1. The System in Three Equations
Start from the GDP expenditure identity, collect autonomous expenditure as A(t) = G(t) + X(t) - M(t) and decompose it into its mean and its fluctuation, A(t) = A + [A(t) - A]. Close the identity with two behavioral equations, both linear in lagged income: consumption with a one-period Robertson lag, and gross investment as an unrestricted two-coefficient accelerator:
(1)\quad Y(t)=C(t)+I(t)+A(t) \\
(2)\quad C(t)=c_0+c\,Y(t-1)+e(t) \\
(3)\quad I(t)=b_0+b_1Y(t-1)+b_2Y(t-2)+v(t)
In (3), b1 and b2 are estimated separately. The textbook pure-accelerator restriction b2 = -b1 is a testable hypothesis, not an assumption.
2. One Substitution
Every right-hand term of (2) and (3) is a constant, a lag of Y, or a disturbance, so the identity is already closed in two lags of Y. One substitution gives the reduced-form propagation equation:
(4)\quad Y(t)=a_0+d_1Y(t-1)+d_2Y(t-2)+u(t),
\quad a_0=c_0+b_0+A, \qquad d_1=c+b_1, \qquad d_2=b_2 \\
\quad u(t)=e(t)+v(t)+\bigl[A(t)-A\bigr]
The parameter map carries the interpretation: d1 confounds the marginal propensity to consume with the first accelerator coefficient, d2 is the second accelerator coefficient alone, and u(t) collects the behavioral disturbances plus the fluctuation of autonomous expenditure about its mean. The lack of structure (no policy variables) in equation (4) is disturbing.
3. Another Objection
An economist raised on Harrod (1939) and Domar (1946) will inspect (4) and find the engine of growth missing. Gross investment – the variable that carries the warranted rate g = s/v, the variable with the dual character of demand today and capacity tomorrow – appears nowhere on the right-hand side. The reduced form appears to write investment out of the growth equation. I confess the disappearance disconcerted me on first derivation. The disconcert deserves a direct answer, and the answer comes in two parts: an algebraic part, and an empirical part the algebra makes testable.
4. Absorption Is Not Absence
The algebraic part first. The second lag belongs to investment alone: d2 = b2. Consumption with a one-period lag cannot reach t-2; only the accelerator can. The entire second-order character of (4) – the characteristic equation, the possibility of complex roots, the intrinsic cycle – exists if and only if b2 differs from zero. Investment has not vanished from the reduced form; investment is the reduced form’s dynamics. Nor is anything lost to the reduction that the structural system does not return: the first accelerator coefficient is recoverable as b1 = d1 - c, given an estimate of the marginal propensity to consume from (2).
What does fold invisibly into (4) is autonomous investment b0, absorbed into the intercept a0. The temptation is to dress this intercept as animal spirits and reinstate it as a separate driver of GDP. The temptation should be resisted: an intercept is a constant, not a variable, and relabeling it tells Harrod-Domar nothing testable. Nor can gross investment I(t) be placed on the right-hand side of a regression for Y(t): by the identity (1), I(t) is a component of Y(t), and such an equation would be an accounting tautology contaminated by simultaneity. The reduction is the discipline the accelerator hypothesis imposes: investment entirely induced by lagged income leaves no independent footprint in levels – only in dynamics.
5. The Cycle Investment Makes
The homogeneous part of (4) has characteristic equation:
(5)\quad r^2-d_1r-d_2=0
with roots
(6)\quad
r=\frac{d_1\pm\sqrt{d_1^2+4d_2}}{2}Complex roots require d1^2 + 4 d2 < 0, hence d2 = b2 < 0: the accelerator’s second coefficient must be negative for the economy to oscillate. In that regime, the modulus R and period of the cycle are:
(7)\quad
R=\sqrt{-d_2}=\sqrt{-b_2}\quad
\cos(\theta)=\frac{d_1}{2R},
\qquad
\text{period}=\frac{2\pi}{\theta}The modulus – whether shocks damp quickly or persist – is the square root of the accelerator’s second coefficient and nothing else. Samuelson (1939) is the special case I(t) = beta [C(t) - C(t-1)] with C(t) = alpha Y(t-1), whence d1 = alpha (1 + beta) and d2 = -alpha beta, and the same conclusion holds: an economy oscillates because firms adjust capital to lagged income. Harrod-Domar’s engine survives the reduction not as a regressor but as the propagation mechanism.
6. The Asymmetry: Debt Survives the Reduction
Equations (1)-(4) collected G(t) + X(t) - M(t) into autonomous expenditure and demoted its fluctuation to the impulse. The memorandum’s full system does not. Government expenditure is closed by the lagged stock of public debt, imports are proportional to current income, and exports remain exogenous. Carrying the same substitution through the full system yields:
(8)\quad Y(t)=a+d_1Y(t-1)+d_2Y(t-2)+d_3D(t-1)+d_4X(t)+u(t)
Note the asymmetry because it is the point. Gross private investment, being induced by lagged income, is absorbed into the propagation pair (d1, d2) and leaves no level footprint. Accumulated public debt, being policy-determined rather than a function of current income, survives the reduction as an explicit regressor, D(t-1). The reduced form treats the classical engine and the modern engine differently because they are different: one works through the private capital-adjustment mechanism, the other through debt-financed demand injected from outside the income loop. And because both channels sit in one estimable equation, the data can adjudicate between them.
7. The Regime Change
Two ratios frame the adjudication. Private nonresidential equipment investment as a share of GDP has oscillated in a narrow band, roughly 5 to 7 percent, for four decades – dipping in recessions, recovering, exhibiting no trend. Federal debt held by the public has risen from about 25 percent of GDP in the 1970s to 98 percent in fiscal 2025. The first ratio is the Harrod-Domar savings term read on its transaction-verified core; equipment and structures are audit-verifiable, whereas the intellectual property products component of measured investment is an imputation no auditor can confirm, and it is excluded from the engine on evidentiary grounds. The divergence between a stagnant investment share and a debt ratio grown several-fold marks a substitution in the source of measured GDP growth: from private capital accumulation to debt-financed demand.
This is why the growth is misfiring. In the Harrod-Domar accounting, investment raises demand today and capacity tomorrow; a debt-financed transfer raises Y(t) on the demand side without raising K(t), buying measured growth today at the price of no warranted growth tomorrow. Domar (1944) supplied the law of motion for the debt ratio itself,
\quad
d(t)=\frac{1+i}{1+g}\,d(t-1)+f(t)where d is the debt-to-GDP ratio, f the primary deficit ratio, i the interest rate, and g the growth rate: the ratio converges only if g > i. An engine that erodes its own convergence condition is not a perpetual substitute for the one it replaced.
The hypothesis is testable in (8). If the Keynes-Kalecki-Kahn investment driver still governs, the propagation pair (d1, d2) carries the explanatory weight and d3 is small; if debt accumulation has replaced it, d3 dominates – particularly in the split samples 1950-1979 and 1980-2025, and in the encompassing comparison of the accelerator against the credit impulse of Biggs, Mayer, and Pick (2010) and Mian, Sufi, and Verner (2017). The Harrod-Domar economist’s objection to (4) is therefore answered twice over: algebraically, investment never left; empirically, the question is whether it still shows up for work.
8. Impulse and Propagation
The composite disturbance u(t) is the Frisch (1933)-Slutsky (1937) impulse: behavioral shocks plus whatever autonomous fluctuation the specification has not promoted to a regressor. The coefficient pair (d1, d2) is the propagation. Serially patternless impulses, filtered through second-order propagation, emerge as recurrent but irregular cycles. No exogenous cycle-maker is required, and none is invoked.
9. Estimation Note
Equations (4) and (8) contain lagged dependent variables, so the Durbin-Watson statistic is biased toward 2 and Durbin’s h (1970) is the appropriate test for residual autocorrelation. We estimate by OLS with Newey-West standard errors (Bartlett kernel, L = 3) and report 68% confidence intervals.
References
Biggs, M., Mayer, T., and Pick, A. (2010), “Credit and Economic Recovery: Demystifying Phoenix Miracles,” SSRN Working Paper. The credit impulse.
Domar, E. D. (1944), “The ‘Burden of the Debt’ and the National Income,” American Economic Review 34(4), 798-827. The debt-ratio law of motion.
Domar, E. D. (1946), “Capital Expansion, Rate of Growth, and Employment,” Econometrica 14(2), 137-147. Investment’s dual character: demand today, capacity tomorrow.
Durbin, J. (1970), “Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors Are Lagged Dependent Variables,” Econometrica 38(3), 410-421. The h statistic.
Elaydi, S. (2005), An Introduction to Difference Equations, 3rd ed., Springer. Roots, modulus, and stability of second-order equations.
Frisch, R. (1933), “Propagation Problems and Impulse Problems in Dynamic Economics,” in Economic Essays in Honour of Gustav Cassel, Allen & Unwin, London. The impulse-propagation distinction.
Harrod, R. F. (1939), “An Essay in Dynamic Theory,” Economic Journal 49(193), 14-33. The warranted rate of growth.
Kahn, R. F. (1931), “The Relation of Home Investment to Unemployment,” Economic Journal 41(162), 173-198. The employment multiplier of investment.
Mian, A., Sufi, A., and Verner, E. (2017), “Household Debt and Business Cycles Worldwide,” Quarterly Journal of Economics 132(4), 1755-1817. Debt-driven demand and subsequent growth reversal.
Samuelson, P. A. (1939), “Interactions Between the Multiplier Analysis and the Principle of Acceleration,” Review of Economics and Statistics 21(2), 75-78. The canonical multiplier-accelerator model.
Slutsky, E. (1937), “The Summation of Random Causes as the Source of Cyclic Processes,” Econometrica 5(2), 105-146. Cycles from summed random causes.