ISSN (Print) 0473-453X

Discussion Paper No. 1061

ISSN (Online) 2435-0982

A SIMPLE AGGREGATE DEMAND ANALYSISWITH DYNAMIC OPTIMIZATIONINA SMALL OPEN ECONOMY
Ken-ichi Hashimoto
Yoshiyasu Ono

July 2019

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

July 31, 2019

A simple aggregate demand analysis with dynamic optimization
in a small open economy*by†‡
Ken-ichi Hashimoto and Yoshiyasu Ono

Abstract
We develop an aggregate demand analysis of a small open economy based on all agents’dynamic optimization. Murota and Ono (2015) present a simple Keynesian cross analysis withdynamic optimization. This paper extends it to a small-country setting with two factors and two
commodities, of which the structure is as simple as the conventional Keynesian cross analysis.
We apply the model to examine the effects of changes in various parameters, such as the termsof trade, foreign asset holdings and government purchases, on aggregate demand. They arequite different from those under full employment and those of the Mundell-Fleming model.

Keywords: aggregate demand shortage, unemployment, small open economy
JEL Classification Number: E12, E13, E21, E24, F32, F41* This research was financially supported by the program of the Joint Usage/Research Center for “Behavioral
Economics” at ISER, Osaka University, and Grants-in-Aid for Scientific Research (A), (C) and (S) from the JSPS:
grant numbers 15H05728, 16H02016, 16K03624, and 19K01638.
†

Graduate School of Economics, Kobe University, 2-1 Rokko-dai, Nada, Kobe 657-8501, JAPAN; E-mail:hashimoto@econ.kobe-u.ac.jp‡

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

1. Introduction
This paper presents a simple aggregate demand analysis with dynamic optimization of a
two-factor two-commodity small open economy that suffers from secular stagnation of
aggregate demand, and examines the effects of changes in fiscal and monetary policies, theterms of trade, foreign asset holdings, and preference and technology parameters, on
consumption and aggregate demand.
A typical classical analysis on aggregate demand fluctuations in a small open economy is
the “Mundell-Fleming” model (Mankiw, 2010; Ch.12). It is an extension of the conventional
Keynesian model to an open-economy setting. Although it has widely been used in policy
making thanks to its simplicity, probably, it neither considers the optimal firm and householdbehavior nor gives the dynamics of economic variables. Instead, it begins with assuming such
ad-hoc functions as the consumption, investment, net export, and liquidity demand functions,and ignores the current account adjustment. Thus, it is essentially a short-run analysis.
Dornbusch (1980) introduced price and exchange-rate dynamics into Mundell-Fleming model,
but still ignored optimal firm and household behavior.
Since the Lucas critique (1976), some micro-foundations have always been required for
macroeconomic analyses and those behavioral functions must endogenously be derived from
optimizing behavior of agents. Unfortunately, however, most of the recent researches onmacroeconomic dynamics with micro-foundations, such as RBC and DSGE models (e.g.,
Kydland and Prescott, 1982; Christiano, Eichenbaum and Evans, 1999; Hayashi and Prescott,2002; Walsh, 2017), do not treat aggregate demand shortages, which many countries are now
facing, but analyze dynamic adjustment processes without aggregate demand shortages.
A dynamic optimization model of secular stagnation due to aggregate demand shortages
was first presented by Ono (1994, 2001) in a closed-economy setting. He showed that if themarginal utility of money (or wealth) holding is insatiable, aggregate demand deficiency and1

deflation can persistently emerge. Although real money balances expand under deflation, themarginal rate of substitution between money and consumption does not change because of theinsatiability, and thus the wealth effect of real money expansions on consumption disappears.
While lower prices as a result of deflation will not stimulate consumption, deflation itself
makes it more advantageous for people to save more and consume less, leading to a steady statewith secular deflation and stagnation. 1 Using the model Murota and Ono (2015) obtain aconsumption function with similar mathematical properties to the conventional Keynesian
consumption function and examine the multiplier effect of macroeconomic policies onaggregate demand.
In this paper we extend the new Keynesian cross analysis by Murota and Ono (2015) to asmall open economy setting with two factors and two commodities and propose a simple
analytical framework like the Mundell-Fleming model. Using it we obtain the effects ofchanges in various policy, preference and technological parameters on consumption andaggregate demand, and show that they are quite different from those under full employment
and those of the Mundell-Fleming model. There are open-economy extensions of the dynamic
stagnation model, e.g. Ono (2006, 2007, 2014, 2018), Johdo and Hashimoto (2009) andHashimoto (2011, 2015), but they use a two-country setting and have much more complicatedframeworks. The present analysis treats a small country case and has a simple structurecomparable to the Mundell-Fleming model.

1This model has widely been used in various analyses under persistent stagnation in a closed-economy setting.
For example, Matsuzaki (2003) studies the effect of a consumption tax on effective demand in the presence ofpoor and rich people. Hashimoto (2004) examines the intergenerational redistribution effects in an
overlapping-generations framework with the present type of stagnation. Johdo (2006) considers the relationshipbetween R&D subsidies and unemployment. Rodriguez-Arana (2007) examines the dynamic path with public
deficit in the present stagnation case and compares it with that of the neoclassical case. Johdo (2009) introduce
habit formation preference on consumption with non-satiated liquidity preference, and Murota and Ono (2011)
introduce status preference for asset holdings. Hashimoto and Ono (2011) examine the effects of various
pro-population policies under this type of stagnation. Murota and Ono (2012) find the properties of zero interest
rate in the stagnant economy. Illing, Ono and Schlegl (2018) analyze financial market imperfections in a stagnant
economy. See Ono and Hashimoto (2012) for various extensions of this stagnation model. Recently, thisstagnation mechanism has been discussed also by Michaillat and Saez (2014) and Michau (2018).

2

The remainder of this paper proceeds as follows. Section 2 outlines the model structure.
After discussing the case of full employment as a benchmark in section 3, we consider the casewhere secular unemployment and stagnation occur and examine effects of changes in the termsof trade and various policy, preference and technological parameters in section 4. It is foundthat those effects are opposite to those under full employment and significantly differ fromthose in the Mundell-Fleming model. The final section summarizes our findings and concludes
the paper.

2. The Model
We consider a small open economy with two factors, labor and capital, and two
commodities, 1 and 2, in a continuous infinite-time setting. The home country specializes incommodity 1 while the rest of the world produces both commodities, which are tradable. 2 The
nominal home prices of the two commodities are 𝑃𝑃1 and 𝑃𝑃2 (= 𝜀𝜀𝑃𝑃2∗ ), where 𝜀𝜀 is the nominal

exchange rate and 𝑃𝑃2∗ is the international price of the foreign commodity in terms of the foreign

currency. Due to the small-country assumption, the international relative price 𝜔𝜔� ofcommodity 2 in terms of commodity 1 is exogenously given to the home country and thus thenominal exchange rate 𝜀𝜀 changes so that it always satisfies
𝜀𝜀𝑃𝑃∗

𝑃𝑃

𝜔𝜔
� ≡ 𝑃𝑃 2 = 𝑃𝑃2.
1

1

Capital and assets freely move into, and out of, the country and thus the real interest rateequals the exogenously given world interest rate 𝑟𝑟̅ , which satisfies
𝑟𝑟 = 𝑅𝑅 − 𝜋𝜋 = 𝑟𝑟̅ ,(1)

where 𝜋𝜋 ≡ 𝑃𝑃̇/𝑃𝑃 represents the inflation (or deflation if negative) rate of the home consumer
2

Under free capital movement across countries, which we assume soon, a small country must specialize in oneof the two commodities. Using a dynamic 2x2x2 model with capital accumulation Ono and Shibata (2006) provethat under free asset trade instead of free capital movement a country with a much smaller population specializes
in one of the two commodities while the other country produces both commodities.

3

price index 𝑃𝑃 and 𝑅𝑅 is the nominal interest rate in terms of the home currency. The
no-arbitrage condition requires

𝑅𝑅 = 𝜀𝜀̇/𝜀𝜀 + 𝑅𝑅� ,(2)

where 𝑅𝑅� is the nominal interest rate in terms of the foreign currency, which is given to thehome country.

2.1. Households
The population of home households is normalized to unity and their labor endowment is
one unit, which is inelastically supplied. The lifetime utility of the representative household is∞

𝑈𝑈 = ∫0 [𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) + 𝑣𝑣(𝑚𝑚)]exp(−𝜌𝜌𝜌𝜌)d𝑡𝑡,

where 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) represents homothetic utility of consumption, 𝑐𝑐𝑗𝑗 (𝑗𝑗 = 1,2) is consumption ofcommodity 𝑗𝑗, 𝜌𝜌 is the subjective discount rate and 𝑣𝑣(𝑚𝑚) is utility of real money holdings 𝑚𝑚. It
is maximized subject to the flow budget equation and asset constraint:𝑎𝑎̇ = 𝑟𝑟̅ 𝑎𝑎 + 𝑤𝑤𝑤𝑤 − 𝑝𝑝1 𝑐𝑐1 − 𝑝𝑝2 𝑐𝑐2 − 𝑅𝑅𝑅𝑅 − 𝜏𝜏,𝑎𝑎 = 𝑚𝑚 + 𝑏𝑏,(3)

where 𝑎𝑎 is total asset consisting of real balances 𝑚𝑚 and international asset-capital 𝑏𝑏, 𝜏𝜏 is thereal lump-sum tax, 𝑤𝑤 ≡ 𝑊𝑊/𝑃𝑃 is the real wage, 𝑝𝑝𝑗𝑗 (𝑗𝑗 = 1,2) is the real price of commodity 𝑗𝑗,
and 𝑥𝑥 is the employment rate. The employment rate 𝑥𝑥 is positive and may be lower than 1:
1 ≥ 𝑥𝑥 ≥ 0,

because involuntary unemployment may appear. Because 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) is homothetic, real prices
𝑝𝑝𝑗𝑗 (𝑗𝑗 = 1,2) depend on only the relative price 𝜔𝜔� and satisfy
𝑃𝑃

𝑝𝑝𝑗𝑗 ≡ 𝑃𝑃𝑗𝑗 = 𝑝𝑝𝑗𝑗 (𝜔𝜔�); 𝑝𝑝1′ (𝜔𝜔�) < 0 , 𝑝𝑝2′ (𝜔𝜔�) > 0 .
R

R

Given the current value Hamiltonian function H of the household optimizing behavior:
𝐻𝐻 = 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) + 𝑣𝑣(𝑚𝑚) + 𝜆𝜆[𝑟𝑟̅ 𝑎𝑎 + 𝑤𝑤𝑤𝑤 − 𝑝𝑝1 (𝜔𝜔�)𝑐𝑐1 − 𝑝𝑝2 (𝜔𝜔�)𝑐𝑐2 − 𝑅𝑅𝑅𝑅 − 𝜏𝜏],4

where 𝜆𝜆 is the costate variable for 𝑎𝑎, the first-order optimal conditions are𝑢𝑢�1 (𝑐𝑐1 , 𝑐𝑐2 ) = 𝜆𝜆𝑝𝑝1 , 𝑢𝑢�2 (𝑐𝑐1 , 𝑐𝑐2 ) = 𝜆𝜆𝑝𝑝2 , 𝑣𝑣′(𝑚𝑚) = 𝜆𝜆𝜆𝜆, 𝜆𝜆̇ = (𝜌𝜌 − 𝑟𝑟̅ )𝜆𝜆.

Because 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) is homothetic, from (4) one has
𝑝𝑝2
𝑝𝑝1

�2 (𝑐𝑐1 ,𝑐𝑐2 )
𝑢𝑢

(= 𝜔𝜔
�) =

�1 (𝑐𝑐1 ,𝑐𝑐2 )𝑢𝑢(4)� (1,𝑐𝑐 /𝑐𝑐 )𝑢𝑢= �2 (1,𝑐𝑐2 /𝑐𝑐1),
𝑢𝑢1

2

1

implying that 𝑐𝑐2 /𝑐𝑐1 is a function of only 𝜔𝜔�. Therefore, the ratio of consumption expenditure
allocated to each commodity depends only on the relative price:𝑝𝑝1 (𝜔𝜔
�)𝑐𝑐 , 𝑝𝑝2 (𝜔𝜔�)]𝑐𝑐 ,
�)𝑐𝑐1 = 𝛿𝛿(𝜔𝜔�)𝑐𝑐2 = [1 − 𝛿𝛿(𝜔𝜔RR

𝑐𝑐 = 𝑝𝑝1 𝑐𝑐1 + 𝑝𝑝2 𝑐𝑐2,

(5)

where 𝑐𝑐 is real total consumption and 𝛿𝛿(∙) is the ratio of consumption expenditure allocated toRcommodity 1.

Because of the homothetic utility, 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) in which 𝑐𝑐1 and 𝑐𝑐2 take the optimal levels

given by (5) must be independent of the relative price 𝜔𝜔� so that one can exhibit 𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) as𝑢𝑢(𝑐𝑐):

𝛿𝛿1−𝛿𝛿

𝑢𝑢(𝑐𝑐) ≡ 𝑢𝑢� �𝑝𝑝 𝑐𝑐, 𝑝𝑝 𝑐𝑐�.
1

From (5) and this equation we obtain

�
𝑢𝑢

2

�
𝑢𝑢

𝑢𝑢′ (𝑐𝑐) = 𝛿𝛿 𝑝𝑝1 + (1 − 𝛿𝛿) 𝑝𝑝2 = 𝜆𝜆.
12From (1), (4) and the above property, we obtain the Euler equation and the money demand
function:𝑐𝑐̇𝜂𝜂 𝑐𝑐 = 𝑅𝑅 − 𝜋𝜋 − 𝜌𝜌 = 𝑟𝑟̅ − 𝜌𝜌,𝑣𝑣 ′ (𝑚𝑚)

𝑅𝑅 = 𝑢𝑢′ (𝑐𝑐) ,where 𝜂𝜂 ≡ −𝑢𝑢′′(𝑐𝑐)𝑐𝑐/𝑢𝑢′(𝑐𝑐) is the elasticity of marginal utility of consumption.
As is standard in small open economy models, we assume𝜌𝜌 = 𝑟𝑟 = 𝑟𝑟̅ ,
5

(6)
(7)

since otherwise this country will eventually be a large country (if 𝜌𝜌 < 𝑟𝑟̅ ) or disappears (if𝜌𝜌 > 𝑟𝑟̅ ). By applying this property to (7) we find
𝑐𝑐̇
𝑐𝑐

= 0,

(8)

implying that consumption 𝑐𝑐 is constant over time and that access to the international

asset-capital market makes consumption completely smoothed. Finally, the transversality
condition is
lim𝑡𝑡→∞ 𝜆𝜆𝜆𝜆exp(−𝑟𝑟̅ 𝑡𝑡) = 0.
2.2. Firms
The home country specializes in commodity 1 and the technology is of constant returns toscale with respect to labor 𝑥𝑥 and capital 𝑘𝑘 as follows:𝑥𝑥𝜃𝜃𝜃𝜃(𝑥𝑥, 𝑘𝑘) = 𝜃𝜃𝑓𝑓(𝑛𝑛)𝑘𝑘, 𝑛𝑛 ≡ 𝑘𝑘,

where 𝜃𝜃 is the total productivity. Because capital freely moves into the country, the first-order

optimal conditions to maximize profits 𝑝𝑝1 (𝜔𝜔�)𝜃𝜃𝑓𝑓(𝑛𝑛)𝑘𝑘 – 𝑤𝑤𝑤𝑤 – 𝑟𝑟̅ 𝑘𝑘 are

𝑝𝑝1 (𝜔𝜔�)𝜃𝜃𝑓𝑓 ′ (𝑛𝑛) = 𝑤𝑤, 𝑝𝑝1 (𝜔𝜔�)𝜃𝜃[𝑓𝑓(𝑛𝑛) − 𝑛𝑛𝑛𝑛 ′ (𝑛𝑛)] = 𝑟𝑟̅ .

(9)Because the international relative price 𝜔𝜔� and the real interest rate 𝑟𝑟̅ (= 𝜌𝜌) are given to thehome country, 𝑤𝑤 is fully determined by exogenous parameters 𝜃𝜃 and 𝜔𝜔� and satisfies
𝑤𝑤 = 𝑤𝑤(𝜃𝜃, 𝜔𝜔�);𝑤𝑤𝜃𝜃 =� )𝑓𝑓(𝑛𝑛)𝑝𝑝1 (𝜔𝜔
𝑛𝑛> 0,𝑤𝑤𝜔𝜔� =� )𝜃𝜃𝑓𝑓(𝑛𝑛)𝑝𝑝1′ (𝜔𝜔
𝑛𝑛

< 0,

(10)

where a subscript of 𝑤𝑤 represents a partial derivative with respect to it. An increase in the laborproductivity obviously increases the real wage. An increase in the relative price of the foreigncommodity works as if the productivity of the home commodity decreases and thus the real
wage declines.

2.3. Government

6

Suppose that the government expands the money supply 𝑀𝑀 𝑆𝑆 at a constant rate 𝜇𝜇 ≡𝑀𝑀̇ 𝑆𝑆 /𝑀𝑀 𝑆𝑆 and purchases 𝑔𝑔1 and 𝑔𝑔2 of the two commodities, respectively. Then, thegovernment’s budget constraint is𝜏𝜏 +𝜇𝜇𝑀𝑀𝑆𝑆𝑃𝑃= 𝑔𝑔(≡ 𝑝𝑝1 (𝜔𝜔�)𝑔𝑔1 + 𝑝𝑝2 (𝜔𝜔�)𝑔𝑔2 ),

(11)

where 𝑔𝑔 represents aggregate government purchases. 3
2.4. Market adjustments

Because the real money holdings 𝑚𝑚 always satisfies
𝑀𝑀𝑆𝑆

𝑚𝑚 = 𝑃𝑃 ,the dynamics of 𝑚𝑚 is

𝑚𝑚̇
𝑚𝑚

(12)

= 𝜇𝜇 − 𝜋𝜋,

where 𝜇𝜇 is the money growth rate. Substituting this equation and the government budget

constraint (11) into the flow budget equation in (3) yields the dynamics of international
asset-capital holdings:
𝑏𝑏̇ = 𝑟𝑟̅���������
𝑏𝑏 + 𝑤𝑤(𝜃𝜃, 𝜔𝜔�)𝑥𝑥 − (𝑐𝑐 + 𝑔𝑔),

(13)

national income

where international asset-capital 𝑏𝑏 is the sum of domestic capital 𝑘𝑘 and foreign asset-debt 𝑏𝑏 𝑓𝑓

and hence𝑏𝑏 = 𝑘𝑘 + 𝑏𝑏 𝑓𝑓 ,𝑏𝑏̇ = 𝑘𝑘̇ + 𝑏𝑏̇ 𝑓𝑓 .

Thus, (13) is equivalent to the current account equation:
𝑏𝑏̇ 𝑓𝑓 = 𝑟𝑟̅ 𝑏𝑏 𝑓𝑓 + ���������
𝑟𝑟̅ 𝑘𝑘 + 𝑤𝑤(𝜃𝜃, 𝜔𝜔�)𝑥𝑥 −

where 𝑖𝑖 is real domestic investment 𝑘𝑘̇.

GDP

3

(𝑐𝑐 +
𝑖𝑖 +��
𝑔𝑔) .
��
���

aggregate demand

The balanced budget is assumed merely for simplicity. Because we take into account the flow budget
equation, the Ricardian equivalence holds. Thus, even if the government issues public bonds and adopts a deficit
budget, the following analysis is valid.

7

The nominal wage adjustment in the labor market is perfect upward but sluggish
downward and follows
𝑊𝑊̇𝑊𝑊= 𝛼𝛼(𝑥𝑥 − 1) if 𝑥𝑥 < 1,

where 𝛼𝛼 represents the adjustment speed of the nominal wage. 4 It is because workers resist a

decline in 𝑊𝑊 but welcome a rise in 𝑊𝑊 no matter how fast it is. Because real wage 𝑤𝑤 is constantover time from (10), this wage adjustment yields the inflation rate of the commodity price asfollows:𝑃𝑃̇𝑊𝑊̇

𝜋𝜋 ≡ 𝑃𝑃 = � 𝑊𝑊𝜇𝜇

= 𝛼𝛼(𝑥𝑥 − 1)

if 𝑥𝑥 < 1,

if 𝑥𝑥 = 1.

(14)This implies that in the presence of aggregate demand shortages 𝑃𝑃 follows the movement of 𝑊𝑊

while under full employment 𝑊𝑊 follows the movement of 𝑃𝑃. The asymmetry in the inflation
process is a fundamental element of stagnation models including among others thecontributions of Eggertsson, Mehrotra and Robbins (2017), Schmitt-Grohé and Uribe (2016,2017), Michau (2018) and Illing, Ono and Schlegl (2018). Stagnation typically results fromsome form of downward nominal wage rigidity that arises in case of unemployment.

3. Full employment
Using the model presented in the previous section, we will propose a simple analyticalframework of aggregate demand fluctuations that can replace the conventional
Mundell-Fleming model, and apply it to examine the effects of changes in various policy,
preference and technological parameters and the terms of trade. Before doing so, this section
treats the case of full employment as a benchmark and shows policy implications, which will
4

This assumption is imposed so that the possibility of unemployment is not intrinsically avoided. Obviously,
this assumption does not eliminate the possibility of full employment steady state. Ono and Ishida (2014) give a
micro-foundation of wage adjustment under which the adjustment converges to this form if stagnation occurs insteady state.

8

be later compared with those under secular stagnation.
From (13) in which 𝑥𝑥 = 1, we find𝑏𝑏̇ = 𝑟𝑟̅ 𝑏𝑏 + 𝑤𝑤(𝜃𝜃, 𝜔𝜔�) − (𝑐𝑐 + 𝑔𝑔),

which is unstable with respect to 𝑏𝑏. Furthermore, from (8), 𝑐𝑐 is constant over time. Hence, 𝑐𝑐

must initially jump so that 𝑏𝑏̇ = 0 and then,

𝑐𝑐 = 𝑐𝑐𝐹𝐹 ≡ 𝑟𝑟̅ 𝑏𝑏0 + 𝑤𝑤(𝜃𝜃, 𝜔𝜔�) − 𝑔𝑔,(15)

where 𝑏𝑏0 is the initial holding of international asset-capital. Because 𝜋𝜋 = 𝜇𝜇 under full
employment from (14), we replace 𝑐𝑐 and 𝜋𝜋 in (6) and (7) by 𝑐𝑐𝐹𝐹 in (15) and 𝜇𝜇 respectively andobtain𝑣𝑣 ′ (𝑚𝑚)𝜌𝜌 + 𝜇𝜇 = 𝑅𝑅 = 𝑢𝑢′ (𝑐𝑐 ),
𝐹𝐹

(16)

which gives the steady state level of 𝑚𝑚. The full-employment consumer price index 𝑃𝑃 moves
in parallel with 𝑀𝑀 𝑆𝑆 so that 𝑚𝑚 satisfies (12).

Noting that 𝑤𝑤(𝜃𝜃, 𝜔𝜔�) satisfies (10), from (15) we find the effect of changes in theparameters on the full-employment consumption 𝑐𝑐𝐹𝐹 :𝜃𝜃↑, 𝜔𝜔
�↓, 𝑏𝑏0 ↑ ⇒ 𝑐𝑐𝐹𝐹 ↑;𝑔𝑔↑ ⇒ 𝑐𝑐𝐹𝐹 ↓,𝑀𝑀 𝑆𝑆 or 𝜇𝜇 has no effect on 𝑐𝑐𝐹𝐹 .

(17)

An increase in productivity 𝜃𝜃, an improvement in the terms of trade (𝜔𝜔�↓), and a larger 𝑏𝑏0

naturally raise national income and hence increases consumption while an increase ingovernment purchases 𝑔𝑔 crowds out consumption. As for monetary policy, the super neutralityof money holds: Neither an instantaneous jump of money supply 𝑀𝑀 𝑆𝑆 nor an increase in themonetary expansion rate 𝜇𝜇 affects consumption.
From (2) and (16), we obtain

𝜀𝜀̇

𝜀𝜀

= 𝜌𝜌 + 𝜇𝜇 − 𝑅𝑅�.

(18)

Thus, an expansion in 𝜇𝜇 raises the depreciation speed of the home currency while changes in9(𝜔𝜔�, 𝑏𝑏0 , 𝜃𝜃, 𝑔𝑔) have no effect on it.

4. Secular demand stagnation
Let us now explore the possibility of persistent unemployment and secular stagnation. We
will obtain the condition under which the steady state with full employment cannot be reached
in a small open economy, and show that a liquidity trap plays a crucial role in leading the
economy to this situation. Secular stagnation that arises under a liquidity trap in a dynamic
optimization setting was first analyzed by Ono (1994, 2001) in a closed-economy setting.. We
apply the model to the present setting.

4.1. Steady state with secular stagnation
A liquidity trap in the present setting arises if the desire for money holding is insatiable:
lim𝑚𝑚→∞ 𝑣𝑣 ′ (𝑚𝑚) = 𝛽𝛽 > 0,

where 𝛽𝛽 is a positive constant. 5 Then, the shape of the money demand curve represented by (7)is as illustrated in Figure 1. In this case it is clear that the solution of 𝑚𝑚 in (16) does not exist if
𝑐𝑐𝐹𝐹 is so large as to satisfy𝛽𝛽𝜌𝜌 + 𝜇𝜇 < 𝑢𝑢′ (𝑐𝑐 ) (<𝐹𝐹

𝑣𝑣 ′ (𝑚𝑚)𝑢𝑢′ (𝑐𝑐𝐹𝐹 )for any 𝑚𝑚).

(19)

From (15) 𝑐𝑐𝐹𝐹 equals home national income minus government purchases, and thus (19) implies

that a richer country tends to fall in secular stagnation.

Now we obtain the steady state with secular stagnation and involuntary unemployment. In
the presence of unemployment (𝑥𝑥 < 1) nominal wages and prices continue to decline in the5

Ono (1994: 4–8) gives an exstensive survey on the insatiable utility of money in the history of economic
thought (e.g., Veblen, Marx, Simmel, Keynes). Ono (1994: 34–8) uses the GMM (generalized method of
moments) to show the validity of this property while Ono, Ogawa and Yoshida (2004) apply parametric andnonparametric methods to support this property.

10way represented by (14), which keeps expanding real balances and lead to 𝑣𝑣 ′ (𝑚𝑚) = 𝛽𝛽. 6 Thus,

from (6), (7) and (14), we obtain𝛽𝛽𝑢𝑢′ (𝑐𝑐)= 𝜌𝜌 + 𝛼𝛼(𝑥𝑥 − 1).

(20)

Because consumption c is constant over time, as proven in (8), c initially jumps to the steadystate value making 𝑏𝑏̇ = 0 and stays there. Otherwise, 𝑏𝑏 keeps either expanding if initially𝑏𝑏̇ > 0 , violating the transversality condition, or decreasing if initially 𝑏𝑏̇ < 0 , which isinfeasible.

𝑅𝑅𝛽𝛽𝑢𝑢′ (𝑐𝑐)O

Figure 1: Money demand with a liquidity trap𝑚𝑚

As there is no dynamics of 𝑏𝑏, the international asset-capital remains at the initial level 𝑏𝑏0

and hence from (13) we always have

which gives𝑏𝑏̇ = 𝑟𝑟̅ 𝑏𝑏0 + 𝑤𝑤(𝜃𝜃, 𝜔𝜔�)𝑥𝑥 − 𝑐𝑐 − 𝑔𝑔 = 0,

6This deflation path satisfies the transversality condition although real balances keep expanding because thenominal interest rate 𝑅𝑅 = 𝛽𝛽/𝑢𝑢′(𝑐𝑐) is positive and hence 𝑚𝑚̇/𝑚𝑚 = 𝜇𝜇 − 𝜋𝜋 = 𝜇𝜇 − 𝑅𝑅 + 𝑟𝑟̅ < 𝑟𝑟̅ as long as 𝜇𝜇 is smaller
than 𝑅𝑅.

11

𝑐𝑐+𝑔𝑔−𝑟𝑟̅ 𝑏𝑏

Substituting this 𝑥𝑥 to (20) yields

0 < 𝑥𝑥 = 𝑤𝑤(𝜃𝜃,𝜔𝜔�)0 < 1.
𝛽𝛽

(21)

𝑐𝑐+𝑔𝑔−𝑟𝑟̅ 𝑏𝑏

𝛷𝛷(𝑐𝑐) ≡ 𝑢𝑢′ (𝑐𝑐) − �𝜌𝜌 + 𝛼𝛼 � 𝑤𝑤(𝜃𝜃,𝜔𝜔�)0 − 1�� = 0,(22)which gives the equilibrium level of 𝑐𝑐.
From (15), (19) and (22) we find𝛽𝛽𝛷𝛷(𝑐𝑐𝐹𝐹 ) = 𝑢𝑢′ (𝑐𝑐 ) − 𝜌𝜌 > 0,𝐹𝐹

in the present case. Thus, in order for the equilibrium 𝑐𝑐 given by (22) to exist, the following

inequality must be satisfied,𝛽𝛽

𝑐𝑐 +𝑔𝑔−𝑟𝑟̅ 𝑏𝑏

𝛷𝛷(𝑐𝑐𝐿𝐿 ) = 𝑢𝑢′ (𝑐𝑐 ) − �𝜌𝜌 + 𝛼𝛼 � 𝐿𝐿𝑤𝑤(𝜃𝜃,𝜔𝜔�) 0 − 1�� < 0,
𝐿𝐿

𝑐𝑐𝐿𝐿 ≡ max{0, 𝑟𝑟̅ 𝑏𝑏0 − 𝑔𝑔},

where 𝑐𝑐𝐿𝐿 is the lowest possible value of consumption, which is either zero or the disposable

income in the case where employment is zero. Furthermore, if the equilibrium is unique, 𝛷𝛷(𝑐𝑐)must be positively inclined in the neighborhood of the equilibrium 𝑐𝑐 –i.e.,𝛷𝛷′ (𝑐𝑐) =−𝛽𝛽𝑢𝑢′′ (𝑐𝑐)2
�𝑢𝑢′ (𝑐𝑐)�

𝛼𝛼− 𝑤𝑤(𝜃𝜃,𝜔𝜔�) > 0.

(23)

4.2. Revised Keynesian cross in a small open economy
We now reformulate the present model to a small-country version of the new Keynesiancross analysis of Murota and Ono (2015). Let us define 𝑦𝑦 to be aggregate demand 𝑐𝑐 + 𝑔𝑔:
𝑦𝑦 = 𝑐𝑐 + 𝑔𝑔,

where real investment 𝑖𝑖 does not appear because the economy is always in steady state, as
mentioned above (21). Replacing 𝑐𝑐 + 𝑔𝑔 in (22) by 𝑦𝑦 yields
𝛽𝛽𝑢𝑢′ (𝑐𝑐)𝑦𝑦−𝑟𝑟̅ 𝑏𝑏= 𝜌𝜌 + 𝛼𝛼 �𝑤𝑤(𝜃𝜃,𝜔𝜔�0) − 1� ⇒ 𝑐𝑐 = 𝑐𝑐(𝑦𝑦; 𝑤𝑤(𝜃𝜃, 𝜔𝜔�), 𝛼𝛼, 𝛽𝛽, 𝑏𝑏0 ).

(24)This is the revised consumption function with dynamic optimization, showing the relationship12between aggregate demand 𝑦𝑦 and consumption 𝑐𝑐. If 𝑏𝑏0 = 0, this function reduces to𝛽𝛽𝑢𝑢′ (𝑐𝑐)𝑦𝑦= 𝜌𝜌 + 𝛼𝛼 �𝑤𝑤 − 1�,

which is the same as the consumption function in the closed-economy model of Murota andOno (2015). Using the conditions in (23) we obtain the following properties:
1 > 𝑐𝑐𝑦𝑦 =𝑐𝑐(𝑦𝑦𝐿𝐿 ; 𝑤𝑤(𝜃𝜃, 𝜔𝜔�), 𝛼𝛼, 𝛽𝛽, 𝑏𝑏0 ) = 𝑢𝑢′−1

�)𝛼𝛼/𝑤𝑤(𝜃𝜃,𝜔𝜔−𝛽𝛽𝑢𝑢′′ /(𝑢𝑢′ )2

�

𝛽𝛽

> 0,

�𝑏𝑏
𝑦𝑦 −𝑟𝑟𝜌𝜌+𝛼𝛼� 𝐿𝐿 � 0 −1�
𝑤𝑤(𝜃𝜃,𝜔𝜔)� > 0, 𝑦𝑦𝐿𝐿 ≡ max{0, 𝑟𝑟̅ 𝑏𝑏0 },(25)

which are mathematically the same as those of the conventional Keynesian consumptionfunction: (i) the derivative with respect to 𝑦𝑦 is positive and smaller than 1, and (ii) the
consumption level is positive when 𝑦𝑦 = 𝑦𝑦𝐿𝐿 .

However, the implication completely differs between the two consumption functions. The

conventional consumption function regards 𝑦𝑦 as income and assumes that a household spends

only a part of income on consumption in the same period. Thus, a tax straightforwardly reduces

consumption while a subsidy expands it. However, this assumption is severely criticizedbecause of its ad-hocness. The present consumption function, in contrast, is derived fromhousehold and firm optimizing behavior, market equilibrium conditions and wage and priceadjustments. In this function 𝑦𝑦 is not income but aggregate demand. It creates employment,
raises inflation (or mitigates deflation) and thereby stimulates consumption.

Because aggregate demand 𝑦𝑦 equals 𝑐𝑐 + 𝑔𝑔, using the revised consumption function givenby (24) we find𝑦𝑦 = 𝑐𝑐(𝑦𝑦; 𝑤𝑤(𝜃𝜃, 𝜔𝜔�), 𝛼𝛼, 𝛽𝛽, 𝑏𝑏0 ) + 𝑔𝑔,(26)

which determines the equilibrium levels of 𝑦𝑦 and 𝑐𝑐. From (25) and (26) we find the multipliereffect of government purchases like the conventional one and the effects of the otherparameters:

13

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

1

𝑑𝑑𝑑𝑑

= 1−𝑐𝑐 > 1,

𝑑𝑑𝑑𝑑

𝑑𝑑𝑀𝑀𝑆𝑆

𝑑𝑑𝜇𝜇

𝑦𝑦

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑

𝑐𝑐

= 𝑑𝑑𝑑𝑑 = 1−𝑐𝑐𝑧𝑧

= 0,

𝑦𝑦

𝑑𝑑𝑑𝑑

𝑐𝑐𝑦𝑦

= 1−𝑐𝑐 > 0;
𝑦𝑦

= 0;

for 𝑧𝑧 = 𝑤𝑤(𝜃𝜃, 𝜔𝜔�), 𝛼𝛼, 𝛽𝛽, 𝑏𝑏0 ,Noting that 𝑥𝑥 < 1 under stagnation, from (24) we have
𝛼𝛼𝛼𝛼

𝑥𝑥−1

𝑐𝑐𝑤𝑤 = − −𝑤𝑤𝑤𝑤𝑢𝑢′′ /(𝑢𝑢′ )2 < 0, 𝑐𝑐𝛼𝛼 = −𝛽𝛽𝑢𝑢′′ /(𝑢𝑢′ )2 < 0,𝑢𝑢′

𝑐𝑐𝛽𝛽 = − −𝛽𝛽𝑢𝑢′′ < 0,𝜌𝜌𝛼𝛼/𝑤𝑤𝑐𝑐𝑏𝑏0 = − −𝛽𝛽𝑢𝑢′′ /(𝑢𝑢′ )2 < 0.

(27)

Because 𝑤𝑤 increases as 𝜃𝜃 is larger and decreases as 𝜔𝜔� rises, as shown in (10), the effects of theparameter changes are summarized as follows:𝑔𝑔↑ ⇒ 𝑦𝑦↑, 𝑐𝑐↑;

𝑀𝑀 𝑆𝑆 or 𝜇𝜇 has no effect on 𝑦𝑦, 𝑐𝑐;𝜃𝜃↑, 𝜔𝜔
�↓ ⇒ 𝑤𝑤↑, 𝑦𝑦↓, 𝑐𝑐↓;𝛼𝛼↑, 𝛽𝛽↑, 𝑏𝑏0 ↑ ⇒ 𝑦𝑦↓, 𝑐𝑐↓.

(28)

As shown in (17), when full employment prevails, government purchases crowd out
consumption 𝑐𝑐𝐹𝐹 while an increase in productivity 𝜃𝜃, an improvement in the terms of trade 𝜔𝜔�↓and an increase in international asset-capital holdings 𝑏𝑏0 expand 𝑐𝑐𝐹𝐹 . Thus, the results in (28)are opposite to the results under full employment. They are also quite different from those of
the Mundell-Fleming model. In the Mundell-Fleming model a fiscal expansion has no effect on
consumption or national income while a monetary expansion expands both of them (seeMankiw, 2010, Ch.12).
Using (26) we propose a Keynesian-cross-like analysis that diagrammatically gives theresults in (28). Figure 2 illustrates the 45 degree line showing the left-hand side of (26) and thecurve of 𝑐𝑐 + 𝑔𝑔 given by the right-hand side of (26). From (26) and (27), we find that increases

in 𝑔𝑔 and 𝜔𝜔� shifts the right-hand side of (26) upward while increases in 𝜃𝜃, 𝛼𝛼, 𝛽𝛽 and 𝑏𝑏0 shifts it
14

downward, and hence the results on 𝑦𝑦 given in (28) obtain in the figure.

𝑦𝑦

45 degree line
𝑔𝑔 ↑, 𝜔𝜔
�↑𝑐𝑐(𝑦𝑦; 𝑤𝑤(𝜃𝜃, 𝜔𝜔� ), 𝛼𝛼, 𝛽𝛽, 𝑏𝑏0 ) + 𝑔𝑔𝜃𝜃 ↑, 𝛼𝛼 ↑, 𝛽𝛽 ↑, 𝑏𝑏0 ↑O𝑦𝑦(= 𝑐𝑐 + 𝑔𝑔)

𝑦𝑦

Figure 2: Keynesian cross in a small open economy

Let us intuitively discuss the implications of the above results. An increase in government
purchases worsens the current account given by (13), which creates a depreciation pressure on
the home exchange rate. However, because the country is small, even an infinitesimal currency
depreciation increases demand for the home commodity and expand employment, which
mitigates deflation and thereby stimulates consumption. Note that the present multiplier effectof government purchases on consumption emerges although the government budget isbalanced while in the conventional Keynesian model it does only under a deficit budget. 7 An
increase in the money stock 𝑀𝑀 𝑆𝑆 or the monetary expansion rate 𝜇𝜇 has no impact on any
7

In the present model the multiplier effect emerges regardless of whether the government adopts a balanced
budget or a deficit budget because the Ricardian equivalence holds.

15

variable.
A deterioration in the terms of trade ( 𝜔𝜔�↑ ) increases world demand for the homecommodity and expands employment, which mitigates deflation and stimulates consumption.
An increase in the wage adjust speed (𝛼𝛼) in the presence of involuntary unemployment worsens

deflation and lowers consumption. An increase in the liquidity preference (𝛽𝛽) directly shifts theconsumption function downward and thus worsens deflation, which further reduces
consumption. An improvement in the productivity (𝜃𝜃) widens the deflationary gap and worsens
deflation, urging households to reduce consumption. An increase in international asset-capital
holdings 𝑏𝑏0 raises the interest earnings and improves the current account. It yields anappreciation pressure on the home currency and worsens home employment. Consequently,
deflation worsens and consumption declines.
Finally, we obtain the effect on the nominal exchange rate 𝜀𝜀. From the no-arbitrage

condition of the international asset-capital in (2), the depreciation speed of the home currency
is
𝜀𝜀̇

𝜀𝜀

𝛽𝛽
= 𝑢𝑢′ (𝑐𝑐) − 𝑅𝑅� ,𝜀𝜀̇−𝛽𝛽𝑢𝑢′′ (𝑐𝑐)i.e., the depreciation speed changes in the same direction as 𝑐𝑐:
𝑑𝑑 �𝜀𝜀� = (𝑢𝑢′ (𝑐𝑐))2 𝑑𝑑𝑑𝑑.

Therefore, a parameter change stimulating consumption has a positive impact on the
depreciation speed of the home currency. This result is also quite different from that under full
employment. Under full employment the depreciation speed of the home currency depends
only on the monetary expansion rate 𝜇𝜇 and a larger 𝜇𝜇 increases it, as presented in (18). Under
secular stagnation, however, 𝜇𝜇 has no effect on the depreciation speed because it affects neither
𝑐𝑐 nor the nominal interest rate 𝑅𝑅(= 𝛽𝛽/𝑢𝑢′ (𝑐𝑐)).

16

4.3. Numerical example
To get the intuition of the magnitudes of the multiplier effects let us present quantitative
examples of the effects of changes in various parameters, such as government purchases 𝑔𝑔,
international asset-capital holdings 𝑏𝑏0 , total productivity 𝜃𝜃 and the relative price (or the termsof trade) 𝜔𝜔�. To do so, we specify the production and utility functions as follows:𝜃𝜃𝜃𝜃(𝑥𝑥, 𝑘𝑘) = 𝜃𝜃𝑥𝑥 𝛾𝛾 𝑘𝑘1−𝛾𝛾 , 𝜃𝜃𝜃𝜃(𝑛𝑛) = 𝜃𝜃𝑛𝑛𝛾𝛾 ,𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) = ln(𝜅𝜅1 𝑐𝑐1𝜎𝜎 + 𝜅𝜅2 𝑐𝑐2𝜎𝜎 )1/𝜎𝜎 .

(29)

By applying the production function in (29) to the optimizing condition of firms in (9), we
find the real wage rate given in (10) to be
𝑤𝑤 = 𝑤𝑤(𝜃𝜃, 𝜔𝜔�) = �𝜃𝜃𝑝𝑝1 (𝜔𝜔
�)𝛾𝛾

𝛾𝛾 (1

1

1−𝛾𝛾 −(1−𝛾𝛾) 𝛾𝛾

− γ)

𝑟𝑟̅

� .

(30)

The consumption utility function in (29) gives the consumer price index 𝑃𝑃, the real price𝑝𝑝1 (𝜔𝜔
�), and the expenditure share δ(𝜔𝜔�) of commodity 1 as follows:
1/(1−𝜎𝜎) −𝜎𝜎/(1−𝜎𝜎)1/(1−𝜎𝜎)𝑃𝑃1+ 𝜅𝜅2(ε𝑃𝑃2∗ )−𝜎𝜎/(1−𝜎𝜎) �𝑃𝑃 = �𝜅𝜅1

1/(1−𝜎𝜎)𝑝𝑝1 (𝜔𝜔�) = �𝜅𝜅1

δ(𝜔𝜔
�) =

1/(1−𝜎𝜎)+ 𝜅𝜅2

1/(1−𝜎𝜎)𝜅𝜅1

𝜔𝜔
� −𝜎𝜎/(1−𝜎𝜎) �

1/(1−𝜎𝜎)𝜅𝜅11/(1−𝜎𝜎)+𝜅𝜅2

� −𝜎𝜎/(1−𝜎𝜎)𝜔𝜔−(1−𝜎𝜎)/𝜎𝜎(1−𝜎𝜎)/𝜎𝜎

,

,

.

(31)

Substituting 𝑐𝑐1 and 𝑐𝑐2 given in (5) into the consumption utility (29) and applying 𝑝𝑝1 (𝜔𝜔�) andδ(𝜔𝜔�) in (31) to the result yields𝑢𝑢�(𝑐𝑐1 , 𝑐𝑐2 ) = ln(𝜅𝜅1 𝑐𝑐1𝜎𝜎 + 𝜅𝜅2 𝑐𝑐2𝜎𝜎 )1/𝜎𝜎 = ln𝑐𝑐 (≡ 𝑢𝑢(𝑐𝑐)).

Thus, the consumption function given in (24) turns to be1

𝑦𝑦−𝑟𝑟̅ 𝑏𝑏

𝑐𝑐 = 𝑐𝑐̂ (𝑦𝑦; 𝑏𝑏0 , 𝑤𝑤(𝜃𝜃, 𝜔𝜔�)) ≡ 𝛽𝛽 �𝜌𝜌 − 𝛼𝛼 + 𝛼𝛼 𝑤𝑤(𝜃𝜃,𝜔𝜔�0)�.

(32)

From the Keynesian cross equation:𝑦𝑦 = 𝑐𝑐̂ (𝑦𝑦; 𝑏𝑏0 , 𝑤𝑤(𝜃𝜃, 𝜔𝜔�)) + 𝑔𝑔,

where 𝑐𝑐̂ is given by (32) and 𝑤𝑤(𝜃𝜃, 𝜔𝜔�) by (30), we obtain the effects on aggregate demand of17

changes in government purchases 𝑔𝑔, international asset-capital holdings 𝑏𝑏0 , total productivity𝜃𝜃, and the terms of trade 𝜔𝜔
�:

𝑑𝑑𝑑𝑑

=
𝑑𝑑𝑑𝑑

𝜃𝜃 𝑑𝑑𝑑𝑑

�𝑦𝑦� 𝑑𝑑𝑑𝑑 = −

1

1−

𝛼𝛼𝛼𝛼

𝑏𝑏

𝛼𝛼 ,

𝛽𝛽𝛽𝛽

1

�𝑏𝑏
𝑟𝑟
𝜌𝜌−𝛼𝛼−𝛼𝛼 0 +𝛽𝛽𝛽𝛽 𝛾𝛾
𝑤𝑤

𝑑𝑑𝑑𝑑

� 𝑦𝑦0 � 𝑑𝑑𝑏𝑏 =
,

0

�
𝜔𝜔

𝑑𝑑𝑑𝑑

�𝑏𝑏0
𝑟𝑟
𝑤𝑤
�𝑏𝑏
𝑟𝑟
𝜌𝜌−𝛼𝛼−𝛼𝛼 0 +𝛽𝛽𝛽𝛽𝑤𝑤� 𝑦𝑦 � 𝑑𝑑𝜔𝜔� =where 1 − 𝛿𝛿 = −𝑝𝑝1 ′(𝜔𝜔�)𝜔𝜔�/𝑝𝑝1 (𝜔𝜔�) from (31).

−𝛼𝛼

𝛼𝛼𝛼𝛼

,
1

�𝑏𝑏
𝑟𝑟
𝜌𝜌−𝛼𝛼−𝛼𝛼 0 +𝛽𝛽𝛽𝛽 𝛾𝛾
𝑤𝑤(1 − 𝛿𝛿).

(33)

To obtain the quantitative results, the parameter values applied in the analysis are derived

from per-capita data in Japan from 2004 to 2013 (the derivation is set out in the appendix). 8
They are𝜌𝜌 = 𝑟𝑟̅ = 0.05, 𝛾𝛾 = 0.6, 𝛼𝛼 = 1.18 × 10−2, 𝛽𝛽 = 1.82 × 10−4,

𝑥𝑥 = 0.956, 𝛿𝛿 = 0.937, 𝑔𝑔 = 32.3, 𝑟𝑟̅ 𝑏𝑏0 = 128.0, 𝑤𝑤 = 183.8.

Applying these values to (33) yields
𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑

= 1.54,𝑏𝑏

𝑑𝑑𝑑𝑑

𝜃𝜃 𝑑𝑑𝑑𝑑

�
𝜔𝜔

𝑑𝑑𝑑𝑑

� 𝑦𝑦0 � 𝑑𝑑𝑏𝑏 = −0.230, �𝑦𝑦� 𝑑𝑑𝑑𝑑 = −0.526, � 𝑦𝑦 � 𝑑𝑑𝜔𝜔� = 0.033.
0

The multiplier of government purchases 𝑔𝑔 on 𝑦𝑦 (= 𝑐𝑐 + 𝑔𝑔) is 1.54 and hence the multiplier on

consumption is 0.54. Increases in 𝑏𝑏0 and 𝜃𝜃 by 1% respectively reduce aggregate demand by
0.230% and 0.526% while a 1% increase in 𝜔𝜔� (viz. a deterioration in terms of trade) expands
aggregate demand by 0.033%.

5. Conclusion
Using a dynamic optimization framework in a closed economy, Murota and Ono (2015)
establishes a new Keynesian cross analysis, which is as simple as the conventional Keynesiancross analysis. This paper extends it to a small open economy setting and proposes a new
Keynesian cross analysis of a small open economy with dynamic optimization.
8

Because the consumption tax was increased in 2014, we use 10-year data of 2004-2013 to avoid its influence.

18

In the new analysis consumption is shown to be a function of aggregate demand, as in theconventional consumption function, but the economic implication is quite different. While theconventional consumption represents the relationship between disposable income andconsumption, the present consumption function is derived from household and firm optimizing
behavior, market equilibrium conditions and wage and price adjustments. It shows therelationship between consumption and the inflation (or deflation if negative) rate determinedby the employment rate, which in turn is a function of aggregate demand. The employment rateis determined so that the current account is balanced. Applying this consumption function to
the Keynesian cross equation (𝑦𝑦 = 𝑐𝑐 + 𝑔𝑔) yields a new Keynesian cross analysis with dynamic
optimization in a small-country setting.

This analytical framework gives quite different economic and policy implications fromthose under full employment and those of the Mundell-Fleming model. An improvement in theterms of trade decreases world demand for the home commodity and decreases employment,
which worsens deflation and reduces consumption and aggregate demand. An increase ininternational asset-capital holdings (due to international transfer, for example) improves thecurrent account. It yields an appreciation pressure on the home currency and lowers home
employment. Consequently, deflation worsens and consumption and aggregate demand
decline. Government purchases worsen the current account, create a depreciation pressure on
the home currency and hence increase employment. Thus, deflation mitigates and consumption
is stimulated.
Those results are opposite to those under full employment. Under full employment an
improvement in terms of trade and an increase in international asset-capital holdings increase
consumption. Government purchase crowds out private consumption. They are also very
different from those of the Mundell-Fleming model. In the Mundell-Fleming model an increasein government purchases reduces net export so that consumption is not affected. Furthermore,19

it cannot deal with the effects of an improvement in the terms of trade and an increase ininternational capital-asset holdings because it does not consider the current accountadjustment.

Appendix: Parameter values for the numerical example
The time preference rate (𝜌𝜌 = 0.05) is the value adopted by Benhabib and Farmer (1996)

and Mankiw and Weinzierl (2006). The labor share in output is 𝛾𝛾 = 0.6, which is presented by

Cooley and Prescott (1995). Following Katagiri and Takahashi (2017), we assume the share ofimported consumption goods to be 1 − 𝛿𝛿 = 0.063. The macroeconomic data used in thefollowing calculations are the averages of 2004-2013 in Japan.

The price adjustment speed 𝛼𝛼 equals 1.18 × 10−2. It is derived by substituting the annual

inflation rate of the consumer price index for all items, 𝜋𝜋 = −5.20 × 10−4 (Ministry ofInternal Affairs and Communications, Statistics Bureau of Japan), and the average of

unemployment rate, 1 − 𝑥𝑥 = 0.044 (Labor Force Survey, Statistics Bureau of Japan), to𝜋𝜋 = 𝛼𝛼(𝑥𝑥 − 1) given by (14).

Under the utility specification (𝑢𝑢′ (𝑐𝑐) = 1/𝑐𝑐) in (29), (20) reduces to 𝛽𝛽𝛽𝛽 = 𝜌𝜌 + 𝜋𝜋. By

substituting 𝜋𝜋 and 𝜌𝜌 given above and the average of actual final consumption expenditure percapita 𝑐𝑐 = 271.5 from National Account for 2017 (Cabinet Office, Government of Japan) into
this equation, we obtain the liquidity preference parameter 𝛽𝛽 = 1.82 × 10−4 .
Under the Cobb-Douglas production function given in (29) we have
𝑤𝑤𝑤𝑤

𝛾𝛾

= 1−𝛾𝛾 = 1.5,𝑟𝑟̅ 𝑘𝑘(A1)

𝑐𝑐 + 𝑔𝑔 − 𝑟𝑟̅ 𝑏𝑏 𝑓𝑓 = 𝑤𝑤𝑤𝑤 + 𝑟𝑟̅ 𝑘𝑘 = 2.5 × 𝑟𝑟̅ 𝑘𝑘.

(A2)

where 𝛾𝛾 = 0.6 (see Cooley and Prescott, 1995). Because 𝑏𝑏 = 𝑏𝑏 𝑓𝑓 + 𝑘𝑘, (21) and (A1) yield

The average of government actual final consumption per capita 𝑔𝑔 (Cabinet Office,20

Government of Japan) and the average of income account per capita 𝑟𝑟̅ 𝑏𝑏 𝑓𝑓 (Ministry of Finance,
Japan) are

𝑔𝑔 = 32.3,

𝑟𝑟̅ 𝑏𝑏 𝑓𝑓 = 10.9.

Because 𝑐𝑐 = 271.5 and 𝑥𝑥(= 1 − 0.044) = 0.956, as stated above, substituting them to (A2)gives𝑟𝑟̅ 𝑘𝑘 = 117.1,

𝑤𝑤 = 183.8, 𝑟𝑟̅ 𝑏𝑏0 (= 𝑟𝑟̅ 𝑏𝑏 𝑓𝑓 + 𝑟𝑟̅ 𝑘𝑘) = 128.0.

References
Benhabib, J. and R. E. A. Farmer (1996) “Indeterminacy and sector-specific externalities”,
Journal of Monetary Economics, 37 (3), pp. 421-443.
Christiano, L., M. Eichenbaum, and C. L.Evans (1999) “Monetary policy shocks: What have
we learned and to what end?” in J. B. Taylor and M. Woodford eds., Handbook of
Macroeconomics, Vol. 1, Amsterdam: Elsevier, pp. 65-148.
Cooley, T. F. and E. C. Prescott (1995) “Economic growth and business cycles,” in T. F.
Cooley eds., Frontiers of Business Cycle Research, Princeton: Princeton University
Press, pp. 1-38.
Dornbusch, R. (1980) Open Economy Macroeconomics, New York: Basic Books.
Eggertsson, G.B., N. R. Mehrotra, and J. A. Robbins (2017), “Model of secular stagnation:
Theory and quantitative evaluation”, Working Paper, Brown University.
Hashimoto, K. (2004) “Intergenerational transfer and effective demand”, Economics Bulletin,
5 (1), pp. 1-13.
Hashimoto, K. (2011) “International outsourcing, the real exchange rate and effective
demand”, Metroeconomica, 62 (2), pp. 305-327.
Hashimoto, K. (2015) “Tariffs, offshoring and unemployment in a two country model”,
Japanese Economic Review, 66 (3), pp. 371-392.
21

Hashimoto, K. and Y. Ono (2011) “Does pro-population policy raise per capita
consumption?”, Japanese Economic Review, 62 (2), pp. 151-169.
Hayashi, F. and E. C. Prescott (2002) “The 1990s in Japan: A lost decade”, Review of
Economic Dynamics, 5 (1), pp. 206-235.
Katagiri, M. and K. Takahashi (2017) “Do term premiums matter? Transmission via exchange
rate dynamics”, Working Paper Series No.17-E-7, Bank of Japan.
Kydland, F. E. and E. C. Prescott (1982) “Time to build and aggregate fluctuations”,
Econometrica, 50 (6), pp. 1345-1370.
Illing, G., Y. Ono, and M. Schlegl (2018) “Credit booms, debt overhang and secularstagnation”, European Economic Review, 108, pp. 78-104.
Johdo, W. (2006) “Geographical space and effective demand under stagnation”, Australian
Economic Papers, 45 (4), pp. 286-298.
Johdo, W. (2009) “Habit persistence and stagnation”, Economic Modelling, 26 (5), pp.
1110-1114.
Johdo, W. and K. Hashimoto (2009) “International relocation, the real exchange rate andeffective demand”, Japan and the World Economy, 21 (1), pp. 39-54.
Lucas, R. (1976) “Econometric policy evaluation: A critique”, Carnegie-Rochester
Conference Series on Public Policy, 1, pp. 19-46.
Mankiw, G. (2010) Macroeconomics, 7th edition, New York: Worth Publishers.
Mankiw, G. and M. Weinzierl (2006) “Dynamic scoring: A back-of-the-envelope guide”,
Journal of Public Economics, 90 (8-9), pp. 1415-1433.
Matsuzaki, D. (2003) “The effects of a consumption tax on effective demand under
stagnation”, Japanese Economic Review, 54 (1), pp. 101-118.
Michaillat, P. and E. Saez (2014) “An economical business-cycle model”, NBER Working
Paper No. 19777.

22

Michau, J. (2018) “Secular stagnation: Theory and remedies”, Journal of Economic Theory,
176, pp. 552-618.
Murota, R. and Y. Ono (2011) “Growth, stagnation and status preference”, Metroeconomica,
62 (1), pp. 112-149.
Murota, R. and Y. Ono (2012) “Zero nominal interest rates, unemployment, excess reserves
and deflation in a liquidity trap”, Metroeconomica, 63 (2), pp. 335-357.
Murota, R. and Y. Ono (2015) “Fiscal policy under deflationary gap and long-run stagnation:
Reinterpretation of Keynesian multipliers”, Economic Modelling, 51, pp. 596-603.
Ono, Y. (1994) Money, Interest, and Stagnation – Dynamic Theory and Keynes’s Economics,
Oxford: Clarendon Press.
Ono, Y. (2001) “A reinterpretation of Chapter 17 of Keynes’s General Theory: Effective
demand shortage under dynamic optimization”, International Economic Review, 42 (1),pp. 207-236.
Ono, Y. (2006) “International asymmetry in business activity and appreciation of a stagnant
country’s currency”, Japanese Economic Review, 57 (1), pp. 101-120.
Ono, Y. (2007) “International transfer under stagnation”, in L. Sajal ed., Theory and Practiceof Foreign Aid, Amsterdam: Elsevier, pp. 155-171.
Ono, Y. (2014) “International economic interdependence and exchange-rate adjustment under
persistent stagnation”, Japanese Economic Review, 65 (1), pp. 70-92.
Ono, Y. (2018) “Macroeconomic interdependence between a stagnant and a fully employedcountry”, Japanese Economic Review, 69 (4), pp. 450-477.
Ono, Y. and J. Ishida (2014) “On persistent demand shortages: A behavioural approach”,
Japanese Economic Review, 65 (1), pp. 42-69.
Ono, Y., K. Ogawa, and A. Yoshida (2004) “The liquidity trap and persistent unemployment
with dynamic optimizing agents: Empirical evidence”, Japanese Economic Review, 55

23

(4), pp. 355-371.
Ono, Y. and A. Shibata (2006) “Long-run specialization”, Review of International Economics,
14 (1), pp. 1-15.
Ono, Y. and K. Hashimoto (2012) Fukyo no Keizai Riron (Economic Theory of Stagnation),
Tokyo: Iwanami, in Japanese.
Rodriguez-Arana, A. (2007) “Inflation and the public deficit when the utility of money isinsatiable”, Japanese Economic Review, 8 (2), pp. 238-254.
Schmitt-Grohé, S. and M. Uribe (2016) “Downward nominal wage rigidity, currency pegs, andinvoluntary unemployment”, Journal of Political Economy 124 (5), pp. 1466-1514.
Schmitt-Grohé, S. and M. Uribe (2017) “Liquidity traps and jobless recoveries”, American
Economic Journal: Macroeconomics 9 (1), pp. 165-204.
Walsh, C. E. (2017) Monetary Theory and Policy, 4th ed., Cambridge MA.: MIT Press.

24