The crucial point is to distinguish between total profit for the (world) economy as a whole and the distribution of profit among firms. The theoretical point to start with is conventional profit theory which asserts that total profit is zero: “The consensus to date has been that it is mathematically impossible for capitalists in the aggregate to make profits.” (Keen, 2010, p. 2)
This assertion is fully accepted by methodologists: “But, from the macro perspective of Walrasian general equilibrium, the total profits in this case cannot be other than zero (otherwise, we would need a Santa Claus to provide the aggregated positive profit) but this does not preclude the possibility of short-run profits and losses of individual firms canceling each other out.” (Boland, 2003, p. 150)
The curious thing is that aggregate profit has been greater than zero for most of the time in most of the known market economies up to the present. This is an empirical fact. Hence there is something wrong with conventional profit theory (Desai, 2008, p. 10). And, clearly, when the profit theory is wrong then distribution theory is wrong too (2014a). This is the main argument against Piketty.
Now for the constructive part.
First of all, it is necessary to show how positive overall profit comes into existence and how it can be maintained over any stretch of time. The structural axiomatic paradigm achieves this in a formally rigorous way (2013; 2011). For the elementary production-consumption economy, the testable macroeconomic Profit Law is given on Wikimedia AXEC29
How Ricardo or Bailey made their individual profits is a historically unique story that unfolds within the framework of structural laws. How universal economic laws and unique historical events play together has been shown in (2014b).
All feathers are subject to the Law of Gravitation, but the trajectory of a flying feather is a unique historical event. Scientists are not primarily interested in the latter but in the underlying law. Likewise in economics, there is the Profit Law (= scientist's real reality) and the history of the distribution of overall profit among individual firms (= laymen's commonsensical reality).
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Boland, L. A. (2003). The Foundations of Economic Method. A Popperian Perspective. London, New York: Routledge, 2nd edition.
Desai, M. (2008). Profit and Profit Theory. In S. N. Durlauf, and L. E. Blume (Eds.), The New Palgrave Dictionary of Economics Online, 1–11. Palgrave Macmillan, 2nd edition. URL
Kakarot-Handtke, E. (2011). The Emergence of Profit and Interest in the Monetary Circuit. SSRN Working Paper Series, 1973952: 1–23. URL
Kakarot-Handtke, E. (2013). Debunking Squared. SSRN Working Paper Series, 2357902: 1–5. URL
Kakarot-Handtke, E. (2014a). The Profit Theory is False Since Adam Smith. What About the True Distribution Theory? SSRN Working Paper Series, 2511741: 1–23. URL
Kakarot-Handtke, E. (2014b). The Synthesis of Economic Law, Evolution, and History. SSRN Working Paper Series, 2500696: 1–22. URL
Keen, S. (2010). Solving the Paradox of Monetary Profits. Economics E-Journal, 4(2010-31). URL