please $$provide $$step $$by $$step $$mathematical $$steps $$so $$i $$can $$see $$the $$process$.$Thank $$you$!$

$3$ Government deficits with the inflation tax

$3\mathrm{.}1$ Inflation and money growth

Consider an economy that is on a steady nominal growth path, along which real variables are

constant.

We denote the level of money by $M_t,$ and the level of real money by $m_t=\frac{M_t}{P_t}.$ Money grows

at a rate ${}_t=\frac{M_t-M_{t-1}}{M_{t-1}},$ which is constant. Inflation is ${}_t=\frac{P_t-P_{t-1}}{P_{t-1}},$

Show that $=.$

$3\mathrm{.}2$ Monetary financing of government spending

Consider that the government needs to finance a constant amount $g$ of real deficit. The only

source of revenue is money printing, and the real revenue is inflation times the real balances:

$g=m.$

The money demand is an inverse relation between real balances and expected inflation:

$m_t=$exp$[-{}_{t+1}^e]$

For now, we focus on a steady state where inflation and real variables are constant.

Show that one can write expected inflation ${}^e$ needed to finance the government spending as a

concave function of the growth rate of money $,$ conditional on $g.$ Draw this line for $g=0\mathrm{.}25.$

$3\mathrm{.}3$ Equilibrium

In a steady state, inflation is fully expected.

Using the figure you drew in the previous point, what are the equilibria in the system?

What is the intuition?

$3\mathrm{.}4$ Feasibility of monetary finance

Show that there is a limit of the amount $g$ that can be financed in this way.

What are the values of $g$ and $$ at this limit$?$

Hint: think of how the various lines in your figure interact at the limit $($in terms of levels and

slopes$).$

$3\mathrm{.}5$ Dynamics of expected and actual inflation

Consider that $g=0\mathrm{.}25,$ and that inflation expectation react in an adaptive way to any error:

${}_{t+1}^e-{}_t^e=({}_t-{}_t^e)$

We analyze the model in terms of deviation around the steady growth path. On that path, real

variables and growth rates are constant: $m_{t.ss}=m_{ss}($where the $t,$ ss subscript indicates the value

at time $t$ along the path$),{}_{t,ss}={}_{ss}=.$

We denote log deviations around the growth path by hatted variables:

hat$(m{)}_t=\frac{m_t-m_{ss}}{m_{ss}},$hat$(M{)}_t=\frac{M_t-M_{t.ss}}{M_{t.ss}},$hat$(P{)}_t=\frac{P_t-P_{t.ss}}{P_{t.ss}}$

hat$({)}_t=\frac{{}_t-}{1+}=$hat$(P{)}_t-$hat$(P{)}_{t-1}$

hat$({)}_t=\frac{{}_t-}{1+}=$hat$(M{)}_t-$hat$(M{)}_{t-1}$

Show that the money demand, $m_t=$exp$[-{}_{t+1}^e],$ is approximated as:

hat$(m{)}_t=-(1+)hat({)}_{t+1}^e$

Show that the dynamics of inflation expectations are $($we set hat$({)}_t=0$ for all $t)$:

hat$({)}_t=-\frac{(1+)}{1-(1+)}$hat$({)}_t^e$

and:

hat$({)}_{t+1}^e-$hat$({)}_t^e=-\frac{}{1-(1+)}$hat$({)}_t^e$

$3\mathrm{.}6$ Stability

If a shock moves inflation expectations away from a long run equilibrium, do they revert to

equilibrium $($in which case expectations ultimately stabilize$)?$

Do the values of parameters matter?

To assess this question, take two values of the growth rate of money: $=0\mathrm{.}357$ and $=2\mathrm{.}153.$

Start with $=0\mathrm{.}357.$ Draw a chart with $3$ panels, each for a different value of $($namely $=0\mathrm{.}2,$

$=0\mathrm{.}5,=1(.$ In each of the $3$ panels show the evolution of inflation and expected inflation after

a shock that puts hat$({)}_t^e=1.$

Then do the same thing for $=2\mathrm{.}153.$

What message emerges from you analysis regarding the policy of financing government spending

with money creation?