Editing Phillips curve (section)

==Mathematics==
There are at least two different mathematical derivations of the Phillips curve. First, there is the traditional or [[Keynesian]] version. Then, there is the new Classical version associated with [[Robert E. Lucas Jr.]]

===The traditional Phillips curve===
The original Phillips curve literature was not based on the unaided application of economic theory. Instead, it was based on empirical generalizations. After that, economists tried to develop theories that fit the data.

====Money wage determination====
The traditional Phillips curve story starts with a wage Phillips curve, of the sort described by Phillips himself. This describes the rate of growth of money wages (''gW''). Here and below, the operator ''g'' is the equivalent of "the percentage rate of growth of" the variable that follows.
:<math>gW = gW^T - f(U)</math>
The "money wage rate" (''W'') is shorthand for total money wage costs per production employee, including benefits and payroll taxes. The focus is on only production workers' money wages, because (as discussed below) these costs are crucial to pricing decisions by the firms.

This equation tells us that the growth of money wages rises with the trend rate of growth of money wages (indicated by the superscript ''T'') and falls with the unemployment rate (''U''). The function ''f'' is assumed to be [[Monotonic function|monotonically]] increasing with ''U'' so that the dampening of money-wage increases by unemployment is shown by the negative sign in the equation above.

There are several possible stories behind this equation. A major one is that money wages are set by ''bilateral negotiations'' under partial [[bilateral monopoly]]: as the unemployment rate rises, ''all else constant'' worker bargaining power falls, so that workers are less able to increase their wages in the face of employer resistance.

During the 1970s, this story had to be modified, because (as the late [[Abba Lerner]] had suggested in the 1940s) workers try to keep up with inflation. Since the 1970s, the  equation has been changed to introduce the role of inflationary expectations (or the expected inflation rate, ''gP''<sup>ex</sup>). This produces the expectations-augmented wage Phillips curve:

:<math>gW = gW^T - f(U) + \lambda gP^\text{ex}.</math>

The introduction of inflationary expectations into the equation implies that actual inflation can ''feed back'' into inflationary expectations and thus cause further inflation. The late economist [[James Tobin]] dubbed the last term "inflationary inertia", because in the current period, inflation exists which represents an inflationary impulse left over from the past.

It also involved much more than expectations, including the price-wage spiral. In this spiral, employers try to protect profits by raising their prices and employees try to keep up with inflation to protect their real wages. This process can feed on itself, becoming a self-fulfilling prophecy.

The parameter '''λ''' (which is presumed constant during any time period) represents the degree to which employees can gain money wage increases to keep up with expected inflation, preventing a fall in expected real wages. It is usually assumed that this parameter equals 1 in the long run.

In addition, the function '''f'''() was modified to introduce the idea of the [[non-accelerating inflation rate of unemployment]] (NAIRU) or what's sometimes called the "natural" rate of unemployment or the
inflation-threshold unemployment rate:
{{NumBlk|:|<math>gW = gW^T-f(U-U^*)+\lambda gP^\text{ex}.</math>|{{EquationRef|1}}}}
Here, ''U*'' is the NAIRU. As discussed below, if ''U'' < ''U''*, inflation tends to accelerate. Similarly, if ''U'' > ''U''*, inflation tends to slow. It is assumed that ''f''(0) = 0, so that when ''U'' = ''U''*, the ''f'' term drops out of the equation.

In equation ({{EquationNote|1}}), the roles of  '''gW<sup>T</sup>''' and '''gP<sup>ex</sup>''' seem to be redundant, playing much the same role. However, assuming that '''λ'''  is equal to unity, it can be seen that they are not. If the trend rate of growth of money wages equals zero, then the case where '''U''' equals '''U*''' implies that '''gW''' equals expected inflation. That is, expected real wages are constant.

In any reasonable economy, however, having constant expected real wages could only be consistent with actual real wages that are constant over the long haul.  This does not fit with economic experience in the U.S. or any other major industrial country. Even though real wages have not risen much in recent years, there have been important increases over the decades.

An alternative is to assume that the trend rate of growth of money wages equals the trend rate of growth of average labor productivity ('''Z'''). That is:
{{NumBlk|:|<math>gW^T=gZ^T.</math>|{{EquationRef|2}}}}

Under assumption ({{EquationNote|2}}), when '''U''' equals '''U*''' and '''λ''' equals unity, expected real wages would increase with labor productivity. This would be consistent with an economy in which actual real wages increase with labor productivity. Deviations of real-wage trends from those of labor productivity might be explained by reference to other variables in the model.

====Pricing decisions====
Next, there is price behavior. The standard assumption is that markets are ''imperfectly competitive'', where most businesses have some power to set prices. So the model assumes that the average business sets a unit price ('''P''') as a mark-up ('''M''') over the [[unit labor cost]] in production measured at a standard rate of capacity utilization (say, at 90 percent use of plant and equipment) and then adds in the unit materials cost.

The standardization involves later ignoring deviations from the trend in labor productivity. For example, assume that the growth of labor productivity is the same as that in the trend and that current productivity equals its trend value:

: '''gZ''' = '''gZ<sup>T</sup>''' and '''Z''' = '''Z<sup>T</sup>'''.

The markup reflects both the firm's degree of market power and the extent to which overhead costs have to be paid. Put another way, all else equal, '''M''' rises with the firm's power to set prices or with a rise of overhead costs relative to total costs.

So pricing follows this equation:

:'''P''' = '''M''' × ('''unit labor cost''') + '''(unit materials cost)'''

:: = '''M''' × ('''total production employment cost''')/('''quantity of output''') + '''UMC'''.

'''UMC''' is unit raw materials cost (total raw materials costs divided by total output). So the equation can be restated as:

:'''P ''' = ''' M ''' × ('''production employment cost per worker''')/('''output per production employee''') + ''' UMC'''.

This equation can again be stated as:

:'''P''' = '''M'''×('''average money wage''')/('''production labor productivity''') + '''UMC'''

:: = '''M'''×('''W'''/'''Z''') + '''UMC'''.

Now, assume that both the average price/cost mark-up ('''M''') and '''UMC''' are constant. On the other hand, labor productivity grows, as before.  Thus, an equation determining the price inflation rate ('''gP''') is:

: '''gP''' = '''gW''' − '''gZ<sup>T</sup>'''.

====Price====
Then, combined with the wage Phillips curve [equation 1] and the assumption made above about the trend behavior of money wages [equation 2], this price-inflation equation gives us a simple expectations-augmented price Phillips curve:

: '''gP''' = −'''f'''('''U''' − '''U*''') + '''λ'''·'''gP<sup>ex</sup>'''.

Some assume that we can simply add in '''gUMC''', the rate of growth of '''UMC''', in order to represent the role of supply shocks (of the sort that plagued the U.S. during the 1970s). This produces a standard short-term Phillips curve:

: '''gP''' = −'''f'''('''U''' − '''U*''')  + '''λ'''·'''gP<sup>ex</sup>''' + '''gUMC'''.

Economist [[Robert J. Gordon]] has called this the "Triangle Model" because it explains short-run inflationary behavior by three factors: demand inflation (due to low unemployment), supply-shock inflation ('''gUMC'''), and inflationary expectations or inertial inflation.

In the ''long run'', it is assumed, inflationary expectations catch up with and equal actual inflation so that '''gP''' = '''gP<sup>ex</sup>'''. This represents the long-term equilibrium of expectations adjustment. Part of this adjustment may involve the adaptation of expectations to the experience with actual inflation. Another might involve guesses made by people in the economy based on other evidence. (The latter idea gave us the notion of so-called [[rational expectations]].)

Expectational equilibrium gives us the long-term Phillips curve. First,  with '''λ''' less than unity:

:'''gP''' = [1/(1 − '''λ''')]·(−'''f'''('''U''' − '''U*''') + '''gUMC''').

This is nothing but a steeper version of the short-run Phillips curve above. Inflation rises as unemployment falls, while this connection is stronger. That is, a low unemployment rate (less than '''U*''') will be associated with a higher inflation rate in the long run than in the short run. This occurs because the actual higher-inflation situation seen in the short run feeds back to raise inflationary expectations, which in turn raises the inflation rate further. Similarly, at high unemployment rates (greater than '''U*''') lead to low inflation
rates. These in turn encourage lower inflationary expectations, so that inflation itself drops again.

This logic goes further if '''λ''' is equal to unity, i.e., if workers are able to protect their wages ''completely'' from expected inflation, even in the short run. Now, the Triangle Model equation becomes:

:- '''f'''('''U''' − '''U*''') = '''gUMC'''.

If we further assume (as seems reasonable) that there are no long-term supply shocks, this can be simplified to become:

: −'''f'''('''U''' − '''U*''') = 0 which implies that  '''U''' = '''U*'''.

All of the assumptions imply that in the long run, there is only one possible unemployment rate, '''U*''' at any one time. This uniqueness explains why some call this unemployment rate "natural".

To truly understand and criticize the uniqueness of '''U*''', a more sophisticated and realistic model is needed. For example, we might introduce the idea that workers in different sectors push for money wage increases that are similar to those in other sectors. Or we might make the model even more realistic. One important place to look is at the determination of the mark-up, '''M'''.

===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.

===New Keynesian version===
The New Keynesian Phillips curve was originally derived by Roberts in 1995,<ref>{{cite journal |last=Roberts |first=John M. |year=1995 |title=New Keynesian Economics and the Phillips Curve |journal=[[Journal of Money, Credit and Banking]] |volume=27 |issue=4 |pages=975–984 |jstor=2077783 |doi=10.2307/2077783 }}</ref> and since been used in most state-of-the-art New Keynesian DSGE models like the one of Clarida, Galí, and Gertler (2000).<ref>{{cite journal |last1=Clarida |first1=Richard |last2=Galí |first2=Jordi |last3=Gertler |first3=Mark |year=2000 |title=Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory |journal=[[The Quarterly Journal of Economics]] |volume=115 |issue=1 |pages=147–180 |doi=10.1162/003355300554692 |citeseerx=10.1.1.111.7984 }}</ref><ref>{{cite book |last=Romer |first=David |year=2012 |chapter=Dynamic Stochastic General Equilibrium Models of Fluctuation |title=Advanced Macroeconomics |publisher=McGraw-Hill Irwin |location=New York |pages=312–364 |isbn=978-0-07-351137-5 |chapter-url=https://books.google.com/books?id=xTovPwAACAAJ&pg=PA312 }}</ref>
:<math>\pi_{t} = \beta E_{t}[\pi_{t+1}] + \kappa y_{t}</math>
where 
:<math>\kappa = \frac{\alpha[1-(1-\alpha)\beta]\phi}{1-\alpha}.</math> 
The current expectations of next period's inflation are incorporated as <math>\beta E_{t}[\pi_{t+1}]</math>.