We consider a standard two-period deterministic OLG model in a production economy. In every period, a unit of individual is born. They can work only when they are young. A unit working hour produces unit goods.
The money supply obeys Lucas’s [
where 
is the nominal money stock per capita that is carried over from the previous period. 
is the gross increase rate of money. In this sense, new money is supplied as its own nominal interest rate.
We make the following alternative assumptions concerning the labor supply: 1) In the representative individual model, the representative individual can chooses his working hours and there is no unemployment problem; 2) In the indivisible employees model, each individual faces the discrete choice of whether to work.
For simplicity, we assume that the representative individual possesses the following utility function
:
where 
is a well-behaved linear homogenous function. 
denote the consumption level sof generation 
during the young and old stages of life, respectively. 
is the hours worked.
has the following properties.
The shape of 
is illustrated in
The assumption that some lower limit 
exists for the disutility of labor is equivalent to the assumption that individuals do not incur any additional disutility by increasing in hours worked to some extent. As the classical economists presume. Its economic meaning of this assumption is that there is an urgent need to produce goods that correspond to the subsistent level, as shown below.
The lifetime budget constraint is
Since the lifetime utility function concerning the consumption stream is concave and homothetic, we obtain the following correspond indirect utility function
:
Moreover, we can ascertain that
holds. This expression implies that, as long as 
is sufficiently small, the equilibrium hours worked always exceeds the subsistent lower limit
, and that the problem of the indivisibility of hours worked never appears in the decision problem.
The optimality conditions are
We assume, according to Lucas [
In addition to the three optimality conditions, there is one independent market equilibrium condition. Here, we consider the condition for the money market equilibrium: that is,
Furthermore, we assume the credibility of money in the sense of Otaki [
There are five endogenous variables
, and five independent Equations (5)-(9). Hence, the model is closed, and the solution consists of a temporary rational expectation equilibrium.
From Equations (8) and (9), it is clear that 
increases with the nominal interest rate of money
Equations (5) and (6) imply that 
is a monotonically decreasing function of the effective inflation rate (the inverse of the real interest rate)
. Thus, 
decreases as 
increases. It is also apparent from Equation (5) that equilibrium working hours 
increases with x3.
To summarize, as long as money is credible, an easy monetary policy increases the real interest, and hence, the representative individual works more to enjoy more future consumption. Accordingly, a monetary expansion advances intertemporal substitution from current consumption and leisure into future consumption by raising the real rate of interest. As such, the expansionary effect of monetary policy is entirely based on the labor supply incentive, not on the expansion of the aggregate demand. In this sense, the representative individual model is inevitably classified as a neoclassical macroeconomic model.
Assume that the representative individual rationally expects that the real effective inflation rate
is kept intact after period
, since no economic environment is changed after period 
4. This assumption implies that
In addition
and 
holds from the money market equilibrium condition (8). Thus, we obtain
That is, future consumption 
becomes time-independent. Hence, from Equations (5)-(7), 
are also time-independent. Consequently, the rational expectation equilibrium characterized by the initial condition 
and the expectation formulation (9) and (10) are stationary.
Here, we assume that labor supply is indivisible, and that each individual has the identical utility function
:
where 
is the same consumption utility function as in Equation (1). 
denotes a definition function that takes the value unity when the individual works unit time and that is zero when the individual does not work.
According to Equation (1), the nominal minimal revenue that individuals decide to work 
is represented as
We must note that firms strictly prefer increasing employment to the upward adjustment in working hours per capita in any interior equilibrium in which unemployment exists and all individuals are indifferent to the decision of whether to work.
The reason is as follows. Even if a unit employment increases, as long as the working hours per capita are fixed, there is no appreciation of nominal wages. However, the concavity of 
requires nominal wages higher than 
to induce working hours that produce the same amount of output as in the case of employment adjustment. Thus, as long as unemployment exists, working hours per capita is fixed to the minimal level
.
Accordingly, we obtain the following difference equation concerning the evolution of price sequence5:
Hence, the equilibrium real interest rate
is independent of 
and takes a constant value.
The equilibrium condition for the money market is
where 
is the marginal propensity to save.
Assuming the credibility of money (i.e.,
)an increase in the monetary growth rate 
increases the current value of money and empowers the purchasing power of old individuals as long as money is a credible asset. As such, the monetary expansion stimulates the economy through the multiplier effect developed by Otaki [
It is not difficult to show the time-independence of the equilibrium. Assume that people rationally believe that the price level 
grows proportionately with the monetary expansion rate
, that is,
Then, the equilibrium real cash balance 
becomes time-independent, so does the real equilibrium GDP 
6.
In addition, as we previously mentioned, the individual employees model is similar to a Keynes’ [