Question

Suppose that a market is described by the following supply and demand equations:

QS = 3P

QD = 400 – P

a. Solve for the equilibrium price and the equilibrium quantity.

b. Suppose that a tax of T is placed on buyers, so the new demand equation is

QD = 300 – (P + T).

Solve for the new equilibrium. What happens to the price received by sellers, the price paid by 

buyers, and the quantity sold?

c. Tax revenue is T x Q. Use your answer to part (b) to solve for tax revenue as a function of T. 

d. The deadweight loss of a tax is the area of the triangle between the supply and demand curves. 

Recalling that the area of a triangle is 1⁄2 x base x height, solve for deadweight loss as a function 

of T. 

e. The government now levies a tax on this good of $200 per unit. Is this a good policy? Why or 

why not? Can you propose a better policy?

Answer 1

(a) The equilibrium occurs when $Q_D=Q_S$:

$$400-P_E=3P_E,$$ $$400=4P_E,$$ $$P_E=\$100.$$

Then, substituting the equilibrium price into the demand function we can find the equlibrium quantity:

$$Q_E=400-100=300.$$

(b) Let's find the new equilibrium price (or the price paid by buyers):

$$300-(P_E+T)=3P_E,$$ $$300-T=4P_E,$$ $$P_E=P_b=75-\dfrac{T}{4}.$$

The sellers received less price:

$$P_b-P_s=T,$$ $$P_s=P_b-T=75-\dfrac{T}{4}-T=75-\dfrac{3}{4}T.$$

Let's find the new equilibrium quantity (the quantity sold):

$$Q_E=3P_E=3\cdot(75-\dfrac{T}{4})=225-\dfrac{3}{4}T.$$

(c)

$$Tax\ Revenue=TQ_E=T(225-\dfrac{3}{4}T)=225T-\dfrac{3}{4}T^2.$$

(d) We can find the deadweight loss as follows:

$$DWL=\dfrac{1}{2}\cdot Tax\cdot(Q_E-Q_{Enew}),$$ $$DWL=\dfrac{1}{2}\cdot T\cdot(300-225+\dfrac{3}{4}T)=\dfrac{1}{2}\cdot T\cdot(75+\dfrac{3}{4}T),$$ $$DWL=37.5T+0.375T^2.$$

(e) It is a bad policy, because in this case the tax revenue is decreasing. A better policy would be to reduce the tax to $150. This results in reducing the DWL.