The difference between PPI and CPI has shown a distinctive pattern of rising before recessions only to fall as slack in the economy increases. Supply shocks impact growth. Now we are facing an extreme in the difference between PPI and CPI with levels never reached before over the entire time series.
The data suggest that CPI will likely stay elevated, and a swift decline will only occur if we have a strong growth slowdown.
The correlation between PPI and CPI is high but has fallen from high levels earlier as described in "The Wedge of the Century: Understanding a Divergence between CPI and PPI Inflation Measures" by Shang-Jin Wei and Yinxi Xie. The decline is associated with lengthening of global production chains.
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