Two regressions are implicit in transfer pricing benchmarking: the linear regression of operating profits on revenue, and the double-log regression of the same two variables. Each produces a slope coefficient. The two coefficients have different meanings, different units, and answer different questions.
This note shows that the product of the two slope coefficients has its own interpretation. It equals the average rate of change of operating profits per dollar of revenue, corrected for the curvature of the underlying relationship. We call this product the adjusted slope, and the rest of this post explains why.
The Two Slopes
Let Y denote operating profits and X denote revenue. The two regressions of interest are:
\text{Linear (no intercept): }
Y = b \cdot X \\
\text{Double-log: }
\ln Y = a + \beta \cdot \ln XThe linear regression estimate b — the IRS and OECD-prescribed slope — has units of Y per unit of X: cents of operating profits per dollar of revenue, in the typical case. It is a representative operating margin for the sample.
The double-log regression estimate β is dimensionless. It is the elasticity: the percentage change in Y associated with a one-percent change in X.
These two coefficients answer different questions. The linear slope tells you, in dollars and cents, how profits scale with revenue on average. The elasticity tells you, in percentages, how profits scale with revenue proportionally. The first has the right units to be a slope. The second has the right curvature to describe the marginal effect when the relationship between profits and revenue is not exactly linear.
One Identity Connects Them
Suppose the underlying relationship between profits and revenue takes the form of a power function:
Y = \alpha \cdot X^{\beta}This is the form that the double-log regression assumes. Take the derivative of Y with respect to X:
\frac{dY}{dX}
=
\beta \cdot \alpha \cdot X^{\beta-1}
=
\beta \cdot \left(\frac{Y}{X}\right)
=
\beta \cdot mwhere m = Y / X is the operating margin. The identity at the right end of this line is the engine of everything that follows. In words: at every point on the power curve, the marginal slope equals the elasticity times the operating margin. The elasticity is the operator that converts a margin (with units) into a marginal slope (also with units). The profit margin furnishes the dimensions; the elasticity furnishes the curvature correction.
What the Linear Slope Actually Measures
Running OLS of Y on X without an intercept, as the U.S. and OECD guidelines specify, produces a slope that is algebraically a weighted average of the individual operating margins in the sample
b
=
\sum w_i \cdot m_i,
\qquad
\text{where }
w_i \propto X_i^2The weights wᵢ are proportional to the square of each company’s revenue. Larger companies receive disproportionately more weight than smaller ones. This is a methodological feature of the OECD-prescribed estimator, not a neutral default, and it is worth naming explicitly. The linear slope b is thus a size-weighted representative margin for the sample.
Why the Product Is the Right Object
The structural identity says that at every firm in the sample,
\left(\frac{dY}{dX}\right)_i
=
\beta \cdot m_iTake the same size-weighted average that produced the linear regression slope b from the margins mᵢ, but apply it to (dY/dX)ᵢ instead. Because β is a constant that does not vary across firms, it passes through the averaging:
\text{size-weighted average of }
\frac{dY}{dX}
=
\beta \cdot
(\text{size-weighted average of } m)
=
\beta \cdot bReplacing β with the elasticity estimated from the double-log regression yields the identity advertised in the lead:
\text{Adjusted slope}
=
\beta \cdot bThe product of the two regression slopes is the size-weighted average marginal rate of change of operating profits with respect to revenue across the sample. It has the units of a slope, and it accounts for the curvature of the underlying relationship.
A Worked Example
The companion EdgarStat blog post on power functions reports two regressions fitted to a panel of five U.S. retailers (167 paired observations of revenue and operating profits):
\text{Linear regression slope: }
b = 0.071 \\
\text{Double-log regression slope: }
\beta = 1.0219The adjusted slope is the product of the two:
\beta \cdot b = 1.0219 \times 0.071 = 0.0726
On average, weighted by the square of revenue across the five retailers, a one-dollar change in revenue is associated with a 7.26-cent change in operating profits. The linear regression alone reports 7.1 cents — a 2.2 percent underestimate of the average marginal effect. The 2.2 percent correction is exactly the elasticity premium (β − 1 = 0.0219), applied multiplicatively to the linear slope.
Had the elasticity come in below one — say β = 0.95 — the correction would have run in the opposite direction, and the adjusted slope would have been smaller than the linear slope. The sign of the correction is the sign of β − 1: positive when the underlying function is mildly convex, negative when concave.
Why This Matters in Practice
Three takeaways for the practitioner.
First. The linear regression slope, on its own, is biased away from the true average marginal effect whenever the underlying relationship is not exactly linear. The bias understates the marginal effect when β > 1 and overstates it when β < 1. The forced-zero-intercept regression specified by the U.S. and OECD guidelines is silent about this bias, but the algebra is not.
Second. The double-log slope, on its own, is dimensionless. It is the right tool for benchmarking proportional changes, but it cannot stand in for a dollars-and-cents marginal effect. A 1 percent change in revenue produces a 1.02 percent change in operating profits, but the practitioner negotiating an intercompany pricing adjustment needs the answer in cents per dollar — and the elasticity does not provide that.
Third. The product β · b carries the dimensions of the linear slope and the curvature correction of the elasticity. It is the natural object of comparison when the question is, How much do operating profits actually change in dollars per dollar of revenue, given that the relationship is not exactly linear? That is the question benchmarking is asking. The adjusted slope is the answer.
A Final Note on the Weighting
The identity Adjusted slope = β · b holds for any consistent summary of the operating-margin distribution. Whichever weighting is used to construct the representative margin b — size-weighted as in the OECD specification, arithmetic mean across firms, harmonic mean, median, or any quartile — the elasticity β converts that summary into the corresponding summary of the marginal slope. The size-weighting is implicit in the OECD linear regression; the harmonic-mean weighting is implicit in the operating-margin ratio Y / X, where a separate bias (treated elsewhere on this blog) arises. What unifies these cases is the structural identity dY/dX = β · m: the elasticity is always the operator that converts a margin into a marginal slope, whatever weighting the practitioner has chosen to summarize the margins. The principal takeaway is that taking the quartiles of comparable profit indicators (assuming zero intercept and no curvature) is a path to audit disaster because this malpractice can’t survive scrutiny.