DBJ Discussion Paper Series, No.1602

A Theory of Quantitative Easing Policy and Negative Interest Policy
Based on the Japanese Experience

Masayuki Otaki(Institute of Social Science, The University of Tokyo)

June 2016

Discussion Papers are a series of preliminary materials in their draft form. No
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A Theory of Quantitative Easing Policy and Negative Interest Policy
Based on the Japanese Experience
Masayuki Otaki1
Institute of Social Sciences
The University of Tokyo
ohtaki@iss.u-tokyo.ac.jp
Abstract
Using a two-period overlapping-generations model, I elucidate how quantitative easing
policy and negative interest policy affect an economy based on the Japanese experience
under the Abe cabinet. Quantitative easing policy forces a huge amount of money
hoarding. Accordingly, the rate of return for money is required to rise. This implies thatdisinflation and/or deflation are accelerated in Japan, which is in line with reality. On
the other hand, quantitative easing policy stimulates the aggregate demand, whichbrings about a mild recovery in business. The business upturn tightens the foreign
market because of an increase in imports and causes the home currency to depreciate.
Negative interest policy implies there is a tax levied on money hoarding. Hence, aslong as the government expenditure is kept constant, money circulating in an economy
decreases, thereby discouraging business. Such a downturn reduces aggregate income
and imports. This induces excess supply of foreign exchange. Consequently, theexchange rate appreciates to equilibrate the market.
These characteristics of the business cycle in conjunction with changes in the attitudeof the monetary authority are entirely consistent with the current Japanese experience
under Abenomics.

1

I am thankful to Takashi Unayama for his incisive comments. Any remaining errors

are entirely my own. This study is supported by MEXT/JSPS KAKENHI Grant Number
60183761.
1

1. Introduction
It seems that a great many economists are not aware that there are two methods forcontrolling widely defined liquidity (roughly speaking, money stock plus short-term
public debt) by a monetary authority in an era of historically low interest. One is open
market operations targeting long-term public debt. A huge amount of the liquidity isprovided via this method in developed economies, and it is dubbed quantitative easing(QE) monetary policy.
Even though QE policy is significant, the other method for defining widely defined
liquidity is not trivial. Japan experiences arbitrage between interest accruing fromexcess reserves and interest from short-term public debt. When the former is set
negative, the short term interest rate follows suit. Negative reserve interest is the tax
levy on hoarding high-powered money. To put it more generally, lowering the rate ofinterest for excess reserves certainly reduces the quantity of widely defined liquidity
because such a policy lowers the rate of return for near money assets via arbitrage. Thisis the other procedure for controlling the widely defined liquidity.
Accordingly, the policy mix of QE policy and negative reserve interest is mutuallycontradictory as follows. QE policy brings about a mild recovery by the multiplier effect,
which is theorized by Otaki (2015) in conjunction with a higher rate of return for widely
defined liquidity, which implies the acceleration of disinflation and/or deflation. Such a
recovery increases imports; and thus, the foreign exchange market falls in excess supply.
Accordingly, QE policy causes the exchange rate to depreciate. Japan experienced a mild
recovery coinciding with the acceleration of disinflation and a weak Japanese yen underthe Abe cabinet until negative reserve interest policy was enacted.
Negative reserve interest policy, on the other hand, shrinks the quantity of widely
defined liquidity. Consequently, an economy falls into stagnation alongside anappreciation of the home currency and a fall in the real rate of return for money as
experienced in the Japanese economy currently.2
The rest of this paper is organized as follows. Section 2 constructs a model of a small
open economy based on Otaki (2012, 2015). The consequences of QE policy and negativeinterest policy are analyzed in Section 3. Section 4 provides brief concluding remarks.

Whether disinflation and/or deflation are resolved is ambiguous because we cannot
predict how seriously a change in the nominal rate of interest (the interest accruing
from excess reserves) affects the real rate of interest.

22

2. The Model
2.1 The Structure of the Model
Let us consider a two-period overlapping-generations model of the production economy
with infinite time horizon. This small open economy under a flexible exchange rate
system consumes two types of goods: goods, x , in which the economy specializes; goods,

y , which are produced in the rest of the world and are circulated internationally. The
only production resource is labor. For simplicity, let us assume that one unit of laborproduces one unit of goods, x , and that the opportunity to of work, which means theopportunity to earn income, is limited within the young generation.
There are two assets accruing no nominal interest: the home currency and the foreign
currency. Goods in each economy are purchasable only by its own currency. It is
assumed that no individual has an inheritance motive, and the balance of payments
must be equilibrated always. Specifically, the surplus in the balance of payments
implies precisely that members of the old generation of the home economy bequeaths
the foreign currency to their descendant because international loans between the samegenerations are unfeasible by the synchronization of the earning opportunity. In the
opposite case, that is, when the balance of payments falls in deficit, the old generation
becomes bankrupt; and thus, rational young individuals never sell their produced goodsin advance.
2.2 Individuals
The utility function U of the residents of this economy is identical:

U  u (c1t , c2 t 1 )   t  , cit  [ xit ] [ yit ]1  , 0    1,

(1)

where cit is the composition of goods that are consumed by the i-th stage of life during
period t . xit and yit are the consumption of the home and foreign goods, respectively. 
denotes the disutility of labor.  t is a definition function that takes unity value when
the individual is employed and zero when unemployed. u () is a well-behaved andhomothetic utility function.
An elementary calculation leads us to the following expenditure function,  :    ptx   t pty 

1 



,  ptx1   t 1 pty1 

1   g u  , (2)where ptx is the price of the home goods during period t ,and pty is that of the foreign3

goods.  t denotes the nominal exchange rate. From the linear homogeneity of theexpenditure function, Equation (2) can be transformed into  ptx  et 1 

, tx  et 1 

1   g u  ,(3)where et denotes the real exchange rate during period t , and tx is the inflation rate interms of the home goods. Abbreviating the adjustment process, I assume that the home
economy is in a stationary state. That is,et  e, t .

(4)Then, Equation (3) can rewritten as  ptx  e 

1 



 1, tx   g u .

(5)

2.3 Firms
Since every firm is assumed to be a price taker and faces the linear homogenous
production function, profit becomes zero in equilibrium. Accordingly, the following
equality is obtained.

ptx  wtR   e1 

 1,  x   g    1 ; wtR  ptx  e1 

 1, tx   g   ,(6)where wtR is the nominal reservation wage, as clarified by the right-hand equation in (6).
Equation (6) describes the participation constraint for individuals. This equation
implies that whenever the home currency depreciates, disinflation should advance. Thisis because such depreciation impoverishes individuals owing to expensive import goods;
and hence, the inflation rate in terms of the home good should be calmed to compensate
for this disadvantage. Thus, a foreign business cycle diffuses through an adjoiningchange in terms of trade (the inverse of the real exchange rate). In this sense, theemployment isolation effect is imperfect even though the balance of payments is always
in equilibrium as Laursen and Metzler (1950) assert in the more undeveloped model.
2.4 The Dynamics of Money Supply: Mechanism of the Negative Interest Rate
To clarify the role of the central bank in negative interest policy, let the dynamics ofmoney supply be formatted. The dynamics can be denoted as

4

M t  1      M t 1  G ,(7)

where M t is the home currency stock during period t .  is the rate of deduction frommoney hoarding.  is a definition function, of which the value is unity when negativeinterest rate policy is enforced, and zero when the policy is dismissed. G is the nominal
government expenditure financed by printing money. The first term of the right-hand of
Equation (7) is total money after deducting imposed negative interest, which the oldgeneration carries over. The second term is the new issue of money via fiscal
expenditure.
Normalizing Equation (7) by ptx , and assuming the stationarity of the real money
supply, m , I obtain 1      Gt
1  m  g , g  a  const.
x

pt



(8)

Equation (8) contains important information concerning the effect of the negativeinterest policy. For given the real expenditure, g , the corresponding real money supply,m , decreases when   1 . That is, the negative interest policy ultimately causes amonetary contraction. The reason is that, given the volume of new issue, g , as inflow,much money leaks from circulation and is hoarded by the central bank. In other words,
the negative interest policy is a tax levy on private money hoarding. Let the relationshipin Equation (8) be denoted asm  m    , g  ,

m
m
 0,
 0.
[    ]g(9)In addition, it is assumed that the real government expenditure is distributed between
the home and foreign goods by the same proportion as the propensity to expenditure of
the private sector.
2.5 Market Equilibrium
There are four markets in this model: goods, labor, foreign currency, and home currency
markets. The goods market and home currency markets are mutually dependent.
Accordingly, it is necessary and sufficient for analyzing the model to focus on the goods,
labor, and foreign currency markets.
The labor market equilibrium condition is described entirely by participation
constraint (6). The foreign currency market is equilibrated by the following condition:5 ey F  1    y H .

(10)

The left-hand side of Equation (10) represents exports of the home economy (the sum of
the consumption of the younger generation, the old generation and the government in
the foreign economy).  is the expenditure propensity to the home goods, and ey is theFaggregate income of the foreign country in terms of home currency. The right-hand sideof Equation (10) is the aggregated import demand for the foreign goods.3 This increases
proportionately with the real aggregate income, y . 1   is the expenditure propensity
H

toward the foreign goods of the home economy. As discussed in 2.1, unless Equation (9)is upheld, old individuals bequeath some foreign money or become bankrupt.
Accordingly, Equation (10) is the equilibrium condition for the foreign currency.
By using Equations (8) and (9), the equilibrium condition for the home goods market isdenoted ass   x  y H  m     , g  , s   x   1  c   x  , s '   x   0.

(11)

Equation (11) implies that the aggregate savings must be equal to the widely definedmoney supply, which corresponds to the sum of the expenditure of the old generationand the government. It is noteworthy that, by Equation (10), the net saving of foreign
currency is always zero.
Equations (6), (10), and (11) are the structural equations in this model: Equation (6)describes the supply-side condition; Equation (10) represents the non-Ponzi condition in
the foreign currency market; and Equation (11) is also interpreted as the equilibrium
condition for the home currency market. There are three endogenous variables in themodel; y ,  , e . Accordingly, the model is closed.
Hx

3. QE policy and Negative Interest Rate Policy in a Small Open Economy Model
It should be noted that, in general, a two period overlapping-generations modelemphasizes the role of money stock in purchasing goods than in the traditional IS-LM
analysis. There is no direct effect of an increase in money on aggregate demand in the
IS-LM analysis. However, the widely defined liquidity rapidly enlarges via radical QE
policy as the introductory discussion suggests, money cannot only be regarded only as a
See Otaki (2015, p.69) for more detail on the derivation of the export and importfunctions.

36

measure for hoarding value, but also as a transaction medium. This implies that thepurchasing power of money directly affects aggregate demand expressed as in Equation(11). Thus, an overlapping-generations model is a convenient vehicle for analyzing QE
policy.4

From the comparative statics around the equilibrium, yH*,  x* , e*  , I obtained thefollowing results:5

dy H *d  x*
de*
 0,
 0,
 0,
dm
dm
dm
dy H *d  x*
de*
 0,
 0,
 0, if   1.
d
d
d

(12)

The upper inequalities represent the effect of QE policy to an economy. Radicalmonetary expansion stimulates the economy and raises the real income, yH*. On theother hand, disinflation and/or deflation advance (  decreases) because the rate ofx*

return for money must arise to absorb the potential excess supply in money. Meanwhile,
since the business upturn increases the demand for the foreign currency, the realexchange rate depreciates ( e* decreases). This is the acute fact that Japan experienced
under the strengthened QE policy of the Abe cabinet from the beginning of 2013 to thebeginning of 2016.
The lower inequalities in (12) show the effects of the negative interest policy. As
aforementioned, the negative interest rate policy means taxation on money hoarding.
Accordingly, as long as the real government expenditure, g , remains intact, such a
policy causes a monetary contraction and thus, the effects are entirely contrary to thoseof QE policy.
That is, a monetary contraction provoked by the negative interest policy thwarts
business and stagnation ensues. Since such economic stagnation lowers the importdemand and causes excess supply of foreign currency, the real exchange rate
appreciates to equilibrate the market. While it is certain that real effective interest,
defined as,

1

 x*

, falls because widely defined liquidity becomes scarce, it is unclear

whether disinflation/deflation decelerates.

See Otaki (2016) for a precise discussion about the intrinsic difference between the
IS-LM analysis and the two period overlapping-generations model.
5 See the Appendix for more details on this calculation.
4

7

As such, this model succeeds in describing macroeconomic aspects in the current
Japanese economy. As far as my theory possesses relevance to the actual economy, it isapparent that QE policy contradicts negative interest policy.
4. Concluding Remarks
Based on a two period overlapping-generations model of a small open and production
economy, I obtain the following results.
First, a radical QE policy not only stimulates an economy but also accelerates
disinflation and/or deflation. This is because rapid monetary expansion requires ahigher rate of return for widely defined liquidity. In addition, higher income inconjunction with the boom increases import demand. To equilibrate the foreign currency
market, the real exchange rate depreciates.
Second, the negative interest imposed on the excess reserve ultimately shrinks the
supply of widely defined liquidity. This is because the negative interest means the tax
levy on hoarding money. Consequently, all effects --- economic stagnation, appreciationof the home currency, and a fall of the real effective interest rate for money --- arecontrary to those of QE policy.

References
Laursen S, and Metzler, L. “Flexible Exchange Rates and the Theory of Employment,’’
Review of Economic Statistics Vol. 32, 1950, pp.281-299.
Otaki, M. “A Keynesian Model of a Small Open Economy under a Flexible Exchange
Rate,”

Theoretical Economics Letters Vol.2, 2012, pp.278-282.

Otaki, M. Keynesian Economics and Price Theory: Re-orientation of a Theory of
Monetary Economy, 2015, Springer, Tokyo.
Otaki, M. Keynes’s General Theory Reconsidered in the Context of the Japanese
Economy, Springer-Briefs in Economics, Develop Bank of Japan Research Series, 2016,
Springer, Tokyo.

Appendix
By differentiating Equations (11), (10), and (6), I obtain the following simultaneous
equations.

8
s ' yH* s
1  0
1 
 0
 ' e*   H*   dy   dm   d  x*     e*  dy F  ,**    de 


0



1    e   x*| x* .
where    1,   , '    0 y F(13)

Solving this simultaneous equation system, the following reduced form is obtained.

  ' e* 1  y F  dy 
2
 x*  1 * 
 d       1     e  de* * 1  1    ' e H*  s  ' e* 1  s ' 1    e* s 1    e* 1  s ' e* 1 

y F  s ' 1     e* 2y H *  0.

Equation (14) implies the inequalities in (12).

9



yH*

 s ' y H * y F   dm 
 *
F
F 
s y
   e  dy  ,0 s ' 1    y H *  (14)