Tutorial: Power Functions in Transfer Pricing

February 17, 2022 by Ednaldo Silva
About the Author
Ednaldo Silva
(Ph.D. Econ.) is a leading economist with over 25 years’ experience in transfer pricing and the valuation of intangibles.
He is founder and managing director of RoyaltyStat, a boutique data, software and consulting firm specializing in transfer pricing and intellectual property solutions. Dr. Silva was first Sr. Economic Advisor at the IRS Office of Chief Counsel, a drafting member of U.S. 26 IRC section 1.482 transfer pricing regulations, and introduced the “comparable profits method” (CPM in the US and TNMM in the OECD), “best method” rule and the concept of arm’s-length represented by a range of results, rather than point estimate.
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A power function has the form: 

(1) Y = X β 

where the slope parameter β is determined from data comparable to the tested party. Y is the dependent variable and X is the independent (explanatory or predictor) variable.

The contrived linear function: Operating Profits = β Revenue, with the forced zero intercept specified in the U.S. and OECD guidelines, and the quadratic and cubic functions arepower functions. Other power functions include the reciprocal and squared-root functions. 

Typical power functions in economics include: 

  1. Linear: Y = β X, like the U.S. and OECD profit indicators. 
  2. Quadratic: Y = X 2, where β = 2. 
  3. Cubic: Y = X 3, where β = 3. 
  4. Reciprocal: Y = 1 / X, where β = −1. 
  5. Squared root: Y = √ X = X 1/2, where β = 1/2.  

Here, we examine a power function that is prevalent in economics: 

(2) Y = α X β 

and take the first derivative of (2) to obtain the slope coefficient: 

(3) d Y / d X = β α X β – 1 

Slope = β α (X β / X) = β (Y / X). 

Except for the special case in which α = β = 1, we can use the double-logarithms regression to estimate equation (2): 

(4) LN(Y) = LN(α) + β LN(X) 

Now, we consider the best regression fit among two rival operating profits functions using actual company-level annual data: 

(5) Y = α + β X,

versus the power equation (2) or its regression version (4). 

Chart 1 contains the linear regression: Y = 0.071 X, with the Newey-West t-statistics of the slope = 7.8158 and the R2 = 0.8032. The alpha-intercept is not significant, so we don’t report it. 

Chart 1: Linear Regression

Chart 2 contains in this case a superior (more reliable) double-logarithms regression (4): LN(Y) = 1.0219 LN(X) − 2.938, with the Newey-West corrected t-statistics of the intercept = − 8.6162, the t-statistics of the slope coefficient = 26.035, and the R2 = 0.9043. Each regression contains 167 annual paired X and Y observations. 

Chart 2: Double-Logrithms Regression 

 

As takeaway, we must examine the comparable company data using bivariate scatterplots instead of assuming the special (contrived) linear equation without an intercept: 

(6) Y = β X,

where the intercept is forced to be α = 0. 

The regression results above were computed online in EdgarStat using the historical pairs of X = Revenue and Y = Operating Profits (after depreciation and amortization) of five U.S. retailers :  

  1. 73119 Bed, Bath & Beyond (BBBY), 1990-2021 
  2. 27050 Best Buy Inc. (BBY), 1983-2021 
  3. 107748 Conn's Inc. (CONN), 2000-2021 
  4. 5067 Lowe’s Companies Inc. (LOW), 1984-2021 
  5. 36090 Williams Sonoma Inc. (WSM), 1986-2021 
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