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.