Picking Piketty Apart — Part II: His “Laws of Capitalism”

Regarding the laws that govern the universe, what we can say is this: There seems to be no single mathematical model or theory that can describe every aspect of the universe. * * * Though this situation does not fulfill the traditional physicists’ dream of a single unified theory, it is acceptable within the framework of model-dependent realism.” — Stephen Hawking and Leonard Mladinow, The Grand Design, Bantam, N.Y. 2010 (p. 58)

According to model-dependent realism, it is pointless to ask whether a model is real, only whether it agrees with observation. * * * We make models in science, but we also make them in life. Model-dependent realism applies not only to scientific models but also to the conscious and subconscious mental models we all create in order to interpret and understand the everyday world. There is no way to remove the observor — us — from our perception of the world, which is created through our sensory processing and through the way we think and reason. Our perception – and hence the observations upon which our theories are based  – is not direct, but rather is shaped by a kind of lens, the interpretive structure of our inner brain. — Hawking and Mladinow, supra (p. 46)

The above quotations are taken from an early chapter entitled “The Nature of Reality,” a chapter dealing mainly with describing how science works. The core concept is “model-dependent realism.” We are reminded that, as in the natural sciences, no theory in the social sciences can be said to describe “reality” except to the extent that it is supported by observation.

Unlike the models and theories describing the physical universe, those pertaining to economic science relate to how we “interpret and understand the everyday world.” Most of the time, we might think, the workings of the everyday world are much easier to interpret and understand than the vast, mysterious cosmos. Thus, it is hard to believe that geniuses like Newton and Einstein developed their theories through the same (or similar) processes of perception and interpretation that we ourselves use to develop our more mundane, everyday theories. Nonetheless, the problem of understanding “reality” is the same in both cases.

Economic theories fail when the assumptions on which they depend do not comport with reality, or when they are incomplete, overlooking facts that must be observed and accounted for before our “everyday world” can properly be understood. A major problem in economics has been that theories are developed under strict sets of assumptions (like “perfect competition” or “full employment”) that are never met, yet we keep on applying the theories anyway, because they are all we’ve got to go on.

This gives rise to a subtle problem highlighted by the above quotation:

Our perception – and hence the observations upon which our theories are based  – is not direct, but rather is shaped by a kind of lens, the interpretive structure of our inner brain.        

Our interpretive structures are framed by what we are taught and, accordingly, what we think we know. Over time, our minds hold on to them, as we live and work with them, and their elaborate constructs become part of the interpretive structure of our brains — the “lens” we use to interpret our observations of the world. Thus, our perceptions persist, even when the world won’t support them, controlling our thought processes. There is only one way out of this conundrum: We must constantly challenge our beliefs, and abandon them when they are not working. Of course, to do that, we need observations and relevant data. But if we don’t do that, when important new information arrives we simply get confused.   

This is a problem, I suggest, that has always plagued economic science. Over the past several years I have been startled by the degree to which the application of mainstream economic theory to the multitude of new information on income and wealth distribution has resulted in a high level of acknowledged confusion. That confusion is best explained, I submit, as a failure of model-dependent realism. Economists are using decades-old models, with insufficient regard for the limiting assumptions under which these models were originally developed. One key assumption, of course, is that these models properly specify all of the factors which actually determine the data we interpret. When we ignore or overlook the constraints reality imposes on our models, our use of them leads to mistaken interpretations of reality, and the perpetuation of wrongful perspectives. 

The Neoclassical Sidetrack

That, I would argue, is the crux of what has happened to economic science over the last 50-100 years, and why mainstream, neoclassical economics, as recently conceded by its leading young superstar, Raj Chetty (“Yes, Economics is a Science,” The New York Times, October 20, 2013, here) still cannot explain the mechanics of growth. This problem plagues Thomas Piketty as well, as he has bravely set forth a growth model, characterized as “the second fundamental law of capitalism,” which traces back to the “Harrod-Domar” growth model and other production function-based models developed more than a half-century ago. In effect, this “neoclassical” framework has created a lens that frames all macroeconomic issues in specific, constrained ways, rendering minds trained to understand the economic world from within that constrained framework nearly incapable of considering, or even perceiving, alternative perspectives.   

Piketty has earnestly and carefully developed two “fundamental laws of capitalism” that emerge from that limited perspective. He then attempts in the last portion of his book to apply those “laws” to understanding the inequality problem but finds, perhaps to his own surprise, that they are of no help in this regard. His application of the model to existing data sources, including those he himself has painstakingly compiled, has also highlighted important concerns about how we use data to understand growth.

Specifically, “capital” is a concept in classical and neoclassical economics that generally means “capital stock,” or investment in the physical plant needed to produce output. The growth models at issue here originally focused on the growth of capital stock, as Piketty well knows. However, to use available data sources, he has resorted to a long-run “equilibrium” model in which “capital” can be thought of as equivalent to the broader concept of “wealth.” That shift from “capital” to “wealth” contains a major key to understanding why mainstream supply-side growth models cannot help understand the inequality problem, and more generally why the attempt to understand growth via aggregate production functions has been a dismal (and still generally unrecognized) failure. 

The extreme redistribution of wealth, and consequently income, in my view, is the real inequality problem, and it is a problem not comprehended by tracing, as Piketty attempts to do, the growth of the productive capital capacity over many decades:

The central thesis of this book is precisely that an apparently small gap between the return on capital and the rate of growth can in the long run have powerful and destabilizing effects on the structure and dynamics of social inequality. (Capital in the Twenty-First Century, Belknap/Harvard, 22014,  p. 77)

His is a hybrid thesis, an ultimately misleading attempt to connect mainstream growth theory with “the dynamics of social inequality.” It is not an entirely straightforward matter to understand what this conclusion means, or how he reached it. 

To make any real progress in reading Piketty’s book, it is essential to understand the nature of his model and his perspective on growth. In this first of two posts on the substance of his book, I endeavor to explain his “laws” and the model they represent. In a second post, I will shift the focus to the issue of wealth and income inequality, discussing the disconnects between production function-based growth models and the models needed to properly explain the growth of distributional inequality, and how the limitations of the neoclassical framework demonstrate the need for a new distribution-based framework for macroeconomics.

Piketty’s presentation is somewhat disorganized and confusing., and his approach to measuring growth has had a complex and controversial history, which he presents only in part. Therefore, I believe, the best way to piece together a coherent picture of his model is to work from (and with) his equations:      


Piketty’s “First Fundamental Law of Capitalism” is expressed in the equation α = r x β, where:

α = the share of income from capital in national income

r = the rate of return on capital, and

β = the capital/income ratio, i.e., the “stock” of capital (C) divided by the annual income of the economy (I)  (pp. 50-51).

The underlying proposition is fairly straightforward: The percentage of all income attributable to capital (as opposed to labor) equals the total amount of (productive) capital times its rate of return. However, the “capital/income ratio,” which introduces the total amount of productive capital as a percentage of income (output), presents some conceptual difficulties:

The Capital/Income Ratio

The derivation of the capital/income ratio can be the source of some confusion, so let’s break it down:

If we let “R” represent the annual income from capital, then

R = r x C

Dividing both sides of this equation by I, we get:  

R/I  =  (r x C)/I  =  r x (C/I),  or

α =  r x β

Piketty asserts that this formula is “a pure accounting identity” (p. 52). “Though tautological,” he reasons, “it should nevertheless be regarded as the first fundamental law of capitalism, because it expresses a simple, transparent relationship among the three most important concepts for analyzing the capitalist system: the capital/income ratio, the share of capital in income, and the rate of return on capital.” (Ibid.)

Here are my main concerns about the “First Law”:  

False Meaningfulness

One would think that an “accounting identity” would not give rise to differences in perspective, or fundamentally conflicting perceptions of reality. However, in this instance it can do just that:

A “law,” as Hawking and Mlodinow define it, is a model that is found to agree with observation. That means that it must always be true, or at least true in certain specified circumstances. But a law derives its meaningfulness only from its ability to explain reality. If it represents no more than a mere “identity,” it has no explanatory power; to say anything is “itself” provides no useful information. Unfortunately, calling this equation a “law” implies that it may be thought of as more than just a mere accounting identity. Piketty suggests as much when he says the formula expresses a “relationship” among the three variable, implying that the variables are functionally related. 

Although each of these variables reflects data recorded in accounts, the capital/income ratio is a ratio of accounts that are not functionally related. As Piketty acknowledges, the “capital” (and wealth) accounts are “stock” accounts, meaning that they are records of “net worth” existing at given points in time.  National Income (and GDP) accounts, however, are “flow” accounts, representing amounts of transactions over time. The capital/income ratio therefore does not express a “relationship;” it represents merely a comparison of the order of magnitude of accumulated capital with a distinctly different, and independent, concept used to represent the overall size of the economy.

In Macroeconomic Theory (MacMillan, 1963), Gardner Ackley’s preeminent textbook of the early 1960s, Ackley emphasized almost before discussing anything else the importance of the distinction between stock and flow data: 

We need not here list the variables in which macroeconomics is interested. But it is useful, right at the beginning, to stress some characteristic types of variables, and their differences. The most important such distinction (the neglect of which has been the cause of infinite confusion) is between stock and flow variables. A stock variable has no time dimension. A flow variable does. The weight of an automobile is a stock variable; its speed is a flow variable. The population of cars is a stock variable; traffic is a flow. 

* * * All this may seem very obvious; but almost no other source of confusion is more dangerous in economic theory — not only to beginners but sometimes to advanced students in the field. Money is a stock; expenditures or transactions in money a flow; income is a flow, wealth a stock. Saving is a flow. . .; savings is a stock. . . Investment is a flow. . .; the aggregate of investments is a stock; * * * only the context can show whether the author means the flow or the stock.   

Piketty’s bestselling book is meant for, and has apparently reached, a much broader audience than just fellow economists. He cannot have expected the average reader to understand the importance Ackley attributed to the distinction between stocks and flows. Given Piketty’s own recognition of that distinction, however, his use of the terms “relationship” and “identity” to describe an equation in which a stock/flow ratio is a major variable is especially misleading; these are the very descriptions that, he says, argue for characterizing the equation as a “law” of capitalism. Thus, he has conceptually transformed the equation into something is is not: an actual “law.”

Piketty does not shy away from this confusion. He entitled the entire Part Two of his book “The Dynamics of the Capital/Income Ratio,” raising an obvious question: How could a mere “accounting identity” possess dynamic properties? This careless use of terminology engenders significant confusion about the true “model-dependent realism” represented by this stock/flow ratio.   

In a perceptive recent comment (“Nit-Piketty,” May 25, 2014, here), economist Debraj Ray made the point in a more summary fashion:

Piketty’s Laws 1 and 2 can, alas, be dismissed out of hand. (Not because they are false. On the contrary, because they’re true enough to be largely devoid of explanatory power.) 

This is certainly true of Law 1: This equation merely introduces a yardstick (the capital/income ratio) against which the order of magnitude of accumulated capital can be measured. It has no explanatory value, and no dynamic properties. (I’ll get to Law 2 in a moment.)

The Narrowness of Perspective

Piketty’s opinion that this equation contains “the three most important concepts for analyzing the capitalist system” is conclusory, and it preempts all other, competing perspectives. Notably, it prohibits consideration of the impact of wealth concentration itself on growth, and on the concentration of income from capital. It directly and unjustifiably narrows the neoclassical perspective on the macroeconomic significance of inequality. This demonstrates the major shortcoming of neoclassical perspectives I have been pointing out for more than three years. 

Data Reconciliation

Attending this equation is a host of unresolved conceptual issues regarding data: Wealth data for the United States are taken from U.S. net worth accounts which present household assets net of liabilities. These include ownership of corporations and, hence, all of the means of production traditionally referred to as “capital stock,” but they include much more. Wealth is broken down into financial and other assets, which include homes and real estate.

This problem has broad implications for Piketty’s “laws.” Accounting “identities” that begin with overly broad definitions of wealth and income run into serious difficulties when introduced into production function-based growth models: To make an assessment of a nation’s productive capacity, a stock measure including all marketable wealth, including assets that do not produce tangible income and wealth such as mansions, yachts, valuable art works, etc., is far too broad.  Similarly, National Income account data cannot be assumed to be limited, even over the very long run, to amounts spent on consumption and investment. U.S. income account data include trillions of dollars every year that are received as “income,” but do not compensate for tangible value produced or received.

To assess economic growth using models that purport to trace the growth of “capital,” it is essential to isolate active wealth from inactive wealth, and productive from non-productive income. This brings us to Piketty’s 2nd Law, but it is important to leave our discussion of the Piketty’s “First Fundamental Law of Capitalism” with an understanding that standard aggregate income and wealth accounts are not sufficiently specific for this assigned task.    


Piketty’s “Second Fundamental Law of Capitalism” is expressed in the equation  β = s/g, where:

β = the capital/income ratio [C/I], i.e., the “stock” of capital divided by the annual income of the economy. (This is identical to the ratio used in the first equation above.)

=  The percentage of saving out of income.

= the rate of growth of productive capital (or alternatively, in the long run, the rate of growth of income).

It is not immediately intuitive why this might be so. “In the long run,” Piketty asserts, this formula expresses a “simple and transparent” relationship among the three variables. It reflects, he says, “an obvious but important point: a country that saves a lot and grows slowly will over the long run accumulate an enormous stock of capital (relative to income), which can in turn have a significant effect on the social structure and distribution of wealth.” (p. 166)

Nor is it transparently clear how the accumulation of the stock of productive capital, over the long run, affects the distribution of wealth. It certainly does not, all by itself, determine the distribution of wealth, as Piketty carefully explains:

To be sure, the law β = s/g describes a growth path in which all macroeconomic quantities – capital stock, income and output flows [note again, what started out as asset net worth has ended up as capital stock] – progress at the same pace over the long run. Still, apart from the question of short-term volatility, such balanced growth does not guarantee a harmonious distribution of wealth and in no way implies the disappearance or even reduction of inequality in the ownership of capital. (p. 232).

Solving for Growth

Note that there are two equations defining the capital/income ratio. As a matter of “accounting identity”:

α = x β,  therefore  β = α/r  

The hypothesis of the Second Law, that β = s/g can therefore be expressed as an accounting identity, thus:   

 s/g = α/r

In words, this equation tells us that the ratio of the percent of income that is saved to the rate of growth of capital stock is equal to an accounting identity: the ratio of the percentage of total income that is attributable to capital (capital’s share) to rate of return on capital. We can also solve for g, multiplying both sides of the equation by the two denominators:

g x α  =  r x s   

       Hence g = s (r/α),  or

g = r (s/α)

Here is what this formulation means in words: An economy’s income (output) will grow at a rate equal to the rate of return on capital times the savings rate divided by the percentage of total income that is attributable to capital. This is certainly hard to visualize: Technically, the model specifies that g equals the rate of growth of capital stock, not the rate of growth of income (output). But this is a model, thanks to Law #1, in which saving is used as a proxy for investment. It is a bit more comprehensible if we substitute investment (i) for saving (s):


In words, this equation says that an economy’s income (output) will grow at a rate equal to the rate of return on capital times the rate of investment divided by the percentage of total income that is attributable to capital. This is a little more understandable (we’re no longer comparing apples and oranges). 

But here’s the rub: Substantively, this specifies an equilibrium condition where capital resources are fully employed, i.e., an economy in full-capital-stock-utilization equilibrium. Moreover, Piketty warns us that Law #2 is only valid in the long run. And that is true because the neoclassical concept of long-run equilibrium presumes that all savings are eventually invested in productive capital, and it is only this long-run equilibrium state that meets the model’s requirement that

the rate of growth of capital stock necessarily equals the rate of growth of income. 

Confused? Notice that what started out as an “accounting identity” (the capital/income ratio) has blossomed into a rigid “law” that is necessarily valid only in the long run. There are two important points:

1. It has become an empirical proposition that must be verified, but because it is merely hypothetical it cannot be verified. Thus, we have not escaped our concerns about model-dependent realism. (More on this in the next post);

2. Because it is only valid in the long run, the formula is necessarily invalid in the short run, unless we assume that all economic variables grow at exactly the same rate over time. (Don’t laugh — some early, rudimentary aggregate production functions were based on exactly this assumption.) This makes the elusive distinction between the long run and the short run critically important: How are we to account for the fact that any point in time necessarily exists in both the short run and the long run? And if growth rates do in fact vary, how would we ever recognize “long-run equilibrium,” were it ever achieved?

At this point in the review, the import of the “Second Fundamental Law of Capitalism” seems anything but “simple and transparent.” Indeed, as in the famous Buddhist parable of “The Tiger and the Strawberry” (here), there is no apparent way out of these dilemmas.

The Model’s Description of Long-Run Equilibrium   

But also as in the parable, the “strawberry” is sweet. The model presents an image of an observable equilibrium state. And the model has arithmetic consequences: moving from one equilibrium position to another (by whatever means, or hypothetically), reveals that the mechanics of the formula have specific implications for distribution and growth.

Consider the numerical example Piketty presents of his two laws, in which he uses a number for β of 600%, which is near the high end of observed capital/income ratios in recent years:

Accumulated capital is equivalent to six times an economy’s annual income. If for a given country  β = 600%, and the rate of return on capital is 5%, then the portion of the country’s income that comes from capital is 5% x 600% = 30% (Law #1, p.  52). Similarly, if β = 600%, and the rate of saving from income is 12%, then average annual growth of income in the long run is 12% /600% = 2% (Law #2, p. 166).

If the hypothetical country is assumed instead to have a greater volume of accumulated capital, such that β = 800%, at a 5% rate of return the portion of income coming from capital is 5% x 800% = 400%. Given a 12% rate of saving, the long-run growth rate is lower, 12%/800% =  1.5%. 

Given the presumed equivalency in long-run equilibrium not only of savings and investment, but also of capital and wealth, a higher capital/income ratio entails a higher equilibrium level of wealth, which in turn entails higher income inequality. Thus, ceteris paribus (all else equal), this model requires that a higher equilibrium concentration of wealth will entail both higher income inequality and lower growth — for any given level of income.  

But we do not need this elaborate, long-run equilibrium model to reach that conclusion: That higher wealth concentration entails lower growth is a fact that can be directly observed. It has been clear for some time that rising wealth and income concentration in the United States necessarily entails lower growth. Piketty’s model merely measures the trending relationship of accounting identities over time, and that is only helpful if, like Piketty, we regard very long trends of the capital/income ratio (over periods of, say, 100 or 150 years or more) as representing some sort of long-run equilibrium β. But such a focus obscures our perception of the short run growth problem, which averages out as growth varies over time. For the shorter run analysis that we need, of course, the model is superfluous.     

The Model’s Oversimplifications

A production function-based growth model is a vast over-simplification of how an economy actually works. Many more variables are at work determining levels of capital, income, and wealth than are reflected in these simple formulas. Piketty candidly acknowledges such shortcomings of his model. First, he says, “The law β = s/g represents a state of equilibrium toward which an economy will tend if the savings rate is and the growth rate g, but that equilibrium state is never perfectly realized in practice.” (p. 169)

Second, he points out that the law is applicable “only if certain crucial assumptions are satisfied”:

1. The law is “asymptotic, meaning that it is valid only in the long run.” Piketty provides no criteria for knowing where in the “long run” we now are, but appears to assume we are at the beginning: “[I]t will take several decades for the law  β = s/g to become true.” (p. 168);

2. The law is invalid if a significant portion of national “capital” consists of pure natural resources, independent of human improvement. Note that the concept of capital (which is the equivalent of wealth only in long-run equilibrium) is now said to include natural resources, contrary to the basic classical concept of wealth (p. 169);

3. And the law is invalid, Piketty says, if asset prices do not evolve on average in the same way as consumer prices. (Id.)

The over-simplification embodied in the long-run equilibrium concept necessarily leads to over-optimistic expectations. For example, the aggregate saving rate (s) increases when income becomes more concentrated, because wealthy people have a much lower “marginal propensity to spend” than people at lower income levels, who spend all or nearly all of their incomes and reduce any savings they may have when their incomes fall. Statistically, the model will pick up the consequent long-run changes in the capital/income ratio, but it provides no way to evaluate short-run effects, which in the United States have been extreme over the last three decades. 

For many readers this is conceptually difficult material. But the untutored mind is an asset here: I urge readers to stay with me, because after reviewing Piketty’s account of the genesis of his model, and Gardner Ackley’s 1963 analysis of the development of production function-based growth models, we reach some surprising conclusions. If you wish, go now to the “Summary and Conclusions” at the end for a broad overview of all of this.  


Piketty’s Discussion of “Harrod-Domar” and Other Growth Models

It was not until p. 230 that Piketty identified the source of the main ideas for his book, which is the theory that has become known as the “Harrod-Domar” growth model:

When the formula β = s/g was explicitly introduced for the first time by the economists Roy Harrod and Evsey Domar in the late 1930s, it was common to invert it as   g = s/β. Harrod, in particular, argued in 1939, that β was fixed by the available technology (as in the case of a production function with fixed coefficients and no possible substitution between labor and capital), so that the growth rate was entirely determined by the saving rate. (p. 230)

It was also at this stage of the book when Piketty mentioned the term “production function” for the first time. It is important to know what that is:

In economics, a production function relates physical output of a production process to physical inputs or factors of production. The production function is one of the key concepts of mainstream neoclassical theories, used to define marginal product and to distinguish allocative efficiency, the defining focus of economics. * * * 

In macroeconomics, aggregate production functions are estimated to create a framework in which to distinguish how much of economic growth to attribute to changes in factor allocation (e.g. the accumulation of capital) and how much to attribute to advancing technology. Some non-mainstream economists, however, reject the very concept of an aggregate production function. (Wikipedia, here)

The aggregate production function is a purely “supply-side” perspective, which assumes that consumer and investor markets will always clear (at least, as discussed above, in the long run).  

Piketty mentions several early “production functions” and stances on the “capital-labor split,” e.g.: 

Marx: “For Marx, the central mechanism by which ‘the bourgeoisie digs its own grave’ corresponded to what I referred to in the Introduction as ‘the principle of infinite accumulation’: capitalists accumulate ever increasing quantities of capital, which ultimately leads to a falling rate of profit (i.e., return on capital) and eventually to their own downfall.” (pp. 227-228)

Cobb-Douglas: This “production function,” first proposed in 1928, “became very popular after World War II (after being popularized by Paul Samuelson)” (p. 218). This function specifies that “no matter what happens, and in particular what quantities of capital and labor are available, the capital share of income is always equal to the fixed coefficient α, which can be taken as a purely technological parameter.” Thus, “if α = 30 percent, then no matter what the capital/income ratio is, income from capital will account for 30 percent of national income (and income from labor for 70 percent).” (p. 218). [In other words, income from capital is unrelated to the value of capital assets in the C/I numerator.] * * * “[H]istorical reality is more complex than the idea of a completely stable capital-labor split suggests. The Cobb-Douglas hypothesis” is a “useful point of departure for further reflection.” (p. 218) 

Bowley-Keynes: The capital-labor split in Britain remained relatively stable in the period 1880-1913 (Bowley); in 1939, “Keynes took the side of the bourgeois economists, calling the stability of the capital-labor split ‘one of the best-established regularities in all of economic science’.” (p. 220)   

Piketty says little about Harrod’s and Domar’s ideas except to suggest:

Harrod: “If the savings rate is 10 percent and technology imposed a capital/income ratio of 5 (so that it takes exactly 5 units of capital, neither more nor less, to produce one unit of output) then the growth rate of the economy’s productive capacity is 2 percent per year. But since the growth rate must also be equal to the growth rate of the population (and of productivity, which at the time was still ill-defined) it follows that growth is an intrinsically unstable process, balanced ‘on a razor’s edge.’ There is always either too much or too little capital.”

* * * Harrod’s intuition was not entirely wrong, and he was writing in the middle of the Great Depression, an obvious sign of great macroeconomic instability. Indeed, the mechanism he described surely helps to explain why the growth process is highly volatile: to bring savings in line with investment at the national level, when savings and investment decisions are generally made by different individuals for different reasons, is a structurally complex and chaotic phenomenon, especially since it is often difficult in the short run to alter the capital intensity and organization of production.  

Domar: “In 1948, Domar developed a more optimistic and flexible version of the law g = s/β than Harrod’s. Domar stressed the fact that the savings rate and capital/income ratio can to a certain extent adjust to each other.” (p. 231)

(Robert) Solow:  “Even more important was Solow’s introduction in 1956 of a production function with substitutable factors, which made it possible to invert the formula and write β = s/g. In the long run, the capital/income ratio adjusts to the savings rate and structural growth rate of the economy rather than the other way around. Controversy continued, however, in the 1950s and 1960s between economists based primarily in Cambridge, Massachusetts (including Solow and Samuelson, who defended the production function with substitutable factors) and economists working in Cambridge, England (including Joan Robinson, Nicholas Kaldor, and Luigi Passinetti), who (not without a certain confusion at times) saw in Solow’s model a claim that growth is always perfectly balanced. thus negating the importance Keynes had attributed to short-term fluctuations. It was not until the 1970s that Solow’s so-called neoclassical growth model definitively carried the day.” (p. 231)

This selective and limited set of ideas and partial theories is inherently confusing. But I urge everyone, especially economists, to step back and take a moment to reflect on the upshot of all this, and ask some basic questions such as these:

1. How can an idea, especially a mere kernel of an idea, that was so extensively debated more than fifty years ago, suddenly emerge now in the 21st Century as “the Second Fundamental Law of Capitalism?”

2. How can such a simple formula have different “versions”? 

3. Why has Piketty presented a simple Harrod-Domar model as the basis for his discussion of production functions, when he apparently favors a different model proposed by Robert Solow?

4. Wasn’t Keynes’ General Theory mainly concerned with short-term fluctuations in aggregate demand, and if so why did his views on short-term fluctuations come up in discussions about the long-run production function?

Ackley’s Evaluation of the Harrod-Domar and Other Growth Models

Ackley’s text was divided into four parts: (1) Concepts and Measurements, (2) The Classical Macroeconomics, (3) The Keynesian Macroeconomics, and (4) Some Extensions. In parts 2 and 3, he provided basic, functional models of classical and Keynesian systems, which showed how they accounted for growth and change of major variables, as augmented by Keynes’s introduction of the concept of effective demand and the “consumption function.” Part 4 contained four chapters which, we might say, dealt with gaps in the coverage of the basic theories: XVII, The Theory of Investment; XVIII, Economic Growth, the Problem of Capital Accumulation; XIX, Selected Problems of Nonproportional Growth; and XX Macroeconomics and Microeconomics. Other than to note that the topic of income and wealth distribution was nascent in the U.S. at that time, given low levels of inequality and high and growing middle-class prosperity, Ackley had nothing to say on that score. However, he did provide a systematic overview of the various growth models that had been proposed and debated over the previous few decades. His discussion was limited to economies “already employing productive techniques and highly-developed economic institutions,” employing a “free-enterprise system of organization.”  (p. 505)

Baumol’s “Magnificent Dynamics”: Ackley discussed William Baumol’s summary (Economic Dynamics, 1951) of the “magnificent dynamics” of the early classical school’s growth model in which “per capita income is just sufficient to permit the population to reproduce itself at the physical (or cultural) minimum level of subsistence,” and should per capita income exceed subsistence, there would be “a margin which would be divided between (a) payment of wages in excess of subsistence, thus encouraging population growth, and (b) profits in excess of the capitalists’ living expenses, a difference that can (and will) be invested to equip the growing population with the necessary tools (or at least to provide the enlarged investment . . . associated with a larger working force.” (p. 507) This model “is too simplistic, not very relevant to Western society.” (p. 509)

Keynes and the stagnationists: Because of the highly inelastic marginal productivity of capital in highly developed countries, “the growth of capital through investment must ultimately lead toward capital ‘saturation,’ a deficiency of investment opportunities relative to full-employment saving, and a necessary decline in income and employment necessary to eliminate the excess of saving.” (p. 509) * * * “Keynes’ view recognizes what the simple ‘magnificent dynamics’ model just reviewed had missed. Namely, it recognizes that capital is more than a means of employing labor. It is itself productive, and an increase in capital even with no increase in labor (or… greater than the increase in labor) can yield a positive, although diminishing return. (p. 510)

“[T]he basic error in this Keynesian position” is “a failure to realize that a growth of income — a growth which the very act of investment permits — can prevent capital saturation.” (p. 511) * * * “However, the causal link from population growth to investment, clear enough for public investment and perhaps even for housing and basic utilities, is far from obvious with respect to private investment in facilities to produce ordinary consumer goods. An increase in population increases potential consumption, and thus the potential size of the economy and the capital stock which it can use without reducing rate of return. But perhaps only if the investment first occurs and incomes rise as a result can the potential consumption, be translated into actual demand and thus provide a justification for the investment.” (p. 512)

“To the extent that the stagnationist position rested on Keynes’s failure to see that the size of the capital stock can only be considered ‘large’ or ‘small’ in relation to the size of the national income, and that it is possible for the two to grow together, the position embodied an analytical error.” * * * “If [the stagnationist position] argued merely that stagnation is a possible state for a wealthy economy, it was arguing little more than Keynes had already demonstrated, quite without reference to long run capital accumulation.” (p. 512)

Domar: “An understanding of one fundamental relationship between capital accumulation and growth stems perhaps most clearly from the work of Every Domar. Domar starts from Keynes’ recognition that today’s investment competes, at least initially, not only with yesterday’s but with tomorrow’s investment. It provides new productive capacity, which, if it is not adequately used, will discourage further investment tomorrow. This will increase the surplus of idle capital. But unlike Keynes Domar saw that there was nothing inevitable about this outcome. If total demand tomorrow should be sufficiently greater than today’s demand, the newly added productive capacity could be fully employed, and there would be room for new investment again tomorrow.” * * * Domar asked at what rate demand would have to grow – and how might this growth come about – in order to make full use of the rising productive capacity provided by capital accumulation.” (p. 513)

Domar’s formula was not the same as Piketty’s (American Economic Review, 37, March 1947). As Ackley explained, Domar was concerned with the degree of utilization of new capacity: The amount of investment (i) times the amount of added capacity per $ of investment (µ) =  the amount of added capacity in a period (µi).  The math for his formula is set forth at pp. 513-514. Here is Ackley’s intriguing observation:

 “Thus, if investment grows at a constant percentage rate, αµ, productive capacity, although continually growing, will be fully used.” Should investment grow “at a lesser rate, added productive capacity would not be fully utilized; instead an increasing margin of idle capacity would accumulate. Thus we have the paradox that if only productive capacity grows fast enough, no idle capacity will develop. But too small a growth of capacity will produce a surplus of capacity.”

This bizarre result, which is due to the fact that actual output from additional capacity is growing even more slowly than capacity, is illustrated and confirmed with a numerical example (p. 516). Here is Ackley’s overview of Domar: 

“Domar did not pretend to provide a theory of growth, but only to indicate one significant aspect of the problem of growth, and to compute, on simplified assumptions, what the necessary rate of growth would have to be in order to avoid the accumulation of excess capacity which would inhibit growth. That is, Domar described an equilibrium growth path, but indicated little about what might cause the economy to follow or to depart from that path. This equilibrium growth path is defined by the condition that all of the capital provided by previous investment is utilized, yet neither is there any capital shortage.” (p. 517)

Harrod: “Harrod had a more ambitious aim. Not only did he recognize the problem of growth but also he tried to provide a theory which explained how steady growth occurred in an economy (“An Essay in Dynamic Theory,” Economic Journal XLIX , March 1939); and also how, if this growth were interrupted — if growth once diverged from its equilibrium path — the economy might either ‘explode’ into too rapid growth, producing inflation, or cease to grow altogethre, producing depression. * * * Harrod’s own presentation leaves certain points quite unclear; consequently, in summarizing his argument, we are necessarily going somewhat beyond his own formulation. * * * Whereas Domar had no theory of what investment would be (but only what it must be for growth to be sustainable), Harrod adopts the acceleration principle as a theory of investment.” (p. 518)

(The acceleration principle rests on the idea, “[a]bstracting from all other influences,” that “the necessary stock of capital (in physical terms) depends on the rate (in physical terms) of demand for final output. * * * [I]f income changes, by a positive or negative amount, investment (or disinvestment) will occur” at a rate  depending on the degree of change in income (p. 486).)

“The equilibrium growth rate is a constant. * * * If the rate of growth happens to diverge even once from its equilibrium path, it will thereafter diverge increasingly.” (p. 520) 

“[L]et  us see if we can express in words what this is all about. Harrod’s vision is of a growing, expanding economy, in which businessmen are always, in effect, “betting” on growth, but not always sure how much growth to count on. Since they must produce in advance of sale, they have no choice but to make a ‘bet.’ Having made their production decisions, the carrying out of these decisions (a) generates consumer incomes and, through consumer spending, a market for part of the output they have decided to produce; and (b) requires additions to productive capacity in the form of capital goods and extra inventories, the magnitude of which additions depends  on the growth of output they have decided (collectively) to provide.” 522 

“There is one . . . rate of growth of output, but only one, which is correct, in the sense that it will generate just enough demand to permit them to sell all that they have produced. This is the “equilibrium” or “warranted” rate. If the collective bets of sellers happen to hit this rate, all is well.” (The correct rate will perpetuate itself.) “But if the collective ‘bets’ should involve an output increase which exceeds this warranted rate, demand will be generated which is even greater, so that shortages appear, etc.* * * But if they are insufficiently optimistic, and make production plans involving insufficient growth, their pessimism will be more than confirmed.” 522  

Two “aspects of all of this” are “particularly implausible”:

1. All of this rests on an “empirical generalization that producers behave in the manner described in the equation”;  i.e., they will repeat last period’s growth when they find it just right;

2. “Second, the notion that production plans come first, and that these, then, through the accelerator, determine investment, quite reverses the more usual (and a priori more plausible?) sequence.” (p. 523)

“In summary, Harrod tried to do considerably more than Domar. Domar defined a sustainable growth path in which all of the capital provided by previous investment is utilized, yet without any deficiency of capital. Harrod’s warranted growth rate also embodies this concept of equilibrium. His use of the accelerator … necessarily precludes either deficient or surplus capital. Rather his warranted growth rate is additionally concerned with another kind of equilibrium: that between demand and supply for current output. Harrod assumes, with little apparent foundation, that producers always expect sales to grow by the same percent as they have been growing. * * * An equilibrium between demand and supply of current output is the crucial element of his growth theory.” (525-526)

“[I]t should be stressed that either [formulation of the Harrod] model does extreme violence to reality. Either model, in strict form, implies a greatly oversimplified theory of expectations.” (p. 529)

Solow: “[E]ven without any technological change the accumulation of capital at a faster rate than the growth of labor would tend to raise the K/y [capital/income] ratio, and, of course, the average productivity of labor. That is, the data we observe are the result of a number of simultaneous changes, and we must attempt to sort out that part which is due to technological change. Among others, an interesting contribution to this task has recently been made by Robert Solow, “Technical Change and the Aggregate Production Function,”Review of Economics and Statistics XXXIX (August 1957), 312-320. 

Domar Revisited: Ackley noted in his summary (pp. 534-535 that all of the growth models he discussed (Keynes, Harrod, and Duesneberry) relate to the “demand approach to economic growth.” These models were concerned with the issue of generating sufficient aggregate demand to permit continued growth. He returned to Domar, because he had tried to determine what the necessary rate of growth would have to be in order to avoid the accumulation of excess capacity which would inhibit a sustainable level of growth. Then he proposed a new approach:

“In the first section of the next chapter, we return, in effect, to the simpler problem posed by Domar: what kind of a rate of growth of demand (if it did occur) would fully utilize an economy’s growing productive facilities? We do not worry about demand. Either we assume that it is naturally buoyant, or that government can make it so. The important question, then, is how fast does capacity grow? But now, for the first time, we are prepared to recognize that an economy’s productive capacity depends on something more than just the size of its capital stock.” (p. 535)

This is a curious reformulation of the issue of growth. Although he had concluded that the size of an economy’s productive capacity depends on “something more than just the size of its capital stock,” by inviting brainstorming that simply assumes needed demand will be there (which was, I must note, the original stance of classical theory, one that Keynes worked tirelessly to overcome with his insistence on the introduction of the demand function in his General Theory) Ackley appeared to be opening an inquiry as to whether any sensible “supply-side” model could be developed. After thirty pages of miscellaneous brainstorming, though, he pretty much gave up the effort:

“Now all of this is by way of framework or background for a theory of growth, and, in itself, provides little insight. Only as we develop further empirical hypotheses can we hope to contribute to growth economics. Some hypotheses are possible and plausible. A very simple one relates to the demand for money in a growing economy.”

Some Observations:

Much thanks to anyone still here after wading through this difficult section. Just now, with all of this in front of us, some clarifying observations emerge: Notice that Ackley’s more detailed discussion of these production capacity-based growth models reveals that they were all short-run models, and that they all attempted to reconcile growth of capital stock with aggregate demand, a singularly Keynesian perspective. Ackley saw, in the development of these models, a consistent recognition “that an economy’s productive capacity depends on something more than just the size of its capital stock.” But he also found all of these models wanting, either unable to live up to expectations that the development of capital stock would provide stable growth, or reflecting unwarranted assumptions about human expectations and behavior.

Piketty’s discussion of consumption function-based growth models avoids reference to these issues.  Instead, Piketty mentions model development in passing, while reducing the issue to the simple Domar question Ackley identified:

“We do not worry about demand. Either we assume that it is naturally buoyant, or that government can make it so. The important question, then, is how fast does capacity grow?”

Piketty proceeds carefully, however, realizing that abandoning “the demand approach to economic growth” requires limiting the question of growth to one of identifying a long-run “equilibrium.” This leads to a conclusion that Piketty does not deny, but appears loathe to frankly acknowledge: His model is incapable of predicting growth.


Here is a brief “executive summary” of the points I find significant with respect to Piketty’s two alleged “fundamental laws of capitalism”:

About What’s Important

1. The point I will be emphasizing in my next post is that, from the standpoint of concerns about inequality, Piketty has raised the wrong issue. Inequality growth is directly related to the concentration of wealth and income, factors that directly destroy prosperity and create poverty, but Piketty’s two laws address a different topic;

2. More precisely, Piketty’s two “fundamental laws of capital” relate to the concentration of capital stock invested in the means of production. “Capital intensity” has almost nothing to do with the distribution of wealth and income (inequality);

3. Although he earnestly seeks to find implications from his “laws” for inequality, Piketty concedes that capital intensity does not determine how wealth is distributed in an economy, and that the two variables move independently;     

4. Because his treatment of the capital accumulation problem relies on dubious neoclassical “equilibrium” assumptions that incorrectly assume productive capacity and wealth are equivalent in the “long run,” Piketty raises the specter of a conclusion he knows is wrong, that there is a long-run equilibrium level of inequality;     

About Production Function Growth Theory

5. On his premier topic, capital accumulation, Piketty has resolved an old theoretical dispute about the growth of productive capital by converting a “short-run” model into its long-run counterpart, thus (a) over-simplifying the question of capital growth, and (b) stripping the model of explanatory power; 

6. From Domar on down, attempts to develop a short-run growth model based on productive capacity did not go well. Theoreticians could not agree on what simplifying assumptions would best represent reality, and data limitations would make it difficult to adequately test these theories;

7. By 1963, although demand-side theories in macroeconomics were making some progress (thanks to Keynes), Ackley’s review of supply-side aggregate growth theories showed that area to still be in a rudimentary stage of development and badly in need of fresh hypotheses;

8. Production function models, however useful they may be in microeconomics, were never developed sufficiently to provide a realistic explanation of  aggregate economic growth;

9. Piketty has presented a system of “tautologies,” or “accounting identities” that by itself has no explanatory power;

10. Although his growth model intends to describe a long-run rate of capital accumulation, it merely describes a long-run trend of capital accumulation, described as a path toward an eventual “equilibrium” state; and as Piketty freely admits, the model tells us nothing about how or when (if ever) such a state might be achieved;

11. Nonetheless, Piketty assumes, without supporting evidence, that an economy is always trending toward the equilibrium level of capital intensity established by its long history of capital accumulation; instead, the trend of capital intensity keeps moving up or down with a host of factors (like technological change);   

About “Model-Dependent Realism”

12. Piketty refers to the two components of his model as “fundamental laws” when they are not functional laws at all. Together, they merely use accounting identities to describe a theoretical equilibrium condition that cannot ever be perceived in “reality;”

13. To the extent Piketty believes his “fundamental laws of capitalism” explain that “reality” consists of persistent growth toward an equilibrium state of capital accumulation, he has reached a faulty conclusion by virtue of a neoclassical perspective that is not supported by experience. Thus, his model fails the test of “model-dependent realism” elucidated by Hawking and Mladinow;

JMH – 6/27/2014 (“Summary” ed., 6/28/2014)

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