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.

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,

1. 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 \( y\).
2. 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.

Here, I would use the cash in advance model to illustrate some economic phenomena.

Assumptions

Two core assumptions of the cash in advance model. 1. People need cash to purchase goods. 2. Income is received with a lag. The main implication of those two assumptions is that people cannot use the proceeds from the current sales to fund the purchases because people cannot get income back immediately but in the next period (e.g. employees earn wages with a lag).

Market Players

Before talking about the model, I would first illustrate the balance sheet of three main players in the market, the central bank, the government, and the private sector.

If consolidate those two constraints together, then we get the public sector Budget Constraint,

$$ T_t+(\hat{b}_t-\hat{d}_t)(1+i_t)-m_{t-1}=(\hat{b}_{t+1}-\hat{d}_{t+1}-m_t+p_t g_t) $$

If define \( D_t=\hat{d}_t+m_{t-1}-\hat{b}_t \) as the net position of public sector debt, then we get,

$$ \underbrace{T_t}_{taxes}+ \underbrace{(D_{t+1}-D_t)}_{deficit}+\underbrace{m_{t-1}i_t}_{seignorage}=\underbrace{i_t D_t}_{interest}+\underbrace{p_t g_t}_{spending} $$

Or if define \(d_t=\hat{d}_t-\hat{b}_t\) (, which can be considered as the net position of government debt, the net amount runing in private sectors), then

\underbrace{T_t}_{taxes}+ \underbrace{(d_{t+1}-d_t)}_{deficit}+\underbrace{(m_t-m_{t-1})}_{seignorage}=\underbrace{i_t d_t}_{interest}+\underbrace{p_t g_t}_{spending}

P.S. Serignorage behaves like the tax of inflation? See the reading in the end.

Private Sectors: Consider that private sectors have an endowment \(y_t\) each period, and they would sell the endowment to get cash, \(p_t \cdot y_t\), in the subsequent period. Private sectors then have to use those cash to buy endowments (goods and services). The private sectors face a budget constraint as the following,

$$ p_{t-1}y_{t-1}+b_t(1+i_t)+(M_{t-1}-p_{t-1}c_{t-1})-T_t=M_t+b_{t+1} $$

$$ p_t c_t \leq M_t $$

, where \( b_t \) is the government bond and \( M_t \) is the money holding.

If define \( x_{t+1} = M_t – p_t c_t \), which means the excess cash holding, then the budget constraints of private sectors are,

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

x_{t+1} \geq 0

LHS are the source of money at period \(t\), and RHS are how the private sector uses those money. The private sector can use the money to (1) consumer, (2) buy bond and earn interest, and (3) simply hold the money

Private sectors maximise their lifetime utility subject to budget constraints.

$$ \max_{c_t, b_{t+1}, x_{t+1}} \sum_{t=0}^{\infty}\ \beta^{t}\cdot u(c_t) $$

$$ s.t. Two\ Constaints $$

Solve the problem by Lagrangian.

$$ \mathcal{L}= \sum_{t=0}^{\infty} \beta^t \{ u(c_t) \\ – \lambda_t ( x_{t+1}+p_t c_t+b_{t+1}-p_{t-1}y_{t-1}+b_t(1+i_t)+x_{t}-T_t ) \\ -\mu_t x_{t+1} \} $$

Take f.o.c.

\( \frac{\partial \mathcal{L}}{c_t}: \quad u'(c_t)=\lambda_t p_t \)

\( \frac{\partial \mathcal{L}}{b_{t+1}}: \quad \lambda_t=\beta(1+i_{t+1})\lambda_{t+1} \)

\( \frac{\partial \mathcal{L}}{x_{t+1}}: \quad \lambda_t -\mu_t=\beta \lambda_{t+1} \)

Here, let’s focus on the second and the third equation. If \( i_{t+1} =0\), then \(\mu\) has to be zero as well to make them equal. Also, by completementary slackness, if \( \mu =0\), then \(x_{t+1}\) must be greater than zero.

The implication is that private sectors would hold excess cash (hoard cash) even if the interest rate is zero. That is the liquidity trap. Although the government adjusts the interest rate to be zero in order to stimulate the economy, people do not spend that money. Instead, people just hoard the money.

Euler Condition of Private Sectors

Combining three f.o.c., we can get the following Euler condition.

$$ u'(c_t)=\beta(1+i_{t+1})\frac{p_t}{p_{t+1}}u'(c_{t+1}) $$

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

If markets clear, then \(y_t=c_t\).

Competitve Equilibrium

The competitive equilibrium of this problem is a sequence of price \( \{ p_t,i_{t+1} \}^{\infty}_{t=0}\) and allocations \( \{ c_t, b_{t+1}, x_{t+1}, g_t, T_t, d_{t+1}, m_t \} \) such that given price,

1. The sequence \( \{ p_t,i_{t+1} \}^{\infty}_{t=0}\) solves the household’s problem.
2. Bond markets clear, \( b_t =d_t \).
3. Goods markets clear, \( y_t = c_t+g_t \).

Equation of Exhange

We here combine the private sectors and public sectors’ budget constraints and apply the markets clear condition, and then we can get,

$$ p_{t-1}y_{t-1}+x_t+(m_t-m_{t-1})=p_t y_t +x_{t+1} $$

Assume at the beginning period when \( t=0\), \( y_{t-1}=x_0=m_{-1}=0\). Thus,

$$ m_0=p_0 y_0 +x_1 $$

Similarly, in the following period,

$$ m_t = p_t y_t +x_{t+1} $$

The above equation is the equation of exchange, the one I mentioned in the blog: Quantity Theory of Money (QTM). It is called the Fischer equation or quantity equation.

Define \( v_t =\frac{m_t-x_{t+1}}{m_t} \), then we can get the QTM equation.

$$ m_t v_t = p_t y_t $$

Recall the liquidity trap. If in the liquidity trap, then \( i_{t+1}=0 \) and \(x_{t+1}>0\) people hoard excess money. Therefore, the velocity of money \(v_t <1\) .

However, if not in the liquidity trap, then \( i_{t+1}>0\), and \(x_{t+1}=0 \) and \(v_t=1\), so

$$ m_t=p_t y_t\ and\ p_t=\frac{m_t}{y_t} $$

P.S. Here if we take logarithm to the equation of exchange, then we can get the relationship \( i_t \approx \pi_t + r_t \).

Also, if the output is relatively stable \( y_t=y\), then \( p_t=\frac{m_t}{y}\) (price level or is directly affected by money. Or if taking the logarithm, the inflation rate is one-to-one affected by the growth rate of money). P.S. the close to one relationship only works in the long run, see Wen (2006).

The empirical evidence of the relationship between excess reserves and the velocity of money can be found. In the figure, those two variables are negatively correlated.

Government Deficits Cause Inflation

Here, I would apply the equation of exchange and government budget constraint to explain how inflation is generated by government deficits. Recalling the government budget constraint,

\overbrace{p_t g_t}^{Gov Spending} + \overbrace{i_t d_t}^{Interest Payment} = \underbrace{(d_{t+1}-d_t)}_{Increase in Debt Position}+\underbrace{T_t}_{Tax Revenue}+\underbrace{m_t-m_{t-1}}_{Print Money}

devide by \( p_t\) to get the equation in the real term,

$$ g_t+i_t \frac{d_t}{p_t}=\frac{d_{t+1}-d_t}{p_t}+\tau_t+\frac{m_t-m_{t-1}}{p_t} $$

, where \( \tau_t=\frac{T_t}{p_t}\).

By denoting real government debt as \( \hat{d}_t=\frac{d_t}{p_{t-1}}\), and replace \( (1+r_t)=(1+i_t)\frac{P_{t-1}}{P_t}=\frac{1+i_t}{1+\pi_t} \) and \( m_t = p_t y_t \), then we get all variables are in real terms,

$$ g_t – \tau_t +(1+r_t)\hat{d}_t =\hat{d}_{t+1}+\frac{p_t y_t-p_{t-1}y_{t-1}}{p_t}$$

At the steady state \( g_t=g_{t+1}=g, \tau_t=\tau_{t+1}=\tau \) and so on, and thus,

$$ \underbrace{g+r\hat{d}-\tau }_{Growth\ of \ interest\ deficits}= \underbrace{\frac{p_t-p_{t-1}}{p_t}}_{Seignorage} \times y$$

From the above equation, we can find that if inflation increases then it means the RHS increases. The LHS consists of two parts. Government Spendings \( g + r\hat{d}\) and government revenues \( \tau \). That means the government is getting deficits if the LHS rises. Meanwhile, the RHS increases and so inflation grows.

In sum, we find that government deficits, in the long run, would induce inflation. The zero-inflation condition is to make the LHS of the equation equal to zero (government spendings offset government revenue).

Reference

Wen, Y., 2006. The quantity theory of money. Monetary Trends, (Nov).