Saturday, May 9, 2015

The Problem with Human Capital as a Factor of Production

Shaikh's "Humbug" paper should have sounded the death knell for the Cobb-Douglas "production" function. Instead of accepting that the Solow residual (A in the equation below) was identically equal to the share-weighted geometric mean of the factor prices, New Classical economists have turned inward.

$$Y = AK^{\alpha}L^{1 - \alpha}$$

Now, the mainstream theory of growth is in constant search to "endogenize" the Solow residual. Thus, the hunt has been on for additional factors to "explain" the Solow residual as "accumulated" factor productivity.

If you take as given the Shaikhian formulation, then it becomes obvious why "total factor productivity" increases over time. Since average wages have generally been increasing over time while the average rate of profit is roughly constant (per the last two Kaldor facts), the Solow residual will necessarily be increasing over time. Thus anything roughly correlated with the average wage rate should allow for the ad-hoc creation of a "factors only" production function.

The direction taken that is perhaps most aggressive in this style of willed ignorance has been that of using "human capital" as a factor of production. In this version of the production function, human capital - the nebulous and often arbitrary combination of health and education, commodified often at the workers' expense - is added to the Cobb-Douglas function as follows:

$$Y = AK^{\alpha}(hL)^{1 - \alpha}$$

The "factor" $$h$$ is usually represented as some sort of exponential index function. In Caselli for instance, he formalizes this as:

$$h = e^{\phi{}(s)}$$

In this formulation, $$\phi{}(s)$$ is a piecewise linear function of various indices of education and health. Caselli noted that without the use of a human capital index, there were stark "total factor productivity" differences across countries. Poor countries, in his estimation, were "less efficient" than richer countries. Again, in light of Shaikh 1974, this is unremarkable, given the difference in factor prices across countries.

Caselli is likewise delighted to report that to include human capital reduces the total factor productivity gaps across countries. This again should come as no surprise given that wage growth is roughly as unequally distributed across countries as the factors of "human capital." To make this more clear for those of you who are more mathematically inclined, begin with the income identity:

$$Y \equiv W + \Pi$$

Remember, this identity will necessarily hold for any economy at any time. We can use any positive index numbers $$L$$ and $$K$$, with factor prices $$w \equiv \frac{W}{L}$$ and $$r \equiv \frac{\Pi}{K}$$, respectively to get:

$$Y \equiv wL + rK$$

We can further index the average wage rate by an arbitrary positive number $$h$$ such that $$w \equiv \omega{}h$$ and get:

$$Y \equiv \omega{}hL + rK$$

We can make all values "per capita" by dividing by $$L$$. Using $$y \equiv \frac{Y}{L}$$ and $$k \equiv \frac{K}{L}$$, we arrive at:

$$y \equiv \omega{}h + rk$$

Differentiating with respect to time, we see that:

$$\dot{y} \equiv \dot{\omega}h + \omega{}\dot{h} + \dot{r}k + r\dot{k}$$

Dividing by income, we can get the above to be in terms of percentage changes (with the percentage change of the variable $$x$$ represented as $$\hat{x}$$):

$$\hat{y} \equiv \frac{\omega{}h}{y}\hat{\omega} + \frac{\omega{}h}{y}\hat{h} + \frac{rk}{y}\hat{r} + \frac{rk}{y}\hat{k}$$

Since, the factor shares are roughly constant over time, we can let:

$$\frac{rK}{Y} \equiv \frac{rk}{y} \equiv \alpha \qquad \frac{wL}{Y} \equiv \frac{\omega{}h}{y} \equiv 1 - \alpha$$

If we substitute the above identities into the equation before it, we can say that:

$$\hat{y} \equiv (1 - \alpha{})\hat{\omega} + \alpha{}\hat{r} + (1 - \alpha)\hat{h} + \alpha{}\hat{k}$$

If we let $$\hat{A} \equiv (1 - \alpha{})\hat{\omega} + \alpha{}\hat{r}$$, then we can take the time integral to show that:

$$ln(y) \equiv ln(A) + (1 - \alpha{})ln(h) + \alpha{}ln(k)$$

or:

$$y \equiv Ah^{1 - \alpha}k^{\alpha}$$

If we multiply by the labor index, we get:

$$Y \equiv A(hL)^{1 - \alpha}K^{\alpha}$$

As you can see, from simple accounting identities, we can arrive at the exact formula touted as the center of New Classical endogenous growth theory. The only thing necessary to arrive at this result on top of the Shaikh formulation is an arbitrary decomposition of the average wage rate into an arbitrary human capital index and an equally arbitrary wage residual $$\omega$$.

Thus, if the value of human capital minimizes the differences in this residual across countries, we'd expect that including it into the form above would likewise minimize the differences of "total factor productivity" (since in reality $$A \equiv \omega{}^{1 - \alpha}r^{\alpha}$$). Whether we include the dead labor of physical capital or the undead labor of human capital, the Cobb-Douglas function still requires a zombie-like devotion to not recognizing this fundamental truth.

Whether New Classical economists want to recognize it or not, there is no magical law of efficiency governing production. Rather, wage and profit rates "determine" the capital intensity of production only insofar as income is distributed into wages and profits. More precisely, there is no "determination" at all, since this is merely a law of algebra. Any type of production, regardless of the efficiency of its method will necessarily abide this law. It says nothing about efficiency or fairness.

Back to the drawing board, I guess.