Proof that the Marginal Tax Rate Exceeds Average Tax Rates (optional material, p. 1)
Assumptions:
i
2
> i
1
> 0
average tax rate at i
1
= ATR
1
=
t(i
1
) / i
1
average tax rate at i
2
= ATR
2
=
t(i
2
) / i
2
ATR
2
> ATR
1
> 0 (since the tax structure is convex)
marginal tax rate = MTR =
t(i
2
) - t(i
1
)
i
2
- i
1
To Prove:
MTR > ATR
1
and MTR > ATR
2
Proof:
Since t(i
1
) / i
1
< t(i
2
) / i
2
, we can multiply both sides by the positive factor i
1
i
2
, giving:
t(i
1
) i
2
< t(i
2
) i
1
Multiplying both sides by -1 reverses the inequality:
