In a publish final week, I said that the deadweight loss from a tax is proportional to the sq. of the tax price. So doubling the tax price, for instance, quadruples the deadweight loss.
I said on Fb that I used primary algebra to show this to my college students. An economist good friend requested me to do it.
So I did.
Right here’s the proof.
For simplicity, begin with a horizontal provide curve, as within the graph above, and compute the Harberger triangle loss. It’s C within the above graph, which is one half the bottom occasions the peak. The peak is the tax, t. The bottom is computed as delta Q, the drop in quantity demanded. So we have to compute the bottom.
Let elasticity = n.
n = delta Q/Q divided by delta P/P.
So delta Q/Q = n occasions deltaP/P.
delta Q = n occasions delta P/P occasions Q.
However delta P = t.
So delta Q = n occasions t/P occasions Q.
So triangle loss = 1/2 t occasions n occasions t/P occasions Q.
So triangle loss = 1/2 t^2 occasions n/P occasions Q.
QED.
Observe: If, as is often the case, the availability curve is upward-sloping, the method for deadweight loss is extra sophisticated however the sq. relationship nonetheless holds.
