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