Question

The price elasticity of supply for a product is 3, while the price elasticity of demand is -1. In equilibrium, price is 6 (in hundreds of dollars) and quantity consumed is 2 (in thousands of
units).
(a) Assuming supply and demand are linear, reconstruct and draw the supply and demand curves. Label the intercepts.
(b) If a subsidy of $1 per unit is imposed what are P_B and P_S after the subsidy? What is the new equilibrium quantity? Illustrate them on the same graph.
(c) How big is the change in consumer surplus, producer surplus, government expenditure, and deadweight loss?

Answer 1

(a)

(I) Linear demand curve: QD = a - bP

When P = 6, QD = 2

2 = a - 6b

Elasticity of demand = (dQD/dP) x (P/QD)

- 1 = - b x (P/QD)

1 = b x (6/2)

b = 1/3 = 0.33

2 = a - 6 x (1/3) = a - 2

a = 4

Linear demand curve: QD = 4 - (P/3)

When QD = 0, P = 12 (vertical intercept) and when P = 0, QD = 4 (horizontal intercept).

(II) Linear supply curve: QS = a + bP

When P = 6, QS = 2

2 = a + 6b

Elasticity of supply = (dQS/dP) x (P/QS)

3 = b x (P/QD)

3 = b x (6/2)

b = 1

2 = a + 6 x 1 = a + 6

a = - 4

Linear supply curve: QS = - 4 + P

When QS = 0, P = 4 (vertical intercept)

(III) In following Graph, D0 and S0 are demand and supply curves intersecting at point E with price P0 (= 6) and quantity Q0 (= 2). Consumer surplus (CS) is area AEP0 and producer surplus (PS) is area BEP0.

2 9, 2.25 8,

(b)

Assuming subsidy is imposed on sellesrs, new supply function is

QS = - 4 + P + 1

QS = - 3 + P

Equating with QD,

4 - (P/3) = - 3 + P

4P/3 = 7

P = 5.25 (price paid by buyers = PB)

Price received by sellers (= PS) = 5.25 + 1 = 6.25

Q = - 3 + 5.25 = 2.25

In above graph, post-subsidy equilibrium is point F where S1 intersects D0 with price paid by buyers being P2 (= 5.25), price received by sellers being P1 (= 6.25) and higher quantity Q1 (= 2.25). New CS is area AFP2, new PS is area BGP1 and cost of subsidy is area P1GFP2. Deadwight loss is area EFG.

(c)

Increase in CS = area P0EFP2 = (1/2) x (6 - 5.25) x (2 + 2.25) = (1/2) x 0.75 x 4.25 = 1.59375

Increase in PS = area P0EGP1 = (1/2) x (6.25 - 6) x (2 + 2.25) = (1/2) x 0.25 x 4.25 = 0.53125

Government expenditure = 1 x 2.25 = 2.25

Deadweight loss = (1/2) x 1 x (2.25 - 2) = 0.5 x 0.25 = 0.125