In U.S. Corporate Profits, 1950–2024 (Springer, 2026), I estimated the operating profit markup of 71 leading U.S. corporations with a bivariate regression, \mathrm{REVT}=a+b\,\mathrm{XOPR}. The slope b is the markup on operating costs, and the operating margin follows from it by indirect least squares (ILS). The regression is the reduced form of a two-equation system: an accounting identity and a structural profit equation. This note enlarges that system by one variable, lagged net property, plant and equipment, PPENT(-1), and derives the new reduced form. The expanded model nests the book specification, turns its maintained assumption into a testable restriction, and separates the markup on current costs from the recovery of, and return to, operating assets.
1. Compustat mnemonics
The model is written in Compustat mnemonics so that each symbol maps to a data item. Entity i=1,\dots,n; year t=1,\dots,T; sample size N=nT.
| Mnemonic | Compustat item |
|---|---|
| REVT | Revenue – Total |
| XOPR | Operating Expenses – Total (cost of goods sold plus XSGA, excluding depreciation and amortization) |
| OIBDP | Operating Income Before Depreciation = REVT – XOPR |
| DP | Depreciation and Amortization – Total |
| OIADP | Operating Income After Depreciation = OIBDP – DP |
| DFXA | Depreciation of Tangible Fixed Assets |
| AM | Amortization of Intangibles |
| PPENT | Property, Plant and Equipment – Total (Net) |
With \mathrm{DP}=\mathrm{DFXA}+\mathrm{AM}, operating income before amortization of intangibles is defined as:
\quad
(1)\qquad
\mathrm{OIBAM}=\mathrm{OIADP}+\mathrm{AM}=\mathrm{OIBDP}-\mathrm{DFXA}OIBAM charges the depreciation of tangible capital against revenue but excludes the amortization of acquired intangibles. The Compustat relation \mathrm{DP}=\mathrm{DFXA}+\mathrm{AM} holds as an approximation: DP can carry other amortization, and DFXA coverage is incomplete, so the construction of OIBAM needs a data audit firm by firm.
2. The equation system
From (1), the exact per-entity identity is \mathrm{REVT}=\mathrm{XOPR}+\mathrm{DFXA}+\mathrm{OIBAM}. The model replaces reported tangible depreciation with a geometric charge on the opening net capital stock, \mathrm{DFXA}=\delta\,\mathrm{PPENT}(-1)+u, where \delta is the depreciation rate and u absorbs the departure of reported depreciation from the geometric rule. The system is:
\begin{aligned}
&(2)\quad
\mathrm{REVT}(i,t)=\mathrm{XOPR}(i,t)+\delta\,\mathrm{PPENT}(i,t-1)+\mathrm{OIBAM}(i,t)+u(i,t)
\\[8pt]
&\quad
u(i,t)=\mathrm{DFXA}(i,t)-\delta\,\mathrm{PPENT}(i,t-1)
\end{aligned}\quad
(3)\qquad
\mathrm{OIBAM}(i,t)=\alpha+\beta\,\mathrm{REVT}(i,t)+\gamma\,\mathrm{PPENT}(i,t-1)+\varepsilon(i,t)Equation (2) is an identity once u is included; the substitution of \delta\,\mathrm{PPENT}(-1) for DFXA is a behavioral postulate, not an accounting fact. Equation (3) is the structural profit equation: \beta is the marginal operating margin on revenue, \gamma is the effect of the opening capital stock on operating income given revenue, and \alpha is the fixed component. REVT and OIBAM are jointly determined; XOPR and PPENT(-1) are predetermined. The capital stock enters with a one-period lag because it is in place at the start of the year and cannot respond to current-year revenue.
3. The reduced form
Substitute (3) into (2):
\quad
\mathrm{REVT}=\mathrm{XOPR}+\delta\,\mathrm{PPENT}(-1)+\left[\alpha+\beta\,\mathrm{REVT}+\gamma\,\mathrm{PPENT}(-1)+\varepsilon\right]+uCollect the REVT terms on the left and the PPENT(-1) terms on the right:
\begin{aligned}
&\quad
\mathrm{REVT}-\beta\,\mathrm{REVT}=\alpha+\mathrm{XOPR}+\delta\,\mathrm{PPENT}(-1)+\gamma\,\mathrm{PPENT}(-1)+\varepsilon+u
\\[8pt]
&\quad
(1-\beta)\,\mathrm{REVT}=\alpha+\mathrm{XOPR}+(\delta+\gamma)\,\mathrm{PPENT}(-1)+(\varepsilon+u)
\end{aligned}Divide by (1-\beta), which is positive whenever the marginal profit margin is below one:
\quad
\mathrm{REVT}=\frac{\alpha}{1-\beta}+\left[\frac{1}{1-\beta}\right]\mathrm{XOPR}+\left[\frac{\delta+\gamma}{1-\beta}\right]\mathrm{PPENT}(-1)+\frac{\varepsilon+u}{1-\beta}Relabeling the coefficients gives the reduced form:
\quad
(4)\qquad
\mathrm{REVT}(i,t)=\pi_0+\pi_1\,\mathrm{XOPR}(i,t)+\pi_2\,\mathrm{PPENT}(i,t-1)+v(i,t)\begin{aligned}
&\quad
\pi_0=\frac{\alpha}{1-\beta}
&&\qquad
\pi_1=\frac{1}{1-\beta}
\\[8pt]
&\quad
\pi_2=\frac{\delta+\gamma}{1-\beta}
&&\qquad
v=\frac{\varepsilon+u}{1-\beta}
\end{aligned}Recovering the structural parameters
The mapping from (\pi_0, \pi_1, \pi_2) to the structural parameters is one-to-one in \alpha, \beta, and the sum \delta+\gamma:
\begin{aligned}
&\quad
\pi_1=\frac{1}{1-\beta}
\quad\Longrightarrow\quad
1-\beta=\frac{1}{\pi_1}
\quad\Longrightarrow\quad
\beta=1-\frac{1}{\pi_1}=\frac{\pi_1-1}{\pi_1}
\\[8pt]
&\quad
\pi_0=\frac{\alpha}{1-\beta}=\alpha\pi_1
\quad\Longrightarrow\quad
\alpha=\frac{\pi_0}{\pi_1}
\\[8pt]
&\quad
\pi_2=\frac{\delta+\gamma}{1-\beta}=(\delta+\gamma)\pi_1
\quad\Longrightarrow\quad
\delta+\gamma=\frac{\pi_2}{\pi_1}
\end{aligned}The reduced form identifies \delta+\gamma, not \delta and \gamma separately. Separation requires one outside restriction. The plausible source is the depreciation data: estimate \delta from \mathrm{DFXA}/\mathrm{PPENT}(-1) by entity, then set \gamma=\pi_2/\pi_1-\delta.
Identification and estimation
REVT is endogenous in (3): by (4) it contains \varepsilon, so OLS on (3) is inconsistent. XOPR is excluded from (3) and appears in (4), which makes (3) exactly identified by the order condition, with XOPR as the one instrument. In the exactly identified case, indirect least squares (OLS on (4), then the mapping above) and two-stage least squares on (3) yield the same estimates. The error of (4) combines the behavioral error \varepsilon and the depreciation discrepancy u; when DFXA is available, using it in place of \delta\,\mathrm{PPENT}(-1) removes u and makes (2) exact.
4. The book model is a restricted case
Remove capital from the system. The book identity and structural equation are:
\begin{aligned}
&\quad
\mathrm{REVT}=\mathrm{XOPR}+\mathrm{OIBDP}
\\[8pt]
&\quad
\mathrm{OIBDP}=\alpha+\beta\,\mathrm{REVT}+\varepsilon
\end{aligned}Substitution and the same steps as in Section 3 give:
\begin{aligned}
&(5)\quad
\mathrm{REVT}=\frac{\alpha}{1-\beta}+\left[\frac{1}{1-\beta}\right]\mathrm{XOPR}+\frac{\varepsilon}{1-\beta}
\\[8pt]
&\phantom{(5)\quad \mathrm{REVT}}
=a+b\,\mathrm{XOPR}+e,
\qquad
b=\frac{1}{1-\beta},
\qquad
\beta=1-\frac{1}{b}
\end{aligned}Equation (5) is my book regression. It is equation (4) under the restriction \pi_2=0, with the profit concept moved from OIBAM to OIBDP. The markup b and the profit margin \beta are the same reduced-form and structural objects in both systems. The expanded model therefore does not discard my book results; it asks whether they survive the inclusion of fixed capital.
Omitted-variable consequence
If \pi_2 differs from zero and (5) is estimated, the probability limit of the bivariate slope is:
\quad
(6)\qquad
\operatorname{plim} b=\pi_1+\pi_2\lambda,
\qquad
\lambda=\frac{\operatorname{Cov}\left[\mathrm{XOPR},\mathrm{PPENT}(-1)\right]}{\operatorname{Var}\left[\mathrm{XOPR}\right]}where \lambda is the slope of the auxiliary regression of PPENT(-1) on XOPR. Because costs and capital both scale with firm size, \lambda is positive, and the bivariate markup absorbs a capital charge in proportion to \pi_2. For capital-intensive entities the book markup is then a composite of pricing on current costs and the recovery of fixed capital. Equation (4) separates the two.
5. The ratio bias, extended
In earlier notes I showed that for Y=a+bX the ratio Y/X=b+a/X has mean b+a/H, where H is the harmonic mean of X, so that ratio profit level indicators are biased by a/H whenever the intercept differs from zero. The expanded system extends the result. Divide (3) by REVT to obtain the operating profit margin:
\quad
\frac{\mathrm{OIBAM}}{\mathrm{REVT}}=\beta+\frac{\alpha}{\mathrm{REVT}}+\gamma\frac{\mathrm{PPENT}(-1)}{\mathrm{REVT}}+\frac{\varepsilon}{\mathrm{REVT}}Average over the N observations. Since the mean of 1/\mathrm{REVT} is 1/H(\mathrm{REVT}):
\begin{aligned}
&(7)\quad
\operatorname{mean}\left[\frac{\mathrm{OIBAM}}{\mathrm{REVT}}\right]=\beta+\frac{\alpha}{H(\mathrm{REVT})}
\\[8pt]
&\phantom{(7)\quad \operatorname{mean}\left[\frac{\mathrm{OIBAM}}{\mathrm{REVT}}\right]}
+\gamma\operatorname{mean}\left[\frac{\mathrm{PPENT}(-1)}{\mathrm{REVT}}\right]+\operatorname{mean}\left[\frac{\varepsilon}{\mathrm{REVT}}\right]
\end{aligned}The same operation on the reduced form (4), divided by XOPR, gives the markup ratio:
\begin{aligned}
&(8)\quad
\operatorname{mean}\left[\frac{\mathrm{REVT}}{\mathrm{XOPR}}\right]=\pi_1+\frac{\pi_0}{H(\mathrm{XOPR})}
\\[8pt]
&\phantom{(8)\quad \operatorname{mean}\left[\frac{\mathrm{REVT}}{\mathrm{XOPR}}\right]}
+\pi_2\operatorname{mean}\left[\frac{\mathrm{PPENT}(-1)}{\mathrm{XOPR}}\right]+\operatorname{mean}\left[\frac{v}{\mathrm{XOPR}}\right]
\end{aligned}A ratio indicator now carries two distortions: the intercept term, which is the harmonic-mean bias of the bivariate case, and a capital-intensity term, which differs across comparables with different asset structures. Neither vanishes as N grows. Transfer pricing practice responds to the second with ad hoc asset intensity adjustments; the regression estimates it as a parameter.
6. What the expanded model adds
| Feature | Bivariate model (5) | Expanded model (4) |
|---|---|---|
| Regressors | XOPR | XOPR, PPENT(-1) |
| Profit concept | OIBDP (before all D&A) | OIBAM (after tangible depreciation, before amortization) |
| Status of capital | Excluded by assumption | Included; exclusion is the testable restriction \pi_2=0 |
| Markup interpretation | Composite of cost pricing and capital recovery when \pi_2 differs from 0 | Markup on current costs, holding capital fixed |
| Capital parameters | None | \delta+\gamma identified; \gamma recoverable given \delta from DFXA |
| Acquisition accounting | Neutral (before AM) | Neutral (before AM) |
| Ratio bias | \alpha/H | \alpha/H plus a capital-intensity term |
Four advantages follow. First, the book specification becomes a hypothesis rather than a premise. Second, the markup is purged of the capital charge described by (6). Third, a margin indicator and a return-on-capital indicator sit inside one equation, so the choice between them under Treas. Reg. §1.482-5(b)(4) becomes an empirical question about \pi_2 rather than a matter of assertion. Fourth, OIBAM excludes the amortization of acquired intangibles, which neutralizes differences in acquisition history across comparables while still charging the wear of tangible capital.
7. Evidence from the 71 corporations
I estimated (4) on the 71 corporations of the Springer book. The estimates of \pi_1 behave as in the bivariate model. The estimates of \pi_2, the reduced-form coefficient on PPENT(-1), do not: signs change across corporations and industries, and for most corporations the interval \pi_2\pm\mathrm{SE}(\pi_2) contains zero. Five readings of this result deserve separation.
The null may be the finding
Suppose corporations price on current costs, so that revenue given XOPR does not respond to the capital stock. Then, holding REVT and XOPR fixed, identity (2) implies that one more dollar of opening capital raises the depreciation charge by \delta and lowers OIBAM by \delta. The structural equation must reproduce this, so \gamma=-\delta, and \pi_2=(\delta+\gamma)/(1-\beta)=0. Under full-cost pricing in the Kalecki and Hall-Hitch tradition, depreciation is recovered through the markup on current costs, and \pi_2 is centered on zero. An estimate centered on zero changes sign from sample to sample. The restriction \pi_2=0 is then supported, and the bivariate book model is vindicated rather than contradicted.
Scale collinearity
XOPR and PPENT(-1) both scale with firm size. The sampling variance of the estimate of \pi_2 is inflated by the factor 1/(1-r^2), where r is their correlation. For large corporations r is high, and the capital stock moves slowly within a firm, so the data carry little independent information about \pi_2.
Slope heterogeneity
The composite coefficient \delta+\gamma differs between capital-intensive industries (petroleum refining, utilities, railroads) and asset-light ones (software, pharmaceuticals, government contracting). Pooled estimates average slopes of opposite sign, and the average moves with sample composition.
Unrecorded assets (capital)
Research and development and much of XSGA are expensed inside XOPR. The capital that earns the return in asset-light corporations sits outside PPENT, and its return loads on \pi_1 rather than \pi_2.
Measurement of the capital stock
When the depreciation rate varies across or within entities, u=\mathrm{DFXA}-\delta\,\mathrm{PPENT}(-1) is correlated with PPENT(-1), which biases the estimate of \pi_2. Net PPE is also distorted by asset vintage, impairments, and acquisitions, and the 2019 adoption of ASC 842 moved operating lease assets onto the balance sheet, which breaks the homogeneity of PPENT across the sample period.
8. Precedent and novelty
Asset intensity has entered price-cost margin regressions before. Collins and Preston (1968) and Domowitz, Hubbard and Petersen (1986) regress industry price-cost margins on concentration and the capital/output ratio. Those specifications are ratio equations, estimated by OLS on industry aggregates, with no accounting identity closing the system; by (7) and (8), their dependent variables carry the intercept and capital-intensity distortions derived above. De Loecker, Eeckhout and Unger (2020) and Traina (2018) use Compustat cost and capital items to estimate markups, but through a production function rather than a pricing equation. To my knowledge, the specification here is new in combining four elements: levels rather than ratios, firm-level Compustat data, a structural profit equation closed by the operating income identity, and a reduced form whose coefficients map back to the markup, the profit margin, and the operating assets parameters.
9. Next steps
- Replace \delta\,\mathrm{PPENT}(-1) with reported DFXA where coverage permits, making (2) exact, and estimate \delta separately.
- Test the restriction \pi_2=0 entity by entity and pooled, and report the variance inflation factor alongside \pi_2\pm\mathrm{SE}(\pi_2).
- Allow \pi_2 to vary by capital-intensity group, and estimate (3) by 2SLS with XOPR as the instrument as a check on the ILS mapping.
- Compare net PPENT with gross PPE (PPEGT), and split the sample at the ASC 842 adoption.
- Weight by scale (GLS) to address heteroskedasticity proportional to firm size.
References
Araquem Silva, E. (2026). U.S. Corporate Profits, 1950–2024: Operating Profit Markups in Leading U.S. Firms. SpringerBriefs in Economics. Cham: Springer. doi:10.1007/978-3-032-11399-3. — the bivariate markup regression for 71 corporations extended here.
Basmann, R. L. (1957). A generalized classical method of linear estimation of coefficients in a structural equation. Econometrica, 25(1), 77–83. — two-stage least squares.
Belsley, D. A., Kuh, E., & Welsch, R. E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. New York: Wiley. — variance inflation and collinearity diagnostics.
Collins, N. R., & Preston, L. E. (1968). Concentration and Price-Cost Margins in Manufacturing Industries. Berkeley: University of California Press. — capital-output ratio in margin regressions.
De Loecker, J., Eeckhout, J., & Unger, G. (2020). The rise of market power and the macroeconomic implications. Quarterly Journal of Economics, 135(2), 561–644. — production-function markups from Compustat.
Domowitz, I., Hubbard, R. G., & Petersen, B. C. (1986). Business cycles and the relationship between concentration and price-cost margins. RAND Journal of Economics, 17(1), 1–17. — margin equation with capital-output ratio.
Financial Accounting Standards Board (2016). Accounting Standards Update 2016-02, Leases (Topic 842). Norwalk, CT: FASB. — lease capitalization affecting PPENT comparability.
Goldberger, A. S. (1964). Econometric Theory. New York: Wiley. — structural and reduced forms; indirect least squares.
Haavelmo, T. (1943). The statistical implications of a system of simultaneous equations. Econometrica, 11(1), 1–12. — simultaneity bias of OLS on a structural equation.
Hall, R. L., & Hitch, C. J. (1939). Price theory and business behaviour. Oxford Economic Papers, 2, 12–45. — full-cost pricing.
Kalecki, M. (1971). Selected Essays on the Dynamics of the Capitalist Economy, 1933–1970. Cambridge: Cambridge University Press. — markup pricing on prime costs.
Koopmans, T. C. (1949). Identification problems in economic model construction. Econometrica, 17(2), 125–144. — order and rank conditions.
Kuh, E., & Meyer, J. R. (1955). Correlation and regression estimates when the data are ratios. Econometrica, 23(4), 400–416. — spurious results from ratio variables.
Maddala, G. S. (1977). Econometrics. New York: McGraw-Hill. — simultaneous equations estimation.
Traina, J. (2018). Is aggregate market power increasing? Production trends using financial statements. Stigler Center New Working Paper Series No. 17, University of Chicago. — markups with Compustat XOPR.
U.S. Treasury Regulations. 26 CFR §1.482-5, Comparable profits method. — profit level indicators, including return on capital employed.
Weisberg, S. (2014). Applied Linear Regression (4th ed.). Hoboken, NJ: Wiley. — regression and weighted least squares.