Rational Partisan Theory

Recently, I have been intrigued by a fragment of the macroeconomics literature known as rational partisan theory. It looks into the effects of changes in the expectations of government policy that occur during elections. Usually, there are two "parties": one that wants high inflation and another that wants low inflation. Expectations of future policy are contingent upon the probability of each party being in power after the next election.

The simplest way to analyse this is to use the Fisher Equation and assume the central bank follows a rule and targets an inflation rate (both of which are unknown to the agents in the model).

First, the Fisher Equation:

$$ i_t = E_t \pi_{t+1} + \rho $$

where $i_t$ is the nominal interest rate, $\pi$ is the rate of inflation, and $\rho$ is the real interest rate. 

Second, the policy rule:

$$ i_t = \phi (\pi_t - \bar\pi) + \bar\pi + \rho $$

where $\bar\pi$ is the central bank's inflation target.

Here's where the "Partisan" part comes in. Each period, there is an $L_t$ percent chance that the left wing (higher inflation) party will be elected and an $R_t$ percent chance that the right wing (low inflation) party will be elected. So, by definition, $L_t + R_t = 1$. Suppose the right wing party wants the inflation rate to be $\pi^R$ and the left wing party wants the inflation rate to be $\pi^L$.

Since the agents don't know the central bank's policy rule or it's inflation target, there best guess for next period's inflation is:

$$ E_t \pi_{t+1} = L_t\pi^L + R_t\pi^R $$

If there is always a 50% chance that the left or right wing party will win the election, and $\pi^R = 0.01, \pi^L = 0.02$, expected inflation will always be $0.015$. 

Assume that the probability that the right wing party will win the election tomorrow evolves like this:

$$ (R_t - 0.5) = \theta (R_{t-1} - 0.5) + \epsilon_t $$

to close the model.