Read the last post before you read this one; this post builds off of the analysis from that one.

In my analysis, I did not model fiscal policy because I was specifically writing about monetary policy ineffectiveness. Nevertheless, fiscal policy could be used to pin down an equilibrium price level when monetary policy can't. To begin with, let's take the budget constraint from the last post:

$$(1a)\: M_{t-1} + (1+i_{t-1})B_{t-1} + P_t y = P_t c_t + B_t + M_t + T_t $$

Since the household sets $c_t$ equal to $y$, it is possible to rewrite the household's budget constraint as the government's budget constraint:

$$(1b)\: M_{t-1} + (1+i_{t-1})B_{t-1} = B_t + M_t + T_t $$

We can also plug in the consumption Euler equation (equation $5$ in the last post) to express the budget constraint without the nominal interest rate

$$(1c)\: M_{t-1} + \left(\frac{1}{\beta} \frac{P_t}{P_{t-1}}\right)B_{t-1} = B_t + M_t + T_t $$

Assuming that the zero lower bound is binding, the budget constraint can be further reduced to

$$(1d)\: M_{t-1} + B_{t-1} = B_t + M_t + T_t$$

With this budget constraint, it is possible to determine an equilibrium price level from the specification of fiscal policy and monetary policy. Essentially, the government can monopolize on four sources of revenue: issuing government bonds, increasing the supply of money, levying taxes, and inflation. To understand why inflation can be used for revenue, it is useful to divide $1d$ by the price level and get all the variables in real quantities (lower case letters indicated real variables, except for $T_t$ which is changed to $\tau_t$):

$$(1e)\: m_{t-1}\left(\frac{P_{t-1}}{P_t}\right) + b_{t-1}\left(\frac{P_{t-1}}{P_t}\right) = b_t + m_t + \tau_t$$

In order to make $1e$ true, the government can obviously either increase $b_t$, $m_t$, or $\tau_t$. Alternatively, it can increase $P_t$, which will reduce the value of the entire left side of the budget constraint. Because of this, the fiscal authority can essentially choose to be irresponsible and the monetary authority will be forced to comply. Normally this means that the central bank must increase the growth rate of the money supply, but because of the zero lower bound, revenue from the central bank can either be more money or more inflation. Assuming the fiscal authority promises to not pay its debts (technically, this is called non-ricardian fiscal policy), only a money supply growth rule is needed to determine the price level. Given the money growth rule, all fiscal policy has to do is be just irresponsible enough to push inflation onto target.

From equations $4a$ and $4b$ in the last post, we know that $P_t/P_{t-1} = M_t/M_{t-1}$ if $P_t/P_{t-1} > \beta$ and that $P_t/P_{t-1} \leq M_t/M_{t-1}$ if $P_t/P_{t-1} = \beta$. This means that the growth rate of the money supply always represents an upper bound for the rate of inflation. Because of this, it makes sense for the central bank to grow the money supply at exactly the desired rate of inflation throughout the liquidity trap. This way, when the fiscal authority switches to non-recardian policy, the inflation rate has an upper bound and when the inflation rate goes up and the cash-in-advance constraint binds again monetary policy doesn't have to change.

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