DBJ Discussion Paper Series, No.1104

The Role of Money: Credible Asset or Numeraire?

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

January

2012

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The Role of Money: Credible Asset or Numeraire?
Masayuki Otaki
Institute of Social Science, University of Tokyo
E‐mail: ohtaki@iss.u‐tokyo.ac.jp
Abstract
It is well known that money is neutral if (i) people hold the extraneous belief
that it is only a numeraire and does not possess intrinstic value, and (ii) new money is
injected into an economy as its own interest in the OLG model under perfect
information (Lucas [1] Theorem 2). We find that whenever (i) is not satisfied and money
is rationally held to have substance value, it becomes non‐neutral even if we use thesame model as Lucas [1].

1Introduction

This article shows that the multiplicity of rational beliefs concerning the valueof money decides whether money is neutral under perfect information structures in thetwo‐period OLG model. The result is kept intact, even if the new money injected into theeconomy is subject to the model of Lucas [1]. This result contrasts with those of Otani[2] and Otaki [3], [4], [5].
If the nominal rate of interest on money increases (i.e., money supply increases),
people can consistently believe that the purchasing power of money (the inverse of thenext period price level) is retained.1 Then, the real interest rate becomes higher,
thereby matching the supply, and the demand for money increases.
From assumptions concerning the utility function, This situation also implies thereduction of current consumption and leisure. Thus, the monetary expansion increases
current total output, and hence, money becomes non‐neutral.
We must note that the attained equilibrium is stationary in the sense that thevalues of real endogenous variables, such as current/future consumption and leisure,are entirely time‐independent. This assertion holds, since once the real interest rate is
raised because of a change of the nominal interest rate (increment of nominal money
supply per capita), one may expect that the change in the inflation rate will equal that of
the nominal interest rate; the heightened real interest rate is kept intact, and thus, theequilibrium becomes self‐enforcing and stationary.
The rest of paper is organaized as follows. In Section 2, I construct the samemodel as Lucas [1], except for the formation of rational expectations concerning thevalue of money, and proves the non‐neutrality of money. A welfare economics
implication is also analyzed. Section 3 contains brief concluding remarks.

2The Model

2.1

The Structure of the Model

1

Otaki [5] defines such a rational expectation as money being credible.

1

We use essentially the same model as Lucas [1], excluding uncertainty. In every
period a unit individual is born and lives two periods. Each individual has an indenticalutility function U :(1)U  U (c1 , n)  V (c2 )where c1 and c2 are the current and future consumption level, and n denotes thehours worked per individual.
Furthermore, U and V satisfy the following properties:
U c > 0, U n < 0,(2)U cc < 0, U nn  0, U cc  U cn < 0, U cn  U nn < 0,c V (c2 )V (c2 )c2  V (c2 ) > 0, 2
  a < 0,V (c2 )limV (c2 ) = , limV (c2 ) = 0.
c2 02.2c2 (3)(4)(5)

The Maximization Problem of Representative Individual

Each individual maximizes his/her lifetime utility U , subject to the following
budget constraint:mnt  c1t  t 1 , pt 1c2t 1  mt 1 x,pt
 nt  c1t  t c2t 1 ,(6)p(7)
t  t 1 ,pt x
where x denotes the increment of money supply per capita, and t is the inverseof the real interest rate. mt‐1 is the money per capita carried over from the previousperiod.
The Kuhn‐Tucker conditions imply that the optimal decision (c1*t , c2*t , nt* )satisfiesdnt*
U c (c1*t , nt* )= 1,(8)dc1*t
U n (c1*t , nt* )2.3

V (c2*t )= tU c (c1*t , nt* )(9)

nt* = c1*t  t c2*t .

(10)Market Equilibrium

There are two markets in the above model: the money market and goods
market. By Walras' law, we can neglect the equilibrium condition for the goods market.
The money market equilibrium condition is(11)mt 1 x = pt*1c2*t 1.
Instead of the quantity‐theoretic equilibrium price function imposed by Lucas [1],2let us assume that money is credible in the sense of Otaki [5] ‐‐ that is, the rational
1
expectation concerning the current purchasing power of money * is not purturbed
pt 1
by an increse of x :dpt 1= 0.
(12)dx
The general equilibrium of markets is attained by five equations: (8), (9), (10),(11) and (12). Endogenous variables are (c1*t , c2*t , pt* , pt*1 , nt* ) .
The partial equilibrium of labor nt* and the younger‐generation's consumption
c1t* is illustrated by Figure 1. The downward sloping curve AA is the locus of Equation(8), which is easily derived from Assumption (3).
The upward sloping curve BB is the locus of Equation (10), which is combinedwith Equation (9). The procedure is as follows: Substituting Equation (9) into (10), weobtainV.
nt* = c1*t  c2*tUc
In differentiating both sides of the above equation,
dnt* dc1*t d [V c2*t ] U cc dc1*t  U cn dnt*= * 
nt*c1tV c2*tUcdc1*t d [V c2*t ]
 * 

c1t
V c2*tc1*t [U cc  U cn
Ucdnt*]
dc1*t dc1*tc1*tdnt*c1*t [U cc  U cn ] dc1*t d [V c2*t ]=[1] * nt*
Ucc1t
V c2*t(13)

holds. Hence, Curve BB is upward sloping for any fixed c2t* . When the money market
equilibrates, the equilibrium consumption of the younger generation and output isdetermined at the intersection of Curves AA and BB (Point E0 ).
Whenever money is credible, it is facile to depict the property of money market
equilibrium. From Equations (11) and (12), we obtain
dc2*t dx
= .
(14)
c2*tx
Since V c2t* is an increasing function of c2t* from Assumption (4), Equations (13) and(14) imply that Curve BB shifts toward the south‐east like, Curve BB by an increaseof x . Thus, the economy moves from Point E0 to E1 .
Accordingly, as long as money is credible, a monetary expansion increases theoutput nt* and future consumption c2t* , and decreases the younger‐generation's
consumption c1t* . To sum up:

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Theorem 1 . If money is a credible asset, it becomes non‐neutral to the real
economy. An accelaration of monetary growth heightens the real interest of money, andhence stimulates future consumption and output/labor supply, and economizes currentconsumption.
Next, we shall show that the equilibrium depicted above is a stationary rational
expectation equilibrium. Suppose that the economy is located at Point E1 by anincrease of x , and individuals believe that the higher equilibrium real interest rate 1/ 1* prevails thereafter.
Then by the definition of  (Equation (7)),**d  1* dx dpt  j 1 dpt  j=[] = 0, j  1(15) 1*xpt* j 1
pt* jholds. Since mt  j 1 x = pt* j 1c2*t  j 1 and mt  j = mt  j 1 x ,**dc2*t  j 1 dc2*t  jdx dpt  j 1 dpt  j=[ * * ][ * * ]xpt  j 1
pt  jc2t  j 1c2t  jalso holds. Combining Equation (16) with (15), we finally obtain
dc2*t  j 1 dc2*t  j= * . j  1.
c2*t  j 1c2t  j(16)(17)

Thus, the equilibrium consumption of an old individual c2* is time‐independnt.
It is clear from Equations (8) and (13) that the rest of the two endogenous
variables (c1*t  j , nt* j ) are also time‐independent. Consequently, the equilibrium
illustrated by Point E1 is stationary in the sense that every equilibrium value ofendgenous variables is time‐independent. One can thus affirm
Theorem 2.
The rational expectation equilibrium defined by Equations (8), (9),(10), (12), and (15) is stationary (i.e., time‐independent). Hence the heightened realinterest rate caused by an increase of the nominal interest on money x permanently
affects the real variables.

2.4

A Welfare Implication of the Model

By Theorem 1, a monetary expansion (an increase in x ) stimulates theequilibrium real GDP n* through the rise of the real rate of interest. Here, we consider
its welfare economics implication. Let the Lagrangean of individual decision L that isevaluated at the equilibrium value.
Then, applying the envelop theorem, we obtaindU
dL
L== * =  * ( x)c2* ( x) < 0,**dd*where  is the Lagrangean multiplier. Accordingly, a monetary expansion improves
economic welfare, since it makes future goods cheaper.

43Concluding Remarks

This paper shows that money is non‐neutral as long as it is credible even if weobey the money‐supply rule proposed by Lucas [1]. A monetary expansion (anaccelaration of the money growth rate) surely hightens the real rate of interest ofmoney whenever people believe that money is credible. The effect of intertemporal
substitution leads them to work more to prepare for more future consumption, andthus, the aggregate products increases. It also implies that economic welfare is
improved by a monetary expansion.

References[1] R. E. Lucas, Jr., ``Expecations and the Neutrality of Money,'' Journal ofEconomic Theory, Vol. 4, No. 2, 1972, pp. 103‐124.
[2] K. Otani, ``Rational Expectations and Non‐Nuetrality of Money,’’
Weltwirschaftliches, Vol. 121, 1985, pp. 207‐ 216.'[3] M. Otaki, ``The Dynamically Extended Keynesian Cross and theWelfare‐Improving Fiscal Policy,'' Economics Letters, Vol. 96, No. 1, 2007, pp.23‐29.
[4] M. Otaki, ``A Welfare Economic Foundation for the Full‐Employment Policy,''
Economics Letters, Vol.102, No.1, 2009, pp. 1‐3.
[5] M. Otaki, ``A Pure Theory of Aggregate Price Determination,'' TheoreticalEconomics Letters, Vol.1, 2011, pp. 122‐128.

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Figure 16