Question

Suppose that a market is initially in equilibrium. The initial demand curve is P=42-Qd. The initial supply curve is P=0.5Qs. Suppose that the government imposes a $3 tax on this market.

1. Solve for initial Equilibrium (before tax)

2. Solve for price elasticities of demand and supply at equilibrium

3.After imposition of tax solve for following. Pe1: Pd: Pes: Qe1: Consumer surplus post tax $

Answer 1

The initial demand curve is P=42-Qd, = Qd = 42 - P

The initial supply curve is P=0.5Qs, Qs = P/0.5

1) The initial Equilibrium (before tax): initial demand curve = initial supply curve

42 - P = P / 0.5

0.5 (42 - P) = P

21 - 0.5P = P

21 = P + 0.5P

P = 21 / 1.5

P = $14

Qs = P/0.5

= 14 / 0.5

= 28

So, the initial equilibrium before tax is Price = $14 and Quantity = 28 units.

(b) The price elasticity of demand = Percentage change in Quantity Demanded / Percentage Change in Price

= (Change in Quantity/Quantity)/(Change in Price/Price)

= (Price/Quantity)(Change in Quantity/Change in Price)

Qd = 42 - P

Change in Quantity / Change in Price = -1

price elasticity of demand = (Price/Quantity)(Change in Quantity/Change in Price)

= (14/28)(-1)

= - 0.5

The price elasticity of supply = Percentage change in Quantity Supplied / Percentage Change in Price

= (Price/Quantity)(Change in Quantity/Change in Price)

Qs = P / 0.5

Change in Quantity / Change in Price = 1 / 0.5

price elasticity of supply = (Price/Quantity)(Change in Quantity/Change in Price)

= (14/28)(1/0.5)

= 0.25

(c) After imposition of tax, Tax = $3,

Price sellers pay = Price Buyers Pay - Tax

42 - P = (P - T)/0.5

42 - P = (P - 3)/0.5

0.5(42 - P) = P - 3

21 - 0.5P = P - 3

21 + 3 = P + 0.5P

1.5P = 24

P = 24 / 0.5

P = $ 16

Pe1: $16 Pd: $16

Price sellers receive = Price Buyers Pay - Tax

= 16 - 3

= 13

Pes: 13

Qe1: Qs = P / 0.5

= 16 / 0.5  

= 32

Consumer surplus post tax $ = 1/2 * Quantity post tax * (Maximum Price - Price post tax)

= 1/2 * 32 * (42 - 32)

= 1/2 * 32 * 10

= 160

When Price = 42, Qd = 42 - P = 42 - 42 = 0, so the maximum price is 42 where the quantity demanded is zero.