We can utilize payroll and determine the value of intangibles. In David Ricardo (1772-1823), *Principles of Political Economy*, Cambridge University Press, 1951 [1817], the price of a commodity produced in period *t* is determined by a profit markup equation:

(1) P* _{t}* = (1 +

*r*)

*w*L

_{t}where *r* is the profit markup and *w* is the wage rate. The labor input/output coefficient (L_{t }) represents the technology in period *t*, counted backwards. This technology is measured by the amount of labor time per unit of the produced commodity. Payroll (wage and salary bill) in period *t* is W_{t} = *w* L_{t}.

We have one price equation per produced commodity and three unknown variables to be determined: P_{t}, *r* and *w*. Given *w*, we can determine the price and the profit markup with one degree of freedom. Equation (1) is a typical markup price equation that has general applicability. E.g., this is the equation used by accountants and lawyers to bill clients.

Looking back, William Whewell (1794-1866) commented on Ricardo's *Principles* (1831, p. 20): “The machine at the end of last year was worth the wages expended on it, together with profits, that is (1 + *r*) *w* L_{t – 1}.” We changed his notation because Whewell used a prime superscript to represent a dated or past quantity. See William Whewell, *Mathematical Exposition of Some Doctrines of Political Economy*, Augustus Kelley, 1971, p. 20. This is a reprint of economic classics, originally published in *Transactions of the Cambridge Philosophical Society* in 1829, 1831 and 1850.

Whewell criticized Ricardo for excluding "fixed" capital from (1). According to Whewell, “we are to take into account the capital employed as well as the labor. In consequence of this consideration, the prices of commodities will be affected by the proportion and durability of fixed capital requisite for their production.” (*Ibid*., p. 19). Together with other classical economists, including Adam Smith and David Ricardo, Whewell conceived the cost of “fixed capital” in terms of dated quantities of labor time accounting (*Ibid*., p. 20):

(2) C_{t} = (1 + *r*) *w* L_{t} + (1 + *r*)^{2} *w* L_{t – 1} + (1 + *r*)^{3} *w* L_{t – 2} + … + (1 + *r*)^{k} *w* L_{t – k}

In price equation (2), L_{t} is the quantity of labor hours spent this year to produce the fixed capital; L_{t – 1} is the labor hours employed last year; L_{t – 2} is labor hours employed two years ago, etc. Whewell represented L_{t} as “the number of laborers who this year work by means of a machine or any other fixed capital.” (*Ibid*, p. 10.) We believe that the number of labor hours (instead of the number of workers employed) is more à propos.

For simplicity of exposition, Whewell assumed no technical change, which implies that L = L_{t} = L_{t – 1} = L_{t – 2} = … = L_{t – k}*.* This assumption is not harmful because it's not likely that the labor input/output coefficient (technology) will decrease during the making of the commodity being priced. Using this time-invariant technology assumption, we can obtain from (2):

(3) C_{t} = *w* L {(1 + *r*) +(1 + *r*)^{2} + (1 + *r*)^{3} + … + (1 + *r*)^{k}}

(4) C_{t} = *w* L λ = λ W, where

(5) λ = (1 + *r* ){[(1 + *r*)^{k} – 1]/*r*},

and *k* > 1 is the duration of the fixed capital (the time to build any produced asset), including intangibles. In this derivation, we converted a complex variable (C_{t } in (2)) into a simple constant.

Whewell combined the current and past payrolls and obtained a price formula including two instead of Ricardo’s single component (adding equations (1) and (2)):

(6) P_{t} = (1 + *r*) *w* L_{t} + *w* L λ

In contrast to (1), equation (6) includes both current and historical payroll.

Given the “fixed capital” multiplier (λ), the computational direction of equation (6) is from W* _{t}* → P

*. We can apply (6) to measure the current value of specific separable and identifiable intangibles based on the current (R&D or Advertising) payroll attributed to their production or development. The multiplier is based on a calculated profit markup when*

_{t}*k*> 1, the backward time of the payroll expense stream. To gain traction, we can perform a what-if

*sensitivity analysis*based on variations of the profit markup within realistic bounds.

**Ednaldo Silva**(Ph.D.) is founder and managing director at RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: esilva@royaltystat.com

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