_{1}

^{*}

My preceding paper on this topic (Otaki [1]) explored whether the equilibrium existence proof in Lucas [2] is truly complete. We showed that the proof is incomplete that some additional conditions are required to complete the job. In this paper, we explore another ambiguity in Lucas’s model, which has been pointed out by Grammond (see Lucas [3]): can the model transform the joint probability density function of the exogenous environment into one that which includes market equilibrium information? This problem is peculiar to the signal extraction problem compatible with the market equilibrium condition. The result indicates that although Lucas [3] was fundamentally correct in refuting Grammond’s critique, the model contains another crucial assumption concerning the property of the equilibrium function, namely, one-to-one correspondence from the environmental variable to the equilibrium price, which has not been proved by Lucas [2] to date.

The essence of Lucas [

This fact implies that information on the exogenous environment, which is indispensable for optimal decision making, is never extracted without using the equilibrium price function. Mathematically, although it can never be directly observable, the probability distribution of the economic environment (i.e., money supply per capita and population of young generation) can be fixed by assumption. However, the joint and/or conditional distribution of the environment and equilibrium price cannot be defined without an equilibrium function given a-priori (although it must be consistent with the rational expectations equilibrium (REE)). Thus, as suggested by Grammond, there emerges a room of multiplicity in the endogenously determined environment/equilibrium-price probability distribution.

We have succeeded in showing that the joint distribution of the environment/equilibrium-price necessarily becomes multiple as asserted by Grammond, and that the conditional distribution of these variables is free from the specification of the equilibrium price function. In other words, the unique conditional density function, independent of the shape of the equilibrium price function, is consistently obtained in using Lucas’ model [

This paper is organized as follows. Section 2 clarifies the theoretical problem, and establishes the theorem concerning the existence of the unique conditional probability function. Section 3 contains brief concluding remarks.

Lucas’ original maximization [

where the left-hand-side (1) presents the marginal utility derived from current consumption, and the right-handside, that from future consumption. That is, (1) is the compounded Euler equation with the market clearing condition in Lucas’ model [

Lucas specifies the equilibrium-price function as

Substituting (2) into (1) gives the following equation according to Lucas [

where is the current additional money supply unknowable to individuals. The random variable should be strictly distinguished from, which means the realized value of ^{1}. Clearly, the imperfect informational structure of the model, which is the backbone of the Lucas’ model [

The main issue, this article deals with, is whether the transformation from (1) to (3) is independent of the functional form of (2). Since and are assumed to follow independently identically distributed (i.i.d.) processes, the problem thus converges to whether the conditional probability distribution function can be defined independently of the form of the tentatively fixed equilibrium-price function in (2), as expressed by (3).

In this subsection, using Lucas’ model [

Proof. Since are i.i.d. processes and are thereby independent of , we should focus on the relationship between the exogenously given joint density function and. denotes the density function of.

By definition and using the formula of transformation of the distribution function, we obtain

Thus, the transformation between joint distribution functions depends on the shape of as suggested by Grammond. However, since the Jacobean of does not contain, the conditional distribution function of becomes

Hence, as long as the inverse function of is well-defined (i.e., is a one-to-one correspondence), as shown by (5), is independent of the functional form of .

Accordingly, we have succeeded in validating Lucas’ transformation [

Despite the validity of Lucas’ transformation [

where is the inverse function of and . Under an additional restrictive condition proposed by Otaki [

Nevertheless, we must note that there is no guarantee that the corresponding image is also invertible, for any fixed invertible . Consequently, the existence proof of Lucas [

This article examined whether Lucas’ signal extraction problem [

First, Lucas [

However, the invertibility of the equilibrium-price function is crucial to infer signals correctly. The results also indicate that Lucas [