# Gali and van Rens: The Vanishing Procyclicality of Labor Productivity

Let's see how questions like the following are taken here.

Gali and van Rens, 2014 show that empirically, the correlation between $Y, Y/L$ has been declining over time. In fact, it only was a "real thing" when the RBC theory was formed, and is now not statistically different from zero.

Is that an end to RBC theory as we know it? And by RBC I mean anything that sets driven by a TFP shock / uses TFP shocks to explain empirical phenomena: Standard neoclassical models for sure, even Neokeynesian models rely (to a lesser extend) on these. Also the whole Diamonds-Mortensen-Pissarides literature, where the surplus of a match comes from labor productivity, seems to be affected.

Alternatively, one could argue that that the correlation has become smaller, but amplification mechanism are larger now. Or attack the measurement of Gali and van Rens (but their results seem to be pretty robust).

What are potential views on this, what arguments does the literature emphasize?

• (+1) Excellent question. Difficult to answer, but very stimulating, and about a very important matter both for the discipline and for the real world economies. – Alecos Papadopoulos Mar 21 '15 at 13:55

A napkin theoretical exploration of the issue, to find possible directions to look for explanations, could go like this:

According to the paper, the volatility of output and productivity has fallen. So for the correlation to have gone down, the covariance must have fell steeply. Modeling output as

$$Y = AK^a(h\cdot L)^{1-a}$$

where $h$ is the "effort per unit" variable, we have

$${\rm Cov}(Y, Y/L) = E(Y^2/L) - E(Y)E(Y/L)$$

Now

$$Y^2/L = A^2(K/L)^{2a}\cdot L \cdot h^{2(1-a)}$$

$$Y/L = A(K/L)^{a}\cdot h^{1-a}$$

So

$${\rm Cov}(Y, Y/L) = E\left[A^2(K/L)^{2a}\cdot L \cdot h^{2(1-a)}\right] -E\left[A(K/L)^{a}\cdot L\cdot h^{1-a}\right]E\left[A(K/L)^{a}\cdot h^{1-a}\right]$$

We observe ${\rm Cov}(Y, Y/L) \rightarrow 0$. Now, I am in the mood of assuming that the capital-labor ratio is essentially a constant, as is $A$. More over, that the effort-per-unit variable is independent of the amount of labor hours. Under this scenario, we have

$${\rm Cov}(Y, Y/L) = A^2(K/L)^{2a} \cdot E(L)\cdot \big[ E\left[(h^{1-a})^2\right] -\left[E(h^{1-a})\right]^2\big]$$

$$\implies {\rm Cov}(Y, Y/L) \rightarrow 0 \implies {\rm Var}(h^{1-a}) \rightarrow 0$$

So here, the effort variable should be approximately a constant too.

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In a more agnostic approach, using lower case letters to denote logarithms, we would have

$${\rm Cov}(y, y-\ell) = Var(y) - {\rm Cov}(y, \ell)$$

If what we observe is

$${\rm Cov}(y, y-\ell) \approx 0 \implies Var(y) \approx {\rm Cov}(y, \ell)$$

But this means that in a least-squares regression of $\ell$ on a constant $\gamma_0$ and $y$ $$\ell = \gamma_0 + \gamma_1y + u$$ we would obtain $\gamma_1 =1$ and the relation

$$\hat \ell = \hat \gamma_0 + y \implies \approx Y = e^{-\gamma_0}L$$

(with $\gamma_0 <0$ most probably). This is consistent with the previous approach since the constant $e^{-\gamma_0}$, may be seen as containing the $A(K/L)^ah^{1-a}$ terms, which were assumed or emerged as constants for the result of zero-covariance to occur. Here, there is more flexibility since we only require that their product is constant, not each one separately.