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

29

As in Brainard and Perry (2000) the results just described cast doubt on conventional

estimates with the NAIRU model.  However they both treat expectations as adaptive and so

cannot refute Sargent's (1971) criticism that rational expectations are formed differently and that

the coefficient on properly measured expectations might be 1.0.  We now address this issue by

using direct measures of expected inflation as explanatory variables in place of distributed lags of

actual inflation rates, while maintaining our division of the sample into periods of high and low

inflation.  The other explanatory variables are the same as those used in the regressions behind

figures 3 and 4.  We used the two direct measures of expected rates of inflation that are available

over our sample period: one from the Survey of Consumer Finances and the other the Federal

Reserve’s Livingston Surveys.  Figures 5 and 6 plot the estimated coefficients on expected

inflation for the variously specified wage and price regressions respectively.  As with the results

using adaptive expectations, the coefficients on expected inflation are substantially different in the

low- and high-inflation periods.  For 288 wage equations the low- and high-period means are 0.29

and 0.85.  For 144 price equations the means are 0.25 and 1.00.

Figures 5 and 6 about here

These results support our general hypothesis even more convincingly than the results with

adaptive expectations. Not only do they address the point that the relevant coefficient for natural

rate theory is not necessarily the coefficient estimated with adaptive expectations, but the results

are as clear about price inflation as they are about wage inflation. 

One possible objection to the results presented here and in the next section is that the

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lower coefficients on inflationary expectations during periods of low inflation are an artifact of

measurement error. For example, if the variance of measurement error is constant while the

variance of true inflationary expectations are higher in times of high inflation, then the coefficient

on expectations could be biased towards zero more in times of low inflation than high inflation.

We investigated this possibility. While it is true that the variance of expectations is higher in

periods of high inflation, it is also true that the sampling error in both the SCF and the Livingston

surveys are also higher. In fact, the sampling error is so much higher that the computed bias is

higher in the low inflation periods imparting a bias against our finding that the coefficient on

expectations is lower in periods of low inflation. Sampling error may not be the only source of

error in the survey expectations. Neither survey may be asking the right people with the right

weights. In an attempt to approximate how much error this problem might introduce we computed

the bias that would be caused if the measurement error variance in expectations was equal to the

variance of the residual of a regression of one of our survey expectations on the other. Again we

found that the “measurement error” variance grew faster than the conditional variance of the

expectations so that the bias caused would work against our finding that the coefficient on

expectations was lower when inflation was low. 

Estimating The Model   

Previously we showed how a Phillips Curve type relation can be derived from our

theoretical model (equation 15). In this section we present estimates  of the model and of  the

optimal rate of inflation and the gain in employment that is possible from moving to the optimal

rate. This section will first discuss the specification of the model we estimate, then our benchmark

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results, and finally, an analysis of their robustness.  

Specifications

In theory, with a large enough sample, it would be possible to estimate the full model

presented above.  The elasticity of demand (



), the parameter for the curvature of the unit cost

function (



), and the parameters of the distribution of rationality thresholds (



 and 

)

), all have

different effects on the objective function.  However, in practice, it was impossible to estimate

more than the mean of the distribution of rationality thresholds, and one of the other parameters

because all three of them— the elasticity of demand, the curvature of the unit cost function and

the standard deviation of the distribution of rationality thresholds—act in much the same way to

determine the impact of past rates of inflation on the cumulative normal term. (See equation (15)

above). 

The lack of identification in practice can be understood if we consider a Taylor series

approximation to the argument of the cumulative normal in equation (15) expanded around a

value of zero inflation.  There is no reason to expect that the argument will be exactly zero at zero

inflation so the constant term will likely be present.  As we have shown above, the first derivative

of the firm’s loss function with respect to inflation is zero at zero inflation and very small at most

rates of inflation less than 10 percent.  Thus the first order term of the Taylor series expansion of

the argument of the cumulative normal will also be zero.  Second and higher order terms will be

present, but analysis we have conducted of the loss function suggests that with  inflation between

zero and ten percent, with elasticity of demand between 2 and 10, with curvature of the unit cost

function from .05 to .95, and any value of the standard deviation of the distribution of rationality

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