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To my understanding, post-keynesian economics generally see price as a product of profits, wage expenses, interest payments, and inventory costs. To express it mathematically:

\begin{gather*} p\ =\ \frac{( 1+\Upsilon )[ \sigma c_{-1} s+( 1-\sigma )( 1+r) cs]}{s}\\ \\ p\ =\ ( 1+\Upsilon )[ \sigma c_{-1} +( 1-\sigma )( 1+r) c] \end{gather*}

$\displaystyle \begin{array}{{>{\displaystyle}l}} Where:\\ \\ p\ =\ price\\ \Upsilon \ =\ profit\ share\ of\ revenu\\ \sigma \ =\ proportion\ of\ sales\ coming\ from\ inventory\\ ( \sigma \ -\ 1) \ =\ proportion\ of\ sales\ coming\ from\ current\ period\ production\\ r\ =\ interest\ rate\\ c\ =\ unit\ cost\ \\ \end{array}$

In that case, the model has only two fundamentally exogenous variables: the rate of interest and the profit share. All other variables can be found from accounting identities. At a certain point, we can make the interest rate endogenous, assuming that banks charge the maximum rate they possibly can. But the case is more difficult with the profit share, since it is socially determined by the impact of regulations, social movements, cultural norms, and negotiations with unions.

My questions are the following:

(a) How can we make the profit share endogenous in a manner coherent with PKE

(b) What are the historical, institutional, and social foundations of your conclusion?

(c) How can this answer be formalized mathematically?

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Answer for a three sectors economy based on SFC modeling

So, after more research, I'va found a first answer. I'll consider this a starting point, since I still critical of it. However, the current answer provide an answer for post-keynesian texts.

a) How can we make the profit share endogenous in a manner coherent with PKE

To make the profit share endogenous, we'll need to integrate two sectors to the model: (1) the government sector and (2) the household sector. To simplify the presentation, we'll make abstraction of the banking sector, removing interest payments from the picture. More sectors (private banks, central bank, etc.) can be added afterward. Including households and government, we're left with the following equations (see Atesoglu 1999 for a slightly simpler version of this):

(1) \begin{equation*} G\ -\ T\ =\ H \end{equation*}

Governement exempense (G) mines taxes (T) is equal to the surplus or deficit (H).

(2) \begin{equation*} C\ =\ H\ +\ a_{1} W*N\ +\ a_{2} M_{-1} \end{equation*}

Household expenses are equal to the governement deficit (H), plus a portion of wage, plus a portion of savings.

(3) \begin{gather*} S\ =\ F\ +\ HWC\\ HWC\ =\ s\sigma UC_{-1} +s( 1-\sigma ) UC_{-1}\\ C\ =S \end{gather*}

Those three equations detail the firm accounts and the connection with households. It indicates that sales are equal to profit plus historical wage costs.The historical wage costs are basically the cost of goods sold, and differ with the evolution of wages and inventories. The last equation says that sales are equal to consumption, which can be detailed in the following way:

(4) \begin{gather*} C\ =\ S\\ H\ +\ a_{1} W*N\ +\ a_{2} M_{-1} \ =\ F\ +\ HWC \end{gather*}

Since the profit share is equal to F/HWC, we can simply isolate it:

\begin{gather*} \delta HWC\ +\ HWC\ =\ ( H\ +\ a_{1} W*N\ +\ a_{2} M_{-1})\\ HWC( \delta \ +\ 1) \ =\ ( H\ +\ a_{1} W*N\ +\ a_{2} M_{-1})\\ \\ \delta \ =\ \frac{( H\ +\ a_{1} W*N\ +\ a_{2} M_{-1})}{HWC} \ -\ 1 \end{gather*}

b) What are the historical, institutional, and social foundations of your conclusion?

Here's an interpretation of the precedent equation:

Profits essentially come from effective demand, which can be compensated by government deficit. This would be in accordance with Sherman and Sherman (2018) who suggest that the capitalist economy is essentially defined be an antagonism between wages and profits (since HWC grow faster than consumption). The authors will suggest that inequality is the essential factor behind business cycles: period of equalization between the wage and profit share are marked by growth, period of inequality precede recessions and crisis. In the case of a recession/crisis, caused by the contraction of effective demand, government can intervene on the short-run, but only a reduction of inequality can really remediate the situation.

Reference

Atesoglu, Sonmez. 1999. "A Post Keynesian Explanation of Employment in the United States". Journal of Post Keynesian Economics 21 (4): 603-610.

Sherman, Howard, and Paul Sherman. 2018. Inequality, Boom, and Bust. Routledge.

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