LadyGeek wrote: Epsilon Delta wrote:
It probably would not be helpful to point out the "marginal tax rate" is really a partial derivative.
I'm also partial to derivatives, care to expand?
I'll try to explain, but it can be tricky because I'm not sure what you already understand. I'm going to assume you already know some calculus and have read some economics but never put them together.
The first point is that when a economist says "marginal" he means a derivative of some sort. If you know calculus this can give insight into concepts like marginal demand and marginal supply. You can bring what you learned about derivatives to bear and skip over the pages of the economics text that derive differential calculus from first principles.
The equivalence of marginal and derivatives is just a matter of definition. Going from words to symbols to the rigor of calculus.
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Increase your income from say $20,000 to $20,100 you have $100 in extra income and pay $15 in extra tax the marginal rate is $15/$100 or 15%. So your marginal tax rate at an income of $20,000 is 15%.
tax(20,000 + 100) - tax(20,000)
marginal tax rate at income of $20,000 = -------------------------------
(20,000 + 100) - (20,000)
tax( I + Δ ) - tax( I )
marginal tax rate(I) = -----------------------
df(x) f(x+δ) - f(x)
----- = lim -------------
dx δ→∞ δ
Now to the partial
derivative. This just says that tax is not simply a function of total income, you have to consider the different types of income, lets consider just wages and capital gains. So we have not a function of one variable tax(income)
but a function of two variables tax(wage,gain)
A partial derivative of a function of two or more variables is just the derivative if we change one variable and leave the others constant. This is how we usually calculate marginal tax rates. For example at an income of about $50,000 the marginal tax rate on wages is 15% because an extra $100 of wages increase taxes by $15 while the marginal tax rate on gains is 0% because an extra $100 of capital gains results in $0 extra tax.
Now lets consider a hedge fund employee who can choose to convert some of his wages into gains. Both wages and gains are a function of yet another variable, the proportion he chooses to allocate to gains, call it x. So we have tax(wage(x),gain(x))
. If we look at how his tax changes as he changes his allocation we have a full
derivative in this case of tax with respect to x. (formally d tax(x)/dx
I will now wish a good day to my one remaining reader.