Suppose the market for cigarettes is characterized by the following information:

Question

Suppose the market for cigarettes is characterized by the following information: Qd=70–5P [Demand] Qs=3P–10 [Supply]

[Note: P = price per unit; Qd = thousands of units demanded; Qs = thousands of units supplied] Suppose the government imposes a sales tax of $2 per unit. Answer questions (i) through (v) below:

i) Calculate the magnitude of the consumer surplus and producer surplus in the pre-tax equilibrium.

ii) Calculate the tax revenue in the post-tax equilibrium.

iii) Calculate the change in consumer surplus due to the sales tax.

iv) Calculatethechangeinproducersurplusduetothesalestax.

v) Calculate the Dead-Weight-Loss due to the sales tax.

Explanation / Answer

Qd = 70 - 5P

Qs = 3P - 10

(i)

When Qd = 0, P = 14 [Vertical intercept of demand curve]

When Qs = 0, P = 10/3 = 3.33 [Vertical intercept of supply curve]

Equilibrium occurs when Qd = Qs

70 - 5P = 3P - 10

8P = 80

P = 10

Q = 3P - 10 = (3 x 10) - 10 = 30 - 10 = 20

Consumer surplus (CS) is the area between demand curve & price:

CS = (1/2) x (14 - 10) x 20 = 40

Producer surplus is the area between supply curve & price:

PS = (1/2) x (10 - 3.33) x 20 = 66.7

(ii) After tax of $2 per unit is imposed:

Demand function remains unchanged.

Qd = Q = 70 - 5P

5P = 70 - Q

P = 14 - 0.2Q

But producers receive a price that is lower by $2 which has to be paid to government. For producers, P = P - 2

Since Qs = Q = 3P - 10,

3P = Q + 10

P = 0.33Q + 3.33

After imposition of tax,

P = P - 2

P - 2 = 0.33Q + 3.33

P = 0.33Q + 5.33

Equating demand with supply:

14 - 0.2Q = 0.33Q + 5.33

0.53Q = 8.67

Q = 8.67 / 0.53 = 16.36

P = 14 - 0.2Q = 14 - (0.2 x 16.36) = 10.73

Tax revenue = Post-tax Q x Unit tax

= 16.36 x $2 = $32.72

(iii)

Post-tax CS = (1/2) x (14 - 10.73) x 16.36 = (1/2) x 3.27 x 16.36 = 26.75

Change in CS = 40 - 26.75 = 13.25

(iv)

Post-tax PS = (1/2) x (10.73 - 3.33) x 16.36 = (1/2) x 7.4 x 16.36 = 60.53

Change in PS = 66.7 - 60.53 = 6.17

(v)

Deadweight loss = (1/2) x Change in P x Change in Q

= (1/2) x (10.73 - 10) x (20 - 16.36) = (1/2) x 0.73 x 3.64 = 1.33

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.