Are prices part of total factor productivity? 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 &quot;absorb&quot; prices into total factory productivity and rewrite everything as:
$$\pi = Y\times p = pAK^{\alpha}L^{\beta} = \tilde{A}K^{\alpha}L^{\beta} $$ where $\tilde{A} = p\times A$?
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.