Editing Phillips curve (section)

===New classical version===
The Phillips curve equation can be derived from the (short-run) [[Lucas aggregate supply function]]. The Lucas approach is very different from that of the traditional view. Instead of starting with empirical data, he started with a classical economic model following very simple economic principles.

Start with the [[aggregate supply]] function:

:<math>Y = Y_n + a (P-P_e) \, </math>

where '''''Y''''' is log value of the actual [[output (economics)|output]], <math>Y_n</math> is log value of the "natural" level of output, <math>a</math> is a positive constant, <math>P</math> is log value of the actual [[price level]], and <math>P_e</math> is log value of the expected [[price level]]. Lucas assumes that <math>Y_n</math> has a unique value.

Note that this equation indicates that when expectations of future inflation (or, more correctly, the future price level) are ''totally accurate'', the last term drops out, so that actual output equals the so-called "natural" level of real GDP.  This means that in the Lucas aggregate supply curve, the ''only'' reason why actual real GDP should deviate from potential—and the actual unemployment rate should deviate from the "natural" rate—is because of ''incorrect expectations'' of what is going to happen with prices in the future. (The idea has been expressed first by [[John Maynard Keynes|Keynes]], ''[[The General Theory of Employment, Interest and Money|General Theory]]'', Chapter 20 section III paragraph 4).

This differs from other views of the Phillips curve, in which the failure to attain the "natural" level of output can be due to the imperfection or incompleteness of markets, the stickiness of prices, and the like. In the non-Lucas view, incorrect expectations can contribute to aggregate demand failure, but they are not the only cause. To the "new Classical" followers of Lucas, markets are presumed to be perfect and always attain equilibrium (given inflationary expectations).

We re-arrange the equation into:
:<math> P = P_e + \frac{Y-Y_n}{a} </math>
Next we add unexpected exogenous shocks to the world supply <math>v</math>:
:<math> P = P_e + \frac{Y-Y_n}{a} + v </math>
Subtracting last year's price levels <math>P_{-1}</math> will give us inflation rates, because
:<math> P-P_{-1}\ \approx \pi </math>
and
:<math> P_e- P_{-1}\ \approx \pi_e </math>
where <math>\pi</math> and <math>\pi_e</math> are the [[inflation]] and expected inflation respectively.

There is also a negative relationship between output and unemployment (as expressed by [[Okun's law]]). Therefore, using
:<math>\frac{Y-Y_n}{a} = -b(U-U_n) </math>
where <math>b</math> is a positive constant, <math>U</math> is unemployment, and <math>U_n</math> is the [[natural rate of unemployment]] or [[NAIRU]], we arrive at the final form of the short-run Phillips curve:
:<math> \pi = \pi_e - b(U-U_n) + v.</math>
This equation, plotting inflation rate <math>\pi</math> against unemployment <math>U</math> gives the downward-sloping curve in the diagram that characterizes the Phillips curve.