Question 3.12: The demand and supply functions for a good are given as Dema......
The demand and supply functions for a good are given as
Demand function: P_d = 100 − 0.5Q_d (3.17)
Supply function: P_s = 10 + 0.5Q_s (3.18)
(a) Calculate the equilibrium price and quantity.
(b) Assume that the government imposes a fixed tax of £6 per unit sold.
(i) Write down the equation of the supply function, adjusted for tax.
(ii) Find the new equilibrium price and quantity algebraically and graphically.
(iii) Outline the distribution of the tax, that is, calculate the tax paid by the consumer and the producer.
(a) The equilibrium quantity and price are 90 units and £55, respectively.
Remember
The equilibrium price of £55 (with no taxes) means that the price the consumer pays is equal to the price that the producer receives.
(b) The tax of £6 per unit sold means that the effective price received by the producer is (P_s – 6). The equation of the supply function adjusted for tax is
P_s − 6 = 10 + 0.5Q
P_s = 16 + 0.5Q (3.19)
The supply function is translated vertically upwards by 6 units (with a corresponding horizontal leftward shift). This is illustrated in Figure 3.8 as a line parallel to the original supply function.
Remember
(i) Translations, Chapter 2.
(ii) The new equilibrium price and quantity are calculated by equating the original demand function, equation (3.17), and the supply function adjusted for tax, equation (3.19):
P_d = P_s
100 − 0.5Q = 16 + 0.5Q equating equations (3.17) and (3.19)
Q = 84
Substitute the new equilibrium quantity, Q = 84, into either equation (3.17) or equation (3.19) and solve for the new equilibrium price:
P = 100 − 0.5(84) substituting Q = 84 into equation (3.17)
P = 58
The point (84, 58) is shown as point E_1 in Figure 3.8.
(iii) The consumer always pays the equilibrium price, therefore the consumer pays £58, an increase of £3 on the original equilibrium price with no tax, which was £55. This means that the consumer pays 50% of the tax. The producer receives the new equilibrium price, minus the tax, so the producer receives £58 – £6 = £52, a reduction of £3 on the original equilibrium price of £55. This also means that the producer pays 50% of the tax.
In this example, the tax is evenly distributed between the consumer and producer.
The reason for the 50:50 distribution is due to the fact that the slope of the demand function is equal to the slope of the supply function (ignoring signs). This suggests that changes in the slope of either the demand or supply functions will alter this distribution.
