1. The equation
The Springer book U.S. Corporate Profits, 1950–2024 estimates, for each of 71 corporations, a bivariate relation between total revenue (REVT) and total operating expense before depreciation (XOPR = COGS + XSGA). The expanded specification adds net property, plant and equipment (PPENT) as a second regressor:
\quad
(1)\qquad
\mathrm{REVT}=\alpha+\beta\,\mathrm{XOPR}+\gamma\,\mathrm{PPENT}+\varepsilonAll variables are Compustat annual mnemonics in nominal dollars; standard errors are Newey–West (HAC). Equation (1) is a revenue equation, but its coefficients are read as profit coefficients through the accounting identity:
\quad
(2)\qquad
\mathrm{REVT}=\mathrm{XOPR}+\mathrm{OIBDP}where OIBDP is operating income before depreciation and amortization. Substituting (2) into (1) and collecting terms:
\quad
(3)\qquad
\mathrm{OIBDP}=\alpha+(\beta-1)\,\mathrm{XOPR}+\gamma\,\mathrm{PPENT}+\varepsilonEquations (1) and (3) are the same regression. The residuals are identical, the intercept is identical, the coefficient on PPENT is identical, and the coefficient on XOPR shifts by exactly one. Nothing is estimated twice; the dependent variable is a matter of presentation. The question of what each coefficient measures is therefore answered once, and the answer holds for both forms.
2. Alpha: the intercept and the ratio bias
Alpha is the component of revenue (equivalently of OIBDP) that does not scale with operating expense or with installed capital. It is not a prediction of revenue at zero cost, which lies outside the sample; it is the vertical displacement of the fitted plane. Its economic importance is indirect but decisive: a nonzero alpha is the condition under which every ratio-based profit indicator is biased.
Take the bivariate case for clarity. If \mathrm{REVT}=\alpha+\beta\,\mathrm{XOPR}, the markup ratio of firm i is,
\quad
(4)\qquad
\frac{\mathrm{REVT}_i}{\mathrm{XOPR}_i}=\beta+\frac{\alpha}{\mathrm{XOPR}_i}The ratio differs from the slope by \alpha/\mathrm{XOPR}_i, a term that varies across firms and across years with the scale of the cost base. Averaging over N observations gives a mean ratio of \beta+\alpha/H, where H is the harmonic mean of XOPR; the quartiles inherit the same displacement through the reciprocal of the quartiles of XOPR. The bias does not vanish with sample size. The same algebra applies to any ratio formed from (3): OIBDP/XOPR, OIBDP/PPENT, or OIBDP/REVT. A significant alpha is a statement that the ratio and the regression coefficient are different numbers, and that only the latter has a standard error.
In the 71-firm sample, alpha is significant for roughly half the corporations. Where it is insignificant, the ratio and the slope coincide within sampling error and the interquartile range of a ratio is defensible as a summary; where it is significant, the ratio is a biased estimator of the markup and the regression coefficient is the object to report.
3. Partial beta: the markup conditional on capital
Beta in (1) is the partial derivative of REVT with respect to XOPR, holding PPENT fixed; \beta-1 in (3) is the corresponding partial markup of OIBDP on operating cost. The word partial carries the content. By the Frisch–Waugh–Lovell theorem, \beta equals the slope from regressing REVT on the residual of XOPR after projecting XOPR on PPENT and a constant: it is the markup on the part of operating cost that is not explained by the size of the fixed capital stock. If XOPR and PPENT were uncorrelated, the partial \beta and the bivariate \beta of the Springer book would coincide; they are correlated within every corporation’s time series, so the two differ, and the difference measures how much of the bivariate markup was a proxy for capital intensity.
The interpretation that follows is the Kaleckian one. \beta-1 is the degree of monopoly expressed as a markup on prime cost, purged of the revenue that accompanies capital in place. For a corporation whose profits are priced on cost, \beta-1 captures the whole of operating profit and \gamma is small; for one whose revenue is tied to installed capacity independently of the current cost base, part of the profit migrates to \gamma.
One point of notation that matters for the reader: XOPR excludes depreciation (DFXA) and amortization (AM). Beta is therefore a markup gross of capital consumption, which is the right measure when the markup is the object: it is unaffected by the useful-life conventions that govern DFXA and by the acquisition accounting that governs AM.
4. Partial gamma: a rent on installed capital, not a return on assets
Gamma is the partial derivative of OIBDP with respect to PPENT, holding XOPR fixed. The temptation is to read it as a return on assets, because its units are dollars of profit per dollar of capital. The reading fails on three counts, each sufficient alone.
Numerator. The accounting return on assets is defined on operating income after depreciation and amortization, \mathrm{OIADP}=\mathrm{OIBDP}-\mathrm{DFXA}-\mathrm{AM}. Gamma is defined on OIBDP. The two numerators differ by the entire charge for capital consumption, which for capital-intensive firms is a large share of OIBDP.
Denominator. The ratio \text{ROA}=\mathrm{OIADP}/(\mathrm{PPENT}+\mathrm{INTANO}) divides by tangible plus intangible operating assets. Gamma is a coefficient on PPENT alone. Intangible assets (INTANO), which for acquisitive corporations may exceed PPENT, are absent from (1).
Form. ROA is an average: total operating income over total assets, intercept and cost channel included. Gamma is a marginal effect with the cost channel held constant. The two coincide only under \alpha=0 and \beta=1, the conditions under which the regression collapses to proportionality; the book’s estimates reject both for every corporation. The relation between them is the same as the relation between the markup ratio and the regression markup in Section 2, with PPENT in place of XOPR in the denominator.
What gamma does measure is the increment to operating profit before depreciation that accompanies an additional dollar of net fixed assets once the current cost base has been paid for. That is closer to a rent on capacity in place than to a return on the book value of assets. A corporation with pricing power over its installed base (a refinery, a pipeline, a fab at full utilization) can show a positive \gamma; one whose plant merely supports a cost-priced output shows a \gamma near zero, and the sign can turn negative when a build-up of PPENT precedes the revenue it is meant to generate.
The empirical result on the 71 corporations is consistent with this reading. Gamma is unstable in sign and insignificant for most firms. That is not a failure of the specification; it is the finding. For the large U.S. corporations in the sample, operating profit is priced on cost, and the fixed capital stock adds little independent explanatory power once cost is in the equation. The result is the firm-level counterpart of the argument against asset-based profit level indicators in transfer pricing: a return on assets presupposes that profit is generated in proportion to assets, and the data for these corporations do not support the presupposition.
5. A regression that would nest ROA
If the object of interest were the return on assets, the regression to estimate would be:
\quad
(5)\qquad
\mathrm{OIADP}=c+\rho\,(\mathrm{PPENT}+\mathrm{INTANO})+uwith \rho the regression counterpart of ROA and c the intercept whose significance decides whether the ratio ROA is biased, by the same harmonic-mean argument as Section 2. Equation (5) and equation (3) are two different behavioral hypotheses: profit priced on assets versus profit priced on cost. They can be nested in a single equation with both regressors, and the data decide which channel carries the weight. Section 4 reports what they decide for the 71 corporations.
6. Summary
Alpha is the displacement that makes ratios biased; its significance is a test of whether any ratio PLI is admissible for the firm.
Beta − 1 is the Kaleckian markup on operating cost, conditional on installed capital, gross of capital consumption.
Gamma is the partial profit effect of net fixed assets with cost held constant: a rent on capacity, defined on OIBDP and PPENT, and not the return on assets, which is defined on OIADP and PPENT + INTANO.
References
Davidson, R. and J. G. MacKinnon (2004), Econometric Theory and Methods, Oxford University Press. Chapter 2 for the Frisch–Waugh–Lovell theorem and the meaning of partial regression coefficients.
Frisch, R. and F. V. Waugh (1933), Partial Time Regressions as Compared with Individual Trends, Econometrica 1(4), 387–401. Original statement of the partialling-out result.
Hall, R. L. and C. J. Hitch (1939), Price Theory and Business Behaviour, Oxford Economic Papers 2, 12–45. Survey evidence that firms price on full cost plus a conventional margin.
Kalecki, M. (1971), Selected Essays on the Dynamics of the Capitalist Economy, 1933–1970, Cambridge University Press. Chapter 5, Costs and Prices, for the markup on prime cost as the degree of monopoly.
Lovell, M. C. (1963), Seasonal Adjustment of Economic Time Series and Multiple Regression Analysis, Journal of the American Statistical Association 58, 993–1010. Extension of Frisch–Waugh to general partitioned regressions.
Newey, W. K. and K. D. West (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica 55(3), 703–708. Standard errors used throughout.
Silva, E. A. (2026), U.S. Corporate Profits, 1950–2024: Operating Profit Markups in Leading U.S. Firms, SpringerBriefs in Economics, Springer. The bivariate model and the 71-firm sample.
Zeileis, A. (2004), Econometric Computing with HC and HAC Covariance Matrix Estimators, Journal of Statistical Software 11(10). Implementation reference cited in the book.