I think that Piketty’s argument is simply that an interest rate greater than the growth rate implies that capital income will grow faster than wage income with a consequent worsening of income distribution. One way to explain Piketty’s argument within the Harrod_Domar framework, however, is presented in the document in attach.

If you are interested in a critical assessment of Piketty's work have a look at the last Acemoglu&Robinson. Maybe this can help to get a complementary view on the issue

http://economics.mit.edu/files/10302

Pasquale Scandizzo,

thank you for a detailed explanation. I find your paper co-authored with Claudio Dicembrino Can Portfolio Diversification increase Systemic Risk? is very interesting and informative. It is a research paper  in 2012. Is it published in one of journals? If it is, please teach me where it appeared.

As for your explanation, I understand how the different theories are related with each other. On this point, your explanation was wonderful.  But there still remains some unsolved points.

My original question is why the income and wealth inequality becomes greater when r > g. If we  suppose  Kaldor-Pasinetti equation,

      g = scr.

In this case, the capitalist or asset holding class spend (1- sc) P and they accumulate ΔK which is equal  to g K and their asset or capital increases at the same rate of the economic growth. Then the the ratio ΔK and W does not change. If the saving rate sc  remains constant, the ratio P and W also remains constant.

Does this mean that even if r > g, the income ratio remain constant? How can Piketty say that r > g is the driving force in the inequality increase?

Thank you. My article with Dicembrino was published in: The IUP journal of financial risk management : IJFRM.- IUP Publ, ISSN 0972-916X, ZDB-ID 26153944. - Vol. 9.2012, 4, p. 52-76.

As for your unsolved point, Pasinetti's result was based on the assumption that capitalists and workers receive the same  rate  of return from their savings, but Piketty explicitly denies this cpndition, since he claims that larger patrimonies will generally command higher  rates of return.  Under the latter condition, , assume that we start from a Pasinetti like Equilibrium and workers,, because of lower transaction costs,  start earning lower rates of return  than capitalists on their accumulated savings, even though there is a single  interest rate  for both.  Aa a consequence, this interest rate will not be sufficient to achieve  warranted growth  since  workers' profits wll become too low to support the amount of savings necessary for warranted growth at the given interest rate (at which capitalists' savings are fixed as they would be without the differential returns). The interest rate will tend to rise, but  the persistence of  differential returns  will cause the proportion of savings to change in favor of capitalists, with an even greater imbalance  . This process of non convergence is characteristic of the knife -hedge nature of any Harrod-Domar Equilibrium. As a consequence, income distribution will tend to deteriorate with a progressive  growth of capital and its concentration in the hands of the capitalists. 

Domenico Rossignoli,

thank you for your information. Acemoglu and Robinson (2014) is famous in Japan too but I had no time to read it. On your invitation, I have read the main text but have omitted reading statistical appendix. I have no ability to read and assess what they do in that part.

The third section "Seeking 21st-Century Laws of Capitalism" was helpful for me in understanding what Piketty wanted to say although I have many points I cannot agree with the authors. I will write on these points another post which follows this one.

I have no time to describe in detail about the points they advanced in the second section "Capital Failures". I have an impression that they did not treated Ricardo and Marx in a way they deserve.

For example, let us take up the alleged Ricardo's Law. Acemoglu and Robinson argue that Ricardo has advanced a law that "predicted a rising share of national income would accrue to land." They add that this is another example of "general laws of capitalism" which "were formulated in an effort to compress the facts and events of their times into a grand theory aiming to be applicable at all times and places." (p.6 of their Working Paper 14-18)

Ricardo indeed argued that landed rent would increase its share in the national income, but with the condition that if England continued to prohibit importing grain or imposing prohibitive tariffs on imported grain. This is the famous Corn Laws problem. It is well known that one of Ricardo's principal aims of writing his Principle was to persuade opinion leaders of his time that the Corn Laws should be repealed. The Corn Laws was repealed in 1846 by the Importation Act 1846 (nearly a quarter century after Ricardo was dead). As a result, the principal cause of rising land rent was lifted and "the share of national income accruing to land started to a monotonic decline and by the 1970s real rents started a rapid fall." (ibid.)

We may discuss when the land rent started to decline but the history of land rent does not contradict Ricardo's theory of rents.

Acemoglu and Robinson ignored (perhaps intentionally) that the laws Ricardo posited had many conditions. If one of major conditions was lifted, the prediction would change as a consequence. What Acemoglu and Robinson call Ricardo’s "law" or prediction is a kind of invention build up by misunderstanding Ricardo's arguments as prediction instead of logical (and theoretical) reasoning. It is quite doubtful if Ricardo had ever imagined in inventing a “law of capitalism" which is applicable “at all times and places” without any conditional clauses.

As for Acemoglu and Robinson's contention that Marx intended to establish "general laws of capitalism" I have no intention to defend his logic in deducting the tendency laws. Fir example, argument in establishing the tendency law of the rate of profit to fall is now known that it contains a logical flaw.

However, Acemoglu and Robinson's claim that Marx ignored "the endogenous evolution of technology" and "the role of institutions and politics" (p.4) is a half lie. Marx is one the first persons to observe mutual interaction between technology and economies (See for example Nathan Rosenberg's Inside the Black Box: Technology and Economics). Marx's Capital (Volume 1 to 3) is descriptions and theoretical observations on how capitalist economy works. However, he had a writing plan which comprises much wider subject matters than he could fully develop in his life.

It is the plan indicated in the Critique of the Political Economy. In fact the plan listed following 6 pillars:

(a) capital in general,

(b) land property,

(c) wage labour

(d) the state

(e) foreign trade

(f) world market

Marx dealt with first 3 pillars in the Capital but could not develop later 3 pillars in any systematic way.

See for example K. Matsui (1970) Marx's plan in the "Critique of Political Economy" and the crisis in the world market, Kyoto University Economic Review 40(1): 28-56.

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/125508/1/ecb0401_028.pdf

Marx deemed the state as the main arena of class struggle which takes the form of politics of all kinds. Before he comes to consider on the State, he wanted to develop a theory of economic processes which he thought compose the basis of political struggles. He might be wrong in considering the social revolution and the abolition of private ownership of the means of production would solve major problems of capitalism but he had always in mind the importance of political struggles.

I could not understand why the two authors wanted to discuss long in blaspheming Marx and Ricardo and condemned alleged "general laws of capitalism", because the biggest contribution of Piketty and what he is doing in his book are not extending general laws of capitalism but are to give a full account he and his collaborators have found.

Pasquale Scandizzo,

thank you for reference information.

I am now interested in financial stability and instability. Systemic risk is one of most important questions we must attack. I will study your paper closely.    

I worked myself a bit on Piketty's reasoning. He proposes two fundamental laws:

First law: 　　α = r β 　　　(1)

where α is the capital share of national income, β the capital income ratio, and r the rate of return on capital.

Second law: 　β = s / g 　　(2)

where s is the saving ratio and g is the growth rate of GDP.

Piketty calls these two equations fundamental laws but they are in fact accounting identities. (Piketty knows this and put it explicitly.)

It is not difficult to confirm this. Let Y be the income and K be the capital of the economy. The capital share P of national income Y is equal to r Y and the saving S is s Y by definitions. If saving S is all invested (as it is normally supposed), S = I and the growth rate g is equal to I / K.

The first law is obtained by a series of equations:

P / Y = r K / Y = r (K/Y) = r β.

The second law follows as

K / Y = (I/Y) / (I/K) = (s Y / Y) / (g) = s / g.

Acemoglu and Robinson accused that Piketty wanted to establish fundamental or "general laws of capitalism" but these are in fact simple accounting identities.

The terminology Piketty used (i.e. fundamental law) is rather unfortunate, because he could have avoided an accusation like Acemoglu and Robinson's. It would be more plausible to call them fundamental equations.

From equations (1) and (2) we can derive

  r = α / β　　and 　　g = s / β. 　　(3)

By consequence,

　　r > g if and only if  α > s,

or equivalently

　　r > g  if and only if  P / Y > S / Y. 　　　(4)

The right condition of equivalence (4) simply means

　　　P > S. 　　　　　　　　　　　　　　　　　　　(5)

Let W be the workers share of national income (from wage), and sc and sw be the saving ratio from profit income (return on capital) and saving ratio from wage. Then

　　　S = sc P + sw W. 　　　　　　　　　　　　(6)

The condition (5) can be rewritten as

　　　P > sc P + sw W

or

　　　(1-sc) P > sw W. 　　　　　　　　　　　　(7)

This means that r > g if and only if the consumption from return on capital is bigger than the savings that workers make from wages.

( When workers do not save at all, i.e. sw = 0, condition (7) is equivalent to sc < 1. In other words, return on capital is greater than growth rate if capitalists' saving ratio is positive and saving ratio from wage is 0. This is an example of sufficient conditions that r > g. )

It is important to note that this proposition obtains from defining equations. Any series of growth and fluctuation satisfies this proposition (i.e. conditional equivalence). The condition r > g is equivalent to inequality α > s, inequality (5) and inequality (7).

I have no idea if one of these equivalent conditions implies increasing income inequality. It seems some other implicit conditions are required. Think for example any expanding growth pass in which capital income ratio and saving ratios remain constant. It such a pass exits, then the capital and total amount of wages remain constant and there will be no increasing inequality.

Problem still remains. What will be a sufficient condition such that inherited wealth grows faster than output and income? If we can find such a condition which seems to hold for a wide situation, we can say that inequality increases when r > g and when some unknown condition is fulfilled.

Under the Piketty’s assumption for the same S and Beta, if r>g then alpha>s. For g>0 and S and non-increasing function of g, inequality in distribution of income will emerge: much more for low g and lessen for high g. I think, however, that even the second fundamental Piketty’s law must be considered an accounting ex post identity. So that, I think better to utilize r – g as a measure of inequality of income, that growth faster for low rate of growth g. Going further, but assuming Scandizzo’s reply, r-g may be considered also a measure of divergence between production prices and assets prices; so, even in part, a measure of inequality in distribution of wealth (i.e. financial earnings vs real asset earnings or between top manager earnings and property earnings). Furthermore r-g is shadow measure of social overhead capital not directly considered by depreciation figures of Piketty (environmental losses not accounted and National Account and Capital Account and growing divergence of intergenerational distribution of income). Further ideas on indirect measure of depreciation may be find an approach à la MIMIC contained in the “Real saving and capital growth in western Europe: exploring patterns before and after euro” by Lo Cascio and Aliano in press.

Martino Lo cascio,

thank you for your post. I cannot understand the logic when you say 

〉〉  For g>0 and S and non-increasing function of g, inequality in distribution of income will emerge: much more for low g and lessen for high g.

Would you explain a bit more in detail.