Question

In the local market for gasoline ,supply is given by QS=30P and demand is given by QD=400−10P.

1. (a) Calculate the equilibrium price and quantity in the competitive market. Then, calculate equilib- rium price and quantity if a $1 per unit tax is imposed. Solution: Absentthetax,QS =30P=QD =400−10P⇒40P=400⇒P=10,Q=300.Withthe tax,QS =30PS =QD =400−10PB =400−10(PS+1)⇒40PS =390⇒PS =9.75,PB =10.75. Quantity is then 30(9.75) = 292.5.

   (b) Calculate the deadweight loss associated with the imposition of the $1 tax. Is the deadweight loss from the tax equal to the decrease in consumer + producer surplus? Be sure to justify your answer.

Solution: The deadweight loss is (1/2)(300 − 292.5)(1) = $3.75. The deadweight loss is smaller than the decrease in consumer + producer surplus since the government receives tax revenue equal to 292.5*1=$292.5 (and this is associated with reduced consumer/producer surplus but does not reflect deadweight loss).

(c) Calculate the elasticities of supply and demand at the pre-tax competitive market equilibrium. Then, explain whether consumers or suppliers will pay a higher share of the newly-imposed tax. Relate your finding to the elasticities of supply and demand you calculated.

Solution: Elasticity of demand is −10(10/300) = −1/3 and elasticity of supply is 30(10/300) = 1. Consumers pay a higher share of the tax since consumers pay the share given by (ES/(ES − ED )) = 1/(1 + 1/3) = 3/4 based on the elasticities. We can verify that this is indeed the share paid by consumers since price increases by $0.75.

I need a breakdown on part (c). How did they derive the demand elasticity equation of −10(10/300) = −1/3 and the elasticity for supply as 30(10/300) = 1 in the solution. What is the formula to get these equations.

Thank you.

Answer 1

(c) Breakdown of the answer:

Qd 400 - 10P So, P= 40 - 1/10Qd or P = -1/10 Q+ 40

This is in the form of Y = mx + b; where m is the slope.

According to our calculation, this slope is (-1/10)

We know that the slope of the curve, algebraically, is y/x or in our case ∆P/ ∆Q (how much a change in price would lead to a change in quantity)

elasticity of demand (Ed) = (∆Q/ ∆P) * (P/Q). If ∆P/ ∆Q is (-1/10),   ∆Q/ ∆P would be inverse, i.e. -10

we know from part (a) that P = 10, Q = 300

Plugging in the values in the formula:

Ed = -10 * (10/300) = -1/3

We can also notice that the slope is directly mentioned in the demaand function itself.(Qd = 400 - 10P). We could pick it up from the demand function itself.

Similarly, in the supply function, we have, Qs = 30P. Applying this slope to the formula of elasticity of supply,

Es = (∆Qs/ ∆P) * (P/Qs) = 30 * (10/300) = 1

Verifying the other parts:

(a) Qs =Qd

30P = 400−10P P=10, Q=300.

When there is tax, Qs =30Ps = Qd =400−10Pb = 400−10(Ps+1) --- since Pb = Ps+1

40Ps =390 or Ps =9.75, Pb =10.75. So, Q = 30*(9.75) = 292.5.

(b) Deadweight loss (DWL) = (1/2) *(300 − 292.5)*(1) = $3.75.

DWL is smaller than the decrease in consumer + producer surplus.

reason: tax revenue= 292.5*1=$292.5, which is reduced consumer surplus, not inclusive in DWL.