Abstract

The perpetual inventory method (PIM) for estimating capital stock (PPENT) is attributed to Raymond Goldsmith (1951). Goldsmith provided conceptual descriptions without explicit algebraic derivations—a ceremonial citation. However, the extended PIM formula with geometrically declining weights is the standard solution to a first-order linear difference equation, documented in textbooks and official manuals (OECD 2001; Barro & Sala-i-Martin 2004).

1. The Problem of Ceremonial Citations

Academic economics exhibits a familiar attribution pattern: a canonical early work becomes the ceremonial citation even when it contains no explicit algebra. Later authors present formulae without derivation, assuming readers know the standard result.

The perpetual inventory method exemplifies this pattern. Goldsmith (1951) is cited as the original source for the PIM formula. Charles Hulten, in his NBER chapter of The Measurement of Capital (1990), noted that studies by pioneers such as Kuznets, Goldsmith, Stigler, and Kendrick have laid the conceptual foundation for capital stock measurement. But conceptual foundations are not algebraic derivations.

Direct examination of Goldsmith (1951) confirms that the extended PIM formula with geometrically declining weights is not specified in that work. My critique of the ceremonial citation is valid: Goldsmith provided no formulae.

2. The Extended PIM Formula

The derivation of PIM proceeds from elementary difference equation algebra.

Equation (1): Capital Stock Change Identity

\Delta K_t = X_t - D_t

Equation (2): Depreciation Proportional to Lagged Capital

D_t = \delta K_{t-1}

Equation (3): Recursive Capital Accumulation

K_t = X_t + \beta K_{t-1}

The coefficient β = (1 − δ) is the capital retention rate. I have not seen this simple beta specification before. See e.g. Usher (1980), p. 100.

Equation (4): Extended PIM with Geometric Declining Weights

K_t = X_t + \beta X_{t-1} + \beta^2 X_{t-2} + \dots + \beta^k K_{t-k}

Equation (4) is found in Hulten (1990) and is posited without explicit derivation.

3.1 The OECD Manual (2001) Shows the Steps

The OECD Measuring Capital manual (2001), Section 3.2.1, presents the substitution step by step. It shows the expansion from equation (3) to equation (4).

3.2 The Continuous Case (Solow 1956)

Solow (1956) derived the continuous-time equivalent:

K(t) = \int e^{-\delta (t - s)} I(s)\, ds

This integral form embodies the same logic as the discrete geometric sum. The textbook by Barro and Sala-i-Martin (2004) presents both discrete and continuous forms.

4. Realistic Historical Attribution

5. What Remains Valid

The critique of ceremonial citations remains valid. Goldsmith (1951) is cited as the “original source” of the PIM formula, yet that work contains no explicit algebra. This is a genuflection to a canonical early work, not an accurate attribution.

Pedagogical expositions—such as Silva (2018), which applied PIM to hard-to-value intangibles in the transfer pricing context—serve a legitimate function: making known results accessible to practitioners unfamiliar with the underlying mathematics. The derivation was considered elementary and was thus often skipped in advanced papers, while it was more fully documented in textbooks and official manuals.

6. Conclusion

Scholars and practitioners should cite the actual sources—OECD (2001, 2009) and standard textbooks—rather than perpetuate ceremonial attributions to Goldsmith.

References (Not exhaustive)

Barro, Robert J., and Xavier Sala-i-Martin (2004). Economic Growth, Second Edition. MIT Press.

Goldsmith, Raymond W. (1951). “A Perpetual Inventory of National Wealth.” Studies in Income and Wealth, Volume 14, pp. 5–73. National Bureau of Economic Research.

Hulten, Charles R. (1990). “The Measurement of Capital.” In Ernst R. Berndt and Jack E. Triplett (eds.), Fifty Years of Economic Measurement. NBER Studies in Income and Wealth, Volume 54. University of Chicago Press.

Jorgenson, Dale. (1963). “Capital Theory and Investment Behavior.” American Economic Review, Vol. 53 (May), pp. 247–259.

Koyck, Leendert. (1954). Distributed Lags and Investment Analysis. North-Holland.

OECD (2001). Measuring Capital: OECD Manual. Paris: OECD Publishing.

OECD (2009). Measuring Capital: OECD Manual, Second Edition. Paris: OECD Publishing.

Silva, Ednaldo. (2018). “Hard to Value Intangibles Sans Mystère.” RoyaltyStat Blog, March 4, 2018. Republished: EdgarStat Blog, March 9, 2026.

Solow, Robert. (1956). “A Contribution to the Theory of Economic Growth.” Quarterly Journal of Economics, Vol. 70, No. 1 (February), pp. 65–94.

Usher, Dan. (1980). Editor. The Measurement of Capital. University of Chicago Press.