Question

Consider the following market demand and supply functions:

Qd = 14 - p

Qs = -6 + p

a) Calculate the perfectly competitive equilibrium price, quantity, consumer surplus, producer surplus, and total surplus.

b) If an output tax t = 2 is imposed on consumers, calculate the new price, quantity, consumer surplus, producer surplus, and total surplus. What is the deadweight loss? Total taxes collected? (Note: an output tax shifts the demand curve intercept downward by the amount of the tax.)

c) Graph the results

Answer 1

(a) In pre-tax equilibrium, Qd = Qs.

14 - p = - 6 + p

2p = 20

p = 10

q = 14 - 10 = 4

From demand function, when Qd = 0, p = 14 (Vertical intercept).

Consumer surplus (CS) = Area between demand curve & price = (1/2) x (14 - 10) x 4 = 2 x 4 = 8

From supply function, when Qs = 0, p = 6 (Vertical intercept).

Producer surplus (PS) = Area between supply curve & price = (1/2) x (10 - 6) x 4 = 2 x 8 = 16

Total surplus (TS) = CS + PS = 8 + 16 = 24

(b)

The tax will shift demand curve leftward by 2 units at every output level. New demand function becomes

Qd = 14 - (p + 2) = 14 - p - 2 = 12 - p

Equating with Qs,

12 - p = - 6 + p

2p = 18

p = 9 (Price received by sellers)

Price paid by buyers = 9 + 2 = 11

Q = 12 - 9 = 3

CS = (1/2) x (14 - 11) x 3 = 1.5 x 3 = 4.5

PS = (1/2) x (9 - 6) x 3 = 1.5 x 3 = 4.5

TS = 4.5 + 4.5 = 9

Tax revenue = Unit tax x After-tax quantity = 2 x 3 = 6

Deadweight loss = (1/2) x Unit tax x Change in quantity = (1/2) x 2 x (4 - 3) = 1 x 1 = 1

(c)

In following graph, D0 and S0 are initial demand and supply curves, intersecting at point A with equilibrium price P0 (= 10) and quantity Q0 (= 4). CS equals area P0AD, PS equals area P0AE and TS equals area DAE. After the consumption tax is imposed, D0 shifts leftward to D1, intersecting S0 at point B with price paid by buyers being higher at P1 (= 11), price received by sellers being lower at P2 (= 9) and lower quantity Q1 (= 3). CS is lower at area P1CD (decrease by area P0DCP1), PS is lower at area P2BE (decrease by area P0ABP2), tax revenue is area P1CBP2 and deadweight loss is area ABC.