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Value of Intangibles Based on Historical Payroll

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

(1)     Pt = (1 + rw Lt

where r is the profit markup and w is the average wage rate. The labor input/output coefficient (L) represents the technology used in period t, counted backwards. This technology is measured by the amount of labor time per unit of the produced property. Payroll (wage and salary bill) in period t is Wt = w Lt.

We have one price equation per produced property and three unknown variables to be determined: Ptr 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 for their business services.

Looking back, William Whewell (1794-1866), a Cambridge polymath, 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 + rw Lt – 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)     Ct = (1 + rw Lt + (1 + r)2 w Lt – 1 + (1 + r)3 w Lt – 2 + … + (1 + r)k w Lt – k

The index k backtracks the payroll history attributed to the property being priced. In price equation (2), Lt is the quantity of labor hours spent this year to produce a specific property such as fixed capital; Lt – 1 is the labor hours employed last year; Lt – 2 is labor hours employed two years ago, etc. Whewell represented Lt as “the number of laborers who this year work by means of a machine or any other fixed capital.”  (Ibid, p. 10.)  However, the number of labor hours (instead of the number of workers employed) employed to price the audited property transfer is more à propos.

For simplicity of exposition, Whewell assumed no technical change, which implies that L = Lt = Lt – 1 = Lt – 2 = … = Lt – k. This assumption is not harmful because the labor input/output coefficients (technology) are not likley to decrease during the making of the property being priced. Using this limited time-invariant technology assumption, we can obtain from (2):

(3)     Ct = w L {(1 + r) +(1 + r)2 + (1 + r)3 + … + (1 + r)k}

(4)     Ct = w L λ = λ W, where

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

and k > 1 is the duration or average time necessary to build the produced property, including intangibles.  In this derivation, we converted a complex variable (Cin (2)) into a simple constant in (5).

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)     Pt = (1 + rw Lt + w L λ

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

We can convert (6) into a simple regression equation of Pt versus Wt:

(7)    Pt = β Wt + α,

where the slope coefficient is β = (1 + r) and the intercept is α = w L λ.

In practice, we can’t use regression analysis to determine the slope and intercept of (7) because in this conception Pt is unknown. Thus, (6) is a horse we can ride. The causal direction of (6) is from Wt → Pt. We can apply (6) to measure the current value of separable and identifiable intangibles based on the current (R&D or Advertising) payroll expenses attributed to DEMPE (development, enhancement, maintenance, protection and exploitation) to the intangible transfer under audit. The multiplier is based on a calculated profit markup when > 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.