Let’s continue with the previous blog post The Neutrality of Money.
In the previous model, consumers maximise their utility subject to contraints.
$$ \max_{c_t, b_{t+1}, x_{t+1}} \sum_{t=0}^{\infty}\ \beta^{t} [u(c_t)-v(l_t)] $$
$$ p_{t-1}y_{t-1}+b_t(1+i_t)+x_{t}-T_t=x_{t+1}+p_t c_t+b_{t+1} $$
0 \leq x_{t+1}
$$ 0 \leq l_t \leq 1 $$
We have solved it and get the Euler condition,
v'(y)=\beta u'(y)\frac{1}{\pi}
Here, we would consider the Planner’s Problem that makes social optimal.
Planner’s Problem
In the planner’s problem, we would release the budget constraints and cash-in-advance constraints, because the planner only needs to achieve social optimal. The planner’s problem is as the following.
\max_{c_t, b_{t+1}, x_{t+1}} \sum_{t=0}^{\infty}\ \beta^{t} [u(c_t)-v(l_t)]
$$ s.t. \quad c_t=l_t $$
F.O.C.
$$ u'(c_t)=v'(l_t) $$
Here let’s compare the planner’s Euler equation with the private sector one.
To make them equal, the only thing we need to adjust is to let \( \beta\times\frac{1}{1+\pi}=1\). The implication is that we need \( \pi =\beta -1\). As in the steady state, the discount factor \( \beta = \frac{1}{1+r}\), so the optimal inflation rate should be \( \pi^*=\frac{-r}{1+r}\).
The implication is that the optimal inflation rate is negative and close to the negative real interest rate.
Cash Credit Good Model
Stokey and Lucas (1987) included the cash-credit good into the cash in advance model.
\max_{ \{ c_t,b_{t+1} \}_{t=0}^{\infty} } \sum_{t=0}^{\infty}\ \beta^{t} [u(c_t^1)+u(c_t^2)]
$$ s.t. \quad b_{t+1}+p_t c_t^1+p_{t-1} c_{t-1}^2 =(1+i_t)b_t+p_{t-1}y_{t-1}$$
In equilibrium, markets clear and resources constraints,
$$ y_{t-1}=c_{t-1}^1+c_{t-1}^2 $$
y_{t}=c_{t}^1+c_{t}^2
F.O.C.
$$ u'(c_t^1)=\lambda_t p_t $$
$$ u'(c_t^2)=\beta\lambda_{t+1}p_t $$
$$\lambda_t=\beta \lambda_{t+1}(1+i_{t+1})$$
Combining them we can get
$$ \frac{u'(c_t^1)}{ u'(c_t^2) }=1+i_{t+1}$$
The ratio of marginal utility is equal to one plus the nominal interest rate.
The implication is that people want to consume \(c_t^2\) instead of \(c_t^1\), pay money at the time at \(t\), and hold some bonds and earn the nominal interest rate.
However, the planner problem is that
$$ \frac{u'(c_t^1)}{ u'(c_t^2) }=1 $$
Thus, the optimal rule is to set \(i_{t+1}=0\).
The Euler equation in the steady state (\( c_t^i=c_{t+1}^i=…=c^i \)) is that,
$$ \beta \frac{1+i_{t+1}}{1+\pi_t}=1 $$
By plugging in \(i_{t+1}=0\), \(\pi^*=\beta -1 \), the Friedman rule also holds.