To measure excess returns attributed to intangibles, we divide the “return on assets” into two components, including a return to tangible assets that is “separate and distinct” from a return to intangible assets. We think of tangible assets consisting primarily of property, plant and equipment, net of accumulated depreciation, and intangible assets consisting of a weighted sum of past investments in R&D, software, and advertising. As such, we define total assets composed of these two components, tangibles (index = 1) and intangibles (index = 2):

(1) A ≡ A_{1} + A_{2}, which implies that

##### (2) A_{1} = A – A_{2}

It follows that the total and separate asset returns (*r*, *r*_{1}, *r*_{2}) can be specified as follows:

* *(3)* r* A = *r*_{1} A_{1} + *r*_{2} A_{2}

We substitute equation (2) into (3) and obtain:

* *(4)* r* A = *r*_{1} (A – A_{2}) + *r*_{2} A_{2} then divide by total assets, and obtain:

* * (5)* r* = *r*_{1} + (*r*_{2} – *r*_{1}) (A_{2 }/ A), which can be expressed as a linear regression equation:

* *(6)* r* = α + β X + random error,

where α = *r*_{1} ≥ 0, β = (*r*_{2} – *r*_{1}) ≥ 0, and the independent variable X = (A_{2 }/ A) ≥ 0 is the ratio of intangible assets to total assets.

To simplify exposition, we abstract from two other subscripts, one indicating company and the other time period. We understand that (unless a cross-section and time-series panel of companies is created) equation (6) applies to every comparable company in the sample covering a specific number of years.

E.g., suppose that we apply regression equation (6) to sample data from comparable companies in which A_{2} > 0, and find these estimates: α = *r*_{1} = 10%, β = (*r*_{2} – *r*_{1}) = 12%, so *r*_{2} = (β − *r*_{1}) = 2%. This means that the *excess return or premium* attributed to intangible assets would be *r*_{2} = 2% above the return attributed to “routine” tangible assets, α = *r*_{1} = 10%.

As a curiosum, we know that the intercept of equation (6) (α = *r*_{1}) consists of two elements, an applicable interest rate (*i*) plus an entrepreneurial or “routine” return (π) attributed to the “risk and trouble” of investing in economic activities (manufacturing, wholesale, retail) and assuming the risk of business failure. In symbols,

##### (7) α = *r*_{1} = (*i* + π) ≥ 0

However, for the purpose of estimating *r*_{2} = (β − *r*_{1}) we don't need equation (7). The remuneration for the “entrepreneurial risk and trouble” is assumed to be semi-positive and independent of the applicable interest rate. In other words, “the normal profits of enterprise” is expected to be π ≥ 0. See Massimo Pivetti, *An* *Essay on Money and Distribution* (St. Martin’s Press, 1991), pp. 24-26. The quaint expression remuneration for the risk and trouble of investing is found in David Ricardo, *Principles of Political Economy and Taxation* (Cambridge University Press, 1951 [1818]), p. 122 (“The farmer and manufacturer can no more live without profit, than the laborer without wages. Their motive for accumulation will diminish with every diminution of profit, and will cease altogether when their profits are so low not to afford them an adequate compensation [above an applicable interest rate, *i*] for their trouble, and the risk which they must necessarily encounter in employing their capital”).

See also Nicholas Kaldor, “Capital Accumulation and Economic Growth,” in F. Lutz & D. Hague (eds.), *The Theory of Capital* (Macmillan, 1963), p. 189. Citing the same p. 122 in Ricardo’s book, Kaldor postulates a well-known condition that capital accumulation (economic growth) requires that the profit rate must be equal to or it must exceed the sum of the interest rate and the normal profits of enterprise, *i*.*e*., *r* ≥ (α = *r*_{1} = (*i* + π)). This condition is also expressed in our equation (6) above, where the excess profit is expressed in our inequality β = (*r*_{2} – *r*_{1}) ≥ 0.