I will start to build the model for the IS-LM model for Effective demand. This model incorporates the growth model of effective demand, the concept of an optimal labor share and the monetary policy framework of effective demand.
In the IS-LM model, interest rates are on the y-axis and Real output (GDP) is on the x-axis. Normally Real GDP is a function of consumer spending, investment, net government expenditures and net exports. But there is another way to define real GDP. It can be divided into Capital income and Labor income.
Real GDP = Capital income + Labor income
Here is the model which separates out capital income and labor income for an economy with a real output of $1000.
This graph gives 13% of real GDP to capital income and 87% to labor income. Real capital income is at equilibrium of $130. Real labor income is in equilibrium at $870. The IS curves of capital and labor together add up to the IS curve of the total economy. As well the LM curves of capital and labor add up to the LM curve of the total economy.
The capital and labor markets are in equilibrium for this particular level of Real GDP. Both capital and labor have the same natural rate of interest at 3%. The corresponding IS & LM curves cross at 3%, which represents the natural rate of interest.
But now what would happen if labor share drops? Labor would then be receiving less of national income and capital would be receiving more. The equilibrium of the IS-LM graph above would be upset. When labor has less income than the optimal equilibrium, they have less liquidity. Would real GDP simply drop too because labor has less consumption power? Possibly, but the other possibility is that the natural rate of interest for labor would decrease encouraging labor to convert savings into cash for spending. Likewise, the natural rate of interest for capital would rise to soak up the excess liquidity of capital.
Thus, capital and labor would end up with different natural rates of interest. Capital's new natural rate would seek to reduce capital's excess liquidity. Labor's new natural rate would seek to increase labor's liquidity for spending. Here is a graph simulating the change...
I lowered the labor share from 87% to 75%. The real income of capital has almost doubled from $130 to $250. The real income of labor has fallen from $870 to $750. The red dots show how the different natural rates might move as labor share fell from the optimal level. The natural rate of capital rises to 5.21% . The natural rate of labor declines to 0.79%. The IS and LM curves of capital and labor still add up to the same curves of the total economy. What we have here is another equilibrium state of the economy at a real GDP of $1000.
How would we determine the natural rate of interest for the total economy? You might think it is where the IS & LM curves of the total economy cross, which in graph #2 is at 3%. But there is another option... where the difference in IS & LM curves of labor and capital balance to 0. For example in graph #2, at a 3% interest rate, capital's IS minus capital's LM is $44. Likewise, at a 3% interest rate, labor's IS minus labor's LM is -$44. The difference between capital and labor balances to zero at 3%.
But now let's change the slope of the lines. We will make capital very insensitive to interest rates (inelastic). A change in interest rates has less effect on capital's spending or hoarding of money. We will make labor more sensitive to interest rates. A change in interest rate will affect labor's demand for money and spending much more.
To the left we see that the slopes of capital's IS & LM curves are steep (inelastic). In the middle, we see that the slopes of labor's IS & LM curves are flatter (more elastic). The curves for the total economy now cross at 1.07% (see dashed green line). Just by changing the elasticities of the curves for labor and capital, the natural interest rate of the total economy changed. In this graph, the natural rate dropped from 3% to 1.07%.
We can see a balanced natural interest rate for the total economy at 1.07%, and not at 3%. At 1.07% the difference between capital's curves is -$28. The difference for labor's curves is $28. Capital and labor balance to zero at 1.07%. Thus, at 1.07% the economy has an equilibrium to clear the markets. When we add up the IS curves for capital and labor at 1.07%, they add up to a real output of $1000. But when we add up the IS curves for capital and labor at an interest rate of 3%, they add up to only $897 of real output. Thus it is more efficient for the Fed rate in this instance to be set at 1.07% instead of 3%. The economy may appear to have a natural interest rate at 3%, but the rate is more optimal at 1.07%.
The natural rate of the total economy will always be between the natural rates of capital and labor, unless the sign of the slopes change. The natural rate of interest for the total economy will move toward the natural rate that is relatively more elastic, whether it be capital or labor.