Slope and Elasticity
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.
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.
Category: Economics Posted on
Thursday, January 31, 2008 at 11:44am
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Your statement (I'm paraphrasing) 'the burden of the tax depends on the ratio of the supply and demand slopes' is logically equivalent to the textbook statement 'the burden of the tax depends on the ratio of the demand and supply elasticities'. (This is true at least for small taxes). This is because the ratio of the supply to demand slopes is necessarily equal to the ratio of the demand to supply elasticities.
Proof:
Since elasticity = (dQ/Q)/(dP/P)= (dQ/dP)X(P/Q)
=(1/slope)X(Price/Quantity)
Then ratio of elasticities
= elasticity of demand/elasticity of supply
= [(1/slope of demand)X(P/Q)]/[(1/slope of supply)X(P/Q)]
= (slope of supply/slope of demand)
I think I got the maths right.
In your example, the change in the elasticity of demand (as we move along the constant slope demand curve) is exactly compensated by the change in elasticity of supply (as we shift the supply curve up keeping its slope the same).
However, the section of the text to which I was referring shows straight-line demand and supply curves and says, "In panel (a) the supply curve is elastic and the demand curve is inelastic..." It's a straight-line demand curve; the curve itself is not inelastic. Rather, the supply curve crosses the demand curve in its inelastic portion.
The description of the graphs makes the same error about panel (b), referring to linear demand curve as "elastic" when it really is an elastic portion of the demand curve.
I must say though, your example forced me to think this through. It wasn't immediately obvious that 'ratio of the slopes' was equivalent to 'ratio of the elasticities', and I have been trying it out on my colleagues, and they have to think it through too.
I wonder if we (teachers of Intro Economics) are right in putting so much emphasis on elasticity rather than slope? It's harder to teach 'elasticity' than 'slope', which most first year students already
understand"have been exposed to" in high school.The benefit of course is that elasticity is unit-free, but I wonder if that benefit is worth the cost? Especially since in one of the main cases where we illustrate the usefulness of 'elasticity' (the tax-incidence example), we could get exactly the same results using 'slope'. Maybe 'elasticity' is more useful than 'slope' because of the relationship between elasticity and marginal revenue? Could that relationship be as easily converted into a relationship between slope and marginal revenue?
Still thinking about this question....
Intermediate textbooks discuss the general determinants of price elasticity and first on the list is usually the number and/or quality of substitutes. This suggests that the side of the market with the poorer substitutes will bear more of the burden of the per-unit tax.
That makes good sense: if you have poorer substitutes, you are more likely to sit still and pay the tax, while if you have better substitutes, you are more likely to run away from it.