foreigncommodity 𝜔𝜔).

From (27), the partial derivative of the home current account 𝑏𝑏̇ with respect to a parameter𝑣𝑣 is generally represented as
∂𝑏𝑏̇
𝜕𝜕𝜕𝜕

= 𝜓𝜓𝑣𝑣 + 𝜓𝜓𝑥𝑥 𝑥𝑥�𝑣𝑣 = 𝜓𝜓𝑣𝑣 + (𝑝𝑝1 𝜃𝜃1 − 𝑐𝑐̂𝑥𝑥 )𝑥𝑥�𝑣𝑣 for 𝑣𝑣 = 𝑏𝑏, 𝛼𝛼, 𝛽𝛽, 𝑔𝑔1∗ , 𝜃𝜃2∗ − 𝑔𝑔2∗ , 𝜃𝜃1 , 𝑔𝑔1 , 𝑔𝑔2,

(32)

where 𝜓𝜓𝑣𝑣 and 𝑥𝑥�𝑣𝑣 are obtained from (24) and (27) as follows:𝜓𝜓𝑏𝑏 = 𝜌𝜌, 𝜓𝜓𝛼𝛼 = −𝑐𝑐̂𝛼𝛼 , 𝜓𝜓𝛽𝛽 = −𝑐𝑐̂𝛽𝛽 ; 𝑥𝑥�𝑣𝑣 = 0 for 𝑣𝑣 = 𝑏𝑏, 𝛼𝛼 and 𝛽𝛽;1𝜔𝜔𝜔𝜔𝜓𝜓𝑣𝑣 = 0 for 𝑣𝑣 = 𝑔𝑔1∗ and 𝜃𝜃2∗ − 𝑔𝑔2∗ ; 𝑥𝑥�𝑔𝑔1∗ = 𝜃𝜃 , 𝑥𝑥�𝜃𝜃2∗ −𝑔𝑔2∗ = (1−𝛿𝛿)𝜃𝜃 ;
𝑥𝑥

1

1

1

𝜔𝜔𝜔𝜔

𝜓𝜓𝜃𝜃1 = 𝑝𝑝1 𝑥𝑥, 𝜓𝜓𝑔𝑔1 = −𝑝𝑝1, 𝜓𝜓𝑔𝑔2 = −𝑝𝑝2 ; 𝑥𝑥�𝜃𝜃1 = − 𝜃𝜃 , 𝑥𝑥�𝑔𝑔1 = 𝜃𝜃 , 𝑥𝑥�𝑔𝑔2 = − (1−𝛿𝛿)𝜃𝜃 .
1

1

1

(33)

The first line in (33) shows that 𝑏𝑏, 𝛼𝛼 and 𝛽𝛽 vary the home current account 𝑏𝑏̇ without directly
changing 𝑥𝑥 (𝑥𝑥�𝑣𝑣 = 0), while the second line shows that 𝑔𝑔1∗ and 𝜃𝜃2∗ − 𝑔𝑔2∗ vary 𝑏𝑏̇ only throughchanges in 𝑥𝑥 (𝜓𝜓𝑣𝑣 = 0). The other parameters, 𝜃𝜃1 , 𝑔𝑔1 and 𝑔𝑔2 , vary 𝑏𝑏̇ through both the channels,
as shown on the third line.

From (27), (30) and (32), a change in parameter 𝑣𝑣 affects 𝜔𝜔 as follows:
∂𝑏𝑏̇

∂𝜔𝜔

∂𝑏𝑏̇

= − 𝜕𝜕𝜕𝜕 / 𝜕𝜕𝜕𝜕 = −𝜕𝜕𝜕𝜕𝜓𝜓𝑣𝑣 +(𝑝𝑝1 𝜃𝜃1 −𝑐𝑐̂𝑥𝑥 )𝑥𝑥�𝑣𝑣
𝛿𝛿2 (1−𝛿𝛿)+𝜔𝜔𝛿𝛿′�𝑝𝑝1 𝜃𝜃1 −𝑐𝑐̂𝑥𝑥 �𝑥𝑥�𝜔𝜔 ��𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′

,

(34)

where the denominator is positive from (30). From (24) and (34), the effect on home
employment 𝑥𝑥 is

∂𝑥𝑥
𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

= 𝑥𝑥�𝑣𝑣 + 𝑥𝑥�𝜔𝜔 𝜕𝜕𝜕𝜕 = −From (26) the effect on 𝑐𝑐 is𝛿𝛿(1−𝛿𝛿)2
�𝑝𝑝 𝜃𝜃 𝑥𝑥�𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ 1 1 𝑣𝑣
2′
(1−𝛿𝛿)+𝜔𝜔𝛿𝛿𝛿𝛿��𝑝𝑝1 𝜃𝜃1 −𝑐𝑐̂𝑥𝑥
𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′𝜓𝜓𝑣𝑣 +�

17

.

(35)

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

= 𝑐𝑐̂𝑣𝑣 + 𝑐𝑐̂𝑥𝑥 �𝜕𝜕𝜕𝜕� for 𝑣𝑣 = 𝛼𝛼, 𝛽𝛽;
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

= 𝑐𝑐̂𝑥𝑥 �𝜕𝜕𝜕𝜕�

𝜂𝜂𝜂𝜂

for 𝑣𝑣 = 𝑏𝑏, 𝑔𝑔1∗ , 𝜃𝜃2∗ − 𝑔𝑔2∗ , 𝜃𝜃1 , 𝑔𝑔1 , 𝑔𝑔2;
𝜂𝜂𝜂𝜂

𝜂𝜂𝜂𝜂

𝑐𝑐̂𝑥𝑥 = 𝛼𝛼/(𝑢𝑢′ (𝑐𝑐)𝑐𝑐 ) > 0, 𝑐𝑐̂𝛼𝛼 = (𝑥𝑥 − 1)/(𝑢𝑢′ (𝑐𝑐)𝑐𝑐 ) < 0, 𝑐𝑐̂𝛽𝛽 = − 𝑐𝑐 < 0,(36)

while from (4) and (28) the effect on foreign consumption 𝑐𝑐 ∗ is𝜕𝜕𝑐𝑐 ∗∂𝜔𝜔∗= 𝑐𝑐̂𝑣𝑣∗ + 𝑐𝑐̂𝜔𝜔
� 𝜕𝜕𝜕𝜕 � for 𝑣𝑣 = 𝑏𝑏, 𝑔𝑔1∗ , 𝜃𝜃2∗ − 𝑔𝑔2∗ ,𝜕𝜕𝜕𝜕𝜕𝜕𝑐𝑐 ∗∗
𝑐𝑐̂𝜔𝜔=𝛿𝛿𝑝𝑝2 𝜃𝜃2∗
𝜔𝜔

𝜕𝜕𝜕𝜕

> 0,

∂𝜔𝜔

∗
= 𝑐𝑐̂𝜔𝜔� 𝜕𝜕𝜕𝜕 � < 0 for 𝑣𝑣 = 𝛼𝛼, 𝛽𝛽, 𝜃𝜃1 , 𝑔𝑔1 , 𝑔𝑔2;

𝑐𝑐̂𝑏𝑏∗ = −𝜌𝜌 < 0,𝑐𝑐̂𝑔𝑔∗1∗ = −𝑝𝑝1 < 0,𝑐𝑐̂𝜃𝜃∗2∗ −𝑔𝑔2∗ = 𝑝𝑝2 > 0,(37)

in the neighborhood of (29). The properties in (37) are standard under full employment. An
increase in 𝜔𝜔 implies an improvement in the foreign terms of trade and thus expands foreign
consumption 𝑐𝑐 ∗ . An increase in 𝑏𝑏, or equivalently a decrease in 𝑏𝑏 ∗, lowers foreign interest

earnings and reduces 𝑐𝑐 ∗ . An increase in 𝑔𝑔1∗ decreases foreign disposable income andconsumption 𝑐𝑐 ∗ . An increase in 𝜃𝜃2∗ − 𝑔𝑔2∗ raises foreign disposable income and 𝑐𝑐 ∗ .

Having these properties in mind, let us analyze the effects of various parameters on therelative price 𝜔𝜔, home employment 𝑥𝑥 and consumption 𝑐𝑐, and foreign consumption 𝑐𝑐 ∗ .
Home foreign-asset position 𝒃𝒃, wage adjustment speed 𝜶𝜶 and liquidity preference 𝜷𝜷:These parameter changes directly affect the home current account without varying home
employment 𝑥𝑥, as shown by (32) and (33).

A transfer from the foreign country to the home country, which is represented by anincrease in 𝑏𝑏 , raises home interest earnings 𝜌𝜌𝜌𝜌 and improves the home current account.

Therefore, from proposition 1, the relative price of the home commodity 1/𝜔𝜔 rises, which is

explicitly obtained from (33) and (34). The decline in 𝜔𝜔 reduces world demand for the homecommodity and lowers home employment 𝑥𝑥, as found from (33) and (35). It worsens deflation,18

making the saving desire dominate the consumption desire, and decreases home consumption
𝑐𝑐, as obtained from (36). Foreign consumption 𝑐𝑐 ∗ also decreases because the transfer (viz., adecrease in 𝑏𝑏 ∗ ) reduces the foreign country’s asset holdings and moreover the decline in 𝜔𝜔
worsens the foreign terms of trade, as is clear from (37).

An increase in the wage adjustment speed 𝛼𝛼 worsens deflation and decreases homeconsumption 𝑐𝑐 while an increase in liquidity preference 𝛽𝛽 directly reduces consumption, bothof which improve the home current account. Hence, from proposition 1, the relative price of thehome commodity 1/𝜔𝜔 rises. This property is mathematically obtained from (33) and (34). The

rise in the home relative price 1/𝜔𝜔 reduces world demand for the home commodity and home
employment 𝑥𝑥, as obtained from (33) and (35), worsening deflation. Home consumption 𝑐𝑐

decreases due to the worse deflation and the initial negative effects of increases in 𝛼𝛼 and 𝛽𝛽, as

explicitly represented by (36). Because the decline in 𝜔𝜔 worsens the foreign terms of trade,
foreign consumption 𝑐𝑐 ∗ also decreases, as seen from (37).
The above results are summarized thus:

Proposition 2: A transfer from the foreign to the home country, an improvement in the home
wage adjustment speed and an increase in home liquidity preference improve the home current
account and raise the relative price of the home commodity. Thus, world demand for the homecommodity and home employment decrease, which in turn worsens deflation and decreases
home consumption. Foreign consumption also decreases because of the decline in the relative
price of the foreign commodity.

Foreign productivity 𝜽𝜽∗𝟐𝟐 , foreign government purchases on the home commodity 𝒈𝒈∗𝟏𝟏 andforeign government purchases on the foreign commodity 𝒈𝒈∗𝟐𝟐 : These parameters change
home employment 𝑥𝑥 and thereby vary the home current account 𝑏𝑏̇, as shown by (32) and (33).
19

Because an increase in 𝑔𝑔1∗ raises demand for the home commodity, home employment 𝑥𝑥

increases and hence the home current account improves. An increase in 𝜃𝜃2∗ − 𝑔𝑔2∗ enlarges

foreign disposable income and stimulates foreign demand for the home commodity, which
expands home employment 𝑥𝑥 and improves the home current account. Therefore, fromproposition 1, the relative price of the foreign commodity 𝜔𝜔 declines, which is mathematically
proved from (31), (33) and (34).

The decline in the foreign relative price 𝜔𝜔 reduces world demand for the home commodity

and home employment 𝑥𝑥, while the initial increases in 𝑔𝑔1∗ and 𝜃𝜃2∗ − 𝑔𝑔2∗ expand 𝑥𝑥. From (33)
and (35), the total effects on 𝑥𝑥 are positive; that is, the decline in 𝜔𝜔 is so vast that 𝑥𝑥 eventually

decreases, as mentioned below. Suppose that the decline in 𝜔𝜔 triggered by the policy changes
is large enough to decrease 𝑥𝑥 to the initial level. Then, the current account represented by (27)is still positive in the neighborhood of (29) because for the same 𝑥𝑥 the decline in 𝜔𝜔 makes
home total output 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥 higher than the initial value. Therefore, 𝜔𝜔 further declines so as torecover the current-account balance, making 𝑥𝑥 and 𝑐𝑐 lower than the initial levels.

Foreign consumption 𝑐𝑐 ∗ decreases with an increase in 𝑔𝑔1∗ because an increase in 𝑔𝑔1∗ notonly enlarges foreign tax burdens but also worsens the foreign terms of trade 𝜔𝜔. An increase in𝜃𝜃2∗ − 𝑔𝑔2∗ also worsens the foreign terms of trade but expands foreign disposable income.
Therefore, 𝑐𝑐 ∗ may increase or decrease. The effect on foreign consumption 𝑐𝑐 ∗ is calculated

from (33), (34) and (37) as follows:𝜕𝜕𝑐𝑐 ∗

𝜕𝜕(𝜃𝜃2∗ −𝑔𝑔2∗ )𝜕𝜕𝜕𝜕𝑝𝑝 �𝛿𝛿(1−𝛿𝛿)2 +𝜔𝜔𝛿𝛿 ′ �𝜃𝜃∗𝜔𝜔𝛿𝛿 ′

2
2∗= 𝑐𝑐̂𝜃𝜃∗2∗ −𝑔𝑔2∗ + 𝑐𝑐̂𝜔𝜔= (1−𝛿𝛿)
2 (𝜓𝜓 +𝜓𝜓 𝑥𝑥� ) �𝜃𝜃 � ��𝛿𝛿(1−𝛿𝛿)2 +𝜔𝜔𝛿𝛿 ′ � 𝑝𝑝1 𝜃𝜃1 − 𝑐𝑐̂𝑥𝑥 �.
𝜕𝜕(𝜃𝜃∗ −𝑔𝑔∗ )
2

2

𝜔𝜔

𝑥𝑥 𝜔𝜔

1

Because the coefficient of 𝑝𝑝1 𝜃𝜃1 in the square bracket of the above equation is smaller than thatin (30), as proven below,𝜔𝜔𝛿𝛿 ′

𝛿𝛿(1−𝛿𝛿)2 +𝜔𝜔𝛿𝛿 ′𝛿𝛿 2 (1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝛿𝛿 3 (1−𝛿𝛿)3− 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ = − [𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ ][𝛿𝛿(1−𝛿𝛿)2 +𝜔𝜔𝛿𝛿′ ] < 0,20𝜕𝜕𝑐𝑐 ∗ /𝜕𝜕(𝜃𝜃2∗ − 𝑔𝑔2∗ ) can either be positive or negative under the condition of (36), depending on
the magnitudes of the exogenous parameters that determine 𝜔𝜔.
The above results are summarized as follows:

Proposition 3: An improvement in foreign productivity 𝜃𝜃2∗ and a decrease in foreign

government purchases on the foreign commodity 𝑔𝑔2∗ increase foreign disposable income andforeign import, which improves the home current account. An increase in foreign government

purchases on the home commodity 𝑔𝑔1∗ straightforwardly improves the home current account.
Consequently, they raise the relative price of the home commodity. The rise is so vast thathome employment and consumption eventually decrease. Foreign consumption may increaseor decrease with an improvement in 𝜃𝜃2∗ and a decrease in 𝑔𝑔2∗ while it always decreases with anincrease in 𝑔𝑔1∗ .

Home productivity 𝜽𝜽𝟏𝟏 , government purchases on the home commodity 𝒈𝒈𝟏𝟏 and on theforeign commodity 𝒈𝒈𝟐𝟐 : These parameters vary the home current account both directly andindirectly by changing home employment 𝑥𝑥, as shown by (32) and (33).

A rise in home productivity 𝜃𝜃1 improves the current account, as found from (32) and (33),and hence from proposition 1 the relative price of the home commodity 1/𝜔𝜔 rises, which

worsens home employment 𝑥𝑥. Furthermore, a rise in 𝜃𝜃1 requires less workers to produce thesame amount. These two effects decrease 𝑥𝑥, which is obtained from (33) and (35), and reduce

home consumption 𝑐𝑐, as is clear from (36). The deterioration in the foreign terms of tradedecreases foreign consumption 𝑐𝑐 ∗ , as shown by (37).

An increase in home government purchases on the home commodity 𝑔𝑔1 works in the

opposite manner as an increase in 𝜃𝜃1 . It expands home employment 𝑥𝑥, mitigates deflation andstimulates home consumption 𝑐𝑐, which worsens the home current account and raises 𝜔𝜔. The
21

rise in 𝜔𝜔 expands home employment and consumption. It also improves the foreign terms oftrade and increases foreign consumption.

If full employment prevails in the home country, an increase in home productivity 𝜃𝜃1lowers, and an increase in home government purchases on the home commodity 𝑔𝑔1 raises, therelative price of the home commodity. These properties are opposite to those in the present
case. This is because in the presence of unemployment those parameter changes affect
employment 𝑥𝑥 without varying demand of the home commodity or production (which equals𝜃𝜃1 𝑥𝑥 − 𝑔𝑔1 ) while under full employment they vary it (which equals 𝜃𝜃1 − 𝑔𝑔1) without changing
employment (𝑥𝑥 = 1).

An increase in home government purchases on the foreign commodity 𝑔𝑔2 directly worsens

the home current account by importing more the foreign commodity. It also decreases home
disposable income and hence for a given 𝜔𝜔 total demand for the home commodity declines,
which reduces home employment 𝑥𝑥 and worsens the home current account. These two effects

worsen the home current account, as is clear from (31) and (32). Thus, from proposition 1, therelative price of the home commodity 1/𝜔𝜔 decreases. It makes home employment 𝑥𝑥 increaseso much as to dominate the initial decrease in 𝑥𝑥 and eventually 𝑥𝑥 becomes larger than theinitial level, as mentioned below. Suppose that the rise in 𝜔𝜔 makes 𝑥𝑥 recover the initial level.
Because the rise in 𝜔𝜔 makes 𝑝𝑝1 (𝜔𝜔) lower than the initial level, the home current account in(27) is still negative. Therefore, 𝜔𝜔 further rises and makes 𝑐𝑐 and 𝑥𝑥 larger than the initial levels.

These properties are mathematically obtained from (33), (35) and (36). The rise in 𝜔𝜔 improves
the foreign terms of trade and increases foreign consumption 𝑐𝑐 ∗ , as obtained from (37).
The following proposition summarizes the abovementioned results.

Proposition 4: An expansion in home productivity and decreases in home government
purchases on the home and foreign commodities improve the home current account.
22

Consequently, they raise the relative price of the home commodity (or equivalently lower the
relative price of the foreign commodity) so much that home employment and consumption
eventually decrease. Foreign consumption also decreases as a result of the deterioration in theforeign terms of trade.

Homogeneous commodities: If the home and foreign commodities are perfectly substitutable,
viz. 𝛿𝛿 ′ (𝜔𝜔) = ∞, or equivalently the two commodities are homogenous, then
𝑝𝑝1 = 1, 𝑝𝑝2 = 1, 𝜔𝜔 = 1, 𝑔𝑔 = 𝑔𝑔1 + 𝑔𝑔2 , 𝑔𝑔∗ = 𝑔𝑔1∗ + 𝑔𝑔2∗ .

(38)

In this case the current account given by (27) reduces to𝑏𝑏̇ = 𝜌𝜌𝜌𝜌 + 𝜃𝜃1 𝑥𝑥 − 𝑔𝑔 − 𝑐𝑐̂ (𝑥𝑥; 𝛼𝛼, 𝛽𝛽) = 0,

which determines 𝑥𝑥 and 𝑐𝑐(= 𝑐𝑐̂ (𝑥𝑥; 𝛼𝛼, 𝛽𝛽)). From the total differentiation of the above equation
one derives

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜌𝜌

= − 𝜃𝜃 −𝑐𝑐̂ < 0,
𝑐𝑐̂

1

𝑥𝑥

𝛼𝛼
= 𝜃𝜃 −𝑐𝑐̂
< 0,
𝜕𝜕𝜕𝜕
1

𝑥𝑥

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜃𝜃1

𝑥𝑥

= − 𝜃𝜃 −𝑐𝑐̂ < 0,
𝑐𝑐̂

1

𝑥𝑥

𝛽𝛽
= 𝜃𝜃 −𝑐𝑐̂
< 0,
𝜕𝜕𝜕𝜕
1

𝑥𝑥

𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

1

= 𝜃𝜃 −𝑐𝑐̂ > 0.
1𝑥𝑥

= 0 for 𝑣𝑣 = 𝑔𝑔∗ , 𝜃𝜃2∗ .

Note that these are also obtained from (33) and (35) to which 𝑝𝑝1 = 1 and 𝛿𝛿 ′ (𝜔𝜔) = ∞ are

applied. Because 𝑐𝑐̂ (𝑥𝑥; 𝛼𝛼, 𝛽𝛽) is the same as that in the case of heterogeneous commodities,
properties in (36) still hold. From (28) and (38), foreign consumption 𝑐𝑐 ∗ reduces to
𝑐𝑐 ∗ = −𝜌𝜌𝜌𝜌 + 𝜃𝜃2∗ − 𝑔𝑔∗ ,which yields
𝜕𝜕𝑐𝑐 ∗= −𝜌𝜌 < 0,𝜕𝜕𝜕𝜕𝜕𝜕𝑐𝑐 ∗

𝜕𝜕𝜃𝜃2∗= 1,

𝜕𝜕𝑐𝑐 ∗

= −1;
𝜕𝜕𝑔𝑔∗

𝜕𝜕𝑐𝑐 ∗
𝜕𝜕𝜕𝜕= 0 for 𝑣𝑣 = 𝛼𝛼, 𝛽𝛽, 𝜃𝜃1 , 𝑔𝑔1 , 𝑔𝑔2 .

They are the same as the effects on 𝑐𝑐 ∗ given in (37) to which the properties in (38) are applied.

Thus, all the properties obtained above are the same as those in the case of heterogeneouscommodities except that the effects through a change in 𝜔𝜔 disappear.

23

A numerical example: For simplicity, the utility function of consumption is assumed to be𝑢𝑢(𝑐𝑐) = 𝛿𝛿 ln(𝑐𝑐1 ) + (1 − 𝛿𝛿) ln(𝑐𝑐2 ),and hence the elasticity of consumption utility and the expenditure ratio satisfy𝜂𝜂 = 1,𝛿𝛿 ′ (𝜔𝜔) = 0.

Applying these properties and (33) to (36) yields𝜕𝜕𝜕𝜕𝜕𝜕(𝜌𝜌𝜌𝜌)=−

1,𝛽𝛽𝑝𝑝 𝜃𝜃
𝛿𝛿� 1 1 �−1
𝛼𝛼

𝜕𝜕𝜕𝜕

𝜕𝜕(𝑝𝑝1 𝑔𝑔1∗ )

=−

𝜕𝜕𝜕𝜕

𝜕𝜕(𝑝𝑝1 𝑔𝑔1 )
1−𝛿𝛿

=
,

𝛽𝛽𝑝𝑝 𝜃𝜃
𝛿𝛿� 1 1 �−1
𝛼𝛼

𝛿𝛿

,

𝛽𝛽𝑝𝑝 𝜃𝜃
𝛿𝛿� 1 1 �−1
𝛼𝛼

𝜕𝜕𝜕𝜕

𝜕𝜕(𝑝𝑝2 𝑔𝑔2∗ )

=

𝜕𝜕𝜕𝜕

𝜕𝜕(𝑝𝑝2 𝑔𝑔2 )
𝛿𝛿

=

1+𝛿𝛿

,

𝛽𝛽𝑝𝑝 𝜃𝜃
𝛿𝛿� 1 1 �−1𝛼𝛼.

𝛽𝛽𝑝𝑝 𝜃𝜃
𝛿𝛿� 1 1 �−1
𝛼𝛼

(39)

In order to calculate the multipliers in (39), 𝛼𝛼 , 𝛽𝛽 , 𝑝𝑝1 𝜃𝜃1 and 𝛿𝛿 must be numericallydetermined. Japanese data during the period 1994-2013 are used to do so because they areavailable based on 93SNA (benchmark year 2005) and the period is almost that of Japan’slong-run stagnation. The average values of the relevant economic variables during the periodare
𝜋𝜋 = −0.00027,
𝑝𝑝 𝜃𝜃 𝑥𝑥𝑐𝑐 = 279.7,

𝑝𝑝1 𝜃𝜃1 𝑥𝑥 = 490.2,

𝑥𝑥 − 1 �= 𝑝𝑝1 𝜃𝜃1 − 1� = −0.01477, 𝑝𝑝1 𝜃𝜃1 =1 1

𝑝𝑝1 𝜃𝜃1 𝑥𝑥
𝑥𝑥= 497.5,

where 𝜋𝜋 is the average of the annual rate of change in the consumer price index for all items(source: Statistics Bureau, Ministry of Internal Affairs and Communications, Japan), 𝑐𝑐 and𝑝𝑝1 𝜃𝜃1 𝑥𝑥 are the averages of real household consumption expenditure and real GDP respectively(source: National Accounts, Cabinet Office, Government of Japan), and 𝑥𝑥 − 1 is the average ofthe ratio of the output gap to potential output (source: IMF World Economic Outlook Database,

April 2015). Following Benhabib and Farmer (1996) and Mankiw and Weinzierl (2006), 𝜌𝜌 isassumed to be

From (25) and (26),𝜌𝜌 = 0.05.

24𝜋𝜋 = 𝛼𝛼(𝑥𝑥 − 1),𝛽𝛽𝛽𝛽 = 𝜌𝜌 + 𝜋𝜋.

Applying the above values to them yields
𝛼𝛼 = 1.828×10−2 ,𝛽𝛽 = 1.778×10−4 ,𝛽𝛽𝑝𝑝1 𝜃𝜃1𝛼𝛼= 4.839.

Table 1 shows the magnitudes of the multiplier effects given in (39) to which the above
figures of 𝛼𝛼, 𝛽𝛽 and 𝛽𝛽𝑝𝑝1 𝜃𝜃1 /𝛼𝛼 are applied for the cases where 𝛿𝛿 = 0.69 and 0.94. From (24)and (29), 𝛿𝛿 is𝑝𝑝 𝜃𝜃 𝑥𝑥1𝛿𝛿 = 𝑝𝑝 𝜃𝜃 1𝑥𝑥+𝑝𝑝
,
𝜃𝜃∗
1 1

2 2

which is the home country’s GDP share of the two countries’ total GDP. Because thepopulation sizes of the two countries are assumed to be the same, 𝛿𝛿 should be calculated fromper-capita values. The two values of 𝛿𝛿 are calculated from the per-capita GDP values of Japanand Korea and from those of Japan and China, respectively. The per-capita GDP values are theaverages in terms of US dollar during the period 1994-2013 (source: IMF statistics), which are$37,322 for Japan, $16,535 for Korea, and $2,382 for China. Note that the Marshall-Lernercondition (30) is valid if the denominator of the multiplier is positive, which indeed holds true
for the present examples.

Table 1: Multipliers on home consumption 𝑐𝑐𝜌𝜌𝜌𝜌 ↑

𝑝𝑝1 𝑔𝑔1 ↑𝑝𝑝2 𝑔𝑔2 ↑𝑝𝑝1 𝑔𝑔1∗ ↑𝑝𝑝2 𝑔𝑔2∗ ↑𝛿𝛿 = 0.69

𝛿𝛿 =0.94

0.30

0.26

−0.43
0.72

−0.13
0.30

25

−0.28
0.54

−0.02
0.26

The case where 𝛿𝛿 = 1 represents the case of a closed economy. Then, only the second

equation in (39), 𝜕𝜕𝜕𝜕/𝜕𝜕(𝑝𝑝1 𝑔𝑔1 ), is relevant and equals 0.26. This value equals that obtained by
Murota and Ono (2015), where they treat a closed economy and use the same data set as thepresent one.

6. Comparison with the symmetric cases
This section compares the results in the asymmetric case with those in the two symmetriccases: full employment in both countries and secular stagnation in both countries. In Table 2
the first column summarizes the results in the asymmetric case while columns 2 and 3 show theresults in the symmetric case of secular stagnation and that of full employment, respectively.
The results in the symmetric stagnation case have been presented by Ono (2014), whereas
those in the symmetric full-employment case are derived from (13) and (14), as shown below.
Because in the symmetric cases the effects of changes in the parameters of the foreign country,
𝑔𝑔1∗ and 𝜃𝜃2∗ − 𝑔𝑔2∗ , are symmetric to those of the home country, 𝑔𝑔2 and 𝜃𝜃1 − 𝑔𝑔1 , the effects of thehome country’s parameters 𝛼𝛼, 𝛽𝛽, 𝑏𝑏, 𝑔𝑔1 , 𝑔𝑔2 and 𝜃𝜃1 are discussed below.

The effects of changes in the stagnant country’s parameters 𝛼𝛼, 𝛽𝛽, 𝑏𝑏, 𝑔𝑔1 , 𝑔𝑔2 and 𝜃𝜃1 on its

own consumption 𝑐𝑐 and the relative price 𝜔𝜔 are similar whether the other country is fully
employed or underemployed, as is clear from columns 1 and 2. The parameter changes that
improve the home current account 𝑏𝑏̇, which are increases in 𝛼𝛼, 𝛽𝛽, 𝑏𝑏 and 𝜃𝜃1 and decreases in 𝑔𝑔1

and 𝑔𝑔2 , raise the relative price of the home commodity 1/𝜔𝜔 so that 𝑏𝑏̇ returns to zero, andreduce home consumption 𝑐𝑐. The spillover effects on the foreign country, however, differ
between the two cases. If full employment prevails in the foreign country, an increase in 𝜔𝜔works as an improvement in the terms of trade for the foreign country and expands foreign26

consumption 𝑐𝑐 ∗ . However, if the foreign country suffers from secular stagnation, an increase in𝜔𝜔 reduces its employment and consumption 𝑐𝑐 ∗ , as shown in column 2.
Table 2: Comparison
ASYMMETRYSYMMETRYSYMMETRY(Stagnation)(Full employment)𝛼𝛼 ↑𝑐𝑐 ↓, 𝑐𝑐 ∗ ↓, 𝜔𝜔 ↓𝑐𝑐 ↓, 𝑐𝑐 ∗ ↑, 𝜔𝜔 ↓No effects𝑏𝑏 ↑𝑐𝑐 ↓, 𝑐𝑐 ∗ ↓, 𝜔𝜔 ↓𝑐𝑐 ↓, 𝑐𝑐 ∗ ↑, 𝜔𝜔 ↓𝑐𝑐 ↑, 𝑐𝑐 ∗ ↓, no effect on 𝜔𝜔𝑐𝑐 ↑, 𝑐𝑐 ∗ ↓, 𝜔𝜔 ↑𝑐𝑐 ↓, 𝑐𝑐 ∗ ↑, 𝜔𝜔 ↑

𝛽𝛽 ↑
𝑔𝑔1 ↑

𝑔𝑔2 ↑
𝜃𝜃1 ↑

𝑔𝑔1∗ ↑

𝜃𝜃2∗ − 𝑔𝑔2∗ ↑𝑐𝑐 ↓, 𝑐𝑐 ∗ ↓, 𝜔𝜔 ↓𝑐𝑐 ↑, 𝑐𝑐 ∗ ↑, 𝜔𝜔 ↑𝑐𝑐 ↑, 𝑐𝑐 ∗ ↑, 𝜔𝜔 ↑𝑐𝑐 ↓, 𝑐𝑐 ∗ ↓, 𝜔𝜔 ↓𝑐𝑐 ↓, 𝑐𝑐 ∗ ↓, 𝜔𝜔 ↓𝑐𝑐 ↓, 𝑐𝑐 ∗ ↑, 𝜔𝜔 ↓No effects

𝑐𝑐 ↑, 𝑐𝑐 ∗ ↓, 𝜔𝜔 ↑𝑐𝑐 ↓, 𝑐𝑐 ∗ ↓, 𝜔𝜔 ↓𝑐𝑐 ↓, 𝑐𝑐 ∗ ↑, 𝜔𝜔 ↓𝑐𝑐 ↓, 𝑐𝑐 ∗ ↑↓, 𝜔𝜔 ↓—𝑐𝑐 ↑, 𝑐𝑐 ∗ ↑, 𝜔𝜔 ↑

—

—

—

If both countries achieve full employment, from (4), (13), (14) one finds that in theneighborhood of (29)𝑑𝑑𝜔𝜔 𝑓𝑓

𝑑𝑑𝑐𝑐𝑓𝑓 = 𝜌𝜌𝜌𝜌𝜌𝜌 − (1 − 𝛿𝛿)𝑝𝑝1 𝜃𝜃1 𝜔𝜔𝑓𝑓 − [𝑝𝑝1 𝑑𝑑𝑔𝑔1 + 𝑝𝑝2 𝑑𝑑𝑔𝑔2 ],𝑑𝑑𝜔𝜔 𝑓𝑓𝜕𝜕𝜔𝜔 𝑓𝑓 (𝜃𝜃1 ,𝑔𝑔1 ,𝑔𝑔2 )𝜕𝜕𝜃𝜃1

𝑑𝑑𝑐𝑐𝑓𝑓∗ = −𝜌𝜌𝜌𝜌𝜌𝜌 + 𝛿𝛿𝛿𝛿2 𝜃𝜃2∗ 𝜔𝜔𝑓𝑓 ,> 0,𝜕𝜕𝜔𝜔 𝑓𝑓 (𝜃𝜃1 ,𝑔𝑔1 ,𝑔𝑔2 )
𝜕𝜕𝑔𝑔1

< 0,

𝜕𝜕𝜔𝜔 𝑓𝑓 (𝜃𝜃1 ,𝑔𝑔1 ,𝑔𝑔2 )
𝜕𝜕𝑔𝑔2

> 0,

where the foreign parameters are fixed. These properties directly yield the results summarizedin column 3 of Table 2.
First, neither 𝛼𝛼 nor 𝛽𝛽 affects real variables including 𝑐𝑐 and 𝑐𝑐 ∗ . A transfer from the foreigncountry to the home country (𝑏𝑏 ↑) naturally increases home consumption 𝑐𝑐 and decreases
27foreign consumption 𝑐𝑐 ∗ while 𝜔𝜔 is unaffected in the case of homothetic utility. Note that in theasymmetric case such a transfer decreases both 𝑐𝑐 and 𝑐𝑐 ∗ because it improves the home current

account and lowers the foreign relative price 𝜔𝜔, which worsens home employment and theforeign terms of trade.

Second, a decrease in home government purchases on the home commodity 𝑔𝑔1 and anincrease in home productivity 𝜃𝜃1 directly increase home disposable income and expand homeconsumption. They lower the relative price of the home commodity 1/𝜔𝜔 and hence foreign

consumption 𝑐𝑐 ∗ increases due to the improvement in the terms of trade. If the home country
faces a liquidity trap, however, home consumption does not respond and unemployment

deteriorates. The deterioration of unemployment worsens deflation, discourages consumption
and improves the current account, which makes the relative price of the home commodity riseso that the current account returns to zero. It further decreases home employment anddiscourages consumption 𝑐𝑐. Moreover, it worsens the foreign terms of trade and reduces
foreign consumption 𝑐𝑐 ∗ .

Third, an increase in home government purchases on the foreign commodity 𝑔𝑔2 reduces

home consumption 𝑐𝑐 by decreasing home disposable income. Foreign consumption 𝑐𝑐 ∗increases because it raises the relative price of the foreign commodity 𝜔𝜔 , which is an
improvement in the foreign terms of trade. This result is also quite different from that in theasymmetric case. In the asymmetric case an increase in 𝑔𝑔2 deteriorates the home current
account and lowers the relative price of the home commodity 1/𝜔𝜔, which increases not only 𝑐𝑐
by expanding home employment but also 𝑐𝑐 ∗ by improving the foreign terms of trade.

Thus, the own and spillover effects in the symmetric cases are quite different from those in

the asymmetric case.

28

7. Conclusions
A two-country economy in which the home country suffers from secular stagnation of
aggregate demand while in the foreign country full employment prevails is considered.
A stagnant country often attempts to improve production efficiency, expecting that it
would expand total production and home income. However, it improves the current account,which leads the home currency to appreciate, the relative price of the home commodity to riseand home employment and consumption to decrease. In order to return the current account tozero, the rise in the relative price must be so high that home employment and consumption
eventually decrease. It is because even if home employment declines so much as to offset the
improvement in the productivity and maintain the same production level, the current account isstill positive because the reduction in home employment lowers home consumption. Therefore,
the relative price of the home commodity rises and home employment further decreases. The
rise in the relative price of the home commodity in turn worsens the foreign terms of trade.
Therefore, if full employment prevails in the foreign country, the real value of foreign
production declines and hence foreign consumption decreases.
Changes in the home country’s policy and preference parameters that worsen the homecurrent account cause the relative price of the home commodity to decrease and thereby expand
home employment and consumption. It is in turn an improvement in the terms of trade for theforeign country and thus increases foreign consumption. Typical such examples are expansionsof home government purchases on home and foreign commodities. Foreign aid may be anotherimportant example. Foreign aid expenditures tend to be cut when a donor country suffers fromsecular stagnation. However, an expansion in foreign aid in fact benefits the donor country
because it worsens the home current account, lowers the relative price of the home commodity,
increases home employment, decreases deflation, and urges people to consume more.

29Moreover, if the recipient country realizes full employment, it not only receives foreign aid but
also benefits from the improvement in the terms of trade.

Appendix I: Optimal household behavior
From the Hamiltonian function of the optimal household behavior:
𝐻𝐻 (∗) = 𝑢𝑢�𝑐𝑐 (∗) � + 𝑣𝑣�𝑚𝑚(∗) � + 𝜆𝜆(∗) �𝑟𝑟𝑎𝑎(∗) + 𝑤𝑤 (∗) 𝑥𝑥 (∗) − 𝑐𝑐 (∗) − 𝑅𝑅 (∗) 𝑚𝑚(∗) − 𝑧𝑧 (∗) �,

one obtains the first-order optimal conditions: 11𝜆𝜆(∗) = 𝑢𝑢′ �𝑐𝑐 (∗) �, 𝜆𝜆(∗) 𝑅𝑅 (∗) = 𝑣𝑣 ′ �𝑚𝑚(∗) �,

From the Euler equations in (A1),
𝜆𝜆∗

=
𝜆𝜆

𝑢𝑢′ (𝑐𝑐 ∗ )= 𝜅𝜅 = constant over time,𝑢𝑢′ (𝑐𝑐)

where𝑢𝑢′′ (𝑐𝑐)𝑐𝑐𝜆𝜆̇(∗)𝜆𝜆(∗)= 𝜌𝜌 − 𝑟𝑟.

𝑐𝑐 ∗ = 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�,𝑢𝑢′′ (𝑐𝑐 ∗ )𝑐𝑐 ∗𝜂𝜂 = − 𝑢𝑢′ (𝑐𝑐) , 𝜂𝜂∗ = − 𝑢𝑢′ (𝑐𝑐 ∗) .

(A1)

𝜂𝜂𝑐𝑐 ∗

𝑐𝑐̇ ∗ = 𝜂𝜂∗𝑐𝑐 𝑐𝑐̇ ,(A2)(3) and (A1) give the Ramsey equation and the money demand function in (6).

Appendix II: Wealth preference and the zero interest rate
In the text it is assumed that the marginal utility of money 𝑣𝑣 ′ (𝑚𝑚) has a positive lower

bound. This property would rather imply wealth preference than liquidity preference.
Moreover, from (6) the nominal interest rate 𝑅𝑅 equals 𝑣𝑣 ′ (𝑚𝑚)/𝑢𝑢′ (𝑐𝑐), which converges to𝛽𝛽 ⁄𝑢𝑢′ (𝑐𝑐) (> 0) in the steady state. However, the rate observed in a stagnant economy is almost

zero. This appendix explicitly separates liquidity preference and wealth preference and showsthat the result in the text still holds while the nominal interest rate is zero.
If wealth preference 𝜑𝜑(𝑎𝑎) is taken into account, the home utility functional turns to be
by replacing 𝑢𝑢(𝑐𝑐) by 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) one obtains the intratemporal and intertemporal optimal
conditions given by (1) and (A1) all at once.
11 Apparently,30

∞∫0 [𝑢𝑢(𝑐𝑐) + 𝑣𝑣(𝑚𝑚) + 𝜑𝜑(𝑎𝑎)] exp(−𝜌𝜌𝜌𝜌) 𝑑𝑑𝑑𝑑,

in which 𝑣𝑣(𝑚𝑚) now represents the transaction motive and satisfies
lim 𝑣𝑣 ′ (𝑚𝑚) = 0.

𝑚𝑚→∞

(A3)

In this case the first-order optimal conditions of the home household in (A1) are replaced by𝜆𝜆 = 𝑢𝑢′ (𝑐𝑐), 𝜆𝜆𝜆𝜆 = 𝑣𝑣 ′ (𝑚𝑚),Therefore, the Ramsey equation in (6) turns to be𝑐𝑐̇𝜂𝜂 𝑐𝑐 =𝜆𝜆̇
𝜆𝜆= 𝜌𝜌 − 𝑟𝑟 −𝜑𝜑′ (𝑎𝑎).
𝑢𝑢′ (𝑐𝑐)𝑣𝑣 ′ (𝑚𝑚)+𝜑𝜑′ (𝑎𝑎)− (𝜌𝜌 + 𝜋𝜋),𝑢𝑢′ (𝑐𝑐)

while the money demand function in (6) is unchanged –i.e.,
𝑅𝑅 =𝑣𝑣 ′ (𝑚𝑚).
𝑢𝑢′ (𝑐𝑐)(A4)

Because all the other equations in the text are unchanged, the dynamic equations are valid byreplacing 𝑣𝑣 ′ (𝑚𝑚) by 𝑣𝑣 ′ (𝑚𝑚) + 𝜑𝜑 ′ (𝑎𝑎).

In the steady state deflation continues in the home country and 𝑚𝑚 continues to expand.

Therefore, from (A3) and (A4), one finds

𝑣𝑣 ′ (𝑚𝑚)= 0,
𝑚𝑚→∞ 𝑢𝑢′ (𝑐𝑐)

𝑅𝑅 = limi.e., the nominal interest rate in the home country is zero. Furthermore, by assuming thatlim 𝜑𝜑 ′ (𝑎𝑎) = 𝛽𝛽𝑎𝑎 > 0,𝑎𝑎→∞instead of (15), one obtains the same steady-state conditions except that 𝛽𝛽 is replaced by 𝛽𝛽𝑎𝑎 in(26). Therefore, all the results of the parameter changes in the text are valid, while the homenominal interest rate is zero.

Appendix III: Dynamics and the existence of the asymmetric steady stateThis appendix summarizes the dynamic equations and the stability analysis, and proves
the lemma in section 4.
31

Unemployment arises in the home country, and hence from (2), (4), (21) and the firstequation of (22),
𝑊𝑊̇

𝑃𝑃̇

𝜔𝜔̇

𝑝𝑝̇ (𝜔𝜔)𝜒𝜒 − 𝛼𝛼 = 𝑊𝑊 = 𝑃𝑃1 = 𝜋𝜋 − 𝑝𝑝1 (𝜔𝜔) = 𝜋𝜋 − (1 − 𝛿𝛿) 𝜔𝜔.
1

1

(A5)

The time differentiation of 𝑥𝑥 in (24) and that of the first equation in (23) to which 𝑐𝑐 ∗ as afunction of 𝑐𝑐 given in (A2) is applied yield
𝑐𝑐

𝑐𝑐 ∗

𝑐𝑐̇

𝛿𝛿(1−𝛿𝛿)+𝛿𝛿 ′ 𝜔𝜔�𝜂𝜂 + 𝜂𝜂∗� 𝜂𝜂 𝑐𝑐 = �

1−𝛿𝛿

𝜔𝜔̇

� (𝑐𝑐 + 𝑐𝑐 ∗ ) 𝜔𝜔 = 𝑝𝑝1 𝜃𝜃1 𝑥𝑥̇ .

Because 𝑥𝑥̇ and 𝜔𝜔̇ are given above as functions of 𝑐𝑐̇ and from (18) and (A5) 𝜋𝜋 is𝜔𝜔̇

𝜋𝜋 satisfies

𝜋𝜋 = 𝑥𝑥̇ + 𝛼𝛼(𝑥𝑥 − 1) + (1 − 𝛿𝛿) 𝜔𝜔,𝑐𝑐 ∗𝑐𝑐̇ 𝑐𝑐11(1−𝛿𝛿)2𝜋𝜋 = 𝛼𝛼(𝑥𝑥 − 1) + 𝜂𝜂 𝑐𝑐 �𝜂𝜂 + 𝜂𝜂∗� �𝑝𝑝 𝜃𝜃 + �𝑐𝑐+𝑐𝑐 ∗� 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ �.
1 1

In the neighborhood of the steady state, deflation continues and 𝑣𝑣 ′ (𝑚𝑚) sticks to 𝛽𝛽 in the homecountry. Given this property, substituting the above 𝜋𝜋 to the Ramsey equation in (6) yields
𝑐𝑐̇where Ω is𝛽𝛽

Ω 𝑐𝑐 = Δ(𝑐𝑐, 𝜅𝜅) ≡ 𝑢𝑢′ (𝑐𝑐) − 𝜌𝜌 − 𝛼𝛼(𝑥𝑥 − 1),
𝑐𝑐

𝑐𝑐 ∗

1

1

(A6)

(1−𝛿𝛿)2

Ω = 𝜂𝜂 �1 + �𝜂𝜂 + 𝜂𝜂∗� �𝑝𝑝 𝜃𝜃 + �𝑐𝑐+𝑐𝑐 ∗� 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ �� > 0,
1 1

and 𝑥𝑥 is a function of 𝑐𝑐 and 𝜅𝜅 from (23) to which 𝑐𝑐 ∗ = 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)� presented in (A2) isapplied.

Next, the dynamic stability of (A6) is examined in the neighborhood of (29). If 𝑐𝑐 is solarge that 𝑐𝑐 + 𝑐𝑐 ∗ makes 𝑥𝑥 = 1, where 𝑐𝑐 ∗ = 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)� for a given 𝜅𝜅, the first equation of(8) turns to be𝑥𝑥 = 1,𝛿𝛿(𝜔𝜔)

𝑝𝑝1 (𝜔𝜔)�𝑐𝑐 + 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (𝑐𝑐)�� = 𝜃𝜃1 .

32

The 𝜔𝜔 that satisfies the above equation and the second equation of (8) to which 𝑥𝑥 ∗ = 1 issubstituted equals 𝜔𝜔 𝑓𝑓 given by (14) and hence 𝑐𝑐 equals 𝑐𝑐 𝑓𝑓 , the full-employment consumption

for the present 𝜅𝜅. Because 𝑥𝑥 = 1 and 𝑐𝑐 𝑓𝑓 satisfies (16), ∆(𝑐𝑐, 𝜅𝜅) defined by (A6) satisfies
𝛽𝛽∆(𝑐𝑐 𝑓𝑓 , 𝜅𝜅) = 𝑢𝑢′ �𝑐𝑐 𝑓𝑓 � − 𝜌𝜌 > 0.

Therefore, in order for the steady-state level of 𝑐𝑐 to exist in the range of (0, 𝑐𝑐 𝑓𝑓 ), one must have∆(0, 𝜅𝜅) < 0.

When 𝑐𝑐 = 0, 𝑐𝑐 ∗ = 𝑢𝑢′−1 �𝜅𝜅𝑢𝑢′ (0)� = 0 and thus 𝑐𝑐 + 𝑐𝑐 ∗ = 0, yielding 𝑥𝑥 = 0 from the second
equation of (23) in the neighborhood of (29). Therefore, from (A6) to which the property that𝑢𝑢′ (0) = ∞ is applied, one finds ∆(0, 𝜅𝜅) = −(𝜌𝜌 − 𝛼𝛼). In order for this value to be negative, it

must be valid that𝜌𝜌 − 𝛼𝛼 > 0.

(A7)

In this case ∆(𝑐𝑐, 𝜅𝜅) must be positively inclined with respect to 𝑐𝑐 around the steady state,

and thus∆𝑐𝑐 (𝑐𝑐, 𝜅𝜅) > 0,i.e., the dynamics given by (A6) is unstable. Therefore, once 𝜅𝜅 is given, 𝑐𝑐 jumps to the levelthat satisfies∆(𝑐𝑐, 𝜅𝜅) = 0,

and the steady state of (A6) is immediately reached.
Let us next determine 𝜅𝜅. Because 𝜅𝜅(= 𝑢𝑢′ (𝑐𝑐 ∗ )/𝑢𝑢′ (𝑐𝑐)) is constant over time, by obtaining

the steady-state levels of 𝑐𝑐 and 𝑐𝑐 ∗ one can find 𝜅𝜅. Home consumption 𝑐𝑐 in the steady state of(A6) satisfies (26). The home current account 𝑏𝑏̇ in (27) always equals zero because otherwise 𝑏𝑏

continues to either expand or decline at an accelerated pace and the non-Ponzi game conditionis violated. Thus, 𝜔𝜔 that satisfies (27) is immediately reached. Consequently, from the firstequation of (23), (24) and (26), 𝐷𝐷, 𝑥𝑥, 𝑐𝑐 and 𝑐𝑐 ∗ (= 𝐷𝐷 − 𝑐𝑐) are also determined, and hence 𝜅𝜅(=33

𝑢𝑢′ (𝑐𝑐 ∗ )/𝑢𝑢′ (𝑐𝑐) from (A2)) obtains. The dynamic property mentioned in the lemma of section 4summarizes the above results.

Let us finally prove that this 𝜔𝜔 indeed exists in (0, 𝜔𝜔 𝑓𝑓 ) and then 0 < 𝑥𝑥 < 1 and 0 < 𝑐𝑐 <𝑐𝑐 𝑓𝑓 in the neighborhood of (29). Under (A7), when 𝜔𝜔 = 0, 𝑥𝑥� in (24), 𝑐𝑐̂ in (26) and 𝑏𝑏̇ in (27)satisfy𝛽𝛽𝜔𝜔 = 0: 𝑥𝑥�(0; … ) = 0, 𝑐𝑐̂ (0, … ) = 𝑢𝑢′−1 �𝜌𝜌−𝛼𝛼� > 0, 𝑏𝑏̇ = −𝑐𝑐̂ (0, … ) < 0.

(A8)

When 𝜔𝜔 = 𝜔𝜔 𝑓𝑓 given by (14), from (24) 𝑥𝑥 = 1 and from (26) 𝑐𝑐̂ (1; 𝛼𝛼, 𝛽𝛽) = 𝑢𝑢′−1 (𝛽𝛽/𝜌𝜌) .
Substituting these values and 𝑐𝑐 𝑓𝑓 given in (13) into 𝑏𝑏̇ in (27) and taking into account thecondition of (16) yields

𝛽𝛽𝜔𝜔 = 𝜔𝜔 𝑓𝑓 : 𝑥𝑥�(𝜔𝜔 𝑓𝑓 ; … ) = 1, 𝑐𝑐̂ (1, … ) = 𝑢𝑢′−1 �𝜌𝜌 � > 0, 𝑏𝑏̇ = 𝑐𝑐 𝑓𝑓 − 𝑐𝑐̂ (1, … ) > 0.

(A9)

Because 𝑏𝑏̇ < 0 when 𝜔𝜔 = 0 and 𝑏𝑏̇ > 0 when 𝜔𝜔 = 𝜔𝜔 𝑓𝑓 , as shown in (A8) and (A9), there is 𝜔𝜔that makes 𝑏𝑏̇ = 0 within (0, 𝜔𝜔 𝑓𝑓 ):0 < 𝜔𝜔 < 𝜔𝜔 𝑓𝑓 ,(A10)

and around the steady state 𝑏𝑏̇ is an increasing function of 𝜔𝜔, implying the Marshall-Lernercondition given in (30) to hold. Because 0 < 𝜔𝜔 < 𝜔𝜔 𝑓𝑓 , 𝑥𝑥�𝜔𝜔 > 0 from (24), and 𝑐𝑐̂𝑥𝑥 > 0 from(26), using the values given in (A8) and (A9) one finds

0 = 𝑥𝑥�(0; … ) < 𝑥𝑥 < 𝑥𝑥�(𝜔𝜔 𝑓𝑓 ; … ) = 1, 0 < 𝑐𝑐̂ (0, … ) < 𝑐𝑐 < 𝑐𝑐̂ (1, … ) < 𝑐𝑐 𝑓𝑓 .

Therefore, unemployment and aggregate demand deficiency appear in the home country.
In the foreign country, on the other hand, there is 𝑃𝑃 ∗ that satisfies the second equation in(12) when (17) holds. It is because under (17)𝛽𝛽𝛽𝛽𝜌𝜌𝜌𝜌 > 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝜃𝜃2∗ − 𝑢𝑢′−1 �𝜌𝜌 � > 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ − 𝑢𝑢′−1 �𝜌𝜌 �,

in which 𝜔𝜔 satisfies (A10) and hence
𝛽𝛽𝑣𝑣 ′ (𝑀𝑀 ∗ /𝑃𝑃 ∗ )< 𝜌𝜌 �= 𝑢𝑢′ (𝜌𝜌𝜌𝜌+𝑝𝑝 (𝜔𝜔)𝜃𝜃∗ )�,𝑢𝑢′ (𝜌𝜌𝜌𝜌+𝑝𝑝 (𝜔𝜔)𝜃𝜃∗ )
2

2

2

34

2

i.e., there is 𝑃𝑃∗ with which full employment is reached in the foreign country.

Thus, the asymmetric steady state exists when (17) holds, as stated in the lemma.

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