Question

Suppose the government of Kurtzmantonia is considering imposing a tax of $1 per room per night on bed-and-breakfasts (B&B’s). Currently, the demand and supply for rooms in B&B’s are as follows:

Demand: ??=13∙(100 – ??)

Supply: ??= 20 + ??

where ?? and ?? are respectively the quantity of rooms demanded and supplied (measured in rooms/night), ??is the price (in $ per room-night) paid by consumers, and ?? is the price collected by sellers.

a. Find the market equilibrium price and quantity of rooms rented before the tax is imposed.

b. What are the elasticities of supply and demand at that equilibrium?

c. (2) If the tax is imposed, what will be the difference between the price paid by buyers and the price received by sellers? Write this relationship as an equation.

d. Find the equilibrium price and quantity of rooms rented if the tax is imposed. (You should be able to report exact numbers, but if not, draw a NEAT graph showing what you would like to do.)

Answer 1

a. At equilibrium, QD = QS
So, 13∙(100 – ?) = 20 + P
So, 1300 - 13P = 20 + P
So, P + 13P = 14P = 1300 - 20 = 1280
So, P = 1280/14
So, P = 91.43

Q = 20 + P = 20 + 91.43
So, Q = 111.43

b. d(QD)/dP = -13
Elasticity of demand = (dQD/dP)*(P/Q) = -13*(91.43/111.43) = -10.67

d(QS)/dP = 1
Elasticity of supply = (dQS/dP)*(P/Q) = 1*(91.43/111.43) = 0.82

c. Difference between price paid by buyers and receieved by sellers is amount of tax.
So, PD - PS = 1

d. After tax, price received by sellers is P - 1.
So, QS' = 20 + P - 1 = 19 + P
At equilibrium, QD = QS'
So,

13∙(100 – ?) = 20 + P
So, 1300 - 13P = 19 + P
So, P + 13P = 14P = 1300 - 19 = 1281
So, P = 1281/14
So, P = 91.5

Q = 1300 - 13P = 1300 - 13(91.5) = 1300 - 1189.5 = 110.5
So, Q = 110.5