Question

3. The (inverse) equations for the supply and demand for French Champagne are given below. Supply: P = 40 + ¼Q Demand: P = 100 – ½Q [Half point for each question]

a) compute the equilibrium price and quantity of Champagne.

b) Suppose and excise tax (i.e. a tax paid by producers) of $18 per bottle is imposed. What are the equations for the new supply and demand curves? What is the new EQ price and quantity? Specify what the prices paid by consumers and received by producers are?

c) Do the same as (b) for a sales tax (i.e. a tax paid by consumers) of $18 per bottle and explain whether the burden has shifted from producers to consumers along with the incidence of the tax from excise to sales.

d) Calculate the Consumer Surplus and Producer Surplus before and after the tax is put into place. How much revenue is raised? Calculate any deadweight loss from the tax.

Answer 1

Demand: P = 100 - (1/2)Q = 100 - 0.5Q

Supply: P = 40 + (1/4)Q = 40 + 0.25Q

(a) In equilibrium, demand function and supply function equate.

40 + 0.25Q = 100 - 0.5Q

0.75Q = 60

Q = 60/0.75 = 80 (Equilibrium quantity)

P = 100 - (0.5 x 80) = 100 - 40 = $60 (Equilibrium price)

(b) A $18 tax imposed on producers will decrease the price received by producers, which will shift supply curve leftward by $18 at every output level. New supply function is:

P - 18 = 40 + 0.25Q

P = 58 + 0.25Q

Equating (unchanged) demand function with revised supply function,

100 - 0.5Q = 58 + 0.25Q

0.75Q = 42

Q = 56

P = 100 - (0.5 x 56) = 100 - 28 = $72 (Price paid by consumers)

Price received by producers = $(72 - 18) = $54

Tax burden borne by consumers = $(72 - 60) = $12

Tax burden borne by producers = $(18 - 12) = $6

(c) A $18 tax imposed on buyers will increasae the price paid by consumers, shifting the demand curve leftward. New demand function becomes

P + 18 = 100 - 0.5Q

P = 82 - 0.5Q

Equating the new demand function with (unchanged) supply function,

82 - 0.5Q = 40 + 0.25Q

0.75Q = 42

Q = 56

P = 82 - (0.5 x 56) = 82 - 28 = $54 (Price received by producers)

Price paid by consumers = $(54 + 18) = $72

Tax burden borne by consumers = $(72 - 60) = $12

Tax burden borne by producers = $(18 - 12) = $6

Therefore, tax burden borne by consumers and producers remains the same.

(d)

Before tax,

From demand function, when Q = 0, P = $100 (Maximum possible price)

Consumer surplus (CS) = Area between demand curve & equilirbrium price = (1/2) x $(100 - 60) x 80 = 40 x $40

= $1,600

From supply function, when Q = 0, P = $40 (Minimum acceptable price)

Producer surplus (PS) = Area between supply curve & equilibrium price = (1/2) x $(60 - 40) x 80 = 40 x $20

= $800

After tax (imposed on producers):

CS = (1/2) x (Maximum price - Price paid by consumers) x New quantity = (1/2) x $(100 - 72) x 56 = 28 x $28

= $784

PS = (1/2) x (Price received by producers - Minimum price) x New quantity = (1/2) x $(54 - 40) x 56 = 28 x $14

= $392

Tax revenue = Tax per unit x New quantity = $18 x 56 = $1,008

Deadweight loss = (1/2) x Tax per unit x Change in quantity = (1/2) x $18 x (80 - 56) = $9 x 24 = $216