Assume the cash-in-advance constraint always binds \((x_{t+1}=0)\).

Still, private sectors maximise their utility s.t. budget constraint and cash-in-advance constraint. Let’s also include labor as a disutility and assume output is produced by labour.

$$ \max_{c_t, b_{t+1}, x_{t+1}} \sum_{t=0}^{\infty}\ \beta^{t} [u(c_t)-v(l_t)] $$

$$ p_{t-1}y_{t-1}+b_t(1+i_t)+x_{t}-T_t=x_{t+1}+p_t c_t+b_{t+1} $$

0 \leq x_{t+1}

$$ 0 \leq l_t \leq 1 $$

, with \( y_{t-1}=l_{t-1}\)

Now the output is not exogenous anymore but depends on an agent’s willingness to work.

F.O.C.

w.r.t. \(c_t: \quad u'(c_t)=\beta (1+i_{t+1})\frac{p_t}{p_{t+1}}u'(c_{t+1}) \)

w.r.t. \(l_t: \quad v'(l_t)=\beta u'(c_{t+1})\frac{p_t}{p_{t+1}} \)

At the steady state, \( \frac{p_t}{p_{t+1}}=\frac{1}{1+\pi}\) and \(y_t=l_t=c_t=y\) (output is equal to labour’s production in the long run). The output could be calculated as the following equation. (at the steady state means in the long run).

v'(y)=\beta u'(y)\frac{1}{\pi}

Therefore, we can find that,

**Money is netural**: if change \( m \) (stock of money, or money supply), then output is not affected. For example, if money doubles in all time, the fraction \( \frac{p_t}{p_{t+1}}\) keeps constant. No affecting the real term of output \( y\).**Moeny is not super netural**: if change \(\pi\) (inflation rate), then output would change. (y decrases if \(\pi\) increases. That can be analysed by the curvture of \( v\) and \( u\) functions).

Question: First we assume cash-in-advance constraint binds. The QTM states that \( growth rate of money\) and \(inflation \) is one-to-one correlated only if assuming /(y/) is stable (\(m_t=p_t y\)). However, we find the relationship between inflation and output here. There seems a contradiction of whether fixes \(y\) or not. So, how to bridge the connection between inflation and money growth?

Answer: From the demand point of view or the Cash-in-Advance constraint ( \(p_t c_t=M_t, or x_t=0\) by our previous assumption ). At the steady state, consumption is stationary, so \( \frac{M_t}{p_t}=\frac{M_{t+1}}{p_{t+1}}= \frac{M_{t+2}}{p_{t+2}} =…\) imply the stock of money and price level are connected, and so the connection between growth rate of money and inlfation works. The inspiration is the cash-in-advance constraint binds, and we consider the problem by fixing consumption in the long-run stationary condition.

The question and answer also state that the neutrality of money bases the key cash-in-advance assumption.

Empirical study examples are as McCandless and Weber (1995)

#### Reference

McCandless, G.T. and Weber, W.E., 1995. Some monetary facts. *Federal Reserve Bank of Minneapolis Quarterly Review*, *19*(3), pp.2-11.