U.S. transfer pricing regulations about the “rate of return on capital employed” (ROA) are misconceived because they rely on untested assumptions. For example, 26 CFR 1.482-5(b)(4)(ii), states:

“Financial ratios measure relationships between profit and costs or sales revenue. Since functional differences generally have a greater effect on the relationship between profit and costs or sales revenue than the relationship between profit and operating assets, financial ratios are more sensitive to functional differences than the rate of return on capital employed. Therefore, closer functional comparability normally is required under a financial ratio than under the rate of return on capital employed to achieve a similarly reliable measure of an arm’s length result. Financial ratios that may be appropriate include the following –

(A) Ratio of operating profit to sales; and

(B) Ratio of gross profit to operating expenses. Reliability under this profit level indicator also depends on the extent to which the composition of the tested party’s operating expenses is similar to that of the uncontrolled comparables.”

The *premise* that “functional differences *generally have a greater* effect on the relationship between profit and costs or sales revenue than the relationship between profit and operating assets” can be true or false, subject to empirical tests, like the *deduction*, “financial ratios are *more sensitive* to functional differences than the rate of return on capital employed.” Emphases added. The coupled affirmations must be subject to empirical tests; they are not true à priori.

In theory, the *gravitation* of the “rate of return on capital employed” (profit rate) to a within-industry (or worse cross-industry) central value is *assumed* as *deux es machina*. The objective of this assumption is to reduce the number of unknown variables to match the number of equations, leaving one degree of freedom. This means that one unknown variable can be left undetermined — that is, one unknown variable can be determined outside of the equation system.

In economics (maximum eigenvalue of the Perron-Frobenius theorem in matrix algebra), the unknown variable is the *assumed single* profit rate. Schumpeter (1954, p. 569) called-out this theoretical (counting equations and unknowns) subterfuge. In theory, economists are satisfied assuming one degree of freedom, such as the assumed uniform profit rate. See Murata (1977, pp. 140-151) and Pasinetti (1977, pp. 197, 218). However, empirical applications in transfer pricing, based on heterogenous financial accounts, are a different domain in which the abscissa of ROA can’t match idealized theory.

Marx wrote more about the “rate of return on capital” than any other economist, and he *assumed* that “capital flows” from one industry to another. However, capital that flows is investment (Capex), measured in the cashflow statement of enterprises. In contrast, capital stock (measured at the beginning or end of the fiscal period), reported on the balance sheet of enterprises, is the composite book account used as the denominator of the regulatory “rate of return on capital employed.”

Nikaido (1996, Chapter 13) showed (using sophisticated mathematics) that the ideal conditions to achieve gravitation of the rate of return on capital are stringent, difficult to satisfy. See also Kuroki (1986). Confusing ideal conditions with complex corporate accounting reality is a fallacy that cannot be relied upon to determine “true” (arm’s length or comparable) taxable income. The untested empirical affirmations of the U.S. transfer pricing regulations about the superior rate of return on capital employed is mismatched (they are unsupported) by the complexity of certain idealized contrivances, such as counting the unknown variables to match the number of equations.

Errors are committed when flow variables are confused with stock variables. The inter-company movement of stock variables, such as the cross-border transfer of property, plant, or equipment, or the transfer of acquired or self-developed intangibles, involves time-delayed asset purchase or lease contracts, which can trigger complex tax events. The assumed gravitation of the profit rate can’t be confused with the messy reality of enterprise balance sheets.

The regulatory notion in transfer pricing that the “rate of return on capital employed” is less sensitive to the pure numbers (or dimensionless) “financial ratios” (operating profit margin or profit markup) is knotty in theory. The assumption that capital stock flows or is fungible, or that capital employed (stock variable) gravitates to less-sensitive assets-based magnitudes, is made without a cogent theory or empirical support (it’s not based on defensible economic theory or comparable facts and circumstances).

The rate of return on capital employed (favored by the IRS in Coca-Cola and Medtronic) is encapsulated in hubris and is vulnerable to attack *in theory* and empirics. Like Perseus wearing a veil such that monsters couldn’t see him, the assumption that the “rate of return on capital employed” is less sensitive to differences in comparability is misconceived, and must be challenged in theory and fact. Misconceived regulations must be corrected, and analysts shouldn’t use veiled economic concepts like the regulatory “rate of return on capital employed.”

#### References

Nikaido Hukukane, Prices, Cycles, and Growth, MIT Press, 1996. Chapter 13 was published before in Nikaido Hukukane, “Marx on Competition.” Zeitschrift Für Nationalökonomie (Journal of Economics), Vol. 43, No. 4 (1983), pp 337–62 (available at: www.jstor.org/stable/41796439).

Ryuzo Kuroki, “The Equalization of the Rate of Profit Reconsidered” in Willi Semmler (editor), *Competition, Instability, and Nonlinear Cycles*, Springer-Verlag, 1986.

Karl Marx, *Capital*, Vol. 3, Progress Publishers, 1971. Published in 1894 from hand-written notes edited by Frederick Engels, after the death of Karl Marx.

Yasuo Murata, *Mathematics for Stability and Optimization of Economic Systems*, Academic Press, 1977. Chapter 4 of this wonderful book covers non-negative square matrices, the Perron-Frobenius theorem, and stability in economic systems.

Luigi Pasinetti, *Lectures on the Theory of Production*, Columbia University Press, 1977. Survey of the economics of David Ricardo, Karl Marx, and Piero Sraffa in matrix algebra, using the Perron-Frobenius theorem, and assuming a uniform profit rate.

Joseph Schumpeter, *History of Economic Analysis*, Oxford University Press, 1954. Published from posthumous and erudite notes collected after nine years of research.