Fanyu Zhao

Recall the Euler condition in the previous blog post A Cash-in-Advance Model.

$$ u'(y_t)=\beta(1+i_t)\frac{p_t}{p_{t+1}}u'(y_{t+1}) $$

Assumption

For simplification, we assume no government spending, \(g_t\), government debt, \(d_t\), and taxes, \(T_t\). Also, we assume money is stable such that \(m_t=m_{t+1}=m\) (so there is not seignorage). We here consider \(y_t\) is exogenous.

Recall

Suppose that \( y_t=u_{t+1}=…=y\), then

$$\quad 1=\beta(1+i_{t+1})\frac{p_t}{p_{t+1}}$$

Now if guess both \(x_{t+1}=x_{t+2}=0\), then the velocity of money \(v_t=1\).

\( \quad p_t=p_{t+1}=\frac{m}{y}, \quad \) and \(\quad i_{t+1}=\frac{1}{\beta}-1\geq0\)

P.S. if violate the guess \(x_{t+1}=x_{t+2}=0\), then the euler equation shows \(1+\beta (1+i_{t+1})\frac{p_t}{p_{t+1}}\) would be \( p_{t+1}=\beta p_{t}\). So, \( p_{t+1}<p_t\). By QTM \(m \cdot v_t= p_t \cdot y\) (\(m, y\) are constant), \( v_{t+1}<v_T\) must be true to make next-period price level be low than the current price level. Lower velocity means \( x_{t+2}>x_{t+1}\) (people would hoard more money on hand in the next period). The loop begins, and price level would decline in the following periods.

If future outputs decrease,

u'(y_t)=\beta(1+i_t)\frac{p_t}{p_{t+1}}u'(y_{t+1})

If replace \(p_{t+1}=\frac{m_{t+1} v_{t+1}}{y_{t+1}}\),

u'(y_t)=\beta(1+i_t)\frac{p_t \times y_{t+1}}{m_{t+1}v_{t+1}}u'(y_{t+1})

u'(y_t)=\beta(1+i_t)\frac{p_t \times y_{t+1}}{m_{t+1}-x_{t+2}}u'(y_{t+1})

Here, by complementary slackness, \( x_{t+2}\times i_{t+1}=0\).

If replace \(p_{t+1}=\frac{m_{t+1} v_{t+1}}{y_{t+1}}\), =\frac{m_{t+1}}{y_{t+1}}\) by assume not in liquidity trap in the first so \(v_{t+1}=1\). Then we get,

u'(y_t)=\beta(1+i_t)\frac{p_t \times y_{t+1}}{m_{t+1}}u'(y_{t+1})

We, in the following, assume \(x u'(x)\) is decreasing in x.

If the economy experiences a fall in period \( t+1\) output from \(y_{t+1}\) to \(y’_{t+1}\). What happens to the nominal interest rate?

We write it in this way for simplification.

u'(y)=\beta(1+i_t)\frac{p_t }{m}y’u'(y’)

As \(y_{t+1}\) decrease, \(y’u'(y’)\) increase as our assumption. The LHS keeps stable, so the interest rate has to decrease to keep the equality holding. Therefore, \(i_{t+1}\) we’ll eventually hit zero.

As \( i_{t+1}=0\), the economy enters into the liquidity trap, and people start to hoard money ,\(x_{t+1}>0\). Recall the QTM equation, \(p_t=\frac{ m_t-x_{t+1} }{y_t}=\frac{mv_t}{y} \), \(p_t\) would decrease. So, the price level at time \(t\) finally decreases as well.

From the figure, we can find that once the effective federal fund rate (The effective federal funds rate (EFFR) is calculated as a volume-weighted median of overnight federal funds transactions) hits zero, excess reserves increases. Injecting more money would only cause excess money reserves in the liquidity trap.

If future outputs decrease and price is sticky,

An extension. If the price is “sticky” in the short run. In other words, \( \bar{p}_t=\frac{m}{y}\), price cannot fall below a certain threshold. Then, a decrease in \(y_{t+1}\) would end up with decrease in current output \(y_t\). As shown in the following equation,

$$ u'(\hat{y})=\beta \frac{\bar{p}_t}{m}y’u'(y’) $$

Future output decrease, then RHS increases, and so LHS has to increase as well. \(\frac{\partial u'(y)}{\partial y}=u”(y)\) is negative. For example, in the isoelasticity form \( u(c)=\frac{c^{1-\sigma}}{1-\sigma} \), and \(0\leq \sigma \leq1\).

In summary, recession in \(t+1\) would bring down \(y_{t+1}\). Then, firstly, decrease \(i_{t+1}\) to 0; secondly, reduce \(p_t\) to \(\bar{y}\) if price is stikcy; and thirdly, drive \(y_t\) decrease in the end. (All those are based on the guess of \(x_{t+1}=x_{t+2}=0\))

In a liquidity trap with sticky prices, outputs become “demand-driven”. The reason is that the Euler Equation is derived from the private sector, and thus \(u'(y_t)=u'(c_t)\) if not replaced with the markets clearing condition in equilibrium. The equation would then show that the increase in the LHS is driven by a decrease in consumption. A disequilibrium starts. Finally, a recession begins if nothing happened to productive capacity.

Intuition

• Private sectors initially earn income, say 100, and buy goods for100 as well (Normal situation).
• When they receive a “news” that income will decrease in the future from \(y_{t+1}\) to \(y’\), then they all wish to save.
• However, in the aggregate, nobody can save, because noboday want to borrow or invest.
• So the interest rate, as the benefits of saving, decrease to eventally zero, and private sectors start to hoard cash.
• Thus, instead of spending 100, they spend80 and save $20. The demand drives down current outputs.

Role of price stickiness

• Initally, current and future outputs (endownments) are all $100. \(y_{t}=y_{t+1}=100\).
• A news tells us future output decrease to 80. In the current period, we save20 and spend $80. Same as the above process.
• So, current spending is 80 and future spending is100.

If the price is sticky, consume $80 today and price decreases 20% at the same time. Ending up with the same amount of current consumption, \(y_t\). No recession.

If the price is sticky, then agents spend $20 fewer goods in the current. Worse off. And recession.

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.