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I am trying to understand how production is related to income/profit and where do prices enter. Suppose there is a single firm with a Cobb-Douglas production technology: $$Y=AK^{\alpha}L^{\beta}$$

Where do prices come into this? I see this being related to whether $Y$ is expressed in units of output or in monetary value (total revenues?). If the former, I would think this means that only $Y\times p$ gives the firm's profit, if $p$ stands for price of output. But if this is the case, can we "absorb" prices into total factory productivity and rewrite everything as: $$\pi = Y\times p = pAK^{\alpha}L^{\beta} = \tilde{A}K^{\alpha}L^{\beta} $$ where $\tilde{A} = p\times A$?

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In micro-econometric work, prices may or may not be part of TFP.

The literature recognizes two versions:

  1. TFPQ (Quantity-based)
  2. TFPR (Revenue-based)

Clearly, TFPR must include prices (Revenue = quantity*prices).

Many scholars argue that TFPQ is the purer and most correct measure of TFP. In the sense that higher productivity means producing more output (quantity) for given input levels, that sounds appealing. Prices can also capture market power. So looking at revenues may conflate higher productivity (which is desirable) with high market power (which is undesirable). However, there are several problems with TFPQ.

First, how to define quantity? Especially for services, output can be hard to define in terms of quantity. E.g. what's the produced quantity of a food delivery service? Is it the number of deliveries? Is it the delivery distance? Does it depend on weather?

Second, how to compare quantities? Typically we want to say productivity is higher or lower in one firm/place than another. How do we compare the quantity of food delivery (however defined), to the "quantity" of other services, like watching a sports event? Revenues in dollar amounts, on the other hand, are perfectly comparable.

Third, there are data limitations. We just don't have quantity data normally, because lots of data comes from firm balance sheets, which are in values. Also, many firms are multi-product so "output data" is difficult to back out. Some papers by Syverson (see e.g. here and here) manage to use TFPQ for ready-mix concrete, but that's a special case.

Fourth, what about quality? For given inputs, a firm that produces one high-quality rug is arguably more productive than a firm producing one low-quality rug. These quality differences could be easily captured by prices and hence by TFPR. But TFPQ would have a hard time distinguishing these firms.

The debate and methodological issues continue and I'm sure I may have forgotten some. The bottom line is that prices may or may not be part of productivity.

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No actually by default when you use Cobb-Douglas function the output is not even measured in monetary units but rather as output per unit of time. This output per time can be still called income without assigning it any 'monetary' value. For example, in Robinson Crusoe economy if you catch 5 fish then those 5 fish are your income from economic perspective.

Moreover, let us suppose we measure $Y$ by GDP. In that case for calculation of TFP we will always use real GDP not nominal GDP. This is done precisely to get rid of any influences change in price level may have on the measurement. Even though nominal GDP is calculated involving prices and we can express nominal GDP as $YP$ (where $P$ is the aggregate price level) when we calculate real GDP we deflate the nominal GDP by dividing it with the aggregate price level. Hence real GDP is $\frac{YP}{P}=Y$.

The above being said when it comes to TFP it is incredibly hard to do dimensional analysis (see this working paper) and generally $A$ does not have consistent units. To be more specific, of course in each specific case there will be some units but the units will depend on parameters of the model (unless we impose some restrictive assumptions as mentioned in that working paper). However, note this is not problematic from a statistical perspective as statistical analysis generally does not require dimensions to be consistent (like regression analysis for example). Consequently, in empirical work we usually just avoid assigning any particular units to TFP.

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  • $\begingroup$ This is very helpful. As a follow up: since TFP does not really have consistent units, does it ever make sense to talk of "price-adjusted TFP"? In a sense, this refers to the second part of my question: can we redefine terms as such so that $\pi = Y\times p = pAK^{\alpha}L^{\beta} = \tilde{A}K^{\alpha}L^{\beta} $, with $\tilde{A}$ being the "price-adjusted TFP"? Basically, can we absorb price units into A and still talk of income and not output? $\endgroup$
    – Paul
    Oct 9 '20 at 17:13
  • $\begingroup$ @Paul you of course can multiply both sides of the equation by $P$ and call the new expression whatever you want really, but given that in production we care about output not about what the price level (P) is I don't think it makes an economic sense. I can't recall ever seeing something like this done. Furthermore, note that when we actually estimate TFP we very often log linearize the relationship $ \ln Y = \ln A + \alpha \ln K + \beta \ln L$ (see Economic Forecasting and Policy by Carnot et al). Hence during any reasonable estimation of your adjusted TFP the $p$ and $A$ would get separated. $\endgroup$
    – 1muflon1
    Oct 9 '20 at 17:28
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    $\begingroup$ @Paul also note that price adjusted would be very misleading word for it. When we talk about price adjustments in economics we talk about correcting nominal variables for the price level (i.e. deflating nominal GDP so $PY/Y=Y$. In your case you are not adjusting for price level at least not as commonly understood in economics. $\endgroup$
    – 1muflon1
    Oct 9 '20 at 17:31

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