The impact of US productivity growth on unemployment in the time–frequency domain: is AI causing a change in the relationship?

Abstract

US data are used to explore the relationship between the unemployment rate and productivity at different cycle lengths by employing time–frequency methodology. Previous studies suggest that the unemployment rate is positively related to productivity increases in the short-run (high frequency ranges), and negatively related at the longer-intervals (low frequency ranges)—and although we largely reproduce these results, with more recent data, we also provide some important additional insights. The extended results show the relationship between unemployment and productivity in the time–frequency domain using growth in corporate profitability as a conditioning variable, to observe the long-term stability of the productivity-unemployment relationship. We find that when conditioned on the growth in corporate profits, the response of unemployment to the growth in productivity has become much stronger and faster in recent years, with the result that labor market adjustments are no longer keeping pace with the technological change. Not only this, but the usual explanation that short-term losses in employment consequent on growth in productivity is outweighed by long-term gains appears to be weakening and our results suggest that this may already be reversing, perhaps in response to the AI revolution currently underway.

Appendices

Appendix

Partial coherence conditional on unit labor costs

In Fig. 

Fig. 7

Multiple coherency plot with unit labor costs

7, we show the multiple coherency between U3 unemployment, rate of change in business productivity and rate of change of real unit labor costs. Here, we can see a few areas of significant coherence, with only intermittent periods of common cycles between the 3 variables, with a 4 to 6 year cycle emerging in the 1960s and a 12-year cycle that shortens to a 6-year cycle by the early 2000s. Interestingly, there is also an extremely long cycle at around 25-year frequency that emerges as significant in the early 1990s and persists until the end of the series, as well as a 9-year cycle that has emerged as significant since 2012.

2.1 Gain and partial gain conditional on unit labor costs

In Fig. 

Fig. 8

Wavelet full and partial coherency. U3 (Unemployment), BPR (Business Productivity); ULC (Unit Labor Costs)

8, we consider the relationship between U3 unemployment and the rate of change in business productivity. In this analysis, the conditional variable is the growth in ULC instead of the growth in real corporate profits. The focus in on the 4–12 year and 24–48 year cycle ranges. The rationale is partially because the former is the business cycle frequency range but also because these were the 2 frequency ranges that displayed significant coherence levels earlier, and lastly so that there is a point of direct comparison with the results of Gallegati et al. (2016) who use discrete wavelet analysis. The comparison between the 2-variable wavelet coherency plot (U3, BPR) in the left-hand panel, and then, the partial coherency conditional on the rate of ULC in the right-hand panel is striking, as the two plots do not immediately look that similar. In certain areas such as from the mid-1970s to 2000 at the 24–32 year cycle, the partial coherence is much stronger than the original 2 variable coherence. In other places, such as in the early 1970s at the 4–8 year cycle frequency, the original 2 variable coherence shows high levels of coherence, whereas the partial coherence shows virtually no coherence.

These observations suggest that we should look at the bigger picture though: for the basic coherence measure, the significant areas are all stacked up on major economic downturns at the higher frequencies. So the 1974–1978 downturn shows up very clearly at frequencies from 2 to 10 years, and the same thing can be said for the 2007–2010 “great recession” downturn. Once the relationship is conditioned on growth in ULC, then those “stacked” high coherency red patches disappear, leaving very little significant coherence at frequencies higher than 24 years. This suggests that the relationship between unemployment and productivity is a business cycle relationship at higher frequencies, but there is also a more fundamental long-run relationship which exists at the lower frequencies which is not dependent on the business cycle.

The left panel Fig. 8 is identical to that of Fig. 5, but the right panel shows that the partial coherence plot with the growth in corporate profits has a much stronger level of joint cycles than for the change in ULC, with the red areas in the partial coherency plot a much darker red.

The lower part of Fig. 8 also plots the phasing of the relationship between unemployment and the rate of growth of productivity with the growth in ULC, with the 4–12 year frequency band first and then the 24–48 year frequency band in the lower plot. The patterns in phasing are very similar between Figs. 5 and 8 for the higher frequency cycles, but at the lower frequencies for the partial coherences they are quite different. At the lower frequencies, when using the ULC growth as a conditional variable for the partial coherence, U3 and the growth in productivity appear to be almost anti-cyclical, which suggests either extremely long lags or a negative contemporaneous relationship.

2.2 Gain and partial gain conditional on unit labor costs

In Fig. 

Fig. 9

Gain and partial gain plots. UN (Unemployment), BPR (Business Productivity); ULC (Unit Labor Costs)

9, we return to the same variable pairing as Fig. 8, where we used the growth in real ULC as the conditioning variable. From a statistical perspective, the areas of significance are slightly smaller than expected, with the lower frequency cycle petering out by 2008. Other than that, the only other area of significant coherence are the sharp downturns identified in the previous section. When looking at the bivariate gain at the 4–12 year frequency, the coefficient starts out close to zero and then, trends positive throughout the time period under study, but at the 24–48 year cycle the gain is negative. So this suggests that at business cycle frequencies there is a positive relationship between productivity growth and unemployment, but at lower frequency cycles, the relationship is quite negative. Given the gain coefficients shown here, the business cycle relationship was dominated by the longer-term relationship, but this dominance has been weakening over time and the two effects appear to largely cancel each other out by the end of the period in question.

When looking at the partial coherence in Fig. 9, the significant areas are much smaller than for the unconditional version, signifying that the growth in ULC has some explanatory power as a conditioning variable. At the higher frequency range, the partial gain first goes slightly negative and then, grows along the same lines as for the unconditional version, becoming positive in the mid-1960s and then, growing through time throughout the rest of the time period. At the lower frequency range, the partial gain starts at the same level as for the unconditional gain, but then perhaps mirroring the different location of the significant coherence areas in the jet plots, the partial gain falls (rather than rises) to trend more negative through the rest of the time period in question. The lower frequency partial gain clearly more than offsets the shorter frequency partial gain, suggesting that the negative relationship between unemployment and productivity growth dominates throughout the time period.

This result shows up in the results for the gain and partial gain, where at the higher frequency the partial gain has a much larger increase toward the end of the period than does the unconditional gain. Figures 6 and 9 are similar at the lower frequencies where the unconditional gain increases as the significant coherence areas fade in the 1980s, but in contrast to Fig. 6, the partial gain does not increase (is more negative). When conditioning on the growth in unit labor costs, although the coherency is significant during deep economic downturns, the partial coherency has very few significant areas, indicating that the conditioning variable largely accounted for business cycle frequency fluctuations.