3 ms·

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

by 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.

by om2 10y ago

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.