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

17

(10)

shall explore in turn.

As the first implication of (9), those who fail to maximize profits either by ignoring

inflation (a = 0 ), or taking it into account only partially (0<a<1), are near-rational.  When 

%

 is

zero the losses of such producers is zero, as can be seen by the fact that when 

%

 is zero, z is 1. 

Thus according to (9) the losses from being near-rational when z is zero will also be zero.  These

losses will also continue to be small at low levels of inflation, near zero, since the derivative of (9)

with respect to 

%

 is also zero when 

%

 is zero.  

Secondly, (9) serves as the springboard for the completion of the model we will estimate

below, which is based explicitly on the losses that are entailed from near-rational behavior.  To

complete the model it is assumed that firm wage and price setters are willing to tolerate losses

relative to their profits, only up to a given threshold, 



, before they will switch to fully rational

behavior.  We assume that these thresholds are normally distributed with mean 



 

 and standard

deviation 

)

¡

.  The fraction of near-rational price setters accordingly will then be:

where 

0

 is the standard cumulative normal distribution, and 



¢

 

and 

)

¢

 

 are respectively the mean

and standard deviation of the distribution of the thresholds 



.

Finally, (9) also yields benchmark estimates of the size of losses because of near-rational

behavior.  Table 1 shows the fraction of the profits of the fully rational firm sacrificed by the near-

rational firm at different rates of inflation for two different values of a and two different values of

both 



 and 



.  

---

18

Table 1

Percent of the Profits of a Fully Rational Firm Lost by 

Near-Rational Behavior in the Treatment of Inflation

Inflation 

Rate

  

              a = 0  

(Near-rational firms ignore inflation)

             a= .7

(Near-rational firms weight inflation)

       Elasticity of Demand (



)

     Elasticity of Demand (



)

3

10

3

10



=.1



=.75



=.1



=.75



=.1



=.75



=.1



=.75

1%

.009%

.002%

.04%

.01%

.001%

.000%

.004%

.001%

2%

.04%

.01%

.16%

.04%

.003%

.001%

.01%

.004%

3%

.08%

.02%

.36%

.10%

.007%

.002%

.03%

.01%

4%

.14%

.04%

.64%

.18%

.01%

.003%

.05%

.02%

5%

.22%

.06%

1.00%

.27%

.02%

.005%

.08%

.02%

7%

.43%

.12%

1.92%

.53%

.04%

.01%

.16%

.04%

10%

.87%

.24%

3.84%

1.06%

.07%

.02%

.31%

.09%

To put the values in table 1 in perspective, consider the findings of Leonard (1987) and

Davis et. al. (1996) that the typical firm annually experiences shocks to demand that cause it to

adjust its size up or down by roughly 10%.  Failing to adjust capacity to accommodate such a

shock would cost a firm 10% of its profits.  Thus it does not seem hard to believe that for the

typical firm, the issue of how to treat inflation in setting prices is far down the list of items

demanding managerial attention—at least with inflation under 5%. 

 

Implications for the Long-Run and the Short-Run Phillips Curve

The model also allows easy derivation of both a short-run Phillips Curve with given

expectations of price inflation and a long-run Phillips curve where actual and expected inflation

must coincide.  

---

19

(11)

(12)

(13)

(14)

The short-run wage-Phillips Curve is obtained from wage-setting behavior and the

equation for the average wage.  The average wage in this economy will be:

Using the wage setting behavior of the rational and near-rational firms,

which can be rewritten as,

using the definition of the reference wage. Dividing the left hand and the right hand side by w

*

-1

and collecting terms yields the relation:

where 

%

w

 is the rate of wage inflation. Taking the logs of both the right hand side and the left

hand side of (14), approximating ln (1 + 

%

w

) by 

%

w

, ln [1 + 

0

 

%

e

  + ( 1 - 

0

) a 

%

e

)] by [

0

 + ( 1 -

0

) a] 

%

e

, and ln  [A - Cu]/[B(1 - 



)]

1/

 

 by its linear approximation, d - e u, yields the short-run

wage-Phillips Curve:

---