**Zero Intercept 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) Y_{i} = α X_{i} 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) Y_{i} = α X_{i}^{β}

Power functions are pervasive in economic estimates. Equation (2) states that Y_{i} is proportional to X_{i}^{β }. 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.