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Stay with me, sway with me.

− English lyrics by Norman Gimbel, music by Pablo Beltran Ruiz. "Sway (Quién Será)," sung by Dean Martin with the Dick Stabile orchestra (1954). Encore by Michael Bublé (2004).

The economic theory of the operating profit markup (or the operating profit margin) is relevant for determining the most reliable profit indicator in transfer pricing.

The profit markup equation can be posited in different forms, including my preferred method of using an accounting equation and a structural equation. Here, the profit markup equation is posited using the Amoroso-Robinson relation, reflecting corporate oligopoly behavior in which price and quantity supplied are variables.

In competition, the supplier takes the output price as given (parameter) and maximizes profits with respect to variations in the quantity supplied. In a monopoly or oligopoly, the supplier can maximize profits with respect to variations in the quantity supplied or in the output price. Oligopoly theory is not novel and dates from the published mathematical work of Augustin Cournot (1838). Today, oligopoly prevails in most U.S. industries.

Define revenue (R) as the supply price (p) multiplied by the quantity (q), where p = ƒ(q):

(1) R = p q > 0

The derivative of R with respect to q (called “marginal revenue”) is:

(2) ∂R / ∂q = p + q (∂p / ∂q)

Equation (2) applies to the ideal “perfect competion” (where output price is not affected by the quantity supplied, and thus ∂p / ∂q = 0), and consequently:

(3) ∂R / ∂q = p

Equation (2) also applies to more realistic market structures, including seller market concentration under a monopoly or oligopoly. In general, ∂p / ∂q < 0 (see Henderson & Quandt, 1980, p. 176).

Maximum profit behavior (axiom) is obtained in all cases by setting marginal revenue (MR) in equation (2) equal to marginal cost (MC):

(4) MR = MC = c

= p + q (∂p / ∂q)

= p [1 + (q / p) (∂p / ∂q)]

c = p [1 + 1 / μ]

where μ = [(p / q) (∂q / ∂p)] is the price elasticity of demand (Daus & Whyburn, 1958, pp. 79, 82; Henderson & Quandt, 1980, pp. 22, 177-178; Kreps, 1990, pp. 255-256).

Solve equation (4) for p and obtain the fundamental equation that the supply price is proportional to marginal (or average) cost:

(5) p = λ MC

where the profit markup is λ = μ / (1 + μ). This slope coefficient (λ) is obtained by multiplying the numerator and the denominator of [1 / (1 + 1 / μ)] by μ.

Equation (5) can be estimated using the principle of least squares (regression analysis) (see Goldberger, 1998, pp. 12-16, 61 or Kmenta, 1986, pp. 207-214). In this approach, the operating profit margin is not explicit compared to using the Lerner profit index or my favorite two equations method.

In sum, the operating profit markup equation (5) can be derived in multiple ways, including the Amoroso-Robinson relation (Equation (4) derived here), the Lerner profit ratio, or my favorite two equation system consisting of an accounting equation coupled with a behavioral structural profit equation.

Augustin Cournot, *Recherches sur les Principes Mathématiques de la Théorie des Richesses*, Oeuvres Complètes, Tome VIII, Librairie Vrin, 1980 [first published in 1838].

Paul Daus & William Whyburn, *Introduction to Mathematical Analysis (With Applications in Economics)*, Addison-Wesley, 1958. Equation (2) is obtained by using the derivative of a product rule (p. 45): if the product is y = u v, where the variables u and v are functions of variable x, the derivatives are ∂y / ∂x = u (∂v / ∂x) + v (∂u / ∂x).

Arthur Goldberger, *Introductory Econometrics*, Harvard University Press, 1998.

James Henderson & Richard Quandt, *Microeconomic Theory: A Mathematical Approach* (3rd edition), McGraw-Hill, 1980 [1st edition 1958].

Michael Kalecki, *Theory of Economic Dynamics *(2nd edition), Allen & Unwin, 1965 [1st edition 1952].

“In the statistical analysis the *least squares method* [regression analysis] is used. ... the purpose of the statistical analysis here is to show the plausibility of the relations between economic variables arrived at theoretically”. (Foreword, emphases added).

The point that regression equations must derive from economic theoretical specifications has been repeated by the econometrics giants Arthur Goldberger and Jan Kmenta.

Jan Kmenta, *Elements of Econometrics *(2nd edition), Macmillan, 1986 [1st edition 1971].

David Kreps, *A Course in Microeconomic Theory*, Princeton University Press, 1990.

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