Phillips curve

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 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 (${\displaystyle g^W}$). Here and below, the operator ${\displaystyle g}$ is the equivalent of "the percentage rate of growth of" the variable that follows.

${\displaystyle gW=g{W}^T-f(U)}$

The money-wage rate (${\displaystyle U}$) is the wage cost per employee, including benefits and payroll taxes. The focus is on only production workers' money wages, because these costs impact firm pricing choices. The growth of money wages rises with the trend rate of growth of money wages, indicated by the superscript ${\displaystyle T}$, and falls with the unemployment rate, ${\displaystyle U}$. The function ${\displaystyle f}$ is assumed to be monotonically increasing with ${\displaystyle U}$ so that the dampening of money-wage increases by unemployment is shown by the negative sign in the equation above.

There are several possible motivations for the relationship outlined in the above 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^ex). This produces the expectations-augmented wage Phillips curve:

${\displaystyle gW=g{W}^T-f(U)+\lambda g{P}^{\text{ex}}.}$

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 ${\displaystyle \lambda }$ (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 ${\displaystyle f(\cdot )}$ 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:

Here, U* is the NAIRU. As discussed below, if ${\displaystyle U<U^{\ast }}$, inflation tends to accelerate. Similarly, if ${\displaystyle U>U^{\ast }}$, inflation tends to slow. It is assumed that ${\displaystyle f(0)=0}$, so that when ${\displaystyle U=U^{\ast }}$, then ${\displaystyle f}$ is eliminated.

In equation (1), the roles of ${\displaystyle g{W}^T}$ and ${\displaystyle g{P}^{ex}}$ seem to be redundant, playing much the same role. However, assuming that ${\displaystyle \lambda }$ 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 ${\displaystyle U}$ equals ${\displaystyle U^{\ast }}$ implies that ${\displaystyle gW}$ 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:

Under assumption (2), when ${\displaystyle U}$ equals ${\displaystyle U^{\ast }}$ and ${\displaystyle \lambda }$ 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:

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, ${\displaystyle M}$ 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^T.

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^ex.

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^ex + 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^ex. 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 detailed 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 area to examine 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:

${\displaystyle Y=Y_n+a(P-P_e)}$

where ${\displaystyle Y}$ is log value of the actual output, ${\displaystyle Y_n}$ is log value of the "natural" level of output, ${\displaystyle a}$ is a positive constant, ${\displaystyle P}$ is log value of the actual price level, and ${\displaystyle P_e}$ is log value of the expected price level. Lucas assumes that ${\displaystyle Y_n}$ 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 Keynes, General Theory.

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:

${\displaystyle P=P_e+\frac{Y-Y_n}{a}}$

Next we add exogenous shocks to the world supply ${\displaystyle v}$:

${\displaystyle P=P_e+\frac{Y-Y_n}{a}+v}$

Subtracting last year's price levels ${\displaystyle P_{-1}}$ will give us inflation rates, because

${\displaystyle P-P_{-1}\approx \pi }$

and

${\displaystyle P_e-P_{-1}\approx {\pi }_e}$

where ${\displaystyle \pi }$ and ${\displaystyle {\pi }_e}$ 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

${\displaystyle \frac{Y-Y_n}{a}=-b(U-U_n)}$

where ${\displaystyle b}$ is a positive constant, ${\displaystyle U}$ is unemployment, and ${\displaystyle U_n}$ is the natural rate of unemployment or NAIRU, we arrive at the final form of the short-run Phillips curve:

${\displaystyle \pi ={\pi }_e-b(U-U_n)+v.}$

This equation, plotting inflation rate ${\displaystyle \pi }$ against unemployment ${\displaystyle U}$ 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, and since been used in most state-of-the-art New Keynesian DSGE models like the one of Clarida, Galí, and Gertler (2000).

${\displaystyle {\pi }_t=\beta E_t[{\pi }_{t+1}]+\kappa y_t}$

where

${\displaystyle \kappa =\frac{\alpha [1-(1-\alpha )\beta ]\varphi }{1-\alpha }.}$

Here ${\displaystyle \beta }$ is the discount factor, ${\displaystyle \varphi }$ is a value depending on the relationship between output and real marginal cost, and ${\displaystyle \alpha }$ is the probability a firm cannot adjust prices in a Calvo sticky price model.

The current expectations of next period's inflation are incorporated as ${\displaystyle \beta E_t[{\pi }_{t+1}]}$.

NAIRU and rational expectations

In the 1970s, new theories, such as rational expectations and the NAIRU (non-accelerating inflation rate of unemployment) arose to explain how stagflation could occur. The latter theory, also known as the "natural rate of unemployment", distinguished between the "short-term" Phillips curve and the "long-term" one. The short-term Phillips curve looked like a normal Phillips curve but shifted in the long run as expectations changed. In the long run, only a single rate of unemployment (the NAIRU or "natural" rate) was consistent with a stable inflation rate. The long-run Phillips curve was thus vertical, so there was no trade-off between inflation and unemployment. Friedman in 1976 and Phelps in 2006 won the Nobel Prize in Economics, partially for this work. However, the expectations argument was in fact very widely understood (albeit not formally) before Friedman's and Phelps's work on it.

In the diagram, the long-run Phillips curve is the vertical red line. The NAIRU theory says that when unemployment is at the rate defined by this line, inflation will be stable. However, in the short-run policymakers will face an inflation-unemployment rate trade-off marked by the "Initial Short-Run Phillips Curve" in the graph. Policymakers can, therefore, reduce the unemployment rate temporarily, moving from point A to point B through expansionary policy. However, according to the NAIRU, exploiting this short-run trade-off will raise inflation expectations, shifting the short-run curve rightward to the "new short-run Phillips curve" and moving the point of equilibrium from B to C. Thus the reduction in unemployment below the "Natural Rate" will be temporary, and lead only to higher inflation in the long run.

Since the short-run curve shifts outward due to the attempt to reduce unemployment, the expansionary policy ultimately worsens the exploitable trade-off between unemployment and inflation. That is, it results in more inflation at each short-run unemployment rate. The name "NAIRU" arises because with actual unemployment below it, inflation accelerates, while with unemployment above it, inflation decelerates. With the actual rate equal to it, inflation is stable, neither accelerating nor decelerating. One practical use of this model was to explain stagflation, which confounded the traditional Phillips curve.

The rational expectations theory said that expectations of inflation were equal to what actually happened, with some minor and temporary errors. This, in turn, suggested that the short-run period was so short that it was non-existent: any effort to reduce unemployment below the NAIRU, for example, would immediately cause inflationary expectations to rise and thus imply that the policy would fail. Unemployment would never deviate from the NAIRU except due to random and transitory mistakes in developing expectations about future inflation rates. In this perspective, any deviation of the actual unemployment rate from the NAIRU was an illusion.

However, in the 1990s in the US, it became increasingly clear that the NAIRU did not have a unique equilibrium and could change in unpredictable ways. In the late 1990s, the actual unemployment rate fell below 4% of the labor force, much lower than almost all estimates of the NAIRU. But inflation stayed very moderate rather than accelerating. So, just as the Phillips curve had become a subject of debate, so did the NAIRU.

Furthermore, the concept of rational expectations had become subject to much doubt when it became clear that the main assumption of models based on it was that there exists a single (unique) equilibrium in the economy that is set ahead of time, determined independently of demand conditions. The experience of the 1990s suggests that this assumption cannot be sustained.

Theoretical implications

To Milton Friedman there is a short-term correlation between inflation shocks and employment. When an inflationary surprise occurs, workers are fooled into accepting lower pay because they do not see the fall in real wages right away. Firms hire them because they see the inflation as allowing higher profits for given nominal wages. This is a movement along the Phillips curve as with change A. Eventually, workers discover that real wages have fallen, so they push for higher money wages. This causes the Phillips curve to shift upward and to the right, as with B. Some research underlines that some implicit and serious assumptions are actually in the background of the Friedmanian Phillips curve. This information asymmetry and a special pattern of flexibility of prices and wages are both necessary if one wants to maintain the mechanism told by Friedman. However, as it is argued, these presumptions remain completely unrevealed and theoretically ungrounded by Friedman.

Gordon's triangle model

Robert J. Gordon of Northwestern University has analyzed the Phillips curve to produce what he calls the triangle model, in which the actual inflation rate is determined by the sum of

1. demand pull or short-term Phillips curve inflation,
2. cost push or supply shocks, and
3. built-in inflation.

The last reflects inflationary expectations and the price/wage spiral. Supply shocks and changes in built-in inflation are the main factors shifting the short-run Phillips curve and changing the trade-off. In this theory, it is not only inflationary expectations that can cause stagflation. For example, the steep climb of oil prices during the 1970s could have this result.

Changes in built-in inflation follow the partial-adjustment logic behind most theories of the NAIRU:

1. Low unemployment encourages high inflation, as with the simple Phillips curve. But if unemployment stays low and inflation stays high for a long time, as in the late 1960s in the U.S., both inflationary expectations and the price/wage spiral accelerate. This shifts the short-run Phillips curve upward and rightward, so that more inflation is seen at any given unemployment rate. (This is with shift B in the diagram.)
2. High unemployment encourages low inflation, again as with a simple Phillips curve. But if unemployment stays high and inflation stays low for a long time, as in the early 1980s in the U.S., both inflationary expectations and the price/wage spiral slow. This shifts the short-run Phillips curve downward and leftward, so that less inflation is seen at each unemployment rate.

In between these two lies the NAIRU, where the Phillips curve does not have any inherent tendency to shift, so that the inflation rate is stable. However, there seems to be a range in the middle between "high" and "low" where built-in inflation stays stable. The ends of this "non-accelerating inflation range of unemployment rates" change over time.