Author: Fanyu Zhao

Recall the government’s budget constraint again,

$$ p_tg_t+d_t=\frac{d_{t+1}}{1+i_{t+1}}+T_t+(m_t-m_{t-1}) $$

Divide by \(p_t\), and assume for simplicity that \(m_t=m_{t+1}=…=\bar{m}\)

$$ g_t+\frac{d_t}{p_t}=\frac{d_{t+1}}{p_t (1+i_{t+1})}+\tau_t $$

denote \(\tilde{d_t}=d_t/p_t\), and recall that \((1+i_{t+1})\frac{p_t}{p_{t+1}}=1+r_{t+1}\).

Thus,

$$ g_t+\tilde{d_t}=\frac{\tilde{d}_{t+1}}{1+r_{t+1}}+\tau_t $$

Iterate forward, and impose the “no-Ponzi condition”, \( \lim_{s\rightarrow \infty} \frac{\tilde{d_{t+s}}}{\prod_{j=1}^s 1+r_{t+j}}=0\) to get,

$$ \tilde{d_t}=\sum_{s=0}^{\infty} \beta^s (\tau_{t+s}-g_{t+s}) $$

, where the equilibrium condition \( \beta=\frac{1}{1+r}\) has been imposed.

Implication:

$$ \frac{d_t}{p_t}=\sum_{s=0}^{\infty} \beta^s (\tau_{t+s}-g_{t+s}) $$

The “unpleasant arithmetic” stated that if the government has leadership, it can coerce expansions in money.

In contrast, FTPL says that the above restrictions are not a constraint to the CB or government. Instead, it is an equilibrium relation.

As a consequence, the CB and the government may choose policies independent of the above constraint. In the end, the price level \(p_t\) must then adjust such that the equation holds.

Recall A Cash-in-Advance Model, the 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).

• In the monetarist view, the price level is determined by the amount of money in the economy.
• Inflation is everywhere a monetary phenomenon.
• Inflation is too much money chasing too few goods

Assume a model that, instead of using the money for transactions, agents just get happy holding some money in their pockets.

The optimisation problem becomes to be,

$$ \max_{c_t, b_{t+1},M_t} \sum_{t=0}^{\infty} \beta^t [u(c_t)+v(\frac{M_t}{p_t})] $$

$$s.t. \quad p_t c_t +\frac{b_{t+1}}{1+i_{t+1}}+M_t=b_t+\underbrace{p_t y}_{current\ sales\ withou\ lages}+M_{t-1}$$

F.O.C.

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

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

Now assume that money supply follows \(M_{t+1}=(1+\mu)M_t\). Also assume government spending is zero, so all seignorage is redistributed as a (negative) lum-sum tax. ENdowments are constant and given by \(y\).

Then the constraint,

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

can be written as,

$$u'(y)-v'(m_t)=\beta \frac{1}{1+\pi_t}u'(y)$$

, where I impose that \(y=c+g\), with \(g=0\), \(m_t=\frac{M_t}{p_t}\), and define \(1+\pi_t=\frac{p_{t+1}}{p_t}\).

Notice that,

$$ \underbrace{\frac{m_{t+1}}{m_t}}_{ratio\ of\ real\ money\ balance}=\frac{ \frac{M_{t+1}}{p_{t+1}} }{ \frac{M_t}{p_t} }=\frac{M_{t+1}}{M_t}\times \frac{P_t}{P_{t+1}}=\frac{1+\mu}{1+\pi_t} $$

so the Euler equation

$$ u'(y)-v'(m_t)=\beta \frac{1}{1+\pi_t}u'(y) $$

becomes (by replacing \(\frac{1}{1+\pi}=\frac{m_{t+1}}{m_t}\frac{1}{1+\mu}\)),

$$ m_{t+1}=\frac{1+\mu}{\beta}\frac{u'(y)-v'(m_t)}{u'(y)}m_t $$

Showing the relationship between real money supply over periods.

Find Solutions of The Euler Equation

• At the Steady State

Assume\(m\neq0\). Let \(m_t=m_{t+1}=…=m^*\) and \(m^*>0\) at the steady state,

$$ 1=\frac{1+\mu}{\beta}\frac{u'(y)-v'(m^*)}{u'(y)} $$

Clearly, there is a \(m^*>0\) satisfy the equation.

• \(m\rightarrow 0\)

$$ m_{t+1}=\frac{1+\mu}{\beta}\frac{u'(y)-v'(m_t)}{u'(y)}m_t $$

If \( \lim_{m\rightarrow 0}v'(m)m=0\), then \(m^*=0\) is also a steady state equilibrium.

• P.S. \(u'(y)=v'(m”)\)

In addition, there exists an \(m”>0\) such that

$$ u'(y)=v'(m”) $$

which implies that \(m_{t+1}\leq0\) for \(m_t\leq m”\).

The relationship could be expressed by the figure below.

If the initial condition of \(m_t\) is at the right of \(m^*\), then RHS is always greater than the LHS and would result in \(\lim_{t \rightarrow \infty} m_t \rightarrow \infty\). No steady state and violate the transversality condition. Thus, \(m_0\) cannot be at the right of \(m^*\).

If the \(m_0\) at the left of \(m^*\), then \(m_t\) would converge to 0 (as the similar logic above). Therefore, as \(m_t \rightarrow 0\), \(p_t\ rightarrow \infty\) even if \(M_t\) keeps constant. That would lead to hyperinflation.

In that scenario, the unique equilibrium is \(m_t=m^*>0\), the price level is determined. Prices, in this case, must be growing at precisely the same rates as money, which adhere to the monetarist doctrine. However, the economy can also display speculative hyperinflation in which inflation far outpace money growth.

In addition, different forms of the RHS might lead to different results.

E.G. the concavity of RHS, the interaction point at zero, etc would all affect the equilibrium condition.

• No interaction at 0.

• No \(m”\)

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 \(y’\).