J. K. Sengupta and Gerhard Tintner, On Some Aspects of Trend in the Aggregative Models of Economic Growth (1963)

Sengupta and Tintner’s journal article combines theoretical model-building with historical statistical estimation to investigate growth trends more general than the exponential paths of the standard Domar model. Its central proposal is a mixed-logistic trend: growth approaches a limiting level within a given economic regime, while changes in technology, population, or organization can shift that limit. The argument proceeds from aggregate income dynamics through empirical estimates for the United States and Britain to a two-sector account of changing output composition. Its distinctive concern is to explain development as both an increase in economic scale and a transformation of productive structure.

The opening contrast between Harrod–Domar and Haavelmo-type models identifies two routes beyond simple exponential growth: more general production functions and stochastic variation in model parameters. The article principally develops the former, leaving a fuller stochastic theory as a research agenda. In the standard Domar formulation, income, consumption, and investment grow at the same constant proportional rate. This imposes a restriction on what development can mean:

The economy changes in scale, but not in composition.

The sentence motivates the article’s two connected investigations. An adequate trend specification must allow aggregate growth to slow without making its ceiling permanently fixed; it must also permit the investment–consumption ratio to change. The authors therefore replace constant capital productivity with a term declining as capital accumulates and make investment depend positively on income but negatively on existing capital stock. Diminishing returns and the depressive influence of accumulated capital on further investment can restrain initially exponential expansion. Technological change can, in turn, alter the coefficients and offset these restraints.

The resulting nonlinear system supplies candidate equilibrium levels of income and capital. A local linearization is presented as yielding nonoscillatory convergence, although the reported zero characteristic root makes its stability claim less decisive than the prose suggests. The conceptual emphasis falls on changing regimes rather than a single permanent equilibrium:

The exact analogue would be a mixed logistic trend where the upper asymptote changes discretely due to technological change, demographic transitions and other exogenous forces.

Under additional assumptions—removing the direct capital-stock restraint on investment and fixing the capital–income ratio—the authors derive a logistic income equation. Changes in coefficients then generate successive logistic segments with different upper limits. Developmental stages acquire a mathematical interpretation through the curve’s initial level, inflection point, and upper asymptote. The authors also introduce autonomous investment and consumption demands to obtain an evolving investment–consumption ratio with similar limiting properties.

The empirical section asks whether long-run observations are compatible with this specification. Missing capital-stock series prevent estimation of the full structural model, so the authors fit reduced-form trends to American real income per head for 1869–1953 and British industrial output for 1700–1940. Their method regresses the reciprocal of income or output on its lagged value; solving this first-order autoregression and inverting the result produces a logistic curve. This makes estimation tractable, but the assumed independent residuals remain explicitly provisional.

For the United States, four roughly equal subperiods yield successively higher estimated income ceilings, approximately 435, 567, 623, and 1,299. These estimates illustrate shifting limits rather than establish precisely dated structural breaks: the periodization is acknowledged to be arbitrary, confidence intervals are sometimes wide, and the 1912–1932 fit is weaker, plausibly because of interwar cyclical disturbances. Most importantly, statistical fitting does not explain the movement between regimes:

The change in the upper asymptote of the income curve remains, however, to be explained in terms of other structural changes like innovation, population growth, etc.

British industrial-output estimates similarly support rising fitted ceilings across subperiods. Their evidential role remains limited by the use of industrial indices as an approximation to income and by the unresolved interaction of trend and cycles.

The final substantive section gives structural change a more explicit theoretical foundation through Hoffmann’s ratio of producer-goods to consumer-goods output. Hoffmann had used its historical increase to characterize industrialization; Sengupta and Tintner connect it to investment allocation and sectoral productivity. A general two-sector model allows unequal investment-growth rates to change the output ratio. A simplified, integrated model—in which investment-goods output supplies expansion in both sectors—produces a logistic path for the ratio itself.

British estimates give successive upper limits near 0.382, 0.681, and 0.882. The authors interpret these as evidence of changing developmental regimes accompanied by a declining proportional growth rate of the ratio. Yet missing observations, an extrapolated terminal value, and sensitivity to business fluctuations qualify the result. The article’s own concluding assessment is carefully restrained:

On the basis of long-run data on income and its components for the United States and the United Kingdom, the preliminary statistical results show that the hypothesis of a mixed-logistic trend is not refuted by the available data.

The contribution is thus a theoretical and empirical bridge between aggregate growth and stage-based structural development, not a causal explanation of regime changes. Mixed-logistic paths make changing limits and changing composition analytically compatible. Explaining why those limits shift—and integrating stochastic processes and cyclical fluctuations—remains the necessary next step.