Elasticities and the Efficiency Loss of a Tax

Consider the drawing at left, which shows the impact of a unit tax of $T. Initially, the equilibrium price was $P, and the equilibrium quantity was Q. The imposition of the tax caused the equilibrium quantity to fall by D Q, and the price to consumers increased by D P_d while the price to producers fell by D P_s. The efficiency loss of the tax is given by the sum of the two triangles labeled A and B on the diagram. For convenience, call the efficiency loss Z, so Z = A + B. As stated in the text, the size of this loss increases with the elasticities of either supply or demand. This note will develop a formula for the size of the efficiency loss so that we may show the dependency of this area on the two elasticities.

We begin by noting that the area of a triangle is one half the base times the height. In this example, each triangle has base equal to D Q. Triangle A has height D P_d, while triangle B has height D P_s. The efficiency loss is therefore Z = A + B = ½D QD P_d + ½D QD P_s = ½D Q(D P_d + D P_s) = ½D QT. This last equality follows because the tax, T, must be the difference between the price paid by consumers and the price received by producers. The size of the loss clearly depends on T, but we are left with some uncertainty because we do not yet know the size of D Q. We suspect that it relates to the elasticities of supply and demand, so that is our next step.

Recall that the elasticity of demand, E_d, can be written as E_d = = . Suppose we solve this for D P_d as follows: D P_d = . Likewise, we could find that D P_s = . We know that the total change in the two prices, D P_d + D P_s, is equal to the tax, so T = + . If we multiply and divide the first term in this sum by E_s and the second term by E_d, we get a common denominator and can add the two terms to get T = = .

Now what we need is D Q, so we solve this last expression in terms of D Q to get D Q = . As suspected, the change in quantity depends on the size of the tax and the two elasticities. We can now plug this value of D Q into our formula for the efficiency loss, Z = ½D QT = .

Notice that the size of this loss increases with the square of the tax. That is, doubling the size of the tax will increase the size of the efficiency loss by a factor of four. To find how the elasticities affect Z, we can take the partial derivatives of Z with respect to E_d and E_s to find = and = . Clearly these are both positive, so that all else constant, the more elastic is either demand or supply, the greater the size of the efficiency loss.