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This is a quote from Mankiw. No further explaination is given and I'm having trouble working through the logic behind this statement.

Could someone explain the process behind this assertion?

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Mankiw probably refers there to the Expectations Augmented Philips Curve (EAPC) or some related model. According to EAPC the following relationship holds (See Romer Advanced Macro pp 259-262):

$$ \pi_t = E[\pi_t] + \beta(\ln y_t - \ln \bar{y}_t) + \epsilon^s$$

where $\pi$ is inflation, $y_t$ is output, $\bar{y}$ natural level of output and $\epsilon^s$ vector of supply shocks. Natural level of employment is attained when natural level of output is also attained so that part of a model could be interchanged (if we talk about unemployment we would just have to switch sign).

Assuming there are no supply shocks (which Mankiw seems to implicitly assume- or maybe even explicitly since you do not provide source there is no way of checking), then:

$$\frac{1}{\beta} \left(\pi_t -E[\pi_t] \right)= \ln y_t - \ln \bar{y}_t$$

If $\pi_t = E[\pi_t]$ then

$$0 = \ln y_t - \ln \bar{y_t} \implies \ln y_t = \bar{y_t} $$

so using EAPC natural rate of output and output and by extension natural level of employment and employment (or unemployment) will be different only when expected inflation is different from actual one.

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