# How were unit labor costs estimated by Gali and Gertler (1999)?

I am trying to estimate a NKPC like the one Gali & Gerter, 1999 and Gali &Gerter, 2001 estimated. I am having trouble with understangind how they calculated the marginal cost, i.e., which time series they actually used. I tried consulting '''Sbordone, 2002, who estimates the same model using the same (or at the very least a similar) proxy for the marginal costs. As far as I could understand MC = C/Y where MC is the marginal cost, Y is GDP and C are the total compensations given to workers, i.e., the sum of all real wages paid to workers. I would like to know if that is correct and, if it isnt, how Gali & Gertler calculated the marginal costs. In the specific interpretation I have described I dont know if one must use real or nominal GDP. If I use real GDP, it seems to me, that when dividing real salaries by the real GDP the deflators dividing both nominal salaries and nominal GDP will "cancel" each other and I would only get the ratio between nominal salaries and nominal GDP, which does not seem useful given the fact that in the price formation model firms are considered not to suffer money ilussion. I apologize in advance in case my words have not been precise enough and thank you all for your answers. Any help and references to articles or textbooks is greatly appreciated.

In Galı́ & Gertler (1999) the authors state that marginal costs is given by:

$$MC_t = \frac{S_t}{\alpha}$$ where $$\alpha$$ is the labor elasticity (from the Cobb-Douglas prod. function), and $$S_t$$ is the labor share of income (i.e. what portion of output GDP goes to labor). Next they actually get around estimating $$\alpha$$ by expressing everything in percentage changes so they actually use:

$$mc_t = \frac{MC_t - MC_{t-1}}{MC_{t-1}}= \frac{\frac{S_t}{\alpha} - \frac{S_{t-1}}{\alpha}}{\frac{S_{t-1}}{\alpha}}$$

where alpha cancels out so you are left with just change in the labor share of income $$\Delta S_t = s_t$$.

They even state in the paper:

The data we use is quarterly for US over the period 1960:1}1997:4. We use the (log) labor income share in the non-farm business sector for $$s_t$$

So in this case they just seem to have directly taken the labor share of income, which is plausible data for labor shares of income exist without any 'assembly' required (e.g. see this Fred dataset as an example).

• Thank you for your answer. I read again the article published in 1999. Gali and Gertler state that $mc_t$ is the marginal cost deviation from its steady state level. I was wondering if actually, $mc_t = \dfrac{\dfrac{S_t}{\alpha}-\dfrac{S^*_t}{\alpha}}{\dfrac{S^*_t}{\alpha}}$ where $S^*_t$ is the marginal cost steady state for period $t$. The steady state marginal cost is estimated, I supose, using the HP filter, as they did with the potential GDP. Oct 25 '20 at 7:24
• @DanielJoséAguilar You are welcome. Also, no in 1999 paper they show no indication of estimating the S (that is what they do in 2001 paper). They mention HP-detrending it but that is not the same as estimating it. Also, they even specifically use words like "we use a measure of real marginal cost" instead of we estimated mc. When they refer to steady state that does not mean they are referring to some empirical model, theoretical models have steady state as well.
– 1muflon1
Oct 25 '20 at 8:52
• Thanks. What I meant is that perhaps they used the deviation of the labour income share from its steady state level as a proxy for $mc_t$. Then, as a proxy for the "steady state labour income share" they just used the labour income share trend which one can obtain by applying the HP filter to the labour income share "original" time series. It seems that I used the word "estimate" incorrectly. One can proxy the steady state GDP as the GDP trend obtained with the HP filter. Similarly perhaps one can proxy the labour income share steady state level as its trend as obtained with the HP filter. Oct 25 '20 at 17:59