Y ] = dLog[Y]
Note that Y is the sum of all incomes. An income change for an individual does not affect the income changes of others. Assuming that other incomes stay fixed, we find for an individual income dY / dy = 1. If y rises and no other income rises, then the growth of national income dLog[Y] is equal to the growth for the single person weighted by its share in total income:
dLog[Y] = (y / Y) dLog[y]
It follows that the marginal tax for the individual is:
d T[y] / dy = r (1 -
)
Now, since