Why do I multiply labor share by 0.78 to get effective labor share? It is the central tendency line in the plot of capacity utilization with the Business sector labor share index with a y-intercept of 0. But let's apply effective labor share to the graphs for productivity and see what we get.
This first graph of productivity is real GDP (GDPC1) divided by the Nonfarm Business Sector: Hours of All Persons (HOANBS) (2005=100)/100. This line will be our standard for productivity.
Real GDP/total hours index/100 ......... Y/L/100
The next graph adds another line for productivity. It is Real GDP per Hour Worked in the United States (USARGDPH) (2005=100) divided by 100.
But wait, the lines don't match up. They should be the same. They both represent Real GDP divided by an index of total labor hours. Well, now watch. I will divide this line by the same 0.78 that converts labor into effective labor share.
Now the lines match up once we do the "effective" conversion.
Now we add one more line for productivity. This will simply be Business Sector: Real Compensation Per Hour (RCPHBS) / Business Sector: Labor Share (PRS84006173). Both 2005=100.
Whoops, they don't match up again. What if we convert the labor share to effective labor share by multiplying it by 0.78?
OK... the line is closer to the others. It is not a perfect match. The lines match in the late 40's, then come back together in the 90's. In looking at this last blue line of real compensation divided by labor share, we could say that in 60's and 70's, either real compensation per hour was over-valued or labor share was under-valued or both. Or in looking at the other two lines (green and red), we could say that either real GDP was under-valued or total labor hours were over-valued.
Whatever your opinion is, it is good to have lines that double-check the accuracy of other lines. One gets a deeper analysis of the numbers.
One more graph for fun... The three lines since 1st quarter 2005...
All 3 lines are a measure of productivity, yet they each reveal a different story. The red line does not use the 0.78 "effective" conversion, as the others do. Yet they are all within roughly 2% of each other. The official statistical numbers that economists use are not perfect, but with a graph like this one can get a better sense of what productivity is really doing.
The concern which everyone should be having is the stagnation of productivity since the beginning of 2010. If business is investing in machines and robots, then why isn't productivity increasing? It's because productivity is constrained by demand. The blue line has risen since the recession partly due to labor share falling 5%. Lower labor share means less demand for finished products. Yet, productivity is not able to go any higher.
Workers could be more productive if customers had more liquidity to buy the products of their work. Workers would be asked to produce more products, in effect be more productive. But demand has run out... and the economy is being held in a state of stagnation (productivity and effective demand stagnant) by loose monetary policy. It's like monetary policy makes the economy float up to its effective demand constraint and then just supports it there. Once monetary policy begins to back off, the economy is coming down.
The Fed is stuck with a problem. They cannot back off from loose monetary policy. They are the foundation of the economic level. Without them the economy will fall to a lower level. Yet, they can't push the economy any higher due to low demand from low labor share.
One solution: Give the Fed the power to adjust the minimum wage in order to regulate liquidity of labor income. They could make a policy of raising the minimum wage 1% per month every month to build a real foundation for demand in the economy. Then monitor productivity, effective demand and real GDP. The economy would reach a point where better wages could allow the Fed to back off from loose monetary policy.
(note: All these lines can be mulitiplied by 0.78 to put productivity at 2005=100. The equations for the lines look like this.)
(GDPC1)/(HOANBS)/100*0.78 ... red line
(USARGDPH)/100 ... green line
((RCPHBS)*0.78)/((PRS84006173)*0.78) ... blue line
The graph looks like this.