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Currently studying Intro to Macroeconomics, and faced Expectations augmented PC. In my textbook it is not clear how expected inflation affects unemployment. My view so far, is that expected inflation move the Phillips Curve upwards. For the same rate of Unemployment corresponds greater inflation in the short-run. However, unexpected iflation reduces unemployment in the short-run until labour market and companies absorb the unexpected increase of inflation.

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In a New Keynesian framework, expectation-augmented Philips curve is the following:

$$ \pi_t = \beta \;\mathbb{E}\pi_{t+1} + \alpha x_t$$ where $\beta$ and $\alpha$ are constant, $\pi_t$ stands for the inflation rate (actually long-linear approximation of it but doesn't matter intuitively) at time $t$, $\mathbb{E}\pi_{t+1}$ is inflation rate at time $t+1$ anticipated at time $t$, and $x_t$ is the output gap -gap between natural level of output at time $t$ and the actual output at time $t$-.

As the equation above suggests, provided a constant $\pi_t$, an increase in the exptected inflation at time $t+1$ leads to a decrease in the output gap at time t, in other words, total production at time goes down -so unemployment increases-. In this model to obtain a relation between expectations and the current economic decisions, we make some assumptions on technology and preferences. In particular, we assume that firms set nominal prices based on the expectations of future marginal costs, thus output gap today depends on marginal cost of production at the current period and at the following periods.

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