## 30 May 2015

### Two Views of Price Determinacy

Intro:

All the following views of price determination revolve around the consolidated government budget constraint that combines the actions of the central bank and the fiscal authority into one equation. The price level is determined by a combination of fiscal and monetary policy which can result in unconventional effects.

The Model:

The fiscal and monetary authority collectively obey a budget constraint:
$$(1) \: B_t + P_t s_t = R_{t-1}B_{t-1}$$
where $B_t$ is the total stock of government liabilities (including money and government bonds), $s_t$ is the lump sum tax levied by the government, $P_t$ is the price level, and $R_t$ is the gross nominal interest rate on government debt.

The central bank targets the price level and sets $R_t$ according to a policy rule:
$$(2) \: R_t = \frac {P_t^\phi}{\beta}$$
$\phi$ measures is the central banks' reaction to an off-target price level and $\beta$ is the factor at which future actions are discounted by agents.

The fiscal authority targets a level of nominal debt by setting $s_t$ according to a policy rule:
$$(3) \: s_t = \left(\frac {1}{\beta} - 1\right) \bar B + \phi^f (B_t - \bar B)$$
where $\bar B$ is the target level of debt and $\phi^f$ measures the fiscal authority's reaction to an off-target debt level.

The nominal interest rate is related to $\beta$ by the Fisher Equation (which is technically a bond pricing equation drawn from the first order conditions of the consumers maximization problem):
$$(4) \: R_t \beta = \frac {E_t P_{t+1}}{P_t}$$

Monetary and fiscal policy can either be active or passive. Monetary policy is active when $\phi > 1$ and passive when $\phi < 1$. Fiscal policy is active when $\phi^f$ is positive and passive when $phi^f \leq 0$. In order for the price level to be determined, one or both of the government policy instruments must be active.

View One: Neo-Fisherian:

Neo-Fisherians contend that the fiscal authority responds only passively to government debt, so when the nominal interest rate is increased above the rate suggested by the policy rule, the amount of government debt will increase with no fiscal response. This forces the inflation rate to increase. The mechanism by which this occurs is best illustrated by iterating the budget constraint forward in time and dividing by the price level:
$$(5) \: \frac {B_{t-1}}{P_t} = E_t \sum_{j=0}^{\infty} \beta^{j} s_{t+j}$$
The increase in government liabilities, without any fiscal policy response, causes the price level to rise.

View Two: Everyone Else:

For the conventional wisdom ($\uparrow R_t$ causes $\downarrow P_t$) to be the case, the fiscal authority needs to commit to commit to have active fiscal policy ($\phi^f > 0$). When the central bank raises the nominal interest rate and the amount of government debt in (5) goes up, the right side of the equation goes too, and the price level falls.

Alternatively, the total stock of government liabilities can be set to zero forever, as in New Keynesian models, changes in the nominal interest rate have no fiscal effects, so the price level is determined by only two equations:
$$(1a)\: R_t \beta = \frac {E_t P_{t+1}}{P_t}$$
$$(2a)\: R_t = \frac {P_t^\phi}{\beta}$$

Conclusion:

Different views on how the price level is determined depend on how monetary and fiscal policy are seen to be interacting. On one hand, if monetary and fiscal policy are active, then monetary policy effectively has fiscal backing. On the other hand, if fiscal policy is passive, then the Neo-Fisherian hypothesis is correct. A third possible equilibrium is one with passive monetary policy and active fiscal policy. This is a fiscal theory of the price level regime that may have occurred between 1950 and 1970 in the US.

The monetary-fiscal regime that a country is in at a given moment can have an important impact on the price level. For example, the US is likely once again in an active fiscal/passive monetary regime, so fiscal policy plays an important role in the determination of the price level.