Category: Economics

Fiscal Policy also helps to get out of the liquidity trap.

Ricardian Equivalence in the Cash-in-Advance Model

Ricardian Equivalence states that the government finance the government spending, \(g_t\), by debts or taxes is irrelevant.

Consider tax, \(T_t\), as government steals money from the private sectors. Then, borrowing money is the same, because the government needs to finance the repayments by taxes as well. Government repay with one hand and steal (or tax) the same amount with the other hand. Thus, funds lent out to the government will never be given back.

Therefore, borrowing is the same as tax. However, this is not true sometimes, because debts and interest can help smooth consumption.

The Ricardian equivalence cannot be shown in the Cash-in-Advance model because tax and debts won’t be shown in the Euler equation, once markets clear and model achieves the equilibrium.

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

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

and \( x_{t+1}\geq0, i_{t+1}\geq0, x_{t+1}\times i_{t+1}=0\).
Clearly, no taxes and debts appear in the equations ( as tax and debts are represented by the government spending).

Fiscal Policy

Recall the model in a liquidity trap,

$$ u'(\hat{y})=\beta \frac{\hat{p_t}}{m’}y’u'(y’) $$

With fiscal policy this would be,

$$ u'(\hat{y}-\underbrace{g}_{Government\ Spending})=\beta \frac{\hat{p_t}}{m’}y’u'(y’) $$

A interesting phenomena appears that \(\uparrow g \Rightarrow \uparrow\hat{y}\). Also, government spending would induce a 1-to-1 increase in current output, because the RHS is unchanged.

The multiplier must exactly equal one, because of Ricardian equivalence.

Intuition:

• Governemnt taxes or borrows \(\hat{p_t}\times g\) units of money.
• This means that agents reduce hoardings, \(x_{t+1}\), with precisely \(\hat{p_t}\times g\) units.
• The government demands \(g\) units of output, and consumers still demand \(c\).
• Output is then \(\hat{y}=c+g\). (if \(\uparrow g\), then \(\uparrow\hat{y}\).)
• The argument relies on a horizontal Aggregate Supply cureve. That is there is slackness in the economy. The economy is not fully employment (not achieve potential GDP) and also price is sticky in the very short run.
• Under unemployment, increase in government spending would make more people employed, and increase the total output at 1-to-1 level.

If not in a liquidity trap, then \(x_{t+1}=0\) and \(i_{t+1}>0\).

The Euler equation is,

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

A rise in \(g\) leads to a rise in \(i_{t+1}\), and the multiplier is zero. Consider the crowding-out effects that increase in government spending induces an increase in government debts (demands in the loanable fund market increase), and thus reduce the demand of investment from the private sector (increase in interest rate leads to increase in supply along the supply curve but not shift the curve).

Intuition:

• Outside of a liquidity trap. agents do not hoard money. \(x_{t+1}=0\).
• The oruvate sector is happy to do this at a higher interest rate.
• Fully Crowding Out.

P.S. See further study about the multiplier effects and crowding-out effect.

Multiplier > 1 ?

A number of reasons to make the multiplier be greater than one.

One of the reasons is unemployment persistence.

Suppose now the outputs is produced as \(y_t=z_t l_t\). Labour supply is inelastic and is normalised to be one. I would also assume \(z_t=1\) for simplicity.

If there is a demand shock with rigid nominal wages, then we have \(\hat{y}=z_t \times l_t<z_t \times 1\), and there would be unemployment, \(u_t=1-l_t\).

Now we include persistent unemployment/ In particular, I assume that \(l_{t+1}=l_t^{\alpha}\), with \( \alpha \in [0,1)\).

The Euler Equation is then,

$$ u'(\hat{y}-g)=\beta \frac{\bar{p_t}}{p_{t+1}}u'(y’) $$

$$ u'(\hat{y}-g)=\beta \frac{\bar{p_t}}{m’}y’u'(y’) $$

$$ u'(\hat{y}-g)=\beta \frac{\bar{p_t}}{m’}z’l’u'(z’l’) $$

$$ u'(\hat{y}-g)=\beta \frac{\bar{p_t}}{m’}z’l^{\alpha} u'(z’l^{\alpha}) $$

By applying \(z_t=1\), the Euler equation is then,

$$ u'(\hat{y}-g)=\beta \frac{\bar{p_t}}{m’}z'{\hat{y}}^{\alpha} u'(z’ {\hat{y}} ^{\alpha}) $$

Then, we can assume the utility function is isoelastic \(u(c)=\frac{c^{1-\sigma}-1}{1-\sigma}\), and apply the implicit function theorem to the Euler equation, then we can get,

$$ \frac{\partial{\hat{y}}}{\partial{g}}=\frac{1}{1-\alpha (1-\frac{1}{\sigma})} $$

That implies the impacts of government spending on output depends on \(\alpha\) and \(\sigma\). The derivative is greater than one based on our assumptions of parameters.

Intuition:

• Governemnt taxes or borrows \(\hat{p_t}\times g\) units of money.
• This means that agents reduce hoardings, \(x_{t+1}\), with precisely \(\hat{p_t}\times g\) units.
• The government demands \(g\) units of output, and consumers still demand \(c\).
• Output is then \(\hat{y}=c+g\), and the unemployment must fall \(l_t \uparrow\). (P.S. \(\uparrow g \Rightarrow \uparrow \hat{y} \Rightarrow \uparrow l_t\), coz \(y=zl\) ).
• As \( l_t \uparrow\), we have that \(l_{t+1}\uparrow\) and \(y_{t+1} \uparrow \). (by presistent employment.)

Then, the loop starts asthe following:

• Since \(y_{t+1} \uparrow\) the fture looks brighter!
• Individuals want to consume more and hoard less, so \(c_t \uparrow\) and \(\bar{y}\uparrow\)!
• Output increases even more, and the unemployment rates fall again, so \(y_{t+1} \uparrow\) even more.
• This makes the future look even brighter, so there is more spending and so on.

Three monetary regimes, aiming to avoid or mitigate the liquidity trap, are introduced here. They are Inflation Targeting (IT), Price Level Targeting (PLT), and Nominal GDP Targeting (NGDPT).

A brief summary is that NGDPT performs better than PT ( in terms of dealing with the liquidity trap), which in turn performs better than IT.

Before doing the analysis, we modify the model a little bit that makes the price to be “somewhat flexible”.

$$ \bar{p}_t=\frac{m_t}{y_t} $$

In that, if not in the liquidity trap, we assume \( p_t\geq \gamma \bar{p_t}\), with \(\gamma \in (0,1)\). (we previously assume \(p_t\geq \bar{p_t})\).

Therefore, our Euler equation is,

$$ u'(\hat{y})=\beta\gamma \frac{\bar{p_t}}{p_{t+1}}u'(y’) $$

Inflation Targeting

Let the inflation target be \( \frac{p_{t+1}}{p_t}=1+\pi\), then our Euler equation becomes，

$$ u'(\hat{y})=\beta \frac{1}{1+\pi}u'(y’) $$

Therefore, \( \uparrow \pi \Rightarrow \uparrow RHS \Rightarrow \uparrow LHS \Rightarrow \uparrow \hat{y}\). Increase in inflation would raise the current output.

Implication:

1. The fall in current output in the crisis is less severe (as \(\uparrow \hat{y}\)).
2. The economy is less likely to fall into a liquidity trap in the first place, because people know inflation in the future, so they will not likely be strucked in the liquidity trap, coz increasing demans in current time) High inflation means money loses value quickly, and thus agents are reluctant to save using cash.

Price Level Targeting

With the price level targeting, the CB aims to keep the price level on a certain path. This means if the CB fails to meet the target, it will catch up in teh later period. For example, if the price level is 100 at period \(t\), and the CB’s price level target is 2%, then the price level in period \(t+1\) should be 102. However, if the CB fails to do that in \(t+1\), then in \(t+2\) the CB should catch up and keep the price level to be 104.

In that, the CB follows \( p_{t+1}=(1+\mu)\bar{p_t}\), where \( \bar{p_t} \) is the “normal times price level”, and \(\bar{p_t}=\frac{m_t}{y_t}\).

Then, the Euler equation

$$ u'(\hat{y})=\beta \gamma \frac{\bar{p_t}}{p_{t+1}}u'(y’) $$

would become,

$$ u'(\hat{y})=\beta \gamma \frac{1}{1+\mu}u'(y’) $$

The difference between the price level target and inflation target in the Euler equation is the \( \gamma\) term. Therefore,

$$ u'(\hat{y_{IT}})=\beta \frac{1}{1+\pi}u'(y’) $$

$$ u'(\hat{y_{PLT}})=\beta \gamma \frac{1}{1+\mu}u'(y’) $$

If \(\pi=\mu\), then the RHS of the second Euler equation is smaller, and thus \(\hat{y_{PLT}} > \hat{y_{IT}}\). (easy to show in maths by assuming the isoelasicity utility function).

NGDP Targeting

With NGDPT, the CB aims to keep nominal GDP on a certain path. For example, aiming to increase NGDP by 2% per year. A failure in one period means a cathcing up in the next period.

$$\underbrace{ p_{t+1}y_{t+1}}_{nominalGDP_{t+1}}=(1+\mu)\underbrace{\bar{p_t}y_t}_{nominalGDP_t} $$

By assuming not in the liquidity trap, \(m_{t+1}=(1+\mu\)m_t\).

Then, the Euler equation

$$ u'(\hat{y})=\beta \gamma \frac{\bar{p_t}}{p_{t+1}}u'(y’) $$

becomes,

$$ u'(\hat{y})=\beta \gamma \frac{ \frac{m_t}{y_t} }{ \frac{m_{t+1}}{y’} }u'(y’) $$

$$ u'(\hat{y})=\beta \gamma \frac{ \frac{m_t}{y_t} }{ \frac{m_t(1+\mu)}{y’} }u'(y’) $$

u'(\hat{y})=\beta \gamma \frac{y’}{y_t}\frac{1}{1+\mu} u'(y’)

Since \( \frac{y’}{y_t}<1 as y'<y_t, and \gamma \in (0,1)\), so RHS is even smaller than that of the PLT Euler equation. Therefore, \(\hat{y_{NGDPT}}\) is even greater than the \(\hat{y_{PLT}}\). That implies that the cirsis would be less severe for a given \(y’\).

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.