Effects of Specific Tax on elasticity:

In the new equilibrium demand should equal supply, but demand depends on the price that consumers pay, p, supply depends on the price that suppliers receive p − τ, therefore:

D(p(τ)) = S(p(τ) − τ)

As we’ve seen before, the equilibrium price can be considered an implicit function of the environmental variable that changes, in this case τ, hence we write p(τ) to emphasize that the equilibrium price depends (indirectly) on the specific tax rate. The change of the equilibrium price due to a small change in τ is the derivative: dp/dτ and the total change in quantity demanded due to a change in price is dD/dp. By using the so called chain rule we find the total change in quantity demanded due to a small change in τ as: dD/dp . dp/dτ. On the supply side, the total change in quantity supplied due to an increase in the equilibrium price is dS/dp and the change in the price received by suppliers is [d(p(τ)−τ)]/dτ = (dp/dτ) −1, putting these changes together gives: (dS/dp) x [(dp/dτ) – 1]. From one equilibrium to another, the total change in quantity demanded should be equal to the total change in quantity supplied:

Solving for the derivative dp/dτ:

The demand curve is downward sloping so dD/dp < 0 and if the supply curve is upward sloping dS/dp > 0, the denominator is positive and we can conclude that the equilibrium price will increase as the specific tax is increased. If dS/dp < 0, the sign is ambiguous (could be positive or negative). By multiplying all terms in both the numerator and denominator by p/Q, we can express the derivative in terms of supply and demand elasticity:

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