DBJ Discussion Paper Series, No.1203

A Studyon
Lucas' ``Expectations and the Neutrality of Money´´Masayuki Otaki(Institute of Social Science, University of Tokyo)

July

2012

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors inorder to protect the tentative characters of these papers. Any opinions, findings,conclusions or recommendations expressed in these papers are those of the authors and do
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Forthcoming in Theoretical Economics Letters

A StudyonLucas' ``Expectations and the Neutrality of Money´´Masayuki Otaki
Institute of Social Science, University of TokyoAbstract
This short article shows that the functional equation on the equilibrium pricefunction is more complicated than that considered by Lucas [1], and that modification isrequired to complete the proof. Furthermore, we provide a sufficient condition thatguarantees the uniqueness of the equilibrium price function.
Key words: Neutrality of Money,
Functional Equation, Contraction Mapping
A Sufficient Condition for the Unique Equilibrium Price Function
JEL Classification: E31, E41

1Introduction

This study aims to show that an additional condition is necessary for the operator
T in Lucas [1] to become the contraction mapping. This is because transformation
between the functional equations on the equilibrium price function is not equivalent. We
also provide a sufficient condition such that T can become the contraction mapping.
This paper proceeds as follows. Section 2 rewrites the functional equation in thecorrect form and shows that, contrary to the implication of the original paper, thecontraction mapping method cannot be easily applied. A sufficient condition for theuniqueness of the equilibrium price function is provided in Section 3. Section 4 contains
brief concluding remarks.

2Equivalent Transformation

Lucas [2] admits that there is no guarantee that p0 and z exhibit a one‐to‐one
correspondence, and some reservation is necessary for the conclusion. He also recognizesxthat the equilibrium price function p ( m, x,  ) should be specified as m ( ) to
determine the unique equilibrium. Besides these problems, the transformation between
functional equations below is not equivalent. The present study aims to clarify this fact and1

show the rather restrictive condition for supporting the original result.
Lucas [1] first derives the following functional equation as the equilibrium conditionof the money market:
xxxx
xx
x]
= V []h[dG ( , x,   | ),xxxx ( )  ( ) ( )  ( )





where x is the realized value of the increment of money during the current period. denotes the realize value of the population of the young generation. x,   are randomvariables of each exogenous shock during the subsequent period. We must note thex
is available information via theexistence of the random variable  . Although z inverse equilibrium price function, x cannot be directly observed by household. Thus,
when x singly appears in the functional equation, it should be treated as the randomvariable  .
The right‐hand side of (1) represents the marginal utility of the current consumption,and the left‐hand side represents the expected marginal utility of the future consumption.
Namely, functional equation (1) is the Euler equation in this model. Lucas [1] asserts that(1) is equivalently transformed intozz  z   z h[]= V []H ( z ,  ) H ( z , )d dz d .
 ( z)  ( z)  ( z )   ( z)However, (1) and (2) are not equivalent. We shall address this problem. Thistransformation assumes that  = x . Nevertheless, as discussed above, x is a realized
vale (real number) of the random variable  (measurable function). Hence, they cannot
be cancelled out. The equivalent transformation from (1) to (2) iszzz  z  z  z h[]= V []dG ( , x,  | z ).
 ( z)  ( z)  ( z )   ( z )

(1)

(2)

(3)

zz]
, G 1 ( x)  xh( x) and G2 ( x)  V ( x) x . Using ( z)  ( z)these definitions, (1) is transformed intoz  ( z ) = G2 [G1 ( ( z ))]dG ( , x,  | z ).
Let us define  ( z )  h[Then the correct form of the operator T in the Appendix of Lucas [1] becomesz Tf  ln G2 [G1 (e f ( z) )]dG ( , x,  | z ).

Consequently, inequality (A.6) in Lucas' [1] appendix|| Tf  Tg || sup | ln G2 [ G1 (e f ( z ) )]  ln G2 [G1 (e g ( z ) )] | ( A.6)
z , ,

is modified as2(4)|| Tf  Tg || sup sup | ln G2 [zz , ,z 

G1 (e f ( z) )]  ln G2 [z 

G1 (e g ( z) )] | .

(5)

It is noteworthy that z  ,   , and  are functions of z in (5). Let us denote thesefunctions asz  =  ( z ),=  ( z ).

(6)

Accordingly, (5) becomes|| Tf  Tg || sup | ln G2 [ z  ( z )G1 (e f ( ( z )) )]  ln G2 [ z  ( z )G1 (e g ( ( z )) )] | .

(7)

z

Let us define z* , x1 , and x2 asz *  arg sup | ln G2 [ z  ( z )G1 (e f ( ( z )) )]  ln G2 [ z  ( z )G1 (e g ( ( z )) )] | .
zx1  f ( ( z * )), x2  g ( ( z * )).
Consequently, (7) is transformed into|| Tf  Tg ||| ln G2 [ z *  ( z * )G1 (e 1 )]  ln G2 [ z *  ( z * )G1 (e 2 ))] | .
Applying the mean value theorem and Lucas' [1] assumptions (A.2) and (A.3)
xG ( x)xG2 ( x)
0 < 1
< 1 ( A.2), 0 < 1  a < 1 ( A.3)
G1 ( x)G2 ( x)to (8), we finally obtain|| Tf  Tg || (1  a) | x1  x2 | (1  a ) || f ( ( z ))  g ( ( z )) || .
Since  ( z )  z generally, the original paper has not succeeded in proving that theoperator T is the contraction mapping.1 Thus, an additional condition is necessary tocomplete the proof.
x3x(8)(9)

A sufficient condition

Since the difficulty arises from the fact that (5) explicitly depends on z , weassume that the function G2 is multiplicatively separable. Namely, suppose that G2
satisfies
G2 ( xy ) = G2 ( x)G2 ( y ).
(10)
In this case, (5) is modified as1

It is noteworthy that|| f ( ( z ))  g ( ( z )) |||| f ( z )  g ( z ) || .
does not necessarily hold. For example, when  ( z ) = 2 z , g < f , and f is a strictlyincreasing function, g is strictly decreasing, then|| f (2 z )  g (2 z ) |||| f ( z )  g ( z ) || .
holds.

3

|| Tf  Tg || sup | ln G2 [z , ,
G1 (e f ( z) )]  ln G2 [ G1 (e g ( z ) )] | .


(11)

This inequality is essentially identical to (A.6), and thus, T becomes the contraction
mapping.
Nevertheless, the function G2 , which satisfies the functional equation (10), isconfined to power functions (see Small [3]). Hence,xV ( x) x = x   V ( x) =.
(12)

(A.3) also requires that 0 <  < 1 . In sum, the CRRA (Constant Relative Risk Aversion)family, whose relative risk aversion is located within (0,1) , is the only function satisfying
the sufficient condition (10).

4Conclusion

We have shown that the functional equation of the equilibrium price function ismore complicated than that considered by Lucas [1]. Hence, some additional condition isnecessary to employ the contraction mapping method. This study finds that if V belongs
to CRRA family of low relative risk aversion, the uniqueness of the solution is guaranteed.
In sum, Lucas' assertion on the neutrality of money under uncertainty holds only undercertain utility functions that are rather restrictive as compared to those previouslyconsidered.

References[1] R. E. Lucas, Jr., ''Expectations and the Neutrality of Money,'' Journal of
Economic Theory, Vol. 4, 1972, pp. 103‐124.
[2] R. E. Lucas, Jr., ''Corrigendum to `Expectations and the Neutrality of Money,''
Journal of Economic Theory, Vol. 31, 1983, pp. 197‐199.
[3] C.G. Small, Functional Equations and How to Solve Them, Springer
Science+Business Media, New York 2007.

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