Many textbooks, including Greg Mankiw's, argue that if a per unit tax is imposed on a good, the portion of the tax eventually borne by sellers and buyers depends on the comparative price elasticities of demand and supply [pp 135-6 of the 4th Cdn edition].
I think that is incorrect.
I think it depends on the comparative slopes, not comparative elasticities. Here is a graph to illustrate this point (which might also appear, I vaguely recall, using calculus in an old edition of Henderson and Quandt [thanks to Brian Ferguson, I see this material on p154 of the 3rd edition]):
Since my drawing skills are not great, please assume that the upward-sloping lines are supply curves, that all four of them are parallel and that each pair shows the effect of levying the same per-unit (or excise) tax on the sellers of the good.
The demand curve (downward-sloping but unlabeled) is a straight line; it has a constant slope, but the price elasticity of demand varies all along it from greater than one (in absolute value) near the vertical axis to less than one near the horizontal axis, and equal to one at its midpoint.
If the "burden of the tax" (which I take to mean the portion of the per unit tax paid by buyers and sellers, respectively, using partial equilibrium analysis) depends on elasticities, it should vary along this linear demand curve, shouldn't it? But it is easy to see that the portion of the tax paid by consumers and sellers is invariant with the elasticities because the relative slopes are the same for both pairs of supply curves.
Kip notes in the comments:
You can argue the same thing in reverse: use a constant elasticity demand curve (i.e., a rectangular hyperbola) and then draw your parallel pairs of supply curves. You'll see that (delta-P)/T is very different at the two extreme ends of the demand curve.
My initial reaction was that the supply curves are intersected at a different point in the upper left than in the lower right. The slopes are the same, but that doesn't mean the elasticity of supply necessarily takes the same value.
However, the elasticity can be constant along every point of a linear supply curve. For example, consider the supply curve Q = mP where m > 0. dQ/dP = m and P/Q = P/mP = 1/m. So the elasticity of supply is constant at 1 regardless of the P-Q combination.