

In a post last week, I stated that the deadweight loss from a tax is proportional to the square of the tax rate. So doubling the tax rate, for example, quadruples the deadweight loss.
I stated on Facebook that I used basic algebra to prove this to my students. An economist friend asked me to do it.
So I did.
Here’s the proof.
For simplicity, start with a horizontal supply curve, as in the graph above, and compute the Harberger triangle loss. It’s C in the above graph, which is one half the base times the height. The height is the tax, t. The base is computed as delta Q, the drop in amount demanded. So we need to compute the base.
Let elasticity = n.
n = delta Q/Q divided by delta P/P.
So delta Q/Q = n times deltaP/P.
delta Q = n times delta P/P times Q.
But delta P = t.
So delta Q = n times t/P times Q.
So triangle loss = 1/2 t times n times t/P times Q.
So triangle loss = 1/2 t^2 times n/P times Q.
QED.
Note: If, as is usually the case, the supply curve is upward-sloping, the formula for deadweight loss is more complicated but the square relationship still holds.
READER COMMENTS
John Hall
Feb 4 2025 at 8:58am
This assumes linear demand curve (and supply in the upward sloping case), correct?
David Henderson
Feb 4 2025 at 9:28am
Yes. Once you state (as I did) that you’re computing the area of a triangle, you’re necessarily assuming linear demand and linear supply.
If they’re non-linear, you need to take an integral, which is much more complicated. The t^2 result still holds up, though.
Manfred
Feb 5 2025 at 4:15pm
Professor Henderson,
Is there a citation with the full proof of the more general case (non-linearity)?
Thank you, Manfred
Frank Collins
Feb 6 2025 at 2:37am
An amazingly strange article; start with an analogy that has no relevance and try, seemingly desperately, to build a coherent and cogent argument.
The rest of the world functions well, treats its citizens as human beings, most have universal healthcare, has social social financial support, doesn’t have mass shootings every other day, yet Americans can’t grasp such “bizarre” world oddities such as sick leave, four weeks holidays every year, and not working every day for a wage that won’t pay the rent. Relying on tips to make a living is a humiliation no one should suffer.
David Henderson
Feb 6 2025 at 4:18pm
How does any of your comment relate to what I wrote? I don’t see the connection.
Knut P. Heen
Feb 7 2025 at 10:09am
This reminds me of an American professor we hired who got really mad when he realized that the cheap kindergarten we offer in Norway actually meant waiting in line for many months without kindergarten. I hope he stays long enough to learn that universal healthcare does not include a long list of drugs the government has decided are too expensive.