We use essentially the same model as Lucas [
:
where 
and 
are the current and future consumption level, and 
denotes the hours worked per individual.
Furthermore, 
and 
satisfy the following properties:
Each individual maximizes his/her lifetime utility 
subject to the following budget constraint:
where 
denotes the increment of money supply per capita. 
is the inverse of the real interest rate.
The Kuhn-Tucker condition implies that the optimal decision 
satisfies
There are two markets in the above model: the money market and good market. By Walras’ law, we can neglect the equilibrium condition for the good market. The money market equilibrium condition is
Instead of the quatity-theoretic equilibrium price function imposed by Lucas [
is not purturbed by an increse of
:
The general equilibrium of markets is attained by five equations: (8), (9), (10), (11), and (12). Endogenous variables are
.
The partial equilibrium of labor 
and the younggeneration’s consumption 
is illustrated by 
is the locus of Equation (8), which is easily derived from Assumption (3).
The upward sloping curve 
is the locus of Equation (10), which is combined with Equation (9). The procedure is as follows: Substituting Equation (9) into (10), we obtain
Differentiating both sides of the above equation,
holds. Hence Curve 
is upward sloping for any fixed
. When the money market equibrates, the equilibrium consumption of younger generation and output is determined at the intersection of Curves 
and 
(Point
).
Whenever money is credible, it is facile to depict the property of money market equilibrium. From Equations (11) and (12), we obtain
Since 
is an increasing function of 
from Assumption (4), Equations (14) and (15) imply that Curve 
shifts toward the south-east, like Curve
, by an increase of
. Thus, the economy moves from Point 
to
.
Accordingly, as long as money is credible, a monetary
expansion increases the output 
and future consumption
, and decreses the younger-generation’s consumption
. To sum up:
Theorem 1. If money is a credible asset, it becomes non-neutral to the real economy. An accelaration of monetary growth hightens the real interest of money, and hence, stimulates future consumption and output/labor supply, economizes current consumption.
Next we shall show that the equilibrium above depicted is a stationary rational expectation equilibrium. Suppose that the economy is located at Point 
by an increase of
, and individuals believe that the higher equilibrium real interest rate 
prevails thereafter.
Then by the definition of 
(Equation (7)),
holds. That is, individuals consider that the change in the equilibrium inflation rate is equal the acceleration rate of monetary growth because there is no substantial change in the economic environment after period
. Since 
and
,
also holds. Combining Equation (17) with (16), we finally obtain
Thus, the equilibrium consumption of an old individual 
is time-independent.
It is clear from Equations (8) and (14) that the rest of the two endogenous variables 
are also timeindependent. Consequently, the equilibrium illustrated by Point 
is stationary in the sense that every equilibrium value of endgenous variables is time-independent. One can thus affirm Theorem 2. The rational expectation equilibrium defined by Equations (8)-(10), (12), and (16) is stationary (i.e., time-independent). Hence the heightned real interest rate caused by an increase of the nominal interest on money 
permanently affects the real variables.
By Theorem 1, a monetary expansion (an increase in
) stimulates the equilibrium real GDP 
through the rise of the real rate of interest. We here consider its welfare economics implication. Let the Lagrangean of individual decision 
that is evaluated at the equilibrium value.
Then, using the envelop theorem, we obtain
where 
is the Lagrangean multiplier. Accordingly, a monetary expansion improves the economic welfare since it makes future goods cheaper.