Depreciation and Income Shares

I would like now to wade briefly into a debate over the gap between growth in productivity and wages by introducing a bit of modeling fun. It seems clear that— in recent decades— although wage income has grown more slowly than GDP there has been little difference between the growth rate of NDP (GDP net of capital depreciation) and the growth rate of total labor compensation.

On the other hand, inequality of compensation has increased quite a bit— driving a large wedge between pay at the top and pay of the ordinary worker. None of this is news. Somewhat less clear is whether net product is more or less appropriate as a comparison. At first blush, workers still have to produce the whole of output no matter how much investment goes to replacing depreciating capital. It might make sense for labor compensation to rise in step with gross production. But...

Over the same period, rapidly depreciating computers and software have represented an increasing share of investment. If labor compensation were to keep pace with gross production and net production fail to do so, then it would also make sense for labor compensation to represent an increasing share of net product.

Yet labor compensation has been more stable as a share of NDP than of GDP. How might this come about? Let us consider a very simple macro-economy with aggregate production $$ Y=K^\alpha\left(AL\right)^{1-\alpha} $$ where $0 < \alpha < 1$, $\dot{A}=gA$, $\dot{L}=nL$, and $$ \dot{K}=sY-\delta K $$ Relative to effective labor $AL$, this may be rewritten $$ y=\frac{K^\alpha\left(AL\right)^{1-\alpha}}{AL}=\left(\frac{K}{AL}\right)^\alpha=k^\alpha $$ so that ${Y}/{K}={y}/{k}=k^{\alpha-1}$ and $$ \frac{\dot{k}}{k}=\frac{\dot{K}}{K}-\frac{\dot{A}}{A}-\frac{\dot{L}}{L}=s\frac{Y}{K}-\delta-g-n=sk^{\alpha-1}-\delta-g-n $$ If we make another rather restrictive assumption that factors are paid their marginal product, then the wage per unit of effective labor is given by $$ w=\left(1-\alpha\right)\frac{Y}{AL}=\left(1-\alpha\right)k^\alpha $$ resulting in a labor share of gross output $1-\alpha$ but a labor share of net output $$ S=\left(1-\alpha\right)\frac{Y}{Y-\delta K}=\frac{1-\alpha}{1-\delta k^{1-\alpha}} $$ Now for the moment let us assume that $g$ and $n$ are fixed but $s$— the rate of gross investment out of gross output— may vary with $\delta$. For example, we may vary $s$ so that $k$ does not change. That is, $$ s=\left(n+g+\delta\right)k^{1-\alpha} $$ As we see in Figure 1, an increase in the rate of capital depreciation raises investment and the share of net output going to labor.

Figure 1: Simulation with steady capital-income ratio

Alternatively, we may vary $s$ so that the labor share of net income does not change. This requires $$ 0=\frac{\dot{S}}{S}=-\frac{k^{1-\alpha}}{1-\delta k^{1-\alpha}}\dot{\delta}-\frac{\delta\left(1-\alpha\right)k^{-\alpha}}{1-\delta k^{1-\alpha}}\dot{k}=-\frac{k^{-\alpha}}{1-\delta k^{1-\alpha}}\left[k\dot{\delta}+\left(1-\alpha\right)\delta\dot{k}\right] $$ or $$ \frac{\dot{\delta}}{\delta}=-\left(1-\alpha\right)\frac{\dot{k}}{k}=-\left(1-\alpha\right)\left[sk^{\alpha-1}-\left(n+g+\delta\right)\right] $$ Labor share is therefore stable when $$ s=\left[\left(n+g+\delta\right)-\frac{1}{1-\alpha}\frac{\dot{\delta}}{\delta}\right]k^{1-\alpha} $$ Figure 2: Simulation with steady labor share of net output

Neither condition makes any obvious behavioral sense. Why would investment move in such a particular manner? However, in this very restrictive model there does appear that maintaining labor share of net output in the face of a higher rate of depreciation leads to a period of low growth in productivity and wages due to lower investment and a reduced capital stock.

Is this the story of the last 40 years? It is certainly possible that rapid tech-capital depreciation has led to lowered net investment and growth. A nice little result from a stupid little exercise.