“In the early pages of Capital in the Twenty-First Century, Thomas Piketty states a 'fundamental law of capitalism' that α = rxβ, where α is the share of profit in income, β is the capital/output ratio, and r is the rate of return on capital, or the rate of profit. Thus: r = α/β.” (Galbraith, 2014, p. 145)
The fundamental law does not hold because the fundamental theorem of income distribution states: Profit is not a factor income (2012, p. 4).
Piketty does not realize that profit and distributed profit are fundamentally different economic entities. Therefore, there is no such thing as “a share of profit in income” but there is “a share of distributed profit in income”.
The axiomatically correct macroeconomic Profit Law reads
(i) for the elementary production-consumption economy with profit distribution
Qm≡Yd−Sm (2012, p. 4, eq. (5)),
(ii) for the investment economy with profit distribution,
Qm ≡Yd+I−Sm: (2014, p. 8, eq. (18)).
Legend: Qm monetary profit, Yd distributed profit, Sm monetary saving, I investment expenditure
The structural axiomatic Profit Law gets a bit more sophisticated when foreign trade and government is included. Summing up investment expenditures over time and taking depreciation into account (ii) ultimately yields the profit rate (2011b, Sec. 6.2).
Note that profit for the economy as a whole does not depend on productivity. What holds for a single firm does not hold for the economy as a whole. One has to be careful here not to commit the Fallacy of Composition.
Changes in the valuation price of assets are captured by nonmonetary profit Qn. This is a different and lengthy issue (2011a).
From the fact that Piketty's profit theory is not correct follows logically that his First Fundamental Law is not correct either.
Time to get the formal foundations right (see here).
Galbraith, J. K. (2014). Unpacking the First Fundamental Law. real-world economics review, (69): 145–148. URL
Kakarot-Handtke, E. (2011a). Primary and Secondary Markets. SSRN Working Paper Series, 1917012: 1–26. URL
Kakarot-Handtke, E. (2011b). Squaring the Investment Cycle. SSRN Working Paper Series, 1911796: 1–25. URL
Kakarot-Handtke, E. (2012). Income Distribution, Profit, and Real Shares. SSRN Working Paper Series, 2012793: 1–13. URL
Kakarot-Handtke, E. (2014). The Three Fatal Mistakes of Yesterday Economics: Profit, I=S, Employment. SSRN Working Paper Series, 2489792: 1–13. URL
The 'First Fundamental Law of Capitalism' is the Profit Law (see Wikimedia AXEC143d). For details of the big picture see cross-references Profit.