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One thing that always puzzles me about r > g is that it's comparing a first derivative to a second derivative. How can this be meaningful? To explain further,

by om2 10y ago

One thing that always puzzles me about r > g is that it's comparing a first derivative to a second derivative. How can this be meaningful? To explain further, r is return on investment, which approximately means rate of growth of wealth of capital. That is, it is a the first derivative of a stock, or equivalently a flow. g is the rate of growth of income, where income itself is a flow. So it's the second derivative of a stock. How does it even make sense to compare these two quantities when they don't have the same units?

tome 10y ago
> How does it even make sense to compare these two quantities when they don't have the same units? They both have units T^-1, don't they? For example, they could both be measured in "Percentage increase per year".
- om2 10y ago
  It doesn't really make sense to take "percentage increase" as dimensionless, because it leads you to compare growth rate of incomparable things. One of the most basic rules of economics is that you can't compare stocks and flows. Why would it be meaningful to compare their % increase per year? Let's say that for construction company A, the total number of houses it has built increases by 10%. (This is comparable to an r of 10%). And let's say that, each year, construction company B builds 5% more new houses than it built the year before. (This is comparable to a g of 5%). No matter how much of a head start A starts with, under these assumptions, B will always eventually have more total houses built. Or to put it more generally, a stock growing at fixed rate grows linearly, but a flow that grows linearly implies a stock growing at a quadratic rate. O(N^2) will always outpace O(N). It seems to me that all the hoopla about r > g is based on an elementary mathematical error. Or really, more honestly, assumptions about savings rate are doing all the work, and r and g really have almost nothing to do with it.
  - tome 10y ago
    > Or to put it more generally, a stock growing at fixed rate grows linearly, but a flow that grows linearly implies a stock growing at a quadratic rate. O(N^2) will always outpace O(N). I see what you're getting at, but surely you've got the rates wrong. A stock growing at a fixed rate grows exponentially, and O(exp N) will always outpace O(N^2).
    - om2 10y ago
      You're right that I made a mistake, but it's the labor income case that is exponential, not the capital income case. Earning Wr every year is constant income and linear growth in wealth (where W is your starting wealth). Earning I(1+g)^n every year is exponential growth in both income and wealth. Note that r is the return on capital, for an asset worth W, you earn W*r income. It says nothing about whether W increases or decreases. (In fact, if the value of W increases but the income generated is the same, then r goes down.) That would be a capital gain (or capital loss). An asset can produce a return on capital while losing value. If you assume W doesn't go down in value, and furthermore, all the capital income is reinvested, but no labor income is reinvested, then you do indeed get exponential growth of wealth for the capital income case, and no growth of wealth for the labor income case. But then all the work is being done by assumptions about savings rate and depreciation, and none by the comparison between r and g.
TheSpiceIsLife 10y ago
I think the idea of the comparison is that r (return on investment) is greater than g (growth of income) and that this leads to the problem of wealth concentration. The relevance of the comparison is that the working class are dependant on g while the truly wealthy have and control r.
barrkel 10y ago
It's a shorthand for talking about how the ownership of everything (both current stocks and claims on future production) is being distributed between workers and capitalists. When r > g, the share of ownership of capital (by growth in capital) is growing faster than the continuous stream of claims on future production (which is what money is, and what makes up income). If capital is liquid, it can be converted into a claim on future production. That's how the units are made compatible.
- om2 10y ago
  That doesn't really make sense. r isn't a growth rate of the share of ownership on capital. It's a return on capital, i.e. the growth rate of capital. r implies nothing about the change in total value of capital assets. Total capital assets could double or halve and r wouldn't have to change. in fact, under most circumstances, the value of capital assets and the return on capital move in opposite directions.
Mikeb85 10y ago
> One thing that always puzzles me about r > g is that it's comparing a first derivative to a second derivative. How can this be meaningful? It's not. It's Piketty trolling the economics field, which has become overly math-y, and try too hard to distil complex interactions into basic formulae. It's basically his way of saying that capital's share of income is growing faster than labour's share of income, or that the return (r) on capital is outpacing the return on labour, which is represented by the growth rate (g) of the economy. In other words, it's not a real mathematical formula. It's an idea, a basic premise, merely represented by a 'formula' because hey, it's how every economist does it nowadays, right?
- om2 10y ago
  r isn't the growth rate of capital's share of income though. r represent the return on capital (which is proportional to capital's share of income), not the growth rate of the return on capital, or the growth rate of capital's (share of) income. On the other hand, g isn't the return on labor, or labor's share of income. It's (approximately) the growth rate of the return on labor. Would you rather have $100 and then get 200% of it every year (r = 200%) or get an amount that increases at 100% every year (g = 100%) starting with $1? With the second option you'll have twice as much money in just 11 years. So in this picture income eventually wins, even though r > g. It's not just "not a formula" it is total nonsense. It doesn't make sense to compare a growth rate of an amount to a growth rate of an annual increase!
  - Mikeb85 10y ago
    In Piketty's model, r is return on capital. So that means it's the return on things like equities (and btw, interest is compounding on investments, don't know why you represented otherwise), real estate (Piketty is not overly find of the role of rentiers/landlords in the economy), machinery and so on. So the money accrued by the owners of wealth and machinery that is, essentially passive income, or income generated by capital, is r. g is the growth of the economy. We calculate it as GDP, what he's interested in is basically people's combined incomes and economic activity that goes with it. The premise that r > g is saying that people who have wealth, their return on that wealth will always be higher than the growth of the economy. If the economy grows 1%, the return on capital (remember it's all capital in the economic sense) will be more than 1%. Now this isn't true for all years in all places, but it has been shown to be true in the long run, which is how wealth is accrued quicker by those who already have wealth, and how inequality increases. Keep in mind Piketty's book is a massive tome. And btw, comparing a rate that's calculated as %/year to another calculated as %/year is mathematically correct. Your example is nonsense btw, it almost seems like you've confused r with g, because interest certainly compounds with capital, whereas with income, while you'll get increases over time, it generally isn't of the same magnitude and relates to labour, of which you have a finite amount.

by om2 10y ago

tome 10y ago

> How does it even make sense to compare these two quantities when they don't have the same units? They both have units T^-1, don't they? For example, they could both be measured in "Percentage increase per year".

om2 10y ago
It doesn't really make sense to take "percentage increase" as dimensionless, because it leads you to compare growth rate of incomparable things. One of the most basic rules of economics is that you can't compare stocks and flows. Why would it be meaningful to compare their % increase per year? Let's say that for construction company A, the total number of houses it has built increases by 10%. (This is comparable to an r of 10%). And let's say that, each year, construction company B builds 5% more new houses than it built the year before. (This is comparable to a g of 5%). No matter how much of a head start A starts with, under these assumptions, B will always eventually have more total houses built. Or to put it more generally, a stock growing at fixed rate grows linearly, but a flow that grows linearly implies a stock growing at a quadratic rate. O(N^2) will always outpace O(N). It seems to me that all the hoopla about r > g is based on an elementary mathematical error. Or really, more honestly, assumptions about savings rate are doing all the work, and r and g really have almost nothing to do with it.
- tome 10y ago
  > Or to put it more generally, a stock growing at fixed rate grows linearly, but a flow that grows linearly implies a stock growing at a quadratic rate. O(N^2) will always outpace O(N). I see what you're getting at, but surely you've got the rates wrong. A stock growing at a fixed rate grows exponentially, and O(exp N) will always outpace O(N^2).
  - om2 10y ago
    You're right that I made a mistake, but it's the labor income case that is exponential, not the capital income case. Earning Wr every year is constant income and linear growth in wealth (where W is your starting wealth). Earning I(1+g)^n every year is exponential growth in both income and wealth. Note that r is the return on capital, for an asset worth W, you earn W*r income. It says nothing about whether W increases or decreases. (In fact, if the value of W increases but the income generated is the same, then r goes down.) That would be a capital gain (or capital loss). An asset can produce a return on capital while losing value. If you assume W doesn't go down in value, and furthermore, all the capital income is reinvested, but no labor income is reinvested, then you do indeed get exponential growth of wealth for the capital income case, and no growth of wealth for the labor income case. But then all the work is being done by assumptions about savings rate and depreciation, and none by the comparison between r and g.

TheSpiceIsLife 10y ago

I think the idea of the comparison is that r (return on investment) is greater than g (growth of income) and that this leads to the problem of wealth concentration. The relevance of the comparison is that the working class are dependant on g while the truly wealthy have and control r.

barrkel 10y ago

It's a shorthand for talking about how the ownership of everything (both current stocks and claims on future production) is being distributed between workers and capitalists. When r > g, the share of ownership of capital (by growth in capital) is growing faster than the continuous stream of claims on future production (which is what money is, and what makes up income). If capital is liquid, it can be converted into a claim on future production. That's how the units are made compatible.

om2 10y ago
That doesn't really make sense. r isn't a growth rate of the share of ownership on capital. It's a return on capital, i.e. the growth rate of capital. r implies nothing about the change in total value of capital assets. Total capital assets could double or halve and r wouldn't have to change. in fact, under most circumstances, the value of capital assets and the return on capital move in opposite directions.

Mikeb85 10y ago

> One thing that always puzzles me about r > g is that it's comparing a first derivative to a second derivative. How can this be meaningful? It's not. It's Piketty trolling the economics field, which has become overly math-y, and try too hard to distil complex interactions into basic formulae. It's basically his way of saying that capital's share of income is growing faster than labour's share of income, or that the return (r) on capital is outpacing the return on labour, which is represented by the growth rate (g) of the economy. In other words, it's not a real mathematical formula. It's an idea, a basic premise, merely represented by a 'formula' because hey, it's how every economist does it nowadays, right?

om2 10y ago
r isn't the growth rate of capital's share of income though. r represent the return on capital (which is proportional to capital's share of income), not the growth rate of the return on capital, or the growth rate of capital's (share of) income. On the other hand, g isn't the return on labor, or labor's share of income. It's (approximately) the growth rate of the return on labor. Would you rather have $100 and then get 200% of it every year (r = 200%) or get an amount that increases at 100% every year (g = 100%) starting with $1? With the second option you'll have twice as much money in just 11 years. So in this picture income eventually wins, even though r > g. It's not just "not a formula" it is total nonsense. It doesn't make sense to compare a growth rate of an amount to a growth rate of an annual increase!
- Mikeb85 10y ago
  In Piketty's model, r is return on capital. So that means it's the return on things like equities (and btw, interest is compounding on investments, don't know why you represented otherwise), real estate (Piketty is not overly find of the role of rentiers/landlords in the economy), machinery and so on. So the money accrued by the owners of wealth and machinery that is, essentially passive income, or income generated by capital, is r. g is the growth of the economy. We calculate it as GDP, what he's interested in is basically people's combined incomes and economic activity that goes with it. The premise that r > g is saying that people who have wealth, their return on that wealth will always be higher than the growth of the economy. If the economy grows 1%, the return on capital (remember it's all capital in the economic sense) will be more than 1%. Now this isn't true for all years in all places, but it has been shown to be true in the long run, which is how wealth is accrued quicker by those who already have wealth, and how inequality increases. Keep in mind Piketty's book is a massive tome. And btw, comparing a rate that's calculated as %/year to another calculated as %/year is mathematically correct. Your example is nonsense btw, it almost seems like you've confused r with g, because interest certainly compounds with capital, whereas with income, while you'll get increases over time, it generally isn't of the same magnitude and relates to labour, of which you have a finite amount.