# Solving Robert Solow’s Puzzle of Labor Share on Two islands

These two islands are actually one.

Brad DeLong has a video discussing Piketty’s book. In the video, the subject of labor share comes up, which has dropped to new lower levels in advanced countries. Robert Solow tries to make an analogy for labor share being an irrelevant factor. So he describes two islands. (36 to 38 minutes in video) and asks which island you would rather live in.

• Island #1… Labor share is rising because real wages are rising but still faster than productivity.
• Island #2… Labor share is falling because real wages are rising but still slower than productivity.

Which island would you rather live in?…  He wants you to think that labor share is irrelevant. But, it’s an unfair analogy, because there really aren’t two separate islands here. The two islands are actually the same island. I start my explanation using the following equation for profit rate.

Profit rate = (productivity – real wages) * Total labor hours/Capital

Let’s put some real numbers in the equation and compare the two islands. Labor share starts out at 75% (75 real wage/100 productivity). Also we hold capital steady throughout the analysis.

Profit rate starts at 10% = (100 – 75) * 1000/250,000

### The Islands

Let’s look at island #1 first… Let’s raise real wages by 3% and productivity by 1%. Labor share rises to 76.5%.

Profit rate of island #1 falls to 9.5% = (101 – 77.25) * 1000/250,000

In order to raise the profit rate back up to 10%, labor hours would have to rise by 5.3% holding capital stock steady. (Note: Sometimes there is a fast decline in the unemployment rate (like in 2014) in order to support vulnerable profit rates.)

Now island #2… let’s raise real wages by 3% and productivity by 4%. Labor share drops to 74.3%.

Profit rate of island #2 rises to 10.7% = (104 – 77.25) * 1000/250,000

In order to lower the profit rate back down to 10%, labor hours would have to be reduced by 6.5% holding capital steady.

### Discussion

In island #1, business owners are not happy. They are losing profits. So they increase hiring while holding capital stock steady. However, increased hiring has its diminishing returns. So eventually rising real wages would decrease the profit rate, and labor hours would start being reduced to cut losses.

In island #2, business owners are happy. Profit rates are increasing. So they are not forced to change labor hours in relation to capital stock. They could still reduce labor hours by less than 6.5% and still have a higher profit rate. But they will probably increase labor hours in order to increase the profit rate even more, especially since island #2 tends to have more unemployed workers because there is less pressure to raise real wages.

### Analysis

So which island is better? Well, neither is better. They are actually the same island! Each one describes a different phase of a business cycle. So one becomes the other through a business cycle.

Island #1 describes the end phase of a business cycle where productivity is against the effective demand limit with pressures to raise real wages from low unemployment. Eventually a recession occurs as businesses cut losses by reducing labor hours.

Island #2 describes the beginning phase of a business cycle coming out of a recession where productivity is rising and there are little pressures to raise real wages since unemployment is comparably high. Eventually island #2 becomes island #1 as unemployment declines, diminishing returns of labor hours sets in, productivity slows down and pressure increases for higher real wages.

Here is a graph of productivity plotted with real wages as year-over-year percent changes. (link)

You can see island #2 described after a recession where the red line (productivity) tends to be higher than the blue line (real wages).  Then island #1 is described before the next recession where the blue line tends to rise above the red line.

### Conclusion

Robert Solow asks which island is better. Yet they are actually the same island in different stages of a business cycle. His puzzle shows that he still views labor share as an oscillating constant from one business cycle to another. But labor share has truly fallen to a new range of oscillation since the turn of the century. His puzzle gives no useful insight into the significance of this new fall in labor share. Some might say that he is simply not updating his priors.

### A Note on the Current Business Cycle

We see island #2 so far in the current business cycle by a continually falling labor share and rising aggregate profit rates until 2012. Productivity rose more than real wages. Since 2012 labor share has been steady. Profit rates have been supported by increasing labor hours; Unemployment declined faster than expected in 2014, which is a sign of vulnerable profit rates.

Now the expectation is that we will see island #1 come into view where labor share starts rising; Real wages (blue line) are expected to rise faster than productivity (red line). It hasn’t happened yet. Corporations are still enjoying their nice profit rates.