Linear Profit Function
The typical OECD TNMM (CPM in the U.S.) prescribes a linear statistical function to test the arm's length character of “net” profits (Y) in terms of the net sales (X):
(1) Yi = α Xi considering i = 1, 2, …, N comparables
where α is the estimated “net” profit margin. For simplification, we set aside a random error term that is added to equation (1). The controlled taxpayer ("tested party") is the case N + 1.
Non-Linear Profit Function
Instead of equation (1), "net" profits and sales may be represented by a power function:
(2) Yi = α Xiβ
Power functions are pervasive in economic estimates. Equation (2) states that Yi is proportional to Xiβ . In this case, the profit margin is the slope coefficient of equation (2), which below we show is different from α. A power function is appropriate e.g. when the selection of comparables to the tested party includes small and large companies or when the residual variance is not constant.