Near-Rational Wage and Price Setting and the Optimal Rates of Inflation and Unemployment

7

The price Phillips Curve will be of the form:

%

 = c - e u + f 

%

e 

+ (1 - f) h 

u

e

,

where h = -C/[b(1 - 



)], u is current unemployment, and 

u

e

 is the expected change in

unemployment.

20

(15)

where f =  (1 - a) (1 - 

0

).

A price Phillips Curve, which is similar to (15), can also be derived from the model.  The

slight difference between the price Phillips Curve implied by our model and the wage Phillips

Curve (15) is the presence of a change in unemployment term in the price Phillips Curve. This

term enters because changes in the unemployment rate will cause changes in productivity and

hence, via (6), in the price/wage markups.

7

  We take this into account when we estimate the model

by allowing lags on the unemployment rate.  The steady state Phillips Curves with constant

unemployment will be unaffected by  varying markups caused by varying unemployment.  

The short-run Phillips Curve (15) should come as no surprise.  If all inflation had been

included in the mental frames of the firms, which are setting wages and prices in this model, then

the coefficient f would be equal to zero. The near-rational firms, which constitute a fraction 1 -  

0

of all of the firms, ignore a fraction (1 - a) of inflation.  As a consequence, the Phillips Curve (15)

mimics the usual inflation-augmented Phillips Curve, but with a fraction (1 - a)(1 - 

0

) of the

expected inflation ignored. Thus the Phillips Curve of the form (15) is not just an artifact of our

illustrative model of price and wage setting. As long as a fraction of inflation is ignored or under-

weighted in near-rational wage and price setting, that fraction of inflation should fail to enter the

inflation augmentation term.  A whole spectrum of other models in which various combinations

---

21

(16)

of firms and workers are ignoring or underweighting inflation in their mental frames will yield

similar results.

Using (15), the long-term Phillips Curve—where actual and expected inflation are equal—

will be:

where u

n

 is the natural rate of unemployment if all firms are rational.  Its value in this model is

d/e.

The Phillips Curve (16) will be bowed out and then  forward bending.  At zero inflation 

%

is zero and therefore unemployment is at the natural rate.  At very high inflation all firms will

have given up being near-rational. The losses from near-rational behavior will be sufficiently

large that by (10), 

0

 will be close to one—so that f, which is (1 -

0

)(1 - a), will be close to zero. 

Thus at both very high and very low inflation unemployment will be close to the natural rate,

which is the level of unemployment that would occur if all firms were totally rational.  At

inflation above zero, unemployment will always be below the natural rate since f will always be

positive.

Figure 1 portrays the rate of unemployment that corresponds to different levels of inflation

in the long run with bench-mark parameters. We have assumed that near-rational firms completely

ignore inflation (a=0). We chose the parameters describing the distribution of 

0

 so that at least ½

of all firms are always fully rational ( thus 



 

  is zero), and 95 percent of all firms are rational by

the time inflation is 5 percent (which implied a value for 

)

 

 of .002 or .2% of normal profits). We

---

8

Interestingly, our choices of the values of the elasticity of demand (

 

), and the curvature of the productivity function

(

¡

), hardly matter for the shape of the curve in figure 1 or for the optimal rate of inflation and unemployment. Once

we set the fraction of firms that are near-rational at two points we have described the curve for a given value of a. 

This result reflects a finding that will surface again later when we estimate the model, which is discussed in more

detail in the next section—the loss function is very nearly approximated by a constant times the square of inflation so

that the argument of the cumulative normal in our model can be very well approximated with two parameters. 

22

also chose 



 at .1 and an elasticity of demand (



) of four,  though as we will discuss below, these

assumptions hardly matter at all for the shape of figure 1. 

The optimal rate of inflation is the level that maximizes the product of f and 

%

.  This level

of inflation, according to (16) will minimize unemployment. For the parameter values chosen to

create figure 1 that inflation rate is 2.6%. At that rate of inflation the long-run equilibrium rate of

unemployment is 1.7 percentage points lower than at either a rate of inflation of zero or a rate

above 6 percent.

8

[Figure 1 about here]

Why does employment rise with  inflation at low rates of inflation? In our model, inflation

is not underestimated, but instead it is under weighted in the reference wage used for wage setting. 

This has the same consequences as underestimation. Near-rational firms either ignore or fail to

fully project inflation so they set lower wages, and therefore also set lower prices, relative to

nominal demand, than they would if they were fully rational. At these lower prices both output

and employment will be higher. These higher levels would also occur in the slightly different

version of the model in which workers’ underestimate the impact of inflation.

In our model the level of inflation that yields the minimum obtainable unemployment rate

---