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## The NAIRU

Here we’re going to work from two concepts, the natural rate of unemployment and the Expectations-Augmented Phillips Curve.

The natural rate of unemployment is the rate of unemployment where the real wage implied by price-setting equals the real wage implied by wage-setting. The natural rate of unemployment basically corresponds to a natural rate of employment which is associated with the amount of workers needed to produce a natural level of output in the economy (which depends on supply-side conditions in the economy). This was basically the story of the trade-off between inflation and unemployment.

When unemployment is lower than the natural rate of unemployment, then it means output is above the natural rate of output, and more workers need to be hired to produce the extra output.

Because unemployment is now lower than the natural rate, it means workers are becoming more scarce, so firms need to pay more wages to hire them and keep them from moving on to rivals. The labour market becomes more of a ‘sellers’ market – the sellers of labour (workers) have the advantage because workers are scarce and jobs are abundant, so wages are bid up. This leads to an increase in firms’ costs, which are passed on to consumers through the price-setting equation. The increase in prices then leads to an increase in expectations of future prices being higher, which will transfer into higher wage demands at the next round of wage bargaining. So the wage-price spiral causes prices to rise faster.

When unemployment is higher than the natural rate, you have the opposite situation, it means output is below the natural level of output and fewer workers are needed. The labour market is a ‘buyers’ market, the buyers of labour (firms) have the advantage because jobs are scarce and unemployed workers are abundant. This will drive down wages and so firms can pay only small wage increases or maybe no increases at all, which keeps their costs from rising and allows them to keep prices lower (to compete with their rivals). This will act as a brake on inflation, so prices will either be stable, rise slowly (more commonly), or if unemployment is very high compared to normal, prices can be driven lower (deflation).

So we have a basic story:
– When unemployment is below the natural rate of unemployment, inflation will rise.
– When unemployment is above the natural rate of unemployment, inflation will fall.

So that implies that when we are at the natural rate of unemployment, inflation will stay constant. This is why the natural rate of unemployment is often called the Non-Accelerating Inflation Rate of Unemployment, or the NAIRU.

We can derive an expression for the natural rate of unemployment from the Expectations-Augmented Phillips Curve: $\pi_t - \pi_{t-1} = (\mu + z) - \alpha u_t$

When inflation is staying constant from one year to the next, then $\pi_t - \pi_{t-1} = 0$ so $0 = (\mu + z) - \alpha u_n \Rightarrow \frac{(\mu + z)}{\alpha} = u_n$

I am using the notation $u_n$, the natural rate of unemployment, for the case where we have a specific $u_t$ such that inflation does not change. The equation above tells us that the NAIRU depends on the profit mark-up, the ‘catch-all’ capturing conditions in the labour market, and the parameter which shows the responsiveness of inflation to unemployment in that economy.

If we have figures over time for an economy’s inflation and unemployment rates, we can form an estimate as to the specific rate of unemployment that would make inflation stable, and so get an estimate of the NAIRU for that economy. Of course if we observe data over a long time period we may find that our estimate of the NAIRU changes, which suggests that supply conditions in the economy are different, ie those parameters above are changing.

$u_n = \frac{(\mu + z)}{\alpha} \Rightarrow \alpha u_n = \mu + z$ so we can substitute this in to $\pi_t - \pi_{t-1} = (\mu + z) - \alpha u_t$ to get $\pi_t - \pi_{t-1} = \alpha u_n - \alpha u_t$.

This can be simplified to $\pi_t - \pi_{t-1} = \alpha (u_n - u_t)$ which we can flip round to give $\pi_t - \pi_{t-1} = -\alpha (u_t - u_n)$.

This is probably the most common and useful form of the Phillips Curve equation. It says that the change in inflation is equal to a negative parameter multiplied by the difference between unemployment in year t, and the natural rate of unemployment.