1. Introduction
This study aims to show that some additional condition is necessary for the operator 
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 
can become the contraction mapping.
The paper is as follows. Section 2 rewrites the functional equation in the correct form and shows that unlike the original paper, the contraction mapping method cannot be easily applied. A sufficient condition for the uniqueness of the equilibrium price function is provided in Section 3. Section 4 contains brief concluding remarks.
2. Equivalent Transformation
Lucas [2] admits that there is no guarantee that 
and 
is a one-to-one correspondence, and some reservation is necessary for the conclusion. Furthermore, he also recognizes that the equilibrium price function 
should be specified as
for determining the unique equilibrium. However, besides these problems, the transformation between functional equations below is not equivalent. The aim of the study is to clarify that fact and show rather restrictive condition for supporting the original result.
Lucas [1] firstly derives the following functional equation as the equilibrium condition of money market:
(1)
where 
is the realized value of the increment of money during the current period. 
also denotes the realize value of the population of the young generation. 
are random variables of each exogenous shock during the next period. We must note the existence of the random variable
. Although 
is an available information through the inverse equilibrium price function, 
cannot be directly observed alone by household. Thus, when 
singly appears in the functional equation, it should be treated as the random variable
.
The right-hand side of (1) means the marginal utility of the current consumption, and the left-hand side implies 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 into
(2)
However, (1) and (2) is not equivalent. We shall deal with this problem. This transformation assumes
. Nevertheless, as discussed above, 
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) is
(3)
Let us define
,
and
. Using these definitions (1) is transformed into
Then the correct form of the operator 
in the Appendix of Lucas [1] becomes
(4)
Consequently, inequality (A.6) in Lucas’ [1] appendix
is modified as
(5)
It is noteworthy that 
are functions of 
in (5). Let us denote those functions as
(6)
Accordingly, (5) becomes
(7)
Let us define
as
Consequently, (7) is transformed into
(8)
Applying the mean value theorem and Lucas’ [1] assumptions (A.2) and (A.3)
to (8), we finally obtain
(9)
Since 
generally, the original paper has not succeeded in proving that the operator 
is the contraction mapping.1 Some additional condition is necessary for completing the proof.
3. A Sufficient Condition
Since the difficulty arises from the fact that (5) explicitly depends on
, we assume that the function 
is multiplicatively separable. Namely, suppose that 
satisfies
(10)
In this case, (5) is modified as
(11)
This inequality is essentially identical to (A.6), and thus, 
becomes the contraction mapping.
Nevertheless, the function
, which satisfies the functional Equation (10), is confined to power functions (See Small [3]). Hence,
(12)
(A.3) also requires
. To sum up, CRRA (Constant Relative Risk Aversion) family, whose relative risk aversion is located within
, is the only function satisfying the sufficient condition (10).
4. Conclusion
We have shown that the functional equation of the equilibrium price function is more complicated than that considered by Lucas [1]. Hence, some additional condition is necessary to employ the contraction mapping method. This study finds that if 
belongs to CRRA family of low relative risk aversion, the uniqueness of the solution is guaranteed. To sum up, Lucas' assertion on the neutrality of money under uncertainty hold only rather restrictive utility functions than has been considered.
NOTES
does not necessarily hold. For example, when 
is a stricktly increasing function, 
is stricktly decreasing, then
holds.