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.
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.