1.4 Excise tax
Using only the simple techniques developed so far we can obtain some interesting insights into problems in economics. In this section we study the problem of excise tax. Suppose that a government wishes to discourage its citizens from drinking too much whisky. One way to do this is to impose a fixed tax on each bottle of whisky sold. For example, the government may decide that for each bottle of whisky the suppliers sell, they must pay the government $1. Note that the tax on each unit of the taxed good is a fixed amount, not a percentage of the selling price.
Some very simple mathematics tells us how the selling price changes when an excise tax is imposed.
Example In the previous example the demand and supply functions are given by
$q^d(p) = 40 - 5p$, $q^s(p) = \frac{15}{2}(p - 10)$,
and the equilibrium price is $p^* = 4$. Suppose that the government imposes an excise tax of $T$ per unit. How does this affect the equilibrium price?
The answer is found by noting that, if the new selling price is $p$, then, from the supplier's viewpoint, it is as if the price were $p - T$, because the supplier's revenue per unit is not $p$, but $p - T$. In other words the supply function has changed: when the tax is $T$ per unit, the new supply function $q^s_T$ is given by
$q^s_T(p) = q^s(p - T) = \frac{15}{2}(p - T) - 10$.
Of course the demand function remains the same. The new equilibrium values $q^T$ and $p^T$ satisfy the equations
$q^T = 40 - 5p^T$ and $q^T = q^s_T(p^T) = \frac{15}{2}(p^T - T) - 10$.
6 Mathematical models in economics
Eliminating $q^T$ we get
$40 - 5p^T = \frac{15}{2}(p^T - T) - 10$.
Rearranging this equation, we obtain
$(5 + \frac{15}{2})p^T = 50 + \frac{15}{2}T$,
and so we have a new equilibrium price of
$p^T = 4 + \frac{3}{5}T$.
The corresponding new equilibrium quantity is
$q^T = 40 - 5p^T = 20 - 3T$.
For example, if $T = 1$, the equilibrium price rises from 4 to 4.6 and the equilibrium quantity falls from 20 to 17. Unsurprisingly, the selling price has risen and the quantity sold has fallen. But note that, although the tax is $T$ per unit, the selling price has risen not by the full amount $T$, but by the fraction 3/5 of $T$. In other words, not all of the tax is passed on to the consumer.
