Question

Q4 – The demand and supply functions of a good are given by

  P=-2QD+48
P=1/2QS+23

Find the following:

(1)The equilibrium quantity and price (3 points)

(2)The equilibrium quantity and price if the government imposes a fixed tax of $5 on each good. (3 points)

(3)Write down the supply and demand equations after the tax incidence. (3 points)

Answer 1

(1) The demand curve is given as \(Q_{D}=24-0.5 P\), and the supply curve as \(Q_{s}=2 P-46\). The equilibrium price would be where the quantity demanded is equal to the quantity supplied, ie \(Q_{D}=Q_{S}\) or \(24-0.5 P=2 P-46\) or \(P^{*}=28 .\) The equilibrium quantity will be hence \(Q^{*}=2 P^{*}-46\) or \(Q^{*}=2 * 28-46\) or \(Q^{*}=10\)

(2) If the government imposes tax of \(\$ 5\) on each good, supposedly on the seller, then the equilibrium quantity would be where the the excess demand price is equal to the tax amount. The demand price is given as \(P_{D}=-2 Q+48\) and the supply price is given as \(P_{S}=0.5 Q+23\). The excess demand price would be \(P_{D}-P_{S}=-2 Q+48-(0.5 Q+23)\) or \(P_{D}-P_{S}=-2.5 Q+25 .\) The equilibrium quantity would be \(P_{D}-P_{S}=5\) or \(-2.5 Q+25=5\) or \(Q^{*}=8 .\) The equilibrium price would be the price payed by consumers, ie \(P_{D}^{*}=-2 Q^{*}+48\) or \(P_{D}^{*}=32\). The producers however, would receive \(\$ 27\) after paying the tax.

(3) After the tax is imposed on the sellers, the demand curve will be the same as \(Q_{D}=24-0.5 P\), but the supply curve would be changed to \(Q_{S}=2(P-5)-46\) (as sellers would discount the tax from the current price they would charge) or \(Q_{s}=2 P-56\).