GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 327

Exchange-rate adjustment and macroeconomic interdependence
between stagnant and fully employed countries

Yoshiyasu Ono

January 2014; revised November 2014

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

November 2014

Exchange-rate adjustment and macroeconomic interdependence
between stagnant and fully employed countries *byYoshiyasu Ono†

Abstract
This paper presents a two-country two-commodity dynamic macroeconomic model with freeinternational asset trade in which one country achieves full employment and the other suffers
long-run unemployment. Own and spill-over effects of changes in policy, technological andpreference parameters emerge through exchange-rate adjustment. The parameter changes thatworsen the stagnant country’s current account depreciate the home currency, expand home
employment and improve the foreign terms of trade, making both countries better off. The
stagnant country’s foreign aid to the fully employed country also yields the same beneficial
effects on both countries.

JEL classification: F32, F41, F35
Keywords: long-run unemployment, current account, liquidity trap, exchange rate, fiscal
expansion, foreign aid.

*This research is financially supported by the Grants-in-Aid for Scientific Research, the Japan Society for
Promotion of Science (JSPS). A significant part of this research was done while the author visited the Center for
Economic Studies (CES) of the University of Munich (LMU). The author thanks G. Illing and the participants ofthe macroeocnomics seminar held in the Univeristy of Munich for their valuable comments. Helpful comments by
the participants of the INFINITI Conference on International Finance 2014 are also gratefully appreciated.
†

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, an increase in productivity obviously expands national income andincreases welfare 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 country’s terms of trade. An increase in homegovernment purchases works as if the home country’s productivity decreased, and harms bothcountries.
If the two countries suffer long-run economic stagnation due to aggregate demand
deficiency, however, the result is quite different. Ono (2006, 2014) presents a two-country
dynamic model with persistent unemployment that arises under a liquidity trap and shows thatan increase in the home country’s productivity excessively improves the current account andleads the home currency to appreciate so much that home employment and consumption
decrease, which makes the home country worse off. The home currency appreciation in turncauses the foreign country’s employment and national income to increase and benefits theforeign country. An increase in home government purchases worsens the home current accountand leads the home currency to depreciate, which improves home employment and reduces
foreign employment. Consequently, the home country is better off while the foreign country isworse off.
There is yet another important case: one country faces persistent deficiency of aggregate
demand while the other country realizes full employment. It may in particular be relevant
between a developed country that faces persistent stagnation due to a liquidity trap, e.g. Japanfor two decades, and an emerging country that has large demand and enjoys high growth, e.g.
China. This paper focuses on this case and examines the effects on the two countries of changes
in various parameters including productivity and government purchases. The effects of foreign
aid on the two countries are also examined.
1

It is naturally believed that foreign aid makes the donor country worse off and the recipient
country better off. Due to this belief the foreign aid budget is usually cut when a donor country
faces persistent stagnation, as Japan did in the ‘Lost Decades’. However, this belief is true only
if full employment prevails in both countries. If both countries face stagnation, the donor
country is better off while the recipient country is worse off as a result of exchange-rate
adjustment (see Ono, 2014). 1
The benefits of foreign aid for both recipient and donor countries in the asymmetric casewere discussed in some important policy decisions. The Marshall Plan 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 a serious shortage 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 bycreating a market. 2 In the context of the North-South problem, the Independent Commission on
International Development Issues, a panel lead by former German Chancellor Willy Brandt inthe early 1980s (the Brandt commission, 1980, 1983), argued that foreign aid from the North to
the South would benefit the donors as well as the recipients through not only stabilizing
political/security situations but also creating import demand and expanding employment in the
North. 3 This paper examines the validity of those statements in a dynamic macroeconomic
framework and finds that foreign aid makes both the recipient and donor countries better off if

1

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

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.
3

This issue was treated by Ono (2007) but he used CES utility functions. This paper extends that analysis tothe case of general homothetic utility.

2the former achieves full employment and the latter faces persistent deficiency of aggregate
demand.
The stagnation considered in the present analysis is not a temporary one but a persistent
one due to aggregate demand deficiency with a liquidity trap. In the recent literature moststudies regard macroeconomic fluctuations as caused by firms’ and labor unions’ monopolistic
pricing a la Calvo (1983) and explore small perturbations due to unanticipated policy andtechnology shocks around the long-run steady state. They are e.g. Yun (1996), Erceg et al.
(2000), Smets and Wouters (2003, 2005, 2007), Christiano et al. (2005), and Adolfson et al.
(2007, 2008). They never consider any demand deficiency in either the adjustment process or
the steady state. The possibility of market disequilibrium is explicitly analyzed by Krugman(1998) in a closed economy setting and Obstfeld and Rogoff (1995) and Hau (2000) in an open
economy setting. 4 In these models, however, demand deficiency occurs only for a period inwhich an unanticipated exogenous shock occurs and prices are predetermined, and disappears
in all the periods that follow.
While these models may well fit to analyze short-run macroeconomic fluctuations, thepresent paper focuses on such long-run stagnation as Japan’s Lost Decades and EU countries’
Great Recession triggered by the financial crisis of 2008. In the recent IMF annual conference
Summers (2013) warned against relying too much on the DSGE approach in solving economic
crises and emphasized the need to work on long-run recessions rather than short-run businessfluctuations. This paper follows this line of thought and considers a long-run stagnation due toaggregate demand deficiency.

4

The latter is called new open economy macroeconomic models. See Lane (2001) for an extensive survey onsuch models.

3

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

The household sectors of the two countries have the same utility function of the two

commodities. The function is homothetic and hence for a given level of consumption
expenditure 𝑐𝑐 (or 𝑐𝑐 ∗ ) it can be summarized as𝑢𝑢(𝑐𝑐) ≡ 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ), 𝑢𝑢(𝑐𝑐 ∗ ) ≡ 𝑢𝑢�(𝑐𝑐1∗ , 𝑐𝑐2∗ ),where 𝑐𝑐𝑖𝑖 (or 𝑐𝑐𝑖𝑖∗ ) is the consumption of commodity 𝑖𝑖 (𝑖𝑖 = 1, 2) that satisfies
𝑝𝑝1 (𝜔𝜔)𝑐𝑐1 = 𝛿𝛿(𝜔𝜔)𝑐𝑐, 𝑝𝑝2 (𝜔𝜔)𝑐𝑐2 = [1 − 𝛿𝛿(𝜔𝜔)]𝑐𝑐,

𝑝𝑝1 (𝜔𝜔)𝑐𝑐1∗ = 𝛿𝛿(𝜔𝜔)𝑐𝑐 ∗, 𝑝𝑝2 (𝜔𝜔)𝑐𝑐2∗ = [1 − 𝛿𝛿(𝜔𝜔)]𝑐𝑐 ∗,1 > 𝛿𝛿(𝜔𝜔) > 0, 𝛿𝛿′(𝜔𝜔) > 0.

𝜔𝜔 is the relative price and 𝑝𝑝𝑖𝑖 (𝜔𝜔) is the real price of commodity 𝑖𝑖 (𝑖𝑖 = 1, 2):
𝑃𝑃𝑃𝑃∗

𝑝𝑝1 (𝜔𝜔) = 𝑃𝑃1 = 𝑃𝑃1∗,

𝑃𝑃

𝑃𝑃∗

𝑝𝑝2 (𝜔𝜔) = 𝑃𝑃2 = 𝑃𝑃2∗ = 𝜔𝜔𝑝𝑝1 (𝜔𝜔),

(1)

(2)

𝑃𝑃

𝑃𝑃1∗ = 𝜖𝜖1 , 𝑃𝑃2 = 𝜖𝜖𝑃𝑃2∗ ,where 𝑃𝑃 and 𝑃𝑃 ∗ are the two countries’ general price indices, 𝑃𝑃𝑖𝑖 and 𝑃𝑃𝑖𝑖∗ are the home andforeign nominal prices of commodity 𝑖𝑖 measured in each currency, and 𝜖𝜖 is the nominal
exchange rate, which satisfies

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:

4

𝜖𝜖̇

𝑅𝑅 = 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
𝑑𝑑𝑑𝑑

𝑑𝑑𝑃𝑃

1

𝑑𝑑𝑃𝑃∗

𝑑𝑑𝑃𝑃 ∗𝑑𝑑𝑃𝑃= 𝛿𝛿(𝜔𝜔) 𝑃𝑃 1 + (1 − 𝛿𝛿(𝜔𝜔)) 𝑃𝑃 2,
𝑃𝑃

𝑑𝑑𝑃𝑃∗

= 𝛿𝛿(𝜔𝜔) 𝑃𝑃∗1 + (1 − 𝛿𝛿(𝜔𝜔)) 𝑃𝑃∗2 ,
𝑃𝑃 ∗

2

1

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

0 = 𝛿𝛿(𝜔𝜔) 𝑝𝑝1 (𝜔𝜔) + (1 − 𝛿𝛿(𝜔𝜔)) 𝑝𝑝2 (𝜔𝜔),1These two equations yield

𝜔𝜔𝜔𝜔′

2

𝜔𝜔𝜔𝜔′

1 + 𝑝𝑝 1 = 𝑝𝑝 2.
1

2

𝜔𝜔𝜔𝜔′

𝜔𝜔𝑝𝑝′

𝛿𝛿 = 1 + 𝑝𝑝 1 = 𝑝𝑝 2.
1

2

(4)

The home and foreign representative households have the same subjective discount rate 𝜌𝜌and the same liquidity preference 𝑣𝑣(∙). They maximize each utility functional:∞∫0 [𝑢𝑢(𝑐𝑐) + 𝑣𝑣(𝑚𝑚)] exp(−𝜌𝜌𝜌𝜌) 𝑑𝑑𝑑𝑑,∞∫0 [𝑢𝑢(𝑐𝑐 ∗ ) + 𝑣𝑣(𝑚𝑚∗ )] exp(−𝜌𝜌𝜌𝜌) 𝑑𝑑𝑑𝑑,

subject to each flow budget equation and asset constraint:𝑎𝑎̇ = 𝑟𝑟𝑟𝑟 + 𝑤𝑤𝑤𝑤 − 𝑐𝑐 − 𝑅𝑅𝑅𝑅 − 𝑧𝑧,𝑎𝑎̇ ∗ = 𝑟𝑟𝑎𝑎∗ + 𝑤𝑤 ∗ 𝑥𝑥 ∗ − 𝑐𝑐 ∗ − 𝑅𝑅 ∗ 𝑚𝑚∗ − 𝑧𝑧 ∗ ,𝑎𝑎 = 𝑚𝑚 + 𝑏𝑏,

𝑎𝑎∗ = 𝑚𝑚∗ + 𝑏𝑏 ∗ ,(5)where 𝑤𝑤 (or 𝑤𝑤 ∗ ) is the real wage, 𝑧𝑧 (or 𝑧𝑧 ∗ ) is the lump-sum tax, and 𝑥𝑥 (or 𝑥𝑥 ∗ ) is employment.

Real total assets 𝑎𝑎 (or 𝑎𝑎∗ ) consist of real money balances 𝑚𝑚 (or 𝑚𝑚∗ ) and foreign asset 𝑏𝑏 (or 𝑏𝑏 ∗ )while the firm value is zero under the linear technology. The real rate of interest 𝑟𝑟 isinternationally the same, as shown by (3). The two countries’ labor endowments arenormalized to 1 and may not be fully employed. Therefore, each country’s actual employment𝑥𝑥 (or 𝑥𝑥 ∗ ) implies each employment rate.
5

From the Hamiltonian function of each household’s optimization behavior:
𝐻𝐻 = 𝑢𝑢(𝑐𝑐) + 𝑣𝑣(𝑚𝑚) + 𝜆𝜆(𝑟𝑟𝑟𝑟 + 𝑤𝑤𝑤𝑤 − 𝑐𝑐 − 𝑅𝑅𝑅𝑅 − 𝑧𝑧),

𝐻𝐻 ∗ = 𝑢𝑢(𝑐𝑐 ∗ ) + 𝑣𝑣(𝑚𝑚∗ ) + 𝜆𝜆∗ (𝑟𝑟𝑎𝑎∗ + 𝑤𝑤 ∗ 𝑥𝑥 ∗ − 𝑐𝑐 ∗ − 𝑅𝑅 ∗ 𝑚𝑚∗ − 𝑧𝑧 ∗ ),one obtains the first-order optimal conditions: 5

𝜆𝜆 = 𝑢𝑢′ (𝑐𝑐), 𝜆𝜆𝜆𝜆 = 𝑣𝑣 ′ (𝑚𝑚),𝜆𝜆̇𝜆𝜆∗ = 𝑢𝑢′ (𝑐𝑐 ∗ ), 𝜆𝜆∗ 𝑅𝑅 ∗ = 𝑣𝑣 ′ (𝑚𝑚∗ ),From the Ramsey equations in (6), one finds

𝜆𝜆= 𝜌𝜌 − 𝑟𝑟,

𝜆𝜆∗̇

𝜆𝜆∗

= 𝜌𝜌 − 𝑟𝑟.

𝜆𝜆∗ = 𝜅𝜅𝜅𝜅, 𝜅𝜅 = constant over time.

(6)

(7)

From (3), (6) and (7), world total consumption 𝐷𝐷 satisfies

𝐷𝐷 (= 𝑐𝑐 + 𝑐𝑐 ∗) = 𝑢𝑢′−1 (𝜆𝜆) + 𝑢𝑢′−1 (𝜅𝜅𝜅𝜅) → 𝜆𝜆 = 𝜆𝜆(𝐷𝐷, 𝜅𝜅),𝜆𝜆 𝐷𝐷 𝐷𝐷̇where 𝜆𝜆 satisfies

� 𝐷𝐷𝜆𝜆 � 𝐷𝐷 = 𝜌𝜌 + 𝜋𝜋 −

𝜆𝜆𝐷𝐷

𝑐𝑐

𝑐𝑐 ∗

= − 1��𝜂𝜂 + 𝜂𝜂∗� < 0,𝜆𝜆𝑢𝑢′′ (𝑐𝑐)𝑐𝑐

and 𝑅𝑅 and 𝑅𝑅 are𝜅𝜅𝜆𝜆𝜅𝜅
𝜆𝜆𝑣𝑣 ′ (𝑚𝑚),𝑢𝑢′ (𝑐𝑐)
𝑐𝑐 ∗

𝑐𝑐

𝑐𝑐 ∗

= − �𝜂𝜂∗���𝜂𝜂 + 𝜂𝜂∗� < 0,

(8)

(9)

𝑢𝑢′′ (𝑐𝑐 ∗ )𝑐𝑐 ∗𝜂𝜂 = − 𝑢𝑢′ (𝑐𝑐) , 𝜂𝜂∗ = − 𝑢𝑢′ (𝑐𝑐 ∗) ,𝑅𝑅 =𝑣𝑣 ′ (𝑚𝑚),𝑢𝑢′ (𝑐𝑐)

𝑅𝑅 ∗ =

𝑣𝑣 ′ (𝑚𝑚∗ ).
𝑢𝑢′ (𝑐𝑐 ∗ )(10)

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

commodity 𝑖𝑖 (𝑖𝑖 = 1, 2) by the amount of 𝑔𝑔𝑖𝑖 (or 𝑔𝑔𝑖𝑖∗ ). Therefore,𝑧𝑧 = 𝑝𝑝1 (𝜔𝜔)𝑔𝑔1 + 𝑝𝑝2 (𝜔𝜔)𝑔𝑔2, 𝑧𝑧 ∗ = 𝑝𝑝1 (𝜔𝜔)𝑔𝑔1∗ + 𝑝𝑝2 (𝜔𝜔)𝑔𝑔2∗ .

5

(11)

Apparently, by replacing 𝑢𝑢(𝑐𝑐) by 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) one obtains the intratemporal and intertemporal optimal
conditions given by (1) and (6) all at once.

6

From the home and foreign demand functions presented by (1), the world demand forcommodity 1 and that for commodity 2 are 𝛿𝛿(𝜔𝜔)𝐷𝐷/𝑝𝑝1 (𝜔𝜔) and [ 1 − 𝛿𝛿(𝜔𝜔)]𝐷𝐷/𝑝𝑝2 (𝜔𝜔) ,respectively. Commodity prices perfectly adjust in the international competitive market so that𝛿𝛿(𝜔𝜔)

𝑝𝑝1 (𝜔𝜔)𝐷𝐷 + 𝑔𝑔1 + 𝑔𝑔1∗ = 𝜃𝜃1 𝑥𝑥,

1−𝛿𝛿(𝜔𝜔)𝐷𝐷 + 𝑔𝑔2 + 𝑔𝑔2∗ = 𝜃𝜃2∗ 𝑥𝑥 ∗ ,𝑝𝑝2 (𝜔𝜔)(12)

where 𝜃𝜃1 is the home productivity and 𝜃𝜃2∗ is the foreign productivity. Real balances 𝑚𝑚 and 𝑚𝑚∗satisfy𝑀𝑀𝑚𝑚 = 𝑃𝑃 ,𝑀𝑀∗

𝑚𝑚∗ = 𝑃𝑃∗ ,(13)

where 𝑀𝑀 and 𝑀𝑀∗ are nominal money supplies. It is assumed that 𝑀𝑀 and 𝑀𝑀∗ are constant overtime while they may increase in a once-and-for-all manner. Because the above equations yield𝑚𝑚̇ = −𝜋𝜋𝑚𝑚,𝑚𝑚̇∗ = −𝜋𝜋 ∗ 𝑚𝑚∗ ,and 𝑧𝑧 and 𝑧𝑧 ∗ satisfy (11), the flow budget equations in (5) reduce to𝑏𝑏̇ = 𝑟𝑟𝑟𝑟 + 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 𝑥𝑥 − 𝑐𝑐 − [𝑝𝑝1 (𝜔𝜔)𝑔𝑔1 + 𝑝𝑝2 (𝜔𝜔)𝑔𝑔2 ],𝑏𝑏̇ ∗ = 𝑟𝑟𝑏𝑏 ∗ + 𝑝𝑝2 (𝜔𝜔)𝜃𝜃2∗ 𝑥𝑥 ∗ − 𝑐𝑐 ∗ − [𝑝𝑝1 (𝜔𝜔)𝑔𝑔1∗ + 𝑝𝑝2 (𝜔𝜔)𝑔𝑔2∗].

(14)

𝑏𝑏 + 𝑏𝑏 ∗ = 0.

(15)

Foreign assets 𝑏𝑏 and 𝑏𝑏 ∗ always satisfy

3. The condition for the asymmetric steady state to arise
This section presents the condition for the asymmetric case to appear, in which the homecountry faces persistent unemployment while the foreign country achieves full employment.
Dynamics and stability are examined in the next section.
When a steady state is reached, 𝑐𝑐 and 𝑐𝑐 ∗ are constant and hence from (6) and (13),𝜌𝜌 =𝑣𝑣 ′ (𝑀𝑀/𝑃𝑃),
𝑢𝑢′ (𝑐𝑐)𝜌𝜌 =7

𝑣𝑣 ′ (𝑀𝑀∗ /𝑃𝑃 ∗ ).
𝑢𝑢′ (𝑐𝑐 ∗ )(16)Current accounts 𝑏𝑏̇ and 𝑏𝑏̇ ∗ given in (14) are zero. Therefore, if both countries achieve full
employment, i.e.,

𝑥𝑥 = 1, 𝑥𝑥 ∗ = 1,

from (14) and (15) 𝑐𝑐 and 𝑐𝑐 ∗ are

𝑐𝑐 = 𝑐𝑐𝑓𝑓 ≡ 𝜌𝜌𝜌𝜌 + 𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝜃𝜃1 − [𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝑔𝑔1 + 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝑔𝑔2 ],𝑐𝑐 ∗ = 𝑐𝑐𝑓𝑓∗ ≡ −𝜌𝜌𝜌𝜌 + 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝜃𝜃2∗ − [𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝑔𝑔1∗ + 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝑔𝑔2∗ ],(17)

where 𝜔𝜔 𝑓𝑓 satisfies (12) in which 𝑥𝑥 = 1 and 𝑥𝑥 ∗ = 1, and hence𝜔𝜔 𝑓𝑓 𝛿𝛿�𝜔𝜔 𝑓𝑓 �1−𝛿𝛿�𝜔𝜔 𝑓𝑓 �𝜃𝜃 −𝑔𝑔 −𝑔𝑔∗= 𝜃𝜃1∗ −𝑔𝑔1 −𝑔𝑔1∗ .
2

2

2

Substituting these 𝑐𝑐 and 𝑐𝑐 ∗ into (16) gives the steady state levels 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 marginal utility of money 𝑣𝑣 ′ (𝑚𝑚) has apositive lower bound 𝛽𝛽, 6lim𝑚𝑚→∞ 𝑣𝑣 ′ (𝑚𝑚) = 𝛽𝛽 > 0,(18)

because the first two equations in (6) gives the home money demand function:
𝑅𝑅 =𝑣𝑣 ′ (𝑚𝑚)𝛽𝛽(> 𝑢𝑢′ (𝑐𝑐)
′
(𝑐𝑐)
𝑢𝑢

for any 𝑚𝑚),and with (18) 𝑅𝑅 stays strictly positive as 𝑚𝑚 increases, implying a liquidity trap. 7 In this case, if

and only if 𝑐𝑐𝑓𝑓 , the home consumption under full employment given in (17), is so large as tosatisfy𝛽𝛽𝜌𝜌 < 𝑢𝑢′ �𝑐𝑐 � �<
𝑓𝑓

𝑣𝑣 ′ (𝑀𝑀/𝑃𝑃)𝑢𝑢′ �𝑐𝑐𝑓𝑓 �

for any 𝑃𝑃�,

(19)

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 both a parametric and a
non-parametric approach.
If 𝑣𝑣 ′ (𝑚𝑚) remains positive as 𝑚𝑚 expands, 𝑣𝑣(𝑚𝑚) may not represent the transaction motive. It would rather
imply wealth preference. In Appendix A this is taken into account and wealth preference and the narrowly-definedtransaction motive are separately introduced to the present model. It is shown that the following analysis is stillvalid and that the home nominal interest rate is zero in the steady state.
78

there is no 𝑃𝑃 that makes the first equation in (16) valid. Therefore, there is no dynamic pathalong which full employment is reached in the home country. Given that the left- andright-hand sides of (19) respectively represent the time preference rate and the liquidity
premium, (19) 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, (19) is valid in the home country while foreign full-employment consumption 𝑐𝑐𝑓𝑓∗
given in (17) satisfies

𝜌𝜌 >

𝛽𝛽

𝑢𝑢′ �𝑐𝑐𝑓𝑓∗ �

so that there is 𝑃𝑃∗ that satisfies the second equation of (16). From (17), (19) and the above

inequality, the asymmetric case emerges when 8𝛽𝛽𝜌𝜌 < 𝑢𝑢′ �𝜌𝜌𝜌𝜌+𝑝𝑝 �𝜔𝜔𝑓𝑓 �𝜃𝜃 −�𝑝𝑝 �𝜔𝜔𝑓𝑓 �𝑔𝑔 +𝑝𝑝 �𝜔𝜔𝑓𝑓 �𝑔𝑔 �� ,
1

1

1

𝛽𝛽

1

2

2

𝜌𝜌 > 𝑢𝑢′ �−𝜌𝜌𝜌𝜌+𝑝𝑝 �𝜔𝜔𝑓𝑓 �𝜃𝜃∗ −�𝑝𝑝 �𝜔𝜔𝑓𝑓 �𝑔𝑔∗ +𝑝𝑝 �𝜔𝜔𝑓𝑓 �𝑔𝑔∗ �� .
2

2

1

1

2

2

(∗)

From these conditions one finds that the asymmetric case in which 𝑔𝑔𝑖𝑖
appears if𝛽𝛽𝛽𝛽𝜌𝜌𝜌𝜌 > max(𝑢𝑢′−1 �𝜌𝜌 � − 𝑝𝑝1 (𝜔𝜔 𝑓𝑓 )𝜃𝜃1 , −𝑢𝑢′−1 �𝜌𝜌 � + 𝑝𝑝2 (𝜔𝜔 𝑓𝑓 )𝜃𝜃2∗ ).

= 0 (for 𝑖𝑖 = 1, 2)(20)This is valid when home productivity 𝜃𝜃1 is high while foreign productivity 𝜃𝜃2∗ is low and when

the home country owns huge foreign assets (i.e., 𝑏𝑏 is large). It may be plausible when the homecountry is developed and the foreign country is emerging. In the following analysis this case isconsidered.

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.

9

4. Dynamics and local stability
In the asymmetric case the home country cannot reach full employment. In order to deal
with this case, therefore, sluggish wage adjustments in the presence of labor demand shortages
must be introduced. The dominant setting of price and wage adjustments in DSGE models isthat of Calvo (1983). However, it does not suit the present analysis because it allows neither
aggregate demand shortages nor unemployment any time. Aggregate demand shortages are
explicitly analyzed by Krugman (1998) in a discrete-time setting. However, it cannot treat
long-run unemployment because the length of each period is exogenously given and a demand
shortage appears only in the period in which an exogenous shock occurs. In contrast, a dynamic
extension of Akerlof’s fair wage model (1982), presented by Ono and Ishida (2014), can deal
with unemployment as well as full employment during the adjustment process and in the steadystate. Therefore, it is adopted in the following analysis.

4.1. Wage and price adjustments
In each country 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. 9 Therefore, employment 𝑥𝑥 follows

𝑥𝑥̇ = −𝛼𝛼𝛼𝛼 + 𝜒𝜒,(21)

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, their fellow workers’ fair wages (which eventuallyequal their own fair wages) and the unemployment level of the society. More precisely, at the9

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.

10

end of the previous period (𝑡𝑡 − ∆𝑡𝑡 ) each employed worker conceives the rightful wage

𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡), which is the wage that he or she believes fair if full employment prevails. Next, heor she calculates the average of the rightful wages of the employed, which are the same to eachother, and the income of the unemployed, which is assumed to be zero for simplicity. The
average is taken as the fair wage 𝑊𝑊𝐹𝐹 (𝑡𝑡).

Because the numbers of the employed and the unemployed are respectively 𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡) and1 − 𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡), the employed calculate the rightful wage 𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡) from𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡)𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡) + 0 × [1 − 𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡)] = 𝑊𝑊𝐹𝐹 (𝑡𝑡 − ∆𝑡𝑡),(22)

at the timing denoted by (A) in figure 1. The newly hired, in contrast, 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 time 𝑡𝑡, the total number ofworkers that they care is 1 − 𝜒𝜒(𝑡𝑡)∆𝑡𝑡 because the number of the newly hired is 𝜒𝜒(𝑡𝑡)∆𝑡𝑡.

Because the number of the employed is 𝑥𝑥(𝑡𝑡 − ∆𝑡𝑡)(1 − 𝛼𝛼∆𝑡𝑡) at the timing denoted by (B) in

figure 1, the fair wage 𝑊𝑊𝐹𝐹 (𝑡𝑡) turns out to be
𝑊𝑊𝐹𝐹 (𝑡𝑡) =𝜈𝜈(𝑡𝑡−∆𝑡𝑡)𝑥𝑥(𝑡𝑡−∆𝑡𝑡)(1−𝛼𝛼∆𝑡𝑡)
1−𝜒𝜒(𝑡𝑡)∆𝑡𝑡.

From (22) and the above equation, one obtains
𝑊𝑊𝐹𝐹 (𝑡𝑡)−𝑊𝑊𝐹𝐹 (𝑡𝑡−∆𝑡𝑡)∆𝑡𝑡= 𝜒𝜒(𝑡𝑡)𝑊𝑊𝐹𝐹 (𝑡𝑡) − 𝛼𝛼𝑊𝑊𝐹𝐹 (𝑡𝑡 − ∆𝑡𝑡).

Reducing ∆𝑡𝑡 to zero yields the dynamics of 𝑊𝑊𝐹𝐹 :
𝑊𝑊̇𝐹𝐹
𝑊𝑊𝐹𝐹

= 𝜒𝜒 − 𝛼𝛼,(23)implying that the fair wage is updated at the replacement pace of the employed.
If there is unemployment in the home country, the firm will set wage 𝑊𝑊 equal to fair wage

𝑊𝑊𝐹𝐹 because 𝑊𝑊𝐹𝐹 is the lowest wage under which the employees properly work. The commodity

price 𝑃𝑃1 adjusts to 𝑊𝑊𝐹𝐹 ⁄𝜃𝜃1 since there is no commodity supply if 𝑃𝑃1 < 𝑊𝑊𝐹𝐹 ⁄𝜃𝜃1 and excess

commodity supply if 𝑃𝑃1 > 𝑊𝑊𝐹𝐹 ⁄𝜃𝜃1 . If full employment prevails, in contrast, the firm tries to11

pick out workers from rival firms to expand the market share by increasing 𝑊𝑊 from 𝑊𝑊𝐹𝐹 so longas the marginal profits are positive. Therefore, 𝑊𝑊 is higher than the fair wage 𝑊𝑊𝐹𝐹 that follows(23), and is equalized to 𝜃𝜃1 𝑃𝑃1 . The same argument is valid in the foreign country.

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

follows the movement of the fair wage 𝑊𝑊𝐹𝐹 when there is unemployment, and that 𝑊𝑊 follows

the movement of each commodity price regardless of 𝑊𝑊𝐹𝐹 when full employment maintains.
Thus, anyway one has

𝑊𝑊 ∗

𝑊𝑊

𝑤𝑤 = 𝑃𝑃 = 𝜃𝜃1 , 𝑤𝑤 ∗ = 𝑃𝑃∗ = 𝜃𝜃2∗ .
1

2

(24)

4.2. Dynamics and the asymmetric steady state
In the following the asymmetric case represented by (20) is considered and then:𝑥𝑥 < 1,𝑥𝑥 ∗ = 1.

Therefore, 𝑊𝑊 = 𝑊𝑊𝐹𝐹 that follows (23) in the home country while 𝑊𝑊 ∗ and 𝑃𝑃2∗ always adjust sothat full employment maintains in the foreign country. Having such price and wage
adjustments in mind, from (2), (4), the second equation of (12) in which 𝑥𝑥 ∗ = 1, (23) and (24),one obtains

𝑊𝑊̇
𝑊𝑊

𝑃𝑃̇

𝜔𝜔̇

= 𝑃𝑃1 = 𝜋𝜋 − (1 − 𝛿𝛿) 𝜔𝜔 = 𝜒𝜒 − 𝛼𝛼.
1

1−𝛿𝛿(𝜔𝜔)𝐷𝐷 + 𝑔𝑔2 + 𝑔𝑔2∗ = 𝜃𝜃2∗
𝑝𝑝2 (𝜔𝜔)→ 𝜔𝜔 = 𝜔𝜔(𝐷𝐷; 𝜃𝜃2∗ − 𝑔𝑔2 − 𝑔𝑔2∗ ).

(25)

Using (4), the time derivative of the first equation in (12) and that of the second equation in(25), one finds
𝜔𝜔̇

𝐷𝐷̇

1−𝛿𝛿

= �𝛿𝛿(1−𝛿𝛿)+𝛿𝛿′ 𝜔𝜔� 𝐷𝐷 ,
𝜔𝜔
𝐷𝐷

𝐷𝐷̇

𝑥𝑥̇ = �𝑝𝑝 𝜃𝜃 � 𝐷𝐷 .
1 1

Equation (21), the first equation of (25) and the two equations of (26) give

12

(26)

(1−𝛿𝛿)2𝐷𝐷𝐷𝐷̇

𝜋𝜋 = 𝛼𝛼(𝑥𝑥 − 1) + �𝑝𝑝 𝜃𝜃 + 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝐷𝐷 .
1 1

Note that the Poison rate of job separation 𝛼𝛼 is to be the price adjustment speed around thesteady state in which 𝐷𝐷 is constant and that 1/𝛼𝛼 denotes the average duration of employment.

Substituting 𝑥𝑥 obtained from the first equation of (12) into the above expression of 𝜋𝜋 andapplying the result to 𝜋𝜋 in (8) leads to𝐷𝐷(1−𝛿𝛿)2𝜆𝜆 𝐷𝐷 𝐷𝐷̇𝑣𝑣 ′ (𝑚𝑚)�𝑝𝑝 𝜃𝜃 + 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ − 𝐷𝐷𝜆𝜆 � 𝐷𝐷 = 𝜆𝜆(𝐶𝐶,𝜅𝜅) − 𝜌𝜌 − 𝛼𝛼 �1 1𝛿𝛿(𝜔𝜔(𝐷𝐷;𝜃𝜃2∗ −𝑔𝑔2 −𝑔𝑔2∗ ))𝑔𝑔1 +𝑔𝑔1∗
𝐷𝐷
+
− 1�,
𝜃𝜃1
𝑝𝑝1 (𝜔𝜔(𝐷𝐷;𝜃𝜃2∗ −𝑔𝑔2 −𝑔𝑔2∗ ))𝜃𝜃1(27)

where 𝜔𝜔(𝐷𝐷; 𝜃𝜃2∗ − 𝑔𝑔2 − 𝑔𝑔2∗ ) is given by the second equation of (25). From (1) and (9), thecoefficient of 𝐷𝐷̇⁄𝐷𝐷 in (27) is positive. In the neighborhood of the steady state, deflation
continues and 𝑣𝑣 ′ (𝑚𝑚) sticks to 𝛽𝛽, and hence the dynamic equation given by (27) reduces to𝐷𝐷(1−𝛿𝛿)2𝜆𝜆 𝐷𝐷 𝐷𝐷̇𝛽𝛽

�𝑝𝑝 𝜃𝜃 + 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ − 𝐷𝐷𝜆𝜆 � 𝐷𝐷 = 𝜆𝜆(𝐷𝐷,𝜅𝜅) − 𝜌𝜌 − 𝛼𝛼 �
1 1

In Appendix B it is proven that if𝛿𝛿(𝜔𝜔)
𝑔𝑔1 +𝑔𝑔1∗
𝐷𝐷+− 1� ≡ ∆(𝐷𝐷, 𝜅𝜅).
𝜃𝜃1
𝑝𝑝1 (𝜔𝜔)𝜃𝜃1𝜌𝜌 − 𝛼𝛼 > 0,

(28)

(29)

𝐷𝐷 jumps to the level that makes ∆(𝐷𝐷, 𝜅𝜅) = 0 and the steady state is immediately reached, once𝜅𝜅 is given. 10

Moreover, from (6) and (7), 𝜅𝜅 must equal 𝑢𝑢′ (𝑐𝑐 ∗ )/𝑢𝑢′ (𝑐𝑐) where 𝑐𝑐 and 𝑐𝑐 ∗ make 𝑏𝑏̇ (= −𝑏𝑏̇ ∗ )in (14) equal zero because otherwise 𝑏𝑏̇ and −𝑏𝑏̇ ∗ continue to either expand or decline and thenon-Ponzi game condition is violated. Therefore,𝑢𝑢′ (𝑐𝑐 ∗ ) = 𝜅𝜅𝑢𝑢′ (𝑐𝑐),𝑐𝑐 = 𝐷𝐷 − 𝑐𝑐 ∗ ,𝑐𝑐 ∗ = −𝜌𝜌𝜌𝜌 + 𝑝𝑝2 (𝜔𝜔)(𝜃𝜃2∗ − 𝑔𝑔2∗ ) − 𝑝𝑝1 (𝜔𝜔)𝑔𝑔1∗ ,(30)

where 𝜔𝜔 satisfies the second equation of (25). This 𝜅𝜅 depends on only 𝐷𝐷 besides exogenous
parameters. By substituting this 𝜅𝜅 into (28) and equalizing the result to zero,10

If the job-separation rate 𝛼𝛼 is regarded as the rate of death as a worker, it is naturally less than the subjective
discount rate 𝜌𝜌.

13

𝛽𝛽

∅(𝐷𝐷; 𝛼𝛼, 𝛽𝛽, 𝑏𝑏, 𝑔𝑔1 , 𝑔𝑔2 , 𝜃𝜃1 , 𝑔𝑔1∗ , 𝜃𝜃2∗ − 𝑔𝑔2∗ )≡ 𝜆𝜆(𝐷𝐷,𝜅𝜅) − 𝜌𝜌 − 𝛼𝛼 �𝛿𝛿�𝜔𝜔(𝐷𝐷;𝜃𝜃2∗ −𝑔𝑔2 −𝑔𝑔2∗ )�

𝑔𝑔 +𝑔𝑔∗𝐷𝐷 + 1𝜃𝜃 1 − 1� = 0,

𝑝𝑝1 �𝜔𝜔(𝐷𝐷;𝜃𝜃2∗ −𝑔𝑔2 −𝑔𝑔2∗ )�𝜃𝜃1

1

(31)

one obtains the steady-state level of 𝐷𝐷. Substituting this 𝐷𝐷 to the second equation of (25) gives𝜔𝜔, and hence from (30) 𝑐𝑐 ∗ and 𝑐𝑐 (= 𝐷𝐷 − 𝑐𝑐 ∗ ) obtain.

Let us finally prove that this steady state indeed exists in the neighborhood where
𝑔𝑔1 = 𝑔𝑔1∗ = 0, 𝑔𝑔2 = 𝑔𝑔2∗ = 0.

(32)

From (4), the second equation of (25) and the second and third equations of (30), one obtains
𝑑𝑑𝑑𝑑𝛿𝛿 2 (1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′= 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ > 0,𝑑𝑑𝑑𝑑i.e., the steady-state level of 𝑐𝑐 is larger as 𝐷𝐷 increases. If 𝐷𝐷 is small enough to make 𝑐𝑐 equal
zero, 𝐷𝐷 = 𝑐𝑐 ∗ and 𝜆𝜆 = 𝑢𝑢′ (0) = ∞. Thus, under (29) ∅ given by (31) satisfies𝛿𝛿(𝜔𝜔)𝑐𝑐 ∗∅(𝑐𝑐 ∗ ; … ) = −𝜌𝜌 − 𝛼𝛼 �𝑝𝑝 (𝜔𝜔)𝜃𝜃 − 1� < 0.
1

1

If 𝐷𝐷 is so large as to make 𝑐𝑐 equal the full-employment level 𝑐𝑐𝑓𝑓 , from the first equation of (12)in which 𝑥𝑥 = 1, the third term of the right-hand side of (31) is zero. Thus, from (19) ∅ given by(31) satisfies

𝛽𝛽∅�𝐷𝐷𝑓𝑓 ; … � = 𝑢𝑢′ �𝑐𝑐 � − 𝜌𝜌 > 0,
𝑓𝑓

where 𝐷𝐷𝑓𝑓 is the level of 𝐷𝐷 that makes both countries attain full employment. These two

properties guarantee the existence of 𝐷𝐷 that makes ∅ equal zero and 𝑐𝑐 locate within (0, 𝑐𝑐𝑓𝑓 ). It
also implies that around the steady state level of 𝐷𝐷∅𝐷𝐷 (𝐷𝐷; … ) > 0.

(33)

5. Own and spillover effects of policy and parameter changes
This section analyzes the effects of changes in various policy, technological andpreference parameters on the two countries’ consumption 𝑐𝑐 and 𝑐𝑐 ∗ . As shown in the previous
14section, after those parameters change, the new steady state is immediately reached. Therefore,
one can obtain the effects on 𝑐𝑐 and 𝑐𝑐 ∗ of changes in the parameters by ignoring the transitional
process and simply calculating the effects on their steady-state levels.
Totally differentiating (31) yields∅𝐷𝐷 𝑑𝑑𝑑𝑑 = − ∅𝛼𝛼 𝑑𝑑𝑑𝑑 − ∅𝛽𝛽 𝑑𝑑𝑑𝑑 − ∅𝑏𝑏 𝑑𝑑𝑑𝑑 − ∅𝑔𝑔1 𝑑𝑑𝑔𝑔1 − ∅𝑔𝑔2 𝑑𝑑𝑔𝑔2 − ∅𝜃𝜃1 𝑑𝑑𝜃𝜃1− ∅𝑔𝑔1∗ 𝑑𝑑𝑔𝑔1∗ − ∅𝜃𝜃2∗ −𝑔𝑔2∗ 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ ).

(34)

As shown in Appendix C, the partial derivatives in the neighborhood of (32) are derived from(4), (9), (31) to which 𝜔𝜔 in the second equation of (25) and 𝜅𝜅 in (30) are applied. Because ∅𝐷𝐷in (34) is positive, as given in (33), and 0 < 𝛿𝛿 < 1 from (1), they satisfy𝜂𝜂

𝛽𝛽 𝛿𝛿 2 (1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝛼𝛼∅𝐷𝐷 = �𝑐𝑐 � � 𝜆𝜆 � 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ − 𝑝𝑝 𝜃𝜃 > 0,∅𝛼𝛼 = 1 − 𝑥𝑥 > 0,𝛼𝛼∅𝑔𝑔1 = − 𝜃𝜃 < 0,
𝜂𝜂

1

1

∅𝛽𝛽 = 𝜆𝜆 > 0,𝜂𝜂𝛽𝛽

∅𝑔𝑔2 = − �𝑐𝑐 � � 𝜆𝜆 �𝛽𝛽𝛼𝛼

∅𝑔𝑔1∗ = �𝑐𝑐 � � 𝜆𝜆 � 𝑝𝑝1 − 𝜃𝜃 > 0,
1

1 1

𝜂𝜂

𝛽𝛽

∅𝑏𝑏 = 𝜌𝜌 �𝑐𝑐 � � 𝜆𝜆 � > 0,

𝑝𝑝2 𝛿𝛿(1−𝛿𝛿)𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝛼𝛼𝛼𝛼− 𝜃𝜃 < 0,
1

𝜂𝜂

𝛽𝛽

𝛼𝛼𝛼𝛼

∅𝜃𝜃1 = 𝜃𝜃 > 0,𝜔𝜔𝛿𝛿 ′1𝛼𝛼∅𝜃𝜃2∗ −𝑔𝑔2∗ = −𝑝𝑝2 ��𝑐𝑐 � � 𝜆𝜆 � 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ − 𝑝𝑝 𝜃𝜃 �.
1 1

(35)

In Appendix C 𝑑𝑑𝑐𝑐 ∗ and 𝑑𝑑𝑐𝑐(= 𝑑𝑑𝑑𝑑 − 𝑑𝑑𝑐𝑐 ∗ ) are derived from the second equation of (25), thethird equation of (30), (34) and (35). They are𝛿𝛿 2 (1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝜌𝜌𝜌𝜌

∅𝐷𝐷 𝑑𝑑𝑑𝑑 = − 𝑝𝑝 𝜃𝜃 𝑑𝑑𝑑𝑑 − 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ �∅𝛼𝛼 𝑑𝑑𝑑𝑑 + ∅𝛽𝛽 𝑑𝑑𝑑𝑑 + ∅𝑔𝑔1 𝑑𝑑𝑔𝑔1 + ∅𝜃𝜃1 𝑑𝑑𝜃𝜃1 �1 1𝛿𝛿�1−𝛿𝛿 2 �+𝜔𝜔𝛿𝛿 ′𝛼𝛼𝛼𝛼𝛿𝛿(1−𝛿𝛿)2+ �𝑝𝑝 𝜃𝜃 � 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ 𝑝𝑝2 𝑑𝑑𝑔𝑔2 − �𝜃𝜃 � 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ 𝑑𝑑𝑔𝑔1∗
1 1

𝜂𝜂

𝛼𝛼

− �𝑝𝑝 𝜃𝜃 �

𝛽𝛽

𝛼𝛼

1 1

1

𝛿𝛿 2 (1−𝛿𝛿)𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝛿𝛿(1−𝛿𝛿)2

𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ ),(36)∅𝐷𝐷 𝑑𝑑𝑐𝑐 ∗ = −𝜌𝜌 ��𝑐𝑐 � � 𝜆𝜆 � − 𝑝𝑝 𝜃𝜃 � 𝑑𝑑𝑑𝑑 − 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ �∅𝛼𝛼 𝑑𝑑𝑑𝑑 + ∅𝛽𝛽 𝑑𝑑𝑑𝑑 + ∅𝑔𝑔1 𝑑𝑑𝑔𝑔1 + ∅𝜃𝜃1 𝑑𝑑𝜃𝜃1 �+�𝛿𝛿(1−𝛿𝛿)𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′
𝜂𝜂

𝜂𝜂

1 1

𝛽𝛽

𝛿𝛿(1−𝛿𝛿)2𝛿𝛿𝛿𝛿� ��𝑐𝑐 � � 𝜆𝜆 � − 𝑝𝑝 𝜃𝜃 � 𝑝𝑝2 𝑑𝑑𝑔𝑔2 − �𝑝𝑝1 ∅𝐷𝐷 + 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ ∅𝑔𝑔1∗ � 𝑑𝑑𝑔𝑔1∗
𝛽𝛽𝜔𝜔𝛿𝛿 ′1 1𝛼𝛼𝛿𝛿(1−𝛿𝛿)2 +𝜔𝜔𝛿𝛿 ′+ ��𝑐𝑐 � � 𝜆𝜆 � 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ − �𝑝𝑝 𝜃𝜃 � 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ ).
1 1

15

(37)

Note that the coefficient of 𝑑𝑑𝑑𝑑 in (37) is negative while that of 𝑑𝑑𝑔𝑔2 in (37) is positive because∅𝐷𝐷 in (35) is positive and 0 < 𝛿𝛿 < 1 from (1). The signs of the other coefficients in (36) and(37), except that of 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ ) in (37), are clear. Therefore, one finds

𝛼𝛼 ↓, 𝛽𝛽 ↓, 𝑏𝑏 ↓, 𝑔𝑔1 ↑, 𝑔𝑔2 ↑, 𝜃𝜃1 ↓, 𝑔𝑔1∗ ↓ ⇒ 𝑐𝑐 ↑ , 𝑐𝑐 ∗ ↑,𝜃𝜃2∗ − 𝑔𝑔2∗ ↑ ⇒ 𝑐𝑐 ↓ , 𝑐𝑐 ∗ ↑↓.

(38)

From the second equation of (25),

𝑑𝑑𝑑𝑑 = 𝜔𝜔𝐷𝐷 𝑑𝑑𝑑𝑑 + 𝜔𝜔𝜃𝜃2∗ −𝑔𝑔2 −𝑔𝑔2∗ 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2 − 𝑔𝑔2∗ ),

𝐷𝐷

1−𝛿𝛿

𝐷𝐷

𝑝𝑝

2
�𝜔𝜔� 𝜔𝜔𝐷𝐷 = 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ > 0, �𝜔𝜔� 𝜔𝜔𝜃𝜃2∗ −𝑔𝑔2 −𝑔𝑔2∗ = − 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ < 0.

From (34) and the above equations, the effect on 𝜔𝜔 of a change in 𝜃𝜃2∗ − 𝑔𝑔2∗ is
𝐷𝐷∅

𝑑𝑑𝑑𝑑

𝛿𝛿𝑝𝑝

𝜂𝜂

𝛽𝛽

𝛼𝛼

2
� 𝜔𝜔𝐷𝐷� 𝑑𝑑(𝜃𝜃∗ −𝑔𝑔∗ ) = − �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � �� 𝑐𝑐 � � 𝜆𝜆 � − 𝑝𝑝 𝜃𝜃 � < 0.
2

2

1 1

These properties and (35) give𝛼𝛼 ↓, 𝛽𝛽 ↓, 𝑏𝑏 ↓, 𝑔𝑔1 ↑, 𝑔𝑔2 ↑, 𝜃𝜃1 ↓, 𝑔𝑔1∗ ↓, 𝜃𝜃2∗ − 𝑔𝑔2∗ ↓ ⇒ 𝜔𝜔 ↑.

(39)

Let us discuss the implication of the main results summarized in (38) and (39). When thehome country suffers persistent unemployment, a decrease in the Poison rate of job separation𝛼𝛼 (or an increase in the average duration of employment 1/𝛼𝛼) alleviates deflation in the homecountry. Therefore, it makes more advantageous for home households to consume than to holdmoney and hence stimulates home consumption. Consequently, the current account worsens
and the home currency depreciates, which improves the competitiveness of home firms andexpands employment, further alleviating deflation. The home-currency depreciation in turnimproves the terms of trade for the foreign country and makes it better off because full
employment prevails in the foreign country.
A decrease in the liquidity preference parameter 𝛽𝛽 stimulates home consumption andmoderates deflation, which further stimulates consumption. It expands imports, worsens thecurrent account, depreciates the home currency, makes home firms more competitive and16expands home employment. The foreign country is also better off because the depreciation of
the home currency improves the foreign terms of trade. An expansion in 𝑔𝑔1 , home government

purchases on the home commodity, creates new home employment and reduces the
deflationary gap. Therefore, the same positive effects as those of a decrease in 𝛽𝛽 emerge. An
improvement in home productivity 𝜃𝜃1 , in contrast, widens the deflationary gap, which reduces
home consumption, and hence the opposite effects arise in both countries. Home employment

shrinks so much as to dominate the initial improvement in 𝜃𝜃1 and the foreign terms of tradeworsens, which makes both countries worse off.

A transfer from the home country to the foreign country lowers the home foreign-asset
position 𝑏𝑏 and thereby worsens the home current account. It causes the home currency todepreciate and increases home production, employment and consumption. Consequently, thehome country is better off. The foreign country is also better off because it not only receives thetransfer but also benefits from the improvement in the terms of trade. This result may beconsistent with the consequence of the Marshall Plan and the statement of the Brandt
commission, both mentioned in the introduction.
An expansion in 𝑔𝑔2 , home government purchases on the foreign commodity, worsens thehome current account and makes the home currency depreciate. Therefore, home firms become

more competitive and increase employment, which alleviates deflation and stimulates homeconsumption. The foreign country is also better off because of the improvement in the terms oftrade. An increase in 𝑔𝑔1∗ , foreign government purchases on the home commodity, works in the

opposite direction. It improves the home current account, which makes the home currency
appreciate and the real price of the home commodity (𝑝𝑝1) increase. Even if the home currency
appreciates so much that home employment 𝑥𝑥 and home consumption 𝑐𝑐 return to the previous
levels, the current account is still positive because 𝑝𝑝1 is higher than before. Therefore, thehome currency further appreciates and eventually makes 𝑥𝑥 and 𝑐𝑐 lower than the previous
17levels, harming the home country. The foreign country is also harmed because under full
employment not only an increase in government purchases crowds out private consumption but
also the terms of trade worsens.
The above argument may be summarized as follows. If the home country suffers long-run
deficiency of aggregate demand and the foreign country achieves full employment, a parameteror policy change that worsens the home current account generally depreciates the homecurrency, stimulates home production and employment, mitigates deflation and hence
stimulates home consumption. The depreciation of the home currency improves the foreignterms of trade and hence benefits the foreign country.
An increase in 𝑔𝑔2∗ , foreign government purchases on the foreign commodity, raises therelative price of the foreign commodity, which expands home employment and stimulates
home consumption. The rise in the foreign commodity price benefits the foreign country.
However, it is caused by an increase in foreign government purchases on the foreigncommodity, which harms the foreign country. Therefore, the welfare effect on foreign
households is ambiguous while the home country benefits. The effect of an increase in foreign
productivity 𝜃𝜃2∗ is just the opposite.

6. Conclusions
A two-country economy in which the home country suffers persistent unemployment due
to aggregate demand deficiency while the foreign country realizes full employment is
considered. The stagnant country often attempts to improve production efficiency, expectingthat such an effort would expand the world market share of home commodities and increase
national income. However, it improves the current account and thereby appreciates the homecurrency. In the presence of aggregate demand deficiency the appreciation of the home18

currency is so high that employment eventually decreases and national income andconsumption decline, making the home country worse off. It also worsens the foreign terms oftrade and harms the foreign country with full employment.
In contrast, the home country’s changes in policy and preference parameters that worsen
the home current account cause the home currency to depreciate and thereby expand home
employment and consumption. It also improves the foreign terms of trade and makes theforeign country better off. Typical examples are expansions of home government purchases onhome and foreign commodities.
Foreign aid may be another important example. Foreign aid expenditures tend to be cut
when a donor country suffers long-run stagnation. However, an expansion in foreign aid in fact
makes the country better off because it worsens the home current account, depreciates thehome currency, stimulates business, increases home employment, decreases deflation, andurges people to consume more. Moreover, if the recipient country is fully employed, it is also
better off because it not only receives foreign aid but also benefits from the improvement in theterms of trade.

19Appendices
Appendix A: Wealth preference and zero interest rate
In the text 𝑣𝑣 ′ (𝑚𝑚) is assumed to have a positive lower bound. This property would rather

imply wealth preference than liquidity preference. Moreover, from (10) the nominal interest
rate is 𝑣𝑣 ′ (𝑚𝑚)/𝑢𝑢′ (𝑐𝑐), which converges to 𝛽𝛽 ⁄𝑢𝑢′ (𝑐𝑐) (> 0) in the asymmetric steady state, while

the observed rate in a stagnant economy, such as Japan, is almost zero. This appendix explicitly
separates liquidity preference and wealth preference and shows that the result in the text stillholds while the nominal interest rate is zero.
If wealth preference 𝜑𝜑(𝑎𝑎) is considered, the home utility is∞∫0 [𝑢𝑢(𝑐𝑐) + 𝑣𝑣(𝑚𝑚) + 𝜑𝜑(𝑎𝑎)] exp(−𝜌𝜌𝜌𝜌) 𝑑𝑑𝑑𝑑,

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

(A1)

In this case the first-order optimal conditions of the home household in (6) are replaced by𝜆𝜆̇

𝜆𝜆 = 𝑢𝑢′ (𝑐𝑐), 𝜆𝜆𝜆𝜆 = 𝑣𝑣 ′ (𝑚𝑚),𝜆𝜆Therefore, (8) turns to be

𝜆𝜆 𝐷𝐷

𝐷𝐷̇

𝜌𝜌 + 𝜋𝜋 − � 𝐷𝐷𝜆𝜆 � 𝐷𝐷 −=while (10) is unchanged; that is,𝑅𝑅 == 𝜌𝜌 − 𝑟𝑟 −𝜑𝜑′ (𝑎𝑎).
𝑢𝑢′ (𝑐𝑐)𝑣𝑣 ′ (𝑚𝑚)+𝜑𝜑 ′ (𝑎𝑎),𝑢𝑢′ (𝑐𝑐)𝑣𝑣 ′ (𝑚𝑚).
𝑢𝑢′ (𝑐𝑐)(A2)

Because all the other equations in the text are unchanged, by replacing 𝑣𝑣 ′ (𝑚𝑚) with 𝑣𝑣 ′ (𝑚𝑚) +𝜑𝜑 ′ (𝑎𝑎) one obtains the same dynamic equations.

In the symmetric steady state deflation continues in the home country and 𝑚𝑚 continues toexpand. Therefore, from (A1) and (A2), one finds
𝑅𝑅 = lim𝑚𝑚→∞

𝑣𝑣 ′ (𝑚𝑚)= 0,𝑢𝑢′ (𝑐𝑐)20

implying that the nominal interest rate observed in the home country is zero. Furthermore, byassuming that
lim𝑎𝑎→∞ 𝜑𝜑 ′ (𝑎𝑎) = 𝛽𝛽𝑎𝑎 > 0,instead of (18), one obtains the same steady-state condition as (31) in which 𝛽𝛽 is replaced by
𝛽𝛽𝑎𝑎 . Therefore, all the results of policy and parameter changes in the text are valid, while thehome nominal interest rate is zero.

Appendix B: Existence and stability of the dynamics represented by (28)
If 𝐷𝐷 equals 𝐷𝐷𝑓𝑓 and the home country faces full employment, the first equation of (12)turns to be

Then, from (6), (8) and (19),𝛿𝛿(𝜔𝜔)𝐷𝐷𝑓𝑓 + 𝑔𝑔1 + 𝑔𝑔1∗ = 𝑝𝑝1 (𝜔𝜔)𝜃𝜃1 .
𝛽𝛽𝛽𝛽

𝜌𝜌 < 𝑢𝑢′ �𝑐𝑐 � = 𝜆𝜆�𝐷𝐷 ,𝜅𝜅� ,and ∆(𝐷𝐷, 𝜅𝜅) defined by (28) satisfies

𝑓𝑓

𝑓𝑓∆(𝐷𝐷𝑓𝑓 , 𝜅𝜅) > 0.

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

Because 𝑐𝑐 = 0 if 𝐷𝐷 = 0 for a given 𝜅𝜅, as is seen from (6), (7) and (8), 𝜆𝜆(0, 𝜅𝜅) = 𝑢𝑢′ (0) = ∞.
Therefore, from (28), in the neighborhood where 𝑔𝑔1 + 𝑔𝑔1∗ = 0 the above condition reduces to𝜌𝜌 − 𝛼𝛼 > 0,

which is (29). Figure 2 illustrates this case –i.e., 𝑅𝑅𝑚𝑚 (= 𝛽𝛽/𝜆𝜆(𝐷𝐷𝑓𝑓 , 𝜅𝜅)) is located above 𝜌𝜌 and𝜌𝜌 − 𝛼𝛼 > 0.

In this case ∆(𝐷𝐷, 𝜅𝜅) must be positively inclines as 𝐷𝐷 increases around the steady state, E in

the figure, and thus21∆𝐷𝐷 (𝐷𝐷, 𝜅𝜅) > 0.

Therefore, once 𝜅𝜅 is given, 𝐷𝐷 jumps to the level that satisfies∆(𝐷𝐷, 𝜅𝜅) = 0,

and the steady state of (28) is immediately reached.

Appendix C: Derivations of (35), (36) and (37)
From the first two equations of (30) one obtains
𝑢𝑢′′ (𝑐𝑐 ∗ )𝑑𝑑𝑐𝑐 ∗ = 𝜅𝜅𝑢𝑢′′ (𝑐𝑐)(𝑑𝑑𝑑𝑑 − 𝑑𝑑𝑐𝑐 ∗ ) + 𝑢𝑢′ (𝑐𝑐)𝑑𝑑𝑑𝑑.

Substituting 𝜅𝜅 given by the first equation of (30) into the above equation and utilizing 𝜂𝜂 and 𝜂𝜂 ∗defined in (9) leads to

𝑑𝑑𝑑𝑑

𝜂𝜂

𝜂𝜂

𝜂𝜂 ∗

= 𝑐𝑐 𝑑𝑑𝑑𝑑 − �𝑐𝑐 + 𝑐𝑐 ∗ � 𝑑𝑑𝑐𝑐 ∗ .
𝜅𝜅

(A3)

𝑑𝑑𝑐𝑐 ∗ = −𝜌𝜌𝜌𝜌𝜌𝜌 + 𝑝𝑝2′ 𝜃𝜃2∗ 𝑑𝑑𝑑𝑑 − 𝑝𝑝1 𝑑𝑑𝑔𝑔1∗ + 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ ),(A4)

From the third equation of (30), 𝑑𝑑𝑐𝑐 ∗ isin the neighborhood of (32), that is, 𝑔𝑔1∗ = 0 and 𝑔𝑔2∗ = 0. Substituting (A4) into (A3) gives
𝑑𝑑𝑑𝑑

𝜂𝜂

𝜂𝜂

𝜂𝜂 ∗

= 𝑐𝑐 𝑑𝑑𝑑𝑑 − �𝑐𝑐 + 𝑐𝑐 ∗ � [−𝜌𝜌𝜌𝜌𝜌𝜌 + 𝑝𝑝2′ 𝜃𝜃2∗ 𝑑𝑑𝑑𝑑 − 𝑝𝑝1 𝑑𝑑𝑔𝑔1∗ + 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ )].
𝜅𝜅

(A5)

Totally differentiating the second equation of (25), substituting 𝛿𝛿 = 𝜔𝜔𝜔𝜔2′⁄𝑝𝑝2 given in (4)to the result and rearranging it yields
𝜔𝜔

𝐷𝐷𝐷𝐷𝜔𝜔 = �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � [(1 − 𝛿𝛿)𝑑𝑑𝑑𝑑 − 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ − 𝑔𝑔2 )].

(A6)

Substituting (A6) into (A5), and applying 𝛿𝛿 = 𝜔𝜔𝜔𝜔2′⁄𝑝𝑝2 given in (4) and (1 − 𝛿𝛿)𝐷𝐷 = 𝑝𝑝2 𝜃𝜃2∗ ,obtained from the second equation of (25) and (32), to the result leads to
𝑑𝑑𝑑𝑑

𝜂𝜂

𝜂𝜂

𝜂𝜂 ∗

𝛿𝛿(1−𝛿𝛿)2𝜔𝜔𝛿𝛿 ′= 𝑐𝑐 𝑑𝑑𝑑𝑑 − �𝑐𝑐 + 𝑐𝑐 ∗ � ��𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑑𝑑𝑑𝑑 − 𝜌𝜌𝜌𝜌𝜌𝜌 + �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ )𝜅𝜅+�𝛿𝛿(1−𝛿𝛿)𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′Totally differentiating (31) results in� 𝑝𝑝2 𝑑𝑑𝑔𝑔2 − 𝑝𝑝1 𝑑𝑑𝑔𝑔1∗ �.

22

(A7)

𝑑𝑑𝑑𝑑

𝛽𝛽

𝜆𝜆

𝛿𝛿𝛿𝛿

𝑑𝑑𝜃𝜃

𝜆𝜆

𝑑𝑑∅ = 𝜆𝜆 − � 𝜆𝜆 � � 𝜆𝜆𝐷𝐷 𝑑𝑑𝑑𝑑 + 𝜆𝜆𝜅𝜅 𝑑𝑑𝑑𝑑� − (𝑥𝑥 − 1)𝑑𝑑𝑑𝑑
𝛿𝛿𝐷𝐷�𝛿𝛿 ′ −𝛿𝛿𝑝𝑝1′ ⁄𝑝𝑝1 �−𝛼𝛼 �𝑝𝑝 𝜃𝜃 𝑑𝑑𝑑𝑑 − 𝑝𝑝 𝜃𝜃 � 𝜃𝜃 1 � +
1 1

1 1

𝑝𝑝1 𝜃𝜃1

1

𝑑𝑑𝑑𝑑 +

𝑑𝑑𝑑𝑑1 +𝑑𝑑𝑑𝑑1∗
𝜃𝜃1�,where 𝑥𝑥 is given by the first equation of (12). Substituting 𝜆𝜆𝐷𝐷 and 𝜆𝜆𝜅𝜅 in (9) and 𝑝𝑝1′ in (4) to theabove equation gives

𝑑𝑑𝑑𝑑

𝑑𝑑∅ = 𝜆𝜆 + �𝛿𝛿𝛽𝛽𝑐𝑐 𝑐𝑐∗𝜆𝜆� + ∗ �
𝜂𝜂 𝜂𝜂

𝛿𝛿𝛿𝛿

𝑐𝑐 ∗ 𝑑𝑑𝑑𝑑

� �𝑑𝑑𝑑𝑑 + �𝜂𝜂∗� 𝜅𝜅 � − (𝑥𝑥 − 1)𝑑𝑑𝑑𝑑

𝑑𝑑𝜃𝜃

−𝛼𝛼 �𝑝𝑝 𝜃𝜃 𝑑𝑑𝑑𝑑 − 𝑝𝑝 𝜃𝜃 � 𝜃𝜃 1 � +
1 1

1 1

1

𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝑝𝑝1 𝜃𝜃1 𝜔𝜔

𝐷𝐷𝐷𝐷𝐷𝐷 +𝑑𝑑𝑑𝑑1 +𝑑𝑑𝑑𝑑1∗

By substituting (A6) and (A7) to the above equation one obtains
𝑑𝑑𝑑𝑑

𝛿𝛿

𝛿𝛿𝛿𝛿

𝑑𝑑𝜃𝜃

𝑑𝑑∅ = 𝜆𝜆 − (𝑥𝑥 − 1)𝑑𝑑𝑑𝑑 − 𝛼𝛼 �𝑝𝑝 𝜃𝜃 𝑑𝑑𝑑𝑑 − 𝑝𝑝 𝜃𝜃 � 𝜃𝜃 1 � +
𝛽𝛽

1 1

1 1

𝛿𝛿(1−𝛿𝛿)2

𝜂𝜂

1

𝜃𝜃1

�.

(1−𝛿𝛿)𝑑𝑑𝑑𝑑−𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ −𝑔𝑔2∗ −𝑔𝑔2 )𝜔𝜔𝛿𝛿 ′

𝑝𝑝1 𝜃𝜃1+

𝑑𝑑𝑑𝑑1 +𝑑𝑑𝑑𝑑1∗+ � 𝜆𝜆 � �𝑐𝑐 � �𝑑𝑑𝑑𝑑 − �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑑𝑑𝑑𝑑 + 𝜌𝜌𝜌𝜌𝜌𝜌 − �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ )−�𝛿𝛿(1−𝛿𝛿)𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝜃𝜃1

�

� 𝑝𝑝2 𝑑𝑑𝑔𝑔2 + 𝑝𝑝1 𝑑𝑑𝑔𝑔1∗ �.

Rearranging this equation and applying the first equation of (12) and (32) to the result yields
𝑑𝑑𝑑𝑑

𝛽𝛽

𝜂𝜂

𝛿𝛿 2 (1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′

𝛼𝛼

𝛼𝛼𝛼𝛼

𝛽𝛽

𝜂𝜂

𝑑𝑑∅ = 𝜆𝜆 − (𝑥𝑥 − 1)𝑑𝑑𝑑𝑑 + �� 𝜆𝜆 � �𝑐𝑐 � � 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � − 𝑝𝑝 𝜃𝜃 � 𝑑𝑑𝑑𝑑 + 𝜃𝜃 𝑑𝑑𝜃𝜃1 + 𝜌𝜌 � 𝜆𝜆 � �𝑐𝑐 � 𝑑𝑑𝑑𝑑
𝛼𝛼

𝛽𝛽

𝜂𝜂

− 𝜃𝜃 𝑑𝑑𝑔𝑔1 − �� 𝜆𝜆 � �𝑐𝑐 � �
1𝛽𝛽𝛿𝛿(1−𝛿𝛿)𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝜂𝜂𝜔𝜔𝛿𝛿 ′

1 1

𝛼𝛼

𝛽𝛽

1

𝜂𝜂

𝛼𝛼

� + 𝑝𝑝 𝜃𝜃 � 𝑝𝑝2 𝑑𝑑𝑔𝑔2 + �� 𝜆𝜆 � �𝑐𝑐 � 𝑝𝑝1 − 𝜃𝜃 � 𝑑𝑑𝑔𝑔1∗
1 1

1

𝛼𝛼

− �� 𝜆𝜆 � �𝑐𝑐 � �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � − 𝑝𝑝 𝜃𝜃 � 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ ).
1 1

The coefficients of the above equation are the partial derivatives of ∅ presented in (35).

Substituting 𝑑𝑑𝑑𝑑 in (A6) into (A4) and applying (1 − 𝛿𝛿)𝐷𝐷 = 𝑝𝑝2 𝜃𝜃2∗ and 𝛿𝛿 = 𝜔𝜔𝜔𝜔2′⁄𝑝𝑝2 givenin (4) to the result leads to𝛿𝛿(1−𝛿𝛿)2𝛿𝛿(1−𝛿𝛿)𝑑𝑑𝑐𝑐 ∗ = −𝜌𝜌𝜌𝜌𝜌𝜌 + �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑑𝑑𝑑𝑑 − 𝑝𝑝1 𝑑𝑑𝑔𝑔1∗ − �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑝𝑝2 𝑑𝑑𝑔𝑔2
𝜔𝜔𝛿𝛿 ′+ �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ ),𝛿𝛿 2 (1−𝛿𝛿)+𝜔𝜔𝛿𝛿 ′𝛿𝛿(1−𝛿𝛿)𝑑𝑑𝑐𝑐(= 𝑑𝑑𝑑𝑑 − 𝑑𝑑𝑐𝑐 ∗ ) = 𝜌𝜌𝜌𝜌𝜌𝜌 + � 𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑑𝑑𝑑𝑑 + 𝑝𝑝1 𝑑𝑑𝑔𝑔1∗ + �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑝𝑝2 𝑑𝑑𝑔𝑔223𝜔𝜔𝛿𝛿 ′− �𝛿𝛿(1−𝛿𝛿)+𝜔𝜔𝛿𝛿′ � 𝑝𝑝2 𝑑𝑑(𝜃𝜃2∗ − 𝑔𝑔2∗ ).

Because 𝑑𝑑𝑑𝑑 is given by (34) and (35), 𝑑𝑑𝑑𝑑 and 𝑑𝑑𝑐𝑐 ∗ above turn to those in (36) and (37).

24

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27

𝑡𝑡 − ∆𝑡𝑡

𝑊𝑊𝐹𝐹 (𝑡𝑡 − ∆𝑡𝑡)(A)𝜈𝜈(𝑡𝑡 − ∆𝑡𝑡)

(B)

𝑊𝑊𝐹𝐹 (𝑡𝑡)𝑡𝑡𝜈𝜈(𝑡𝑡)

Figure 1: Timing of Fair-Wage Determination

28

𝑅𝑅

𝑅𝑅 =

𝑅𝑅𝑚𝑚
𝜌𝜌

𝛽𝛽
𝜆𝜆(𝐷𝐷, 𝜅𝜅)

𝑅𝑅 = 𝜌𝜌 + 𝛼𝛼 �𝛿𝛿(𝜔𝜔)𝐷𝐷 − 1�
𝑝𝑝1 (𝜔𝜔)𝜃𝜃1E𝜌𝜌 − 𝛼𝛼𝐷𝐷𝑓𝑓O

Figure 2: Stagnation steady state

29

𝐷𝐷