Question

Consider a market with the following demand and supply curves:

𝑄^d =100−2𝑝

𝑄^s=3𝑝

Calculate

a. equilibrium price and quantity.

b. At the equilibrium, find the elasticity of demand and supply and compare.

c. At this equilibrium, calculate the producer’s surplus, consumer’s surplus, and total surplus.

d. Suppose the government would like to decrease the quantity traded in the market by 10%. Calculate the tax that will have to be imposed per unit.

e. Calculate the deadweight loss due to the tax calculated in part d.

Answer 1

Solution:

a.). Equilibrium price and quantity:

At equilibrium: Qd = Qs

100 – 2p = 3p

100 = 3p + 2p

100 = 5p

P = 20

Equilibrium price = 20

Substitute in the quantity demanded to derive quantity:

Qd = 100 – 2p = 100 – 2(20) = 100 – 40 = 60

Equilibrium quantity = 60

 

b.). Elasticity of demand: $\frac{\triangle Qd}{\triangle P}\times \frac{P}{Qd}$

=$-2\times\frac{20}{60} = -0.67$

Elasticity of demand = 0.67

Elasticity of supply: $\frac{\triangle Qs}{\triangle P}\times \frac{P}{Qs}$ ΔQs/ΔP x P/Qs

= $3\times\frac{20}{60} = 0.99 \;or 1$ 3 x 20/60 = 0.99 or 1

Elasticity of supply = 1

 

c.). Consumer surplus =$\frac{1}{2}\times 60\times (50 - 20) = 900$

Producer surplus = $\frac{1}{2}\times 60\times (20 - 0) = 600$

Total surplus = Consumer surplus + Producer surplus

TS = 900 + 600 = 1,500

 

d.). Tax to be imposed per unit:

$=10\% \times 60 = 6$

= 60 – 6 = 54

 = $\frac{60 - 54}{54} =$ 0.1 per unit

 

e.). Deadweight Loss = $\frac{1}{2}\times (60 - 50)\times (35 - 15) = \frac{1}{2}\times 200 = 100$

Deadweight Loss = 100