We explore the relationship between unemployment and inflation in the United States (1949-2019) through both Bayesian and spectral lenses. We employ Bayesian vector autoregression (“BVAR”) to expose empirical interrelationships between unemployment, inflation, and interest rates. Generally, we do find short-run behavior consistent with the Phillips curve, though it tends to break down over the longer term. Emphasis is also placed on Phelps’ and Friedman’s NAIRU theory using both a simplistic functional form and BVAR. We find weak evidence supporting the NAIRU theory from the simplistic model, but stronger evidence using BVAR. A wavelet analysis reveals that the short-run NAIRU theory and Phillips curve relationships may be time-dependent, while the long-run relationships are essentially vertical, suggesting instead that each relationship is primarily observed over the medium-term (2-10 years), though the economically significant medium-term region has narrowed in recent decades to roughly 4-7 years. We pay homage to Phillips’ original work, using his functional form to compare potential differences in labor bargaining power attributable to labor scarcity, partitioned by skill level (as defined by educational attainment). We find evidence that the wage Phillips curve is more stable for individuals with higher skill and that higher skilled labor may enjoy a lower natural rate of unemployment.

Dating back to at least 1926 (Fisher [1]), economists have pondered the relationship between inflation and unemployment. Irving Fisher, citing early statistical work, theorized inflation has a lagged, distributed impact on unemployment due to revenue flexibility and expense rigidity.

In 1958, A. W. Phillips popularized this inflation-employment association in his famous paper The Relationship between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957, detailing the response of wage growth to unemployment rates. Rather than relying on the sort of the Cantillon effect Fisher described, Phillips theorized that employers tend to bid wage rates up rapidly during periods of low unemployment. Conversely, he believed workers are hesitant to reduce wage demands when unemployment is high. The rate of change of wage rates is therefore convex to the unemployment rate. The simplicity of the model combined with the strength of his empirical results generated significant interest in the subject.

Phillips’ theoretical model was quickly generalized by numerous authors to relate movements in the broad price level (i.e., the inflation rate) to the unemployment rate, consistent with Fisher’s approach. Samuelson and Solow [2] considered the possibility of exploiting such a relationship and the policy implications that follow, which is thought to have had significant influence on policy-setting over the subsequent 20 years (this argument is made in Hart and Hall [3]). Such a tradeoff was thought to imply a metaphorical “menu” from which politicians can choose depending on the economic environment they inherit.

In 1968, Milton Friedman etched his own contributions in stone while addressing the American Economic Association (Friedman [4]). The Phillips curve tradeoff may indeed exist, but only by misleading the general public who set their own expectations regarding wage and price inflation. A policy maker may surprise the public in the short-run, increasing inflation to reduce unemployment through the effect on real wages, but eventually the veil of “money illusion” would dissipate. Inflation expectations would rise in response to observed inflation. Unemployment would return to its “natural” level, but with equilibrium inflation higher than before. Further attempts to drive unemployment down will then require accelerating inflation hikes. In the long-run, Friedman insisted, there is only a single rate of unemployment consistent with a steady rate of inflation. This is the nonaccelerating inflation rate of unemployment (“NAIRU,” or sometimes referred to as the natural rate of unemployment). (See Figure 6 in Appendix A for a graphical representation of this theory.)

In the time since, much empirical work has been performed to sort out this theoretical controversy. Emphasis is often placed on the generalized form of the Phillips curve, i.e., the relationship between broad inflation and broad unemployment. Attention has also been paid to the market power implications of Phillips’ analysis (for a few examples, see Eagly [5], Aquilante et al. [6], or Dennery [7]). Some researchers have commented on the role of market power in relation to observed flattening of the Phillips curve, but few have examined the interindustry differences in market power that Phillips alluded to in his original paper (in Phillips [8] pp. 292, Phillips notes “…wage rates rose more slowly than usual in the upswing of business activity from 1893 to 1896 and then returned to their normal pattern of change; but with a temporary increase in unemployment during 1897. This suggests that there may have been exceptional resistance by employers to wage increases from 1894 to 1896, culminating in industrial strife in 1897. A glance at industrial history confirms this suspicion. During the 1890’s there was a rapid growth of employers’ federations and from 1895 to 1897 there was resistance by the employers’ federations to trade union demands for the introduction of an eight-hour working day, which would have involved a rise in hourly wage rates. This resulted in a strike by the Amalgamated Society of Engineers, countered by the Employers’ Federation with a lock-out which lasted until January 1898.” This excerpt, along with others, highlights Phillips’ consideration of bargaining power between employee and employer as a function of the level of unemployment and a determinant of wage growth.).

In this paper, we seek to explore potential differences in bargaining power stemming from labor scarcity/abundance by skill level as defined by educational attainment. We accomplish this using Phillips’ simple hyperbolic functional form to determine the degree of convexity in the wage growth–unemployment relationship and the point of wage stability. We then turn to testing the relationship between inflation acceleration and the unemployment gap as described by NAIRU. We employ BVAR to understand the interrelationships between inflation, unemployment, and interest rates. Lastly, wavelet analysis is performed to test for time dependency in the estimated Phillips curve and NAIRU theory relationships.

The paper is structured as follows. In “Literature Review,” we review the literature relevant to the topics discussed. In “Methodological Overview,” we give a brief overview of the methodology employed (see Appendix F for more detailed discussion). “Data” explains the data we used and the cleaning applied. “Results and Interpretation” summarizes and interprets our findings, and “Conclusion” concludes.

Phillips’ analysis in 1958 regressed nominal wage growth (y) on the unemployment rate (x) . He expressed the relationship as

                                                         ,

or 

Phillips specified  to be where y is the wage rate. The constants a,b and c were estimated by ordinary least squares (“OLS”). The constant a was selected by trial and error to reposition the curve. There were two other factors Phillips considered important for the wage–employment relationship. These included the rate of change of unemployment (dx/dt) and retail/import prices. To accommodate the former, Phillips considers a functional relationship of the following form:

where the constant k is measured by OLS and  is the unemployment rate at the beginning of time . Phillips decides against this form, noting that x is trend-free, meaning (1/) (dx/dt) should be uncorrelated with x. This decision produced Phillips’ famous “loops,” where the data appeared to form neat cycles around the fitted line presumably reflecting the rate of change of the unemployment rate. (See Figure 12 in Appendix B for an example using the 1861–1868 data.)

Fitting the three data sets to Equation (12) and using the mean of each parameter’s posterior distribution results in the following fits (Figure 1):

Figure 1

Visualization of fits of the three data sets to Equation (12) using the mean of each parameter’s posterior distribution.

Less Than High School (LHS): 

High School (HS):

Bachelors (B):

The 95% highest marginal posterior density intervals (“HPDI”) for each of the parameters are in Table 1.

   Table 1

    95% HPDIs.

Initial inspection of the results may reveal sample size difficulties. Unemployment for the Bachelors group is clustered below 5%. Additionally, the shape of the curve for the “Less Than High School” group is heavily influenced by a single observation of -0.89% wage growth corresponding to a 14.3% unemployment rate from 2010. This point significantly enhances the convexity of the Less Than High School curve, as defined by the second derivative of the fitted lines after reversing the logarithm, relative to the other two. It also reduces the error precision and widens the HPDI for this group, making the position and curvature of the Less Than High School curve highly uncertain. (See Appendix C for further discussion on this curve’s sensitivity to individual data points.)

The convexity of the High School and Bachelors groups is quite similar, although the 95% HPDI is significantly tighter for the Bachelors group. In general, as skill level declines, the HPDI becomes increasingly wide for all parameters. This may suggest more stability (and predictability) of wage growth as skill level increases. (See Appendix D for graphical representation of the HPDI for each group.)

The intersection of the mean of the fitted curve with the -axis is heavily dependent on numerous factors including the shape forced by Phillips’ functional form, the time period, and the sample size. Nonetheless, each group intersects the -axis at a different point. This may reflect labor market segmentation by skill level, with each segment having a unique natural rate of unemployment. The Bachelors group has the lowest natural rate of unemployment of the three groups, suggesting increased skill may reduce the natural rate of unemployment. The awkward shape of the Less Than High School curve relative to the other two may signify a different degree of wage rigidity at the lowest end of the skill spectrum and possibly the need for a skill-specific functional form.

To support the assertions of the NAIRU theory, specifically that there is a short-run tradeoff between the change in inflation and the unemployment gap, the value for  in Equation (13) (sourced originally from Equation (9)) should be negative.

The results of the analysis weakly support the NAIRU theory. The mean of the marginal posterior distribution for  is negative (-0.064); however, the results detailed in Table 2 and Figure 2 indicate a nonnegligible portion of the marginal posterior lies in positive territory.