A Guide to Thomas Piketty’s Capital in the Twenty-first Century

(An expanded version of this essay can be found on Amazon.com, Click Here.)

The purpose of this guide is to examine the overall conclusion of Thomas Piketty’s Capital in the Twenty-first Century and to elaborate on the theoretical framework within which this conclusion is derived.

Piketty’s Overall Conclusion

Piketty summarizes the overall conclusion of his analysis at the beginning of his concluding chapter:

The overall conclusion of this study is that a market economy based on private property, if left to itself, contains powerful forces of convergence, associated in particular with the diffusion of knowledge and skills; but it also contains powerful forces of divergence, which are potentially threatening to democratic societies and to the values of social justice on which they are based.

The principal destabilizing force has to do with the fact that the private rate of return on capital, r, can be significantly higher for long periods of time than the rate of growth of income and output, g. (l. 10093, p. 571)

The inequality r > g implies that wealth accumulated in the past grows more rapidly than output and wages. This inequality expresses a fundamental logical contradiction. The entrepreneur inevitably tends to become a rentier, more and more dominant over those who own nothing but their labor. Once constituted, capital reproduces itself faster than output increases. The past devours the future. (l. 10093, p. 571)

This fundamental force for divergence . . . functions as follows. . . . [I]f g = 1% and r = 5%, saving one-fifth of the income from capital (while consuming the other four-fifths) is enough to ensure that capital inherited from the previous generation grows at the same rate as the economy. If one saves more, because one’s fortune is large enough to live well while consuming somewhat less of one’s annual rent, then one’s fortune will increase more rapidly than the economy, and inequality of wealth will tend to increase even if one contributes no income from labor. For strictly mathematical reasons, then, the conditions are ideal for an “inheritance society” to prosper—where by “inheritance society” I mean a society characterized by both a very high concentration of wealth and a significant persistence of large fortunes from generation to generation. (l. 6080, p. 351)

What Piketty is saying here is that if the difference between the rate of return on an individual’s capital r and the overall rate of growth of the economy g is sufficiently large that the individual is able to save a fraction of his or her income equal to (r-g)/r the individual’s fortune will grow at the same rate as the economy, and if he or she is able to save a larger fraction than (r-g)/r the individual’s fortune will grow faster than the economy. As will be shown below, this is just simple arithmetic (or not so simple arithmetic depending on how comfortable one is with arithmetic) within Piketty’s theoretical framework, and, in itself, it tells us nothing about how the economic system will actually behave. All this arithmetic tells us is what will happen if r is sufficiently greater than g for a protracted period of time. For this knowledge to be useful we must examine what we know about the determinants of r and g and the mechanisms that lead to the concentration of wealth and income in the real world.

This is what Piketty does in Capital in the Twenty-first Century, and, in so doing, he comes to the conclusion that there are powerful forces within an unregulated market economy (i.e., a market economy that is “left to itself”) that will cause the rate of growth of income and output g to fall over the next century and that will support the private rate of return on capital r. As a result, he concludes that—in the absence of government intervention—there are reasons to believe that r > g will become the rule at some point during the twenty-first century and that if this is allowed to happen it is likely to lead to dramatic increases in the concentration of income and inherited wealth as a fraction of national income. He also finds that the processes that are likely to bring this about have already begun.

Piketty does not argue that this is inevitable. In fact, he discusses ways in which this can be prevented, and he discusses the kinds of things that have interrupted this process in the past and that could reverse this process or keep it from happening in the future. At the same time, he finds the mere possibility of a dramatic increase in the concentration of income and wealth to be somewhat disturbing:

The consequences for the long-term dynamics of the wealth distribution are potentially terrifying, especially when one adds that the return on capital varies directly with the size of the initial stake and that the divergence in the wealth distribution is occurring on a global scale. (l. 10093, p. 571, emphasis added.)

It is not difficult to understand why Piketty feels this way. As he points out, it is exceedingly difficult to imagine how the democratic political and social institutions of modern civilization can be sustained in a world in which wealth and income become more and more concentrated into the hands of a small minority of the population:

Under such conditions, it is almost inevitable that inherited wealth will dominate wealth amassed from a lifetime’s labor by a wide margin, and the concentration of capital will attain extremely high levels—levels potentially incompatible with the meritocratic values and principles of social justice fundamental to modern democratic societies. (l. 546, p. 26, emphasis added.)

This observation is especially ominous in view of Piketty’s additional observation that the historical reign of r > g culminated at the beginning of the twentieth century in, and was temporarily brought to an end as a result of World War I, the Great Depression, and World War II.

Is Piketty right in his analysis in Capital in the Twenty-first Century? In my attempt to understand the historical roots of the financial crisis and the economic stagnation we are in the midst of today I came to a similar conclusion at the end of Where Did All The Money Go?:

If we . . . continue to follow the failed ideological mantra of lower taxes, less government, and deregulation we are most certainly going to end up right back where we started in the 1930s. And if the political leaders throughout the world continue to follow this failed ideological mantra and refuse to come to grips with the root cause of the worldwide economic catastrophe we face today—namely, the concentration of income at the top of the income distribution—we are likely to end up where we ended up in the 1940s. (l. 5656)

As I argued there, it seems to me that, given the state of mass-production technology within our economic system, full employment of our resources is unsustainable in the face of the increased concentration of income we have experienced over the past thirty-five years. In the absence of government intervention to reverse this trend

our ability to produce will be diminished as our employment problem is solved through the transfer of resources out of those industries that serve the bulk of our population (those that produce for domestic mass markets) and into those that serve the wealthy few, (l. 286)

travel down a road that leads toward our becoming a nation of servants, groundskeepers, security guards, police officers, soldiers, and builders of pyramids. (MS Piketty) In the meantime, our transportation, water and waste treatment facilities, education, regulatory, legal, and other governmental systems will continue to deteriorate, and the standard of living of the vast majority of our population will continue to stagnate or fall as the divisiveness within our society increases. (l. 5649)

It is hard for me to imagine how our basic political and social institutions can be maintained if we continue on this journey:

It seems unlikely to me that Mc Mansion investment will be sufficient to maintain full employment in this situation or that the requisite government expansion needed to provide full employment will come from expanding social services and infrastructure. When I project this dynamic on a global scale I find it impossible to avoid Piketty’s conclusion. Thus, in my view, Piketty most certainly is right. The potential consequences of allowing the dynamic forces of income and wealth to play themselves out in the manner explained in Capital in the Twenty-first Century are, indeed, terrifying.

At the same time, there is no reason to believe there will be a consensus on the issues raised by Piketty. The economic, political, and social implications of Piketty’s analysis are such that it is inevitable there will be a massive effort to refute Piketty's conclusion, much of that effort taking the form of obfuscation. In order to combat this it is essential that the nature of Piketty’s arguments be understood as clearly as possible.

Piketty’s Methodology

Piketty uses massive amounts of historical data that he and his collaborators have collected to examine the economic, political, social, demographic, and other determinants of r and g for a) income groups, b) countries, c) groups of countries, and c) for the entire world in an attempt to understand the historical behavior of these ratios. Having examined the historical record, Piketty then looks to the future and attempts to apply the lessons learned from the past to an examination of the possibilities that will be available in the future. In the process he avoids the explicit use of mathematics whenever possible. Only four equations are emphasized in the text, and Piketty attempts to explain the nature of these equations in words as simply as possible. Those who have a basic understand of percentages and the nature of compound interest should have little difficulty in understanding the economic significance of these equations even in the absence of formal training in economics, and Piketty’s account of the economic history of the seventeenth through the twentieth centuries is riveting even for those who lack a basic understanding of percentages and compound interest.[2]

While Piketty’s minimalist approach toward mathematics in explaining the nature of his equations in words as simply as possible may be beneficial when it comes to those who appreciate this approach, I fear that many who are capable of understanding these equations and how they fit into Piketty’s theoretical framework who are more comfortable with mathematics than with words are going to be put off by the way Piketty introduces his equations without explaining their derivation. This is something I had to stop and think about as I read his book, and while Piketty does, more or less, explain the derivation of his equations in his Online Appendix (and in the various papers and books linked to in this appendix) he does not do so succinctly. My fear is that if, in spite of my background in economics, I had to stop and think about the nature of Piketty’s equations and spend time working them out there may be a number of people out there who just won’t bother and will not fully appreciate Piketty’s argument without some assistance in understanding where his equations come from and how they fit into his theoretical framework.

As a result, most of the rest of this guide is devoted to explaining the derivation of Piketty’s four equations and how they fit into his theoretical framework while, at the same time, attempting to adhere to Piketty’s minimalist approach toward mathematics by explaining the derivation of these equations as simply as possible and using as little mathematics as possible. Before attempting this task, however, I think it may be best to examine Piketty’s definitions of capital, wealth, and income within the context of his theoretical framework in order to make clear what is meant by these terms within this context.

Capital, Wealth, and Income

I begin with a brief explanation of Piketty’s definitions of capital and income. Piketty is quite clear in his explanation of these definitions, but since they are somewhat different from the generally accepted definitions within the discipline of economics, and because they define what Piketty is talking about and play a central role in the logic of his analysis, it is best to summarize them at the beginning in order to avoid confusion on the part of those who are familiar with the generally accepted definitions and to give some concrete meaning to the subject matter of Piketty’s work.

Piketty uses “the words ‘capital’ and ‘wealth’ interchangeably, as if they [are] perfectly synonymous.” (l. 889, p. 47) Formally, he states:

I define “national wealth” or “national capital” as the total market value of everything owned by the residents and government of a given country at a given point in time, provided that it can be traded on some market. It consists of the sum total of nonfinancial assets (land, dwellings, commercial inventory, other buildings, machinery, infrastructure, patents, and other directly owned professional assets) and financial assets (bank accounts, mutual funds, bonds, stocks, financial investments of all kinds, insurance policies, pension funds, etc.), less the total amount of financial liabilities (debt). (l. 911, p. 48, emphasis added.)

Thus, Piketty specifically includes financial as well as physical assets in his definition of capital. Economists generally exclude financial assets from the definition of capital, the reason being that the focus of most economists is on the determination of employment and output. Given this focus, it is only natural to exclude financial assets since financial assets represent only claims against income and wealth and do not contribute directly to the processes of production itself. Since Piketty’s focus is on the distribution of income and wealth, rather than on employment and output, and since financial assets generate income and provide a vehicle for storing wealth just as physical assets provide these functions, financial assets, by necessity, play a crucial role in Piketty’s analysis.

Piketty also includes public (i.e., government) assets as well as privately owned assets in his definition of national wealth or national capital where public wealth, as in any accounting system, is defined as the difference between the value of the assets owned and the value of debts owed. In addition, he states that

total national wealth can always be broken down into domestic capital and foreign capital:

Domestic capital is the value of the capital stock (buildings, firms, etc.) located within the borders of the country in question. Net foreign capital—or net foreign assets— measures the country’s position vis-à-vis the rest of the world: more specifically, it is the difference between assets owned by the country’s citizens in the rest of the world and assets of the country owned by citizens of other countries. (l. 936, p. 47-9, emphasis added.)

Thus, Piketty’s emphasis on the distribution of income and wealth leads him to go beyond domestic private capital to include both public and net foreign capital within his concept of national capital/wealth.

Piketty’s definition of national income is a bit more conventional than his definition of national capital, though not entirely so:

National income is defined as the sum of all income available to the residents of a given country in a given year, regardless of the legal classification of that income. . . .

In order to calculate national income, one must first subtract from GDP the depreciation of the capital that made this production possible. . . . When depreciation is subtracted from GDP, one obtains the “net domestic product,” which I will refer to more simply as “domestic output” or “domestic production,” which is typically 90 percent of GDP. Then one must add net income received from abroad . . . (ll. 818-824, pp. 43-4, emphasis added.))

Thus, national income is equal to net domestic product plus “net income received from abroad” where:

a) “net domestic product” is the aggregate value of goods and services produced in the domestic economy as defined in the National Income and Product Accounts (GDP less depreciation), and

b) “net income received from abroad” is the difference between the value of all income and transfers generated in foreign countries that are paid to domestic entities less the value of all income and transfers generated in the country that are paid to foreign entities. (BEA 4.1)

Again, since Piketty is examining the distribution of income and wealth rather than output produced and the level of employment he must, by necessity, account for the incomes of domestic entities that are paid out to and received from foreign entities in his analysis. Hence, his inclusion of “net income received from abroad” in his definition of national income.

Having summarized these key definitions that underlie Piketty’s theoretical framework, we begin our examination of Piketty’s four equations by examining his First Fundamental Law of Capitalism: α = r × β.

The First Fundamental Law of Capitalism: α = r × β

Piketty explains his First Fundamental Law of Capitalism: α = r×β as follows:

I can now present the first fundamental law of capitalism, which links the capital stock to the flow of income from capital. The capital/income ratio β is related in a simple way to the share of income from capital in national income, denoted α. The formula is

The formula α = r × β is a pure accounting identity. It can be applied to all societies in all periods of history, by definition. Though tautological, 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. (l. 982, p.52)

This law simply states that the share of total income that is received by the owners of capital α is equal to the rate of return on capital r multiplied by the capital/income ratio β:

According to Piketty, this equation shows that the “capital/income ratio β is related in a simple way to the share of income from capital in national income, denoted by α.” At first glance, however, the validity of this equation is not obvious; at least it wasn’t to me, in spite of the fact that it is a direct implication of the definition of what we mean when we talk about “the rate of return on capital.”

If the total value of capital in the economic system as a whole is equal to K, and if the total income earned from this capital by its owners during a given period of time is Y_k, then, by definition, the rate of return on capital r in the economic system during that period of time is given by:

The importance of this definition in Piketty’s analysis is that historical data exists on Y_k and K for various countries, and this data can be used to calculate the historical values for r in these countries. This allows him to examine how the rates of return on capital r in various countries have changed over time.

Once the definition of the rate of return on capital r embodied in the definition 2) is understood, it is just a hop, skip, and a jump through the magic of elementary algebra to get from this definition to Piketty’s First Fundamental Law. If we multiply both sides of the definition in 2) by K, the K will cancel on the right-hand side and we are left with:

Since Y_k/Y is the share of capital income in the national income which Piketty denotes as α (=Y_k/Y), and K/Y is the capital/income ratio which Piketty denotes as β (=K/Y), substituting into 4) and switching the sides around yields:

In other words, Piketty’s First Fundamental Law of Capitalism follows directly from what we mean when we talk about (the definition of) the rate of return on capital r. As Piketty states, this law is an identity—it is true by definition—and the significance of this law is that it provides a mechanism by which it is possible to examine how changes in the rate of return on capital r = Y_k/K and the capital/income ratio β = K/Y affect the share of national income that is paid to the owners of capital α over time.

The Second Fundamental Law of Capitalism: β = s / g

In the long run, the capital/income ratio β is related in a simple and transparent way to the savings rate s and the growth rate g according to the following formula:

. . . This formula, which can be regarded as the second fundamental law of capitalism, reflects 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 its income), which can in turn have a significant effect on the social structure and distribution of wealth. (l. 13513, p.166)

Here, again, we are looking at a formula that, at first glance, may not be intuitively obvious. What does the rate of savings s have to do with the capital/income ratio β?

The connection is made by noting that within the National Income and Product Accounts the amount of savings—that is, the portion of income that is not consumed—is, by definition, always equal to the amount of (net) investment, and investment is, by definition, the amount by which capital K is increased as a result of current saving. This is what is meant by saving and investment in the National Income and Product Accounts, and within Piketty’s theoretical framework savings is that portion of income that is not consumed but, rather, is added to the stock of capital K.[3]

Thus, when we express the capital/income ratio β as the ratio of capital K to income Y,

it becomes apparent that, since the rate of savings (out of income) s determines the amount of savings (i.e., the amount of current income that is not consumed) which is, by definition, the amount of income Y that is added to the capital stock K the rate of savings s must contribute to the rate at which the numerator K of the fraction K/Y changes over time. By the same token, the growth rate g is defined as the rate at which the denominator Y of the fraction K/Y changes over time. Thus, the rate of savings s and the growth rate g must affect the way in which the capital/income ratio β, as defined by Piketty’s Second Law β = s/g, changes over time since they affect the rates at which the numerator and denominator of the capital-income ratio β = K/Y change.

There is nothing mysterious about this, but it’s important to recognize that Piketty’s Second Law is fundamentally different from his first. Piketty’s First Law α = r×β is an identity that defines the relationship between capital’s share of income α, the rate of return on capital r, and the capital income ratio β at all points in time. Piketty’s Second Law,

specifies the long-run, steady-state equilibrium condition for the ratio of capital K to income Y within his theoretical framework—that is, it specifies the long-run, steady-state equilibrium condition for β. As such, this law holds only in the long run and even then only under certain conditions. Specifically, this law cannot hold unless the long-run savings and growth rates are equal to s and g and the prices of assets relative to other prices (as well as many other things) do not change. [4]

What this law tells us is the direction toward which the capital/income ratio β is headed if other things don’t change. Specifically, it tells us that if other things don’t change:

a) if β < s/g the capital stock K will grow more rapidly than income Y, and β will increase,

b) if β > s/g the capital stock K will grow more slowly than income Y, and β will decrease,

c) if β = s/g the capital stock K and income Y will grow at the same rate, and β will not change. [5]

The argument is elementary. Let me illustrate it with an example. In concrete terms: if a country is saving 12 percent of its income every year, and if its initial capital stock is equal to six years of income, then the capital stock will grow at 2 percent a year, thus at exactly the same rate as national income, so that the capital/income ratio will remain stable.

By contrast, if the capital stock is less than six years of income, then a savings rate of 12 percent will cause the capital stock to grow at a rate greater than 2 percent a year and therefore faster than income, so that the capital/income ratio will increase until it attains its equilibrium level.

Conversely, if the capital stock is greater than six years of annual income, then a savings rate of 12 percent implies that capital is growing at less than 2 percent a year, so that the capital/income ratio cannot be maintained at that level and will therefore decrease until it reaches equilibrium.

In each case, the capital/income ratio tends over the long run toward its equilibrium level β = s/g (possibly augmented by pure natural resources), provided that the average price of assets evolves at the same rate as consumption prices over the long run. (ll. 2934-2941, p. 170)

The importance of this law is that it provides a framework within which it is possible to make conditional predictions—the condition being that other things don’t change—as to what will happen to the capital/income ratio β in the future. In so doing, it provides a framework that can be used to analyze the effects of changes in factors other than s and g that affect the capital/income ratio.[6]

It is also worth noting that when Piketty’s First Law α = r × β is combined with Piketty’s Second Law β = s/g the result is the steady-state equilibrium condition toward which capital’s share of income α is headed if things don’t change:

Thus, Piketty’s Second Law provides a framework within which to make conditional predictions as to what will happen to capital’s share of income α in the future as well as to what will happen to the capital/income ratio β.

Finally, it should be noted that Piketty’s Second Law β = s/g is not new with Piketty. It has a long history in the theory of economic growth that goes back to at least 1939.[7]

The Fundamental Force for Divergence: r > g

This fundamental inequality, which I will write as r > g (where r stands for the average annual rate of return on capital, including profits, dividends, interest, rents, and other income from capital, expressed as a percentage of its total value, and g stands for the rate of growth of the economy, that is, the annual increase in income or output), will play a crucial role in this book. In a sense, it sums up the overall logic of my conclusions. . . .

When the rate of return on capital significantly exceeds the growth rate of the economy (as it did through much of history until the nineteenth century and as is likely to be the case again in the twenty-first century), then it logically follows that inherited wealth grows faster than output and income. People with inherited wealth need save only a portion of their income from capital to see that capital grow more quickly than the economy as a whole. Under such conditions, it is almost inevitable that inherited wealth will dominate wealth amassed from a lifetime’s labor by a wide margin, and the concentration of capital will attain extremely high levels—levels potentially incompatible with the meritocratic values and principles of social justice fundamental to modern democratic societies. (l. 546, p. 25-6) [8]

Piketty does not write out explicitly the relationships between national income Y, the rate of growth of national income g, the size of an individual’s fortune (K_i), and the rate of growth of an individual’s wealth (g_ki) in his explanation of why he believes “it is almost inevitable that inherited wealth will dominate wealth,” and I believe it may be helpful to some if these relationships are made explicit.

The place to begin in sorting through these relationships is by defining β_i as the ratio of an individual’s wealth K_i to national income Y:

It is clear from this definition that whether the individual’s wealth/national-income ratio β_i increases, decreases, or remains unchanged depends on whether the rate of growth of the individual’s wealth g_i (i.e., the rate of growth of the numerator K_i of β_i = K_i/Y) is greater than, less than, or equal to the rate of growth of national income g (i.e., the rate of growth of the denominator Y of βi = K_i/Y). [9] Thus, to sort through these relationships we have to examine the determinants of the growth rate of the individual’s wealth g_i within Piketty’s theoretical framework.

Piketty is quite clear on the fact that an individual’s income (Y_i) can be expressed as the sum of two components—income from capital (Y_Ki) and income from labor (Y_Li):

where the individual’s income from capital Y_Ki is defined as the rate of return he or she is able to earn on his or her capital (r_i) multiplied by the amount of capital K_i he or she has:

Thus, if we substitute Y_Ki from 11) into 10) we can express the individual’s income as:

Since, by definition, the amount saved S_i is equal to the amount by which the individual’s capital stock K_i increases as a result of saving, the rate of growth of the individual’s fortune g_ki is given by:

and since, by definition, the amount of the individual’s income that is saved S_i is equal to the individual’s rate of savings out of income (s_i) multiplied his or her income Y_i:

Since the individual’s capital/national-income ratio β_i (=K_i/Y) can increase, decrease, or remain the same if and only if the growth rate of the individuals wealth g_ki (= s_i(r_i + Y_Li/K_i)) is greater than, less than, or equal to the growth rate of income g,[10] it is clear from 16) that:

What we have here are the conditions under which an individual’s wealth relative to national income K_i/Y will change. Piketty argues that if K_i is large enough, “it is almost inevitable” that s_i(r_i + Y_Li/K_i) will be greater than g, and if this is so a) tells us that the individual’s wealth will grow relative to national income Y.

What Piketty finds significant about the situation in which s_i(r_i + Y_Li/K_i) > g, that is, the situation in which “the rate of return on capital significantly exceeds the growth rate of the economy,” is that if an individual’s wealth is such that the rate at which he or she is able to save s_i times the rate of return he or she is able to earn on his or her capital r_i is greater than the rate at which national income grows g (i.e., if s_ir_i > g) his or her wealth K_i will increase relative to national income Y even if he or she has no income from labor (i.e., even it Y_Li = 0)—that is, even if he or she does not work. Hence Piketty’s conclusion:

When the rate of return on capital significantly exceeds the growth rate of the economy . . . it is almost inevitable that inherited wealth will dominate wealth amassed from a lifetime’s labor by a wide margin . . . (l. 546, p. 25)

It is important to note that the forgoing is not a mathematical proof of Piketty’s conclusion. Piketty arrives at this conclusion through an analysis of the historical workings of this dynamic in the past and by projecting this dynamic into a future in which the circumstances are the same or very similar to those that existed in the past.

The Three Forces

In explaining the three forces that determine the annual economic flow of inheritances and gifts, Piketty explains that:

In general, the annual economic flow of inheritances and gifts, expressed as a proportion of national income that we denote by b_y, is equal to the product of three forces:

β is the capital/income ratio (or, more precisely, the ratio of total private wealth, which, unlike public assets, can be passed on by inheritance, to national income), m is the mortality rate, and μ is the ratio of average wealth at time of death to average wealth of living individuals. (l. 6642, p. 383)

Piketty does not refer to this equation as being fundamental, but it deserves comment nonetheless because it deals with the flow of inherited wealth b_y within society, and the justification for Piketty’s expressing this flow as b_y = μ × m × β may not be obvious to some readers.

The justification for this equation can be seen by examining how Piketty has defined the terms in this expression: β is the private-wealth/income ratio (=W/Y), m is the number of deaths each year divided by the population (=D/P), and μ is the ratio of (annual) inherited wealth divided by the number of deaths (=W_h/D) to total wealth divided by the population (=W/P), that is, (W_h/D)/(W/P). When we substitute these definitions into Piketty’s expression b_y = μ×m×β, invert and multiply where necessary, and cancel the terms that appear in both the numerator and denominator we get:

In other words, all Piketty has done here is break down (factor) the annual flow of inherited wealth (divided by income) b_y into three components—μ, m, and β—which when multiplied together yield the annual flow of inherited wealth b_y.

By factoring the annual flow of inherited wealth b_y into these three components Piketty is able to establish two ways in which to estimate this variable. The first is by estimating the amount of inherited wealth each year W_h from tax records then dividing it by national income Y to arrive at the annual flow of inherited wealth b_y. The second is by estimating inherited wealth each year W_h from probate records then combining this estimate with estimates of the population P, deaths D, total private wealth W, and national income Y by way of the formula b_y = μ×m×β and the definitions explained above to arrive at the annual flow of inherited wealth b_y. This makes it possible to check the reliability of each of these estimating methods against the other in countries like France where both kinds of data exist. It also makes it possible to estimate the annual flow of inherited wealth b_y in countries where only probate records are available provided, of course, that data on population P, deaths D, private wealth W, and income Y also exist for those countries, which, as it turns out, are not very many.

Piketty also examines the role played by the individual factors μ, m, and β—each of which is directly related to the size of inheritance relative to income—in determining the annual flow of inherited wealth b_y. It is worth noting, in this regard, that

is the annual inherited wealth W_h divided by private wealth W which, as Piketty notes (l. 6765, p. 389), is the flow of inherited wealth expressed as a percentage of total wealth W (which in Piketty’s notation would be written b_w). Piketty uses μ×m to examine the way in which the flow of inherited wealth expressed as a percentage of private wealth W, (i.e., b_w = W_h/W) is related to the flow of inherited wealth expressed as a percentage of income Y, (i.e., b_y= W_h/Y).

Implicit Relationships

There is a set of implicit relationships that Piketty relies upon throughout his analysis that he does not write out as equations—relationships that I fear may cause some confusion if they are not made explicit. Piketty transitions between discussing national income, per capita income, population, productivity and the growth rates of these variables as if the relationships between them are self evident which, of course, they should be, but I, at least, had to stop and think about them. I found this particularly confusing when I came to the statement: “Recall that g measures the long-term structural growth rate, which is the sum of productivity growth and population growth” (l. 3953, p. 227) only to discover through a digital search that the expression “long-term structural growth” had not yet appeared in the text, and the only previous reference to “the sum of productivity growth and population growth” that I could find was in a footnote on page 167:

Sometimes g is used to denote the growth rate of national income per capita and n the population growth rate, in which case the formula would be written β = s/(g + n). To keep the notation simple, I have chosen to use g for the overall growth rate of the economy, so that my formula is β = s/g. (l. 1 0648, p. 167, ftn. 3)

I did not find this footnote particularly enlightening since it only explains Piketty’s choice of notation in defining the overall growth rate g without explicitly differentiating between the g in β = s/(g + n) and the g in β = s/g which is, of course, obvious (g = g + n) but somewhat confusing at first glance. It would seem that something has been lost in translation here, and it may be helpful to explain these relationships in a bit more detail so they are in the forefront of one’s mind when reading Piketty’s discussion of them.

The relationship between national income Y and per capita income (i.e., average national income per person which I will denote as Y_pc) is, of course, obvious since per capita income Y_pc is, by definition, equal to national income Y divided by the population (P),

This means that national income Y must be equal to the population P multiplied by per capita income Y_pc since, given 19):

Piketty often uses this relationship between population P and per capita income Y_pc implicitly as he explains the behavior of national income Y by describing the behavior of per capita income over time.

The relationship between the overall growth rate g of national income Y and the growth rate of the population P (g_p) and the growth rate of per capita income Y_pc (g_pc) in Piketty’s analysis may not be so obvious. Piketty simply asserts, without explanation, that the latter two growth rates g_p and g_pc sum to the overall growth rate g of national income,

Again, this is, perhaps, because he sees this as intuitively obvious since it is just another (not so) simple arithmetic truth: the growth rate of the product of two numbers is equal to the sum of the individual growth rates of the two numbers.[11] What is not so simple about this relationship is that it only applies to growth rates that assume continuous compounding rather than annual compounding or some other compounding scheme.

That this is so is easily seen by considering a situation in which the population P is equal to 100 and per capita income Y_pc is equal to 50 which means that national income Y must equal 5,000 (=100×50). If the population P increases at a rate of 10% compounded annually and per capita income increases at a rate of 20% compounded annually, after one year P will increase to 110 and Y_pc will increase to 60. As a result, Y will increase to 6,600 (=110×60) which is a 32% (=100×(6600-5000)/5000) annually compounded increase rather than the 30% increase obtained by adding g_p (=10%) and g_pc(=20%).

But if we examine the continuously compounded interest rates implicit in this example we find that a 10% annually compounded rate is equivalent to a 9.531018% continuously compounded rate, and a 20% annually compounded rate is equivalent to an 18.23216% continuously compounded rate. The sum of these two continuously compounded rates is 27.7632% (=9.531018+18.23216) which is equal to the continuously compounded rate that is equivalent to the 32% annually compounded rate by which P×Y_pc increased. [12]

Thus, it should be clear that the breakdown of the overall growth rate g of national income Y into its two components—the population growth rate g_p and per capita income growth rate g_pc—in Piketty’s analysis applies to continuously compounded rates of growth only and not to rates that assume an alternative compounding scheme even if the rates of the alternative compounding scheme are equivalent in the sense that they lead to the same change in the principle to which they are applied as do the continuously compounded rates.

The breakdown of the overall growth rate g into its population growth rate g_p and per capita income growth rate g_pc components plays a crucial role in Piketty’s analysis. By breaking down the overall growth rate g into these two components Piketty is able to separate the effects on the overall growth rate g that arise from changes in the population as measured by the population growth rate g_p and those which arise from changes in prices and productivity (output per capita) as measured by the per capita income growth rate g_pc. This allows him to isolate the effects of the population growth g_p on the capital/income ratio β and capital’s share of income α as well as on the steady state equilibrium capital/income ratio s/g toward which β and α (= r×β=r×s/g) will move if other things don’t change.

Not Too Mathematical Appendix

This appendix attempts to explain the mathematical foundations of Piketty’s theoretical framework using as little math as possible. Unfortunately, I am unable to avoid using math entirely, but I have limited the math in my arguments to elementary high school algebra even when talking about mathematical concepts that go beyond elementary algebra. My purpose in writing this appendix is not to teach the math, but to simply show the way in which Piketty’s verbal arguments and analysis are firmly rooted in sound mathematical reasoning, that is, that Piketty’s theoretical framework is mathematically/logically consistent.

The Quotient Rule of Differential Calculus

The arguments throughout much of Capital in the Twenty-first Century are based on what is known as the quotient rule in differential calculus. This rule is nothing more than an elementary theorem that tells us how the value of the ratio of two variables (functions or numbers) must change as the values of its numerator and denominator change in response to a change in a third variable. What this rule tells us is that if we want to know how a ratio will change as a third variable changes all we have to do is:

a) multiply the variable in the denominator of the ratio times the derivative of the variable in the numerator with respect to the third variable;

b) subtract the value of the variable in the numerator multiplied times the derivative of the variable in the denominator with respect to the third variable, then

For present purposes, all you have to know about a derivative is that it is a mathematical operator that specifies how one variable (number or function) changes when another variable changes.

The reason the quotient rule is relevant to Piketty’s analysis is that Piketty tries to explain (among other things) the behavior of the capital/income ratio β—a ratio of two variables K/Y. In addition, he is attempting to explain this ratio over time as it responds to changes in other variables over time which means that the variables in his analysis are functions of time. As a result, changes in the variables in Piketty’s analysis can be expressed as derivatives with respect to time—that is, in terms of the mathematical operator that defines how his variables change in response to a change in time.

For example, we can apply the quotient rule to see how the capital/income ratio β must change when the value of its numerator K and denominator Y change with respect to a change in time t by a) multiplying the denominator Y of the ratio Y/K times the derivative of its numerator with respect to time dK/dt, b) subtracting the numerator K multiplied times the derivative of the denominator with respect to time dY/dt, and c) dividing this difference by the denominator squared Y² to obtain:

where the expression d(β)/dt that appears in 22) is the derivative of β (= K/Y) with respect to time t. What this expression tells us is how β must change in response to a change in t given the effects of the change in t on K and Y. Similarly, the expression dK/dt is the derivative of K with respect to t, and it tells us how K changes in response to a change in t; dY/dt is the derivative of Y with respect to t, and it tells us how Y changes in response to a change in t.

Dividing both the numerator and the denominator of the fraction on the right-hand side of 22) by K×Y yields:

where (dK/dt)/K is, by definition, the rate of change of the numerator of β and (dY/dt)/Y is, by definition, the rate of change of the denominator of β. These rates of change are the growth rates in Piketty’s theoretical framework, and the significance of expressing the quotient rule in the form given in 24) is that it shows the relationship between changes in the ratio β (=K/Y) and the rates of change of its numerator Y and denominator K.

Specifically, by examining 24) we can see that if Y and K are of the same sign, β will be positive and 24) implies that:

In other words, what the quotient rule, as embodied in equation 24), tells us is that the value of the ratio of any two variables (K/Y) will increase (decrease, remain unchanged) if and only if the growth rate of the numerator ((dK/dt)/K) is greater than (smaller than, equal to) the growth rate of the denominator ((dY/dt)/Y).[13] Piketty uses the mathematical truth embodied in this implication of the quotient rule throughout his analysis.

Piketty’s Second Fundamental Law of Capitalism: β = s / g

The relevance of the quotient rule to Piketty’s arguments can be seen by noting that (dK/dt)/K on the right-hand side of 24) is the (instantaneous) rate of change in capital K with respect to time t, and (dY/dt)/Y is the rate of change of income Y with respect to time. Since g is the rate of change in income Y with respect to time in Piketty’s notation we can substitute g for (dY/dt)/Y in 24) to get:

And since, by definition, the change in the capital stock (with respect to time) dK/dt must be equal to the change in savings (dS/dt) which, in turn, is equal to the rate of savings out of income s times national income Y (i.e., dK/dt =dS/dt = s×Y), by substituting s×Y for dK/dt in 25) we get:

As was noted in the discussion of Piketty’s Second Law above, what this means is that

a) If (s – g×β) > 0 then dβ/dt will be positive, β will increase, and g/β will decrease toward closing the gap between s and g×β.

b) If (s – g×β) < 0 then dβ/dt will be negative, β will decrease, and g/β will increase toward closing the gap between s and g×β.

c) The capital/income ratio β can only be constant (i.e., dβ/dt will only equal 0) if s - g×β = 0 which means β will be constant if and only if β = s/g.

In other words, by applying the quotient rule to the definition of the capital/income ratio β and expressing the overall growth rate of income g and the definitions of savings and investment in terms of the time derivatives of savings dS/dt and investment dK/dt we are able to derive the dynamic mechanisms by which the long-run, steady-state equilibrium condition β = s/g is established within Piketty’s theoretical framework. In so doing we have been able to establish that Piketty’s Second Fundamental Law is little more than a direct implication of the quotient rule of differential calculus.

Piketty’s Fundamental Force for Divergence: r > g

The quotient rule is also the basis for Piketty’s Fundamental Force for Divergence: r > g. Since an individual’s income (Y_i) as the sum of income from capital (Y_Ki) income from labor (Y_Li),

where the individual’s income from capital Y_Ki is defined as the rate of return he or she is able to earn on his or her capital (r_i) multiplied by the amount of capital K_iowned:

Since, by definition, the rate of growth of the individual’s capital g_ki is equal to dK_i/dt divided by the stock of capital K_i:

And since, as we have seen above, by definition, the change in the individual’s capital stock dK_i/dt (in time) must be equal to the change in savings (dS_i/dt) which, in turn, is equal to the rate of savings out of income s_i times the individual’s income Y_i (i.e., dK_i/dt =dS_i/dt = s_i×Y_i), by substituting s_i×Y_i for dK_i/dt in 30) we get:

What we have done here is derive the growth rate g_ki of the numerator of the individual’s capital/national-income ratio K_i/Y in terms of the parameters s_iand r_i and the variables Y_Li and K_i in Piketty’s analytic framework. This makes it possible to apply the quotient rule in comparing this growth rate expressed in terms of these parameters and variables s_i(r_i + Y_Li/K_i) with the growth rate of national income g to conclude that:

In other words, all that Piketty has done in arriving at his Fundamental Force for Divergence: r > g is use one of the most fundamental and basic implications of the quotient rule—that the value of the ratio of any two variable (functions, numbers) will increase (decrease, remain unchanged) if and only if the growth rate of the numerator is greater than (smaller than, equal to) the growth rate of the denominator—to explain the behavior of the individual’s capital/national-income ratio K_i/Y ratio over time within the context of his analytic framework. There are many qualifications to this rule that Piketty examines in detail in the text (at times, in excruciating detail!), but the source of this law is nothing more than a simple and straightforward application of the quotient rule of differential calculus.

The Product Rule and g = g_p + g_pc

There is a second rule of differential calculus that Piketty relies upon in addition to the quotient rule, namely, the product rule. This rule, as with the quotient rule, is nothing more than an elementary theorem of differential calculus except that it deals with the product of two variables (functions or numbers) rather than their ratio. It tells us how the value of the product of two variables must change as the individual values the numbers change in response to a change in a third variable. What this rule tells us is that if we want to know how the value of the product of two variables changes in response to a change in its multiplicands all we have to do is:

a) multiply the first variable of the product by the derivative of the second variable with respect to the third variable and

b) add the value of the second variable multiplied times the derivative of the first variable with respect to the third variable.

The relevance of this rule to Piketty’s analysis can be seen by applying this rule to the relationship between national income Y and per capita income Y_pc:

to find the way in which national income Y changes (dY/dt) as population P and per capita income Y_pc change (dP/dt and dY_pc/dt). We can apply this rule by a) multiplying the first variable in the product P by the derivative of the second Y_pcwith respect to time t (dY_pc/dt) then b) adding the value of the second variable Y_pcmultiplied times the derivative of the first variable P with respect to time t (dP/dt) to obtain:

The result embodied in 35) is the mathematical relationship between the growth rate of national income g (= (dY/dt)/Y) and the growth rates of per capita income g_pc (=(dY_pc/dt)/Y_pc) and population g_p (=(dP/dt)/P). Substituting from these definitions into 35) yields:

As was noted in discussing the Implicit Relationships above, this relationship applies to continuously compounded rates of growth and not to rates that assume an alternative compounding scheme even if the rates are equivalent in the sense that they lead to the same change in the principle to which they are applied.

Endnotes

[2] For those who fall into the latter category or who have an abhorrence of mathematics in general I would recommend that you forget about the rest of this guide and go directly to Piketty’s Capital in the Twenty-First Century if for no other reason than to take advantage of its excellent exposition of economic history.

[3] It should be noted that savings = investment is an identity within the accounting system of the National Income and Product Accounts. It should also be noted that the terms “savings”, “investment”, and “income” in Piketty’s theoretical framework always refer to net saving, net investment, and net income, that is, to gross savings, gross investment, and gross income less the amount of capital consumed in the process of producing income. As a result, savings is always net savings out of net income in Piketty’s theoretical framework and is always equal to net investment which is, by definition, the amount by which the capital stock changes in the National Income and Product Accounts.

[4] It is obvious that this law cannot hold in the long run if either s or g change since a change in these parameters will change the long-run equilibrium value of β as defined in Piketty’s Second Law β = s/g. Similarly, if the prices of assets relative to other prices change (i.e., relative to the prices that determine the value of income Y), this will cause the value of capital K relative to the value of income Y to change independently of savings and investment. If this happens, s and g cannot be the sole determinant of β = K/Y as is required by Piketty’s Second Law β = s/g.

[5] These relationships are a direct implication of the quotient rule of differential calculus, and a proof of their validity is provided in the Appendix below.

[6] Capital gains associated with asset price changes relative to other prices will increase the value of capital K beyond the contribution of savings/investment, and capital losses associated with asset price changes relative to other prices will decreases the value of capital K below the contribution of savings/investment. Thus, the capital/income ratio β will not necessarily move toward its long-run equilibrium value s/g if these relative prices change. Piketty argues that this phenomenon can be particularly relevant in the midst of speculative bubbles and their collapse and in the face of dramatic shocks to the system such as those that resulted from the two World Wars and the Great Depression.

[7] See: Harrod, Henry, “An Essay in Dynamic Theory,” Economic Journal, 1939, vol.49 (193), pp.14-33.

[8] It should, perhaps, be noted in regard to the assertion: “When the rate of return on capital significantly exceeds the growth rate of the economy . . . it logically follows that inherited wealth grows faster than output and income” it is clear from Piketty’s explanation of the circumstances in which wealth will grow in this situation that these circumstances are “logically” determined only in the circular sense that “significantly exceeds” means such that “inherited wealth grows faster than output.” I would also note that my interpretation here is consistent with the tenor of Piketty’s arguments throughout the book. Even though his arguments are logical to the nth degree, he does not argue (as do our Austrian friends) that his conclusions are “logically” determined, but, rather, that they are determined by the relevance of his arguments to real world data.

[9] The fact that the ratio of two variables (functions or numbers) will increase (decrease or remain unchanged) if and only if the rate of growth of its numerator is greater than (less than or equal to) the rate of growth of its denominator is a direct implication of the quotient rule of differential calculus. A proof of the fact that “the ratio of two variables etc.” is provided in the Appendix below.

[10] Again, this is a direct implication of the quotient rule of differential calculus, and a proof of the validity of the relationships that follow is provided in the Appendix below.

[11] This relationship is implied by the product rule of differential calculus and is explained in the Appendix below.

[12] An online calculator that calculates continuously compounded rates of growth (actually rates of interest) that are equivalent to annual rates of growth can be found at: http://www.calculator.net/compound-interest-calculator.html?cinterestrate=10&cincompound=annually&coutcompound=continuously&x=56&y=11.

[13] We can also see that if Y and K are of opposite sign β will be negative and the inequalities on the left will be reversed which, as was footnoted in the discussion of Piketty’s Second Fundamental Law in the main text above, makes no difference to the argument with regard to the semantics as to when β increases or decreases since a negative number increases when its absolute value decreases and decreases when its absolute value increases.

Real-World Economics