*You better stop the things you do. *Jay Hawkins (1929-2000), “I Put a Spell on You.”

US-listed retailers data show that a simple formula can be used to provide reliable estimates of a controlled retailer’s operating profits for transfer pricing purposes.

To enhance tax certainty, I recommend that US state tax authorities allow retailers to use the profit margin based on this formula as a transfer pricing safe harbor.

The regression method proposed here can be applied to any industry, including to provide safe harbors for inbound controlled wholesale distributors or to provide safe harbors for outbound controlled suppliers or for outbound controlled service providers.

#### Determining the Profit Markup

In standard microeconomic theory, a difference in transfer prices charged for the same product in different geographic markets is determined by differences in their elasticities of demand, assuming identical or similar average costs of production, and holding the market shares constant.

In practice, elasticity of demand is difficult to compute because of regression model identification and the lack of product-specific data.

Thus, while standard models of price determination are based on the elasticity of demand, modern versions add a fixed markup to average costs.

Microeconomics textbooks consider only the standard (elasticity of demand) version. See Henderson & Quandt (1980), pp. 176, 178, 181-182 or Layard & Walters (1978), pp. 236-237, 253.

A modern version (without appeal to the elasticity of demand) of markup pricing is found in Kalecki (1971), pp. 43-47.

As a pedagogical device, I elaborate the standard version of markup pricing:

##### (1) R_{i} = p_{i} q_{i}

_{i}

_{i}

_{i}

Accounting equation (1) states that revenue per company product *i* = 1 to N is the individual product price multiplied by the quantity sold.

In a stylized version in which the seller is a price-taker, the equality p* _{i}* = p = market price is assumed. However, I reject this parable and assume the existence of oligopoly with the resulting company-level stable profit markups.

Take the total differentiation of the revenue equation (1) and obtain the Amoroso-Robinson relation between price and the slope of the demand curve of the *i*-th company or product:

##### (2) p_{i} − q_{i} ∂ p_{i} / ∂ q_{i} = c_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

where c* _{i}* is the average cost of the

*i*-th product in consideration.

I employed the product rule of differentiation to derive equation (2) from (1). Among others, see Chiang (1984), pp. 196, 357-358.

The partial derivative ∂ p* _{i}* / ∂ q

*is the slope of the demand function for the company or product*

_{i}*i*. The negative sign results from ∂ p

*/ ∂ q*

_{i}*< 0, because of the inverse relationship between the quantity of product sold and its price. See Henderson & Quandt (1980), p. 176 and Phlips (1983), pp. 96, 257 (endnote 21).*

_{i}Transpose the partial derivative expression to the right-hand side of (2), and obtain the profit markup price equation:

##### (3) p_{i} = c_{i} + κ_{i}

_{i}

_{i}

_{i}

where the profit markup is κ* _{i}* = q

*∂ p*

_{i}*/ ∂ q*

_{i}*whose absolute value is positive.*

_{i}The profit markup is related to the product market share, which can be shown by multiplying kappa by q / q. See Amoroso (1930), equation (3), p. 10, Phlips (1983), pp. 95-98, and Varian (1992), pp. 289-290.

The multiplicative form of the markup price equation is preferable because it is amenable to regression analysis:

##### (4) p_{i} = λ_{i} c_{i} + random error

_{i}

_{i}

_{i}

where the profit markup is λ* _{i}* > 1.

If regression equation (4) is tested by using annual aggregated product data from samples of US-listed companies, the estimated profit markup is found to be stable over time.

Online in the RoyaltyStat website, I ran the regression price equation (4) with grouped product data per company:

##### (5) R_{t} = λ C_{t} + U_{t}

_{t}

_{t}

_{t}

where the aggregate company revenue is R = ∑ R* _{i}* and the aggregate company total cost is C = ∑ C

*.*

_{i}From the estimated operating profit markup, the operating profit *margin* is obtained by using a simple formula:

##### (6) µ = (λ – 1) / λ

I use Standard & Poor’s Compustat mnemonics and define the selected company revenue as R = REVT (same as SALE) and the total company cost as C = XOPR = COGS + XSGA, where COGS denote cost of goods sold and XSGA denotes selling, general and administrative (*i*.*e*., operating) expenses.

Depreciation and the amortization of acquired intangibles (DP) are excluded from total cost because they are deductible *returns* to the prior acquisition of property, plant & equipment (PPENT), and the prior acquisition of external intangible assets (INTAN).

Like everyone else, I’m forced to use the annual aggregate data of the “tested party” (corporate taxpayer under audit) and of the comparable companies’ book accounts because the disaggregated R* _{i}* = p

*q*

_{i}*variables are not available. These disaggregated company book accounts exist only in the imagination of beguiling advocates.*

_{i}The variable U in equation (5) is the unknown uncertainty (or the random error) because (4) is not an exact relationship.

#### Empirical Estimates of the Operating Profit Markup

As examples, I computed the operating profit (OIBDP same as EBITDA) markups of Best Buy and Walmart using all available data in the interactive RoyaltyStat/Compustat database, and obtained:

*Example 1: Best Buy Co* Inc. (GVKEY 2184)

##### (5.1) R_{t} = 1.0669 C_{t} + U_{t}

_{t}

_{t}

_{t}

Counting 37 years of paired company data, the Newey-West corrected *t*-statistics = 298.6, and the correlation coefficient squared = 0.9998.

*Example 2: Walmart* Inc. (GVKEY 11259)

##### (5.2) R_{t} = 1.071 C_{t} + U_{t}

_{t}

_{t}

_{t}

Counting 50 years of paired company data, the Newey-West corrected *t*-statistics = 311.9, and the correlation coefficient squared = 0.9999.

The high Newey-West corrected *t*-statistics suggest that the standard errors of the regression coefficients are minuscule compared to the estimated operating profit markup.

The minuscule standard errors of the regression coefficients suggest that the empirical estimates of the price equation (5) are reliable.

*Example 3: Group of Ten Major U.S. Retailers *

##### (5.3) R_{t} = 1.0713 C_{t} + U_{t}

_{t}

_{t}

_{t}

Counting 434 years of paired company data, the Newey-West corrected *t*-statistics = 344.6 and the correlation coefficient squared = 0.9996.

The ten major US retailers analyzed in this group are: (1) Best Buy Co Inc. (GVKEY 2184), (2) Conn’s Inc. (15614), (3) Costco Wholesalers Corp. (29028), (4) Dillard’s Inc. (3964), (5) Home Depot Inc. (5680), (6) Kohl’s Corp. (25283), (7) Lowe’s Cos Inc. (6829), (8) Macy’s Inc. (4611), (9) Target Corp. (3813), and (10) Walmart Inc. (11259).

The interquartile range of these historical operating profit markups varies from Q1 = 4.824% to Q3 = 8.89%, with median = 7.001%, average = 7.031%, and trimean = 6.929%. The Tukey’s statistical notches of operating profit markups are more reliable than the quartiles, varying from 6.664% to 7.338%.

**A transfer Pricing Safe Harbor for Retailers **

From this exploratory data analysis (EDA), I conclude that the operating profit markups reported by major US retailers are stable.

The wide operating profit markup variations found among cherry-picked comparables in the US retail industry are a statistical fiction created by the artificial truncation (discretionary clipping) of the available data.

Given the empirical stability of the operating profit markups found among major US retailers, the tax authorities can publish stable multiyear profit margins as safe harbors (*i*.*e*., income tax reporting certainty) to eligible corporate taxpayers. Changing the profit margin from year to year is self-defeating, and is not justified by the robust empirical estimates demonstrated herein.

The safe harbors must exclude controlled retailers primarily engaged in lower volume specialty (differentiated) goods and services (because they have elevated market shares). They should exclude also controlled retailers that create marketing intangibles by deducting substantial advertising expenses.

The characterization of the tested party with at least one of these important attributes (high market share, advertising expenses) as a limited function, limited risks, or limited assets retailer is a legal fiction without theoretical or empirical support.

Like the proposed Section 1.163(j) regulations that set safe harbor limits for related-party interest deductions, the internal accounts from the consolidated corporate taxpayer (in which related party transactions are eliminated) can be used to estimate equation (5) in transfer pricing cases. This is a reliable test of reasonableness of the proposed transfer pricing adjustments made under the comparable profits (CPM ≈ TNMM) or the profit split method.

**References**

Luigi Amoroso, “La Curva Statica di Offerta,” *Giornale degli Economisti* (Serie quarta), Vol. 70, No. 1, January 1930. Stable URL: http://www.jstor.org/stable/23227293.

Alpha Chiang, *Fundamental Methods of Mathematical Economics* (3^{rd} edition (1^{st} edition 1967)), 1984. If u and v are variables subject to change, and *d* is the differential operator, the product rule of differential calculus is as follows: *d* (u v) = v *d* u + u *d* v. The derivative of u with respect to v (*i*.*e*., *d* u / *d* v) is defined as the limit of the difference quotient ∆ u / ∆ v as ∆ v → 0. See Chiang, § 6.4 (The concept of limit), pp. 132-133.

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

Michal Kalecki, “Costs and Prices” (1943, 1954) in his *Selected Essays on the Dynamics of the Capitalist Economy*, Cambridge University Press, 1971.

This chapter about markup pricing was first published by Michal Kalecki, *Studies in Economic Dynamics*, George Allen & Unwin, 1943, pp. 9-31 (“If technical progress reduces average costs of all firms in the same proportion, it does not affect the markup (unless technical change alters the conditions of market imperfection and oligopoly”)).

In the price equation (4), the profit markup λ is a stable constant, such that ∆p* _{i}* = λ

*∆c*

_{i}*and ∆c*

_{i}*< 0 is regarded as a cost-reducing technical change.*

_{i}Instead of using the Cournot-Amoroso-Robinson elasticity of demand approach, the markup price equation can be obtained, as I have done herein, by defining an accounting equation (1a) in which average price = average total costs + average operating profits, and then positing that average operating profits are proportional to average total cost (equation (2a)):

##### (1a) p_{i} = c_{i} + κ_{i}

_{i}

_{i}

_{i}

##### (2a) κ_{i} = φ_{i} c_{i}

_{i}

_{i}

_{i}

##### (3a) p_{i} = c_{i} + φ_{i} c_{i}

_{i}

_{i}

_{i}

_{i}

##### (4) p_{i} = λ_{i} c_{i}

_{i}

_{i}

_{i}

where the operating profit markup factor, λ* _{i}* = (1 + φ

*), can be estimated by regression analysis using linear or double-logarithms (power function) specifications.*

_{i}Richard Layard & Alan Walters, *Microeconomic Theory*, McGraw-Hill, 1978. (“This book is based on the final-year graduate course in microeconomics that we have been giving at the London School of Economics.” Preface, p. vi. “Except in markets for financial assets or for ‘commodities’ (like tin) there are very few sellers who do not have some latitude in the prices they can charge.” P. 235).

Louis Phlips, *The Economics of Price Discrimination*, Cambridge University Press, 1983.

Hal Varian, *Microeconomic Analysis* (3^{rd} edition (1^{st} edition 1978)), Norton, 1992.

**Ednaldo Silva**(Ph.D.) is founder and managing director of RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: esilva@royaltystat.com

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