On October 1st of last year, the District of Columbia instituted a tax on health club businesses, sometimes referred to as the so-called “gym tax” or “yoga tax”. Many wondered what the incidence of this tax will be: will the prices rise such that consumers will bear most of the burden, or will gyms and yoga studios find higher prices will drive away too many customers, leaving them to keep prices steady and reducing their profits? Let’s examine how incidence is calculated by using a stylized example with a given demand and supply curve and a per-session tax1 . Suppose the quantity of yoga sessions demanded by consumers is given by the function: Qd = 200 − 2Pc, and the quantity of yoga sessions offered by studios is given by the function: Qs = 25 + Ps 2 , where Qd and Qs are the quantities demanded and quantities supplied respectively and Pc and Ps are the consumers’ and suppliers’ prices. In equilibrium, the quantities and prices will be equal, however when there is a tax, the consumers’ and producers’ prices are defined by the function Pc = Ps + T.
(a) Assume, initially, there is no tax. What is the equilibrium price and quantity in the yoga market?
(b) Assuming that D < 0, determine the elasticity of demand at the original equilibrium price and quantity.
(c) Assuming that S > 0, determine the elasticity of supply at the original equilibrium price and quantity.
(d) Now assume a tax of $5 (T = 5) has been levied on yoga sessions. Determine the impact that a change in T has on producers’ prices with the formula2 : dPs dT = D S − D . Additionally, determine the impact on consumers’ prices with the formula: dPc dT = S S − D.
(e) Based on the above calculations, what will be the new price paid by consumers and price received by studios? Confirm this result by solving for the new equilibrium using the above demand and supply curves and substituting Pc = Ps + 5 into one of the curves.