Every autumn brings an increase in the demand for pumpkin spice lattes (PSL). This fall, the demand for PSLs in a mid‑sized midwestern town is given by ????D=10−0.5P , where P is the price of a PSL in dollars and ???? is thousands of PSLs per day. The corresponding supply is given by ????S=−5+2P . The equilibrium price of a PSL is PE=$6 ; the equilibrium quantity is ????E=7 . Eager to capitalize on the addictive nature of the PSL, the City Council decides to impose a $1 tax on pumpkin spice lattes. "We can use the proceeds to fund an espresso machine for City Hall!" they rave.
a. At equilibrium price and quantity, what is the price elasticity of demand? what is the price elasticity of supply?
b. after the City Council decides to impose a $1 tax on pumpkin spice lattes, what is the share of the tax borne by consumers? what is the share of the tax borne by sellers?
c. After the tax is imposed, What is the expected price buyers will have to pay?

You did not give the new PSL price by autumn but since the question says there's an increase in demand for PSL every autumn, I will assume a new price (which will be higher) of $8 by autumn. Following the laws of economics (demand & supply), if producers know that demand will rise at a certain time of the year, they'll increase price. So the new price is $8.

(A) Price elasticity of demand is calculated thus:

Change in price = $8-$6 =$2

New quantity demanded = 10-0.5(8) = 6 pieces

Change in quantity demanded = 7-6 = 1

PEOD = [(Change in Qd/Old quantity) × 100] ÷ [(Change in Price/Old price) × 100]

PEOD = [1/7 × 100] ÷ [2/6 × 100]

PEOD = 14.2857 ÷ 33.33

= 0.428 ~ 0.43

This figure shows that a 33.3% increase in price (from $6 to $8) results in or effects a 14.3% DECREASE in quantity demanded (from 7 pieces to 6 pieces). This implies that the change in Qd due to a change in price is little or inelastic; as the PEOD point figure is 0.43, which is less than 1.

This explains the high taste that the people of this city have for PSL. A higher price of PSL won't be much of a discouragement when it comes to purchasing product.

(B) Price elasticity of supply is calculated thus:

Change in price = $2

New quantity supplied = -5 + 2(8) = -5 +16 = 11 pieces

Change in QS = 11-7 = 4

PEOS = [4/7 × 100] ÷ [2/6 × 100]

PEOS = 57.143 ÷ 33.33

= 1.7143 ~ 1.71

Here, the reverse is the case; as expected from producers. A 33.3% rise in price causes as much as a 57.1% INCREASE in Qs (from 7 pieces to 11 pieces). This implies that the change in Qs owing to a change in price is elastic, as the PEOS point figure is 1.71, which is greater than 1.