The Calculus of Increasing Returns to Scale and Technological Progress in Macroeconomic Growth

Chinese population

Based on Lecture Notes from Dr. Naish at CSUF (Topics in Economic Analysis, Fall 2009)

In contemporary macroeconomic textbooks the assumption of constant returns to scale and perfect competition are the building blocks of much of the economic growth models and theories.  The models imply that the only method for increasing per capita income is via technological progress.  The following calculations demonstrate that with Increasing Returns to Scale the growth of the population can also be a factor even if there is no technological progress.

Basic Equations and Assumptions:


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The increasing returns to scale are symbolized by the fact that alpha plus beta is greater than one.  It can be shown that this inequality implies that the produciton function, Y, exhibits increasing returns to scale.  The increasing returns to scale imply that there are cost benefits to operating at a larger scale whether this is achieved through increasing the size of firms, opening up to international trade, or simply by increasing the population of a country is another question that should be handled in the context of the resource constraints and the overall economic goals of a nation.  This derivation is to simply demonstrate that there can be advantages to operating at larger scales even without technological progress which is something most macroeconomic textbooks ignore.

There are 2 other variables which need a more rigorous definition and those are the rate of population growth and the growth rate of capital since both are elements in the production function above.


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The rate of growth of the capital stock is equal to I, which is investment per period.  Investment is equal to the marginal propensity to save multiplied by total GDP since most savings is in on way or another an investment.  Money saved in the bank become mortgages, loans, and bond purchases by the banks which essentially means that, roughly speaking, investments equal the proportion of savings in an economy.


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The formula for population growth in this model is an exponential growth model.  The population grows starting from an initial population and grows exponentially as a function of the time multiplied by the growth rate n.  This model of population growth is well suited for developing countries where populations increase at high rates.  A Logistic Model for population growth incorporates both the exponential growth pattern of developing nations as well as the increasing at decreasing rates of growth in industrialized countries.  The exponential model will suit this example well and the marginal benefit of a more accurate population model is less than the cost of using the logistic growth model for most countries in the world.


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The following calculation take advantage of the previous derivations of population and capital growth rates to examine how increasing returns to scale can have a higher growth rate in the capital to labor ratio which in turn increase productivity.  This increase in productivity has nothing to do with increases in technological progress, but comes strictly from the benefits of operating at larger scales.

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In equilibrium the rate of growth in the capital per workers should be constant since any increase would cause an increase in output and thus by definition the system would not be in equilibrium.  This yields the final equation for growth in the capital to labor ration with no technological progress, but with increasing returns to scale:


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Technological progress is not the only way to increase the capital to labor ratio.  The fact that the rate of population growth is included in the rate of capital per worker above in such a manner that demonstrates a positive relationship between increasing rates of population growth and increasing capital to labor ratios is apparent in a country which experiences large increasing returns to scale.  Countries with higher capital to labor ratios have higher GDP per capita and thus higher income.  High income countries tend to have better educated and healthier people so the importance of economies of scale cannot be downplayed much like it has in many books on macroeconomic growth.


2 thoughts on “The Calculus of Increasing Returns to Scale and Technological Progress in Macroeconomic Growth

  1. Thanks for bringing this kind of simple analysis to net. This kind of thing should be table stakes for discussions about the economy but instead we have something far worse: arguments about differential equation solutions using nothing more than intersections of lines on charts, at best! Literally it’s medieval – Leonardo da Vinci engineered with similar techniques more but he didn’t have algebra or calculus in his world yet. What’s our excuse?!?

    Assuming equilibrium is a dangerous and bad assumption. It doesn’t have to be made! Assuming equilibrium creates bizzaro-world models such as systems where having deflation and unemployment is impossible. It also ignores real world violations of the assumption such as “sales cycles are _not_ instantaneous” for large swaths of the economy, especially for critical high technology products. The effect of this is that S and D for either labor or products has a non-ignorable time lag where S!=D. These effects do have impact on macro growth and thus can not be swept under the carpet.

    Also a more elaborate model which include environment and natural resources shows that “free resource growth” has the same impact as population growth or technology/efficiency growth. Similarly a decline (such as the now acknowledged Peak Oil) has as much effect as a demographic population decline (e.g. Japan). These are the “big 3” with stuff like government spending and taxes having only “damped responses” for growth.

    Unfortunately folks in America don’t apparently understand any of this: Peak Oil and Peak “name your resource” with population issues are upon the US just as its largely outsourced every key high technology elsewhere. Brilliant. One can only innovate high tech by manufacturing it oneself – received experience through locality and intimacy are essential components to innovation.

    The nature of globalization also makes it clear that interdependencies between countries and markets do not allow endogenous models to be sufficient – a national endogenous is a coupled system with many dozens of others. An endogenous model is at best a “homogeneous solution” that must be combined with a “forced response solution” from extranational economies to mean anything at all.

    The economies of scale comment is interesting. Scaling really _is_ everything. Does a business or aggregate as a national economy actually scale? Let’s not forget that compounded growth is exponential growth. That’s a non-trivial scaling regime to achieve.

    I agree economists generally do not pay attention to how business or aggregated economic models scale. This seems to be associated with the general innumeracy of most economists (most PhDs in economics know less math than a 2nd year undergrad in engineering). It’s hard to imagine how anyone can ignore scaling when the key metric is growth – growth is entirely dependent on how the scaling works or doesn’t work!

    And the metric for modern economics is compounded growth – the most aggressive scaling regime there is – very few systems in nature are actually exponential over many orders of magnitude. Few systems of man scale at a compounded rate and the ones most people think do actually do not – they are logistic instead. BTW I do see many merits to Islamic finance from a practicality and human-scale design point of view – it doesn’t do “compounded growth”, only “simple growth”.

    Human labor-based work simply does not scale. It appears to scale through the leverage of technology, but it’s the technology used that is scaling, not the human component using the technology. Technology’s scaling is simply passed through. Thus the control of scaling and growth require control of the technology and its innovation. This is reinforced by looking at where most of the investment return revenue comes from in innovation – the leading edge and not the trailing edge. You can’t be on the leading edge if you are not manufacturing the predecessor or a penultimate predecessor.

    It’s useful to remember what came before the industrial revolution and what made the industrial revolution the inevitable successor. The predecessor was a pure knowledge/service-only economy called crafts and guilds. The key factor to industrial inevitability was better scaling through the innovation of interchangeable parts.

    The US knowledge/service economy as a national economic model is very, very wrong because it is not scalable without technology. In fact, if US keeps it, the US is doomed. Without doing manufacturing, it can never control technology innovation itself nor reap most of the technology adoption benefits/revenues to justify putting out the effort required to innovation. In contrast economies that do manufacturing do get multi-order scaling and the leading edge economic return but also have the *choice* to also have knowledge/service (more degrees of freedom of action) in addition.

    The non-scaling of knowledge/service is why we have a health care crisis right now. It’s also the reason Wall Street had to “invent” profits with derivatives, etc. – nothing else could provide the scaling required to deliver expected growth rates.

    1. Hi Jeff,

      Thank you for your comment. I am looking to replace the exponential population growth assumption in this model with a logistic growth model and see the results that I get. It should be interesting and challenging to include the non-linear realities that many economist assume away for the sake of ease of computation.

      I am planning on taking some Numerical Analysis classes as part of my graduate school training so that I can continue use more non-linear differential equations in my research.

      Do you have a account? If you please feel free to connect,

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