A friend asked me for a summary. Here is my understanding of two themes which may be wrong; I made a few mistakes earlier.

I guess the basic idea is simple. That is looking at wealth to income ratio for a country, equivalently the ratio of average wealth to average in come and how it changes over time. Total income is GDP minus depreciation plus foreign income if any. Wealth or capital includes the sum total all non human assets that can be owned or exchanged in a market. it includes real estate, financial and professional capital including plants, patents etc. This ratio is denoted β is C/I where C is capital and I is income. The first grows by savings and the second by growth rate (includes the population growth too and can be measured from time to time, for example each year. Another quantity, he considers is alpha α, share of income from capital in the national income. the first accounting identity is α=rβ, where r is the rate of return on the capital. The growth rate, income, savings rate etc can be measured each year and so also α. This allows us to estimate r which is around 4-5 percent in the west for much of the time except for a period between the wars and then upto 1970 or so. The growth rates have declined, but since this includes population growth, there is some difference between stagnant countries and countries like USA. Savings rate vary, high for Japan (14% or so) and low for USA (around 7). β was around 7 in Europe before the first world war, declined to less than 3 around 1950 and went up to nearly 6 recently. In USA change is less pronounced but has similar U shape. 4.5., 3.5., 4.5. For some reason, Piketty uses percentage signs, I think that these are just numbers. Capital income α also follows a U pattern currently around 30% in most western countries.

Clearly, we do not want α, β to go up, they reinforce each other depending on r. So we need to see with possible numbers. Here is an example from Piketty Chapter 10.

"For example, if g=1%, and r=5%, saving one-fifth of the capital from income (while consuming the other four-fifth) 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 less than 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."

So either g has to increase but by diminishing returns etc, it is expected to stabilize around 1.5 or so for advanced countries. So if we do not want a Marxian apocalypse where α, β do not explode, we have to reduce. Piketty recommends progressive taxes on wealth etc. Some are suggesting minimum wages and other means.

What is impressive is that he and collaborators were able to tease out so much data which even if there are further corrections, gives the broad contours for further discussions.

That was the main message. There is one more formula which Piketty frequently uses asymptotic formula β=s/g, called Harrod-Domar-Solow formula. If we denote the corresponding quantities using subscript _t, what this means is that β_t tends to s_t/g-t,as t gets large which may be 50-60 years or even more. This formula depends on models which imply some sort of stability or convergence for β_t. If we assume convergence, it is easy to see the formula (but this is not really necessary for his main message). To see the formula, there is an easy formula β_(t+1)/β_t=(1+s_wt)/(1+g_t) where s_wt=s_t/ β_t. In general the right hand side has to be multiplied by a capital gains factor. Now if β_t stabilizes β_t tends to β. so the left hand side tends to one. So from the right hand side _t tends to s_t/ β or β in the long run is the limit of s_t/g_t. But in short term it fluctuates. This formula is not really necessary to understand his main message, Which Milanovic also emphasized. The subtle part for me is how wealth is passed onto the next generation. This is in Chapter 12 and I will have to read it again. I think that is also discussed by Gregory Clark from a different perspective in "The Son Also Rises". Criticisms and other supplementary material, I will write later on. This is all the mathematics that there is in the book and are in the paper by Piketty and Zucman "Capital is Back".

Corrections are welcome.

I guess the basic idea is simple. That is looking at wealth to income ratio for a country, equivalently the ratio of average wealth to average in come and how it changes over time. Total income is GDP minus depreciation plus foreign income if any. Wealth or capital includes the sum total all nonhuman assets that can be owned or exchanged in a market. it includes real estate, financial and professional capital including plants, patents etc. This ratio is denoted β is C/I where C is capital and I is income. The first grows by savings and the second by growth rate (includes the population growth too and can be measured from time to time, for example each year. Another quantity, he considers is alpha α, share of income from capital in the national income. the first accounting identity is α=rβ, where r is the rate of return on the capital. The growth rate, income, savings rate etc can be measured each year and so also α. This allows us to estimate r which is around 4-5 percent in the west for much of the time except for a period between the wars and then upto 1970 or so. The growth rates have declined, but since this includes population growth, there is some difference between stagnant countries and countries like USA. Savings rate vary, high for Japan (14% or so) and low for USA (around 7). β was around 7 in Europe before the first world war, declined to less than 3 around 1950 and went up to nearly 6 recently. In USA change is less pronounced but has similar U shape. 4.5., 3.5., 4.5. For some reason, Piketty uses percentage signs, I think that these are just numbers. Capital income α also follows a U pattern currently around 30% in most western countries.

Clearly, we do not want α, β to go up, they reinforce each other depending on r. So we need to see with possible numbers. Here is an example from Piketty Chapter 10.

"For example, if g=1%, and r=5%, saving one-fifth of the capital from income (while consuming the other four-fifth) 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 less than 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."

So either g has to increase but by diminishing returns etc, it is expected to stabilize around 1.5 or so for advanced countries. So if we do not want a Marxian apocalypse where α, β do not explode, we have to reduce. Piketty recommends progressive taxes on wealth etc. Some are suggesting minimum wages and other means.

What is impressive is that he and collaborators were able to tease out so much data which even if there are further corrections, gives the broad contours for further discussions.

That was the main message. There is one more formula which Piketty frequently uses asymptotic formula β=s/g, called Harrod-Domar-Solow formula. If we denote the corresponding quantities using subscript _t, what this means is that β_t tends to s_t/g-t,as t gets large which may be 50-60 years or even more. This formula depends on models which imply some sort of stability or convergence for β_t. If we assume convergence, it is easy to see the formula (but this is not really necessary for his main message). To see the formula, there is an easy formula β_(t+1)/β_t=(1+s_wt)/(1+g_t) where s_wt=s_t/ β_t. In general the right hand side has to be multiplied by a capital gains factor. Now if β_t stabilizes β_t tends to β. so the left hand side tends to one. So from the right hand side _t tends to s_t/ β or β in the long run is the limit of s_t/g_t. But in short term it fluctuates. This formula is not really necessary to understand his main message, Which Milanovic also emphasized. The subtle part for me is how wealth is passed onto the next generation. This is in Chapter 12 and I will have to read it again. I think that is also discussed by Gregory Clark from a different perspective in "The Son Also Rises". Criticisms and other supplementary material, I will write later on. This is all the mathematics that there is in the book and are in the paper by Piketty and Zucman "Capital is Back".

Corrections are welcome.