Home > Macro, Monetary Policy > An equation linking inflation, output growth, unemployment and money growth

An equation linking inflation, output growth, unemployment and money growth

Here we are going to make a single equation out of three other expressions:

1. The Phillips Curve which linked inflation to unemployment: \pi_t - \pi_{t-1} = \alpha (u_n - u_t).

2. Okun’s Law which linked unemployment to output: u_t - u_{t-1} = -\beta (g_{yt} - \bar{g_y})

3. The relationship between nominal money growth, output growth and inflation: g_{yt}=g_{mt}-\pi_t

You can combine the first two to get:

\pi_t - \pi_{t-1} = -\alpha (u_{t-1} - u_n - \beta (g_{yt} - \bar{g_y})).

Now you can substitute the value for output growth given in the third equation into that to get:

\pi_t - \pi_{t-1} = -\alpha (u_{t-1} - u_n - \beta (g_{mt}-\pi_t - \bar{g_y})).

Now for some multiplying out of whats in the brackets:

\pi_t - \pi_{t-1} = -\alpha (u_{t-1} - u_n - \beta g_{mt} + \beta \pi_t + \beta \bar{g_y}).
\Rightarrow \pi_t - \pi_{t-1} = -\alpha u_{t-1} +\alpha u_n + \alpha \beta g_{mt} - \alpha \beta \pi_t - \alpha \beta \bar{g_y}.
\Rightarrow \pi_t + \alpha \beta \pi_t - \pi_{t-1} = -\alpha u_{t-1} +\alpha u_n + \alpha \beta g_{mt} - \alpha \beta \bar{g_y}.
\Rightarrow \pi_t (1 + \alpha \beta ) - \pi_{t-1} = -\alpha u_{t-1} +\alpha u_n + \alpha \beta g_{mt} - \alpha \beta \bar{g_y}.
\Rightarrow \pi_t  = \frac{1}{(1 + \alpha \beta )}[\pi_{t-1} -\alpha u_{t-1} +\alpha u_n + \alpha \beta g_{mt} - \alpha \beta \bar{g_y}].

We can put the right hand side back in brackets to get:
\pi_t  = \frac{1}{(1 + \alpha \beta )}[\pi_{t-1} -\alpha (u_{t-1} - u_n - \beta (g_{mt} - \bar{g_y}))]

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Categories: Macro, Monetary Policy
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