Question 3.22: (a) Show graphically that the equilibrium price and quantity......

(a) The table of points and the graphs of equations (i) and (ii) are shown in Figure 3.19. The equilibrium point for each pair is the same, Q = 27.5, P = 65.

(b) When a tax of 20 is imposed, the price the supplier receives is the (original price – tax); therefore, replace P in the supply functions by (P – 20).

(i) With the tax, the set of equations is now

P_d = 120 − 2Q,               stays the same
P_s − 20 = 10 + 2Q → P_s = 30 + 2Q

Solve for equilibrium at Q = 22.5, P = 75.

The consumer (who always pays the equilibrium price) pays (P_{e_{(tax)}} – P_e) = 75 – 65 = 10 more than before the tax. The producer receives (P_{e_{(tax)}} – tax) = 75 – 20 = 55. This is 10 units less than before the tax was imposed. So when the slopes of the demand and supply functions are equal the distribution of tax is 50:50.
A table of values is set up to plot this pair of graphs with the untaxed supply function. The range of Q-values is selected so that the graph focuses on the original and the new equilibrium points as shown in Figure 3.20

(ii) With the tax, the set of equations is now

P_d = 120 − 2Q
P_s − 20 = 37.5 + Q → P_s = 57.5 + Q

Solve for equilibrium at Q = 20.83, P = 78.33.

The consumer pays (P_{e_{(tax)}}  –  P_e) = 78.33 – 65 = 13.33 more than before the tax. The producer receives (P_{e_{(tax)}} – tax) = 78.33 – 20 = 58.33. This is 6.67 units less than before the tax was imposed. As expected, the function whose slope is greater in magnitude pays the greater share of the tax.
A table of values is set up to plot this pair of graphs with the untaxed supply function. The range of Q-values is selected so that the graph focuses on the original and the new equilibrium points as shown in Figure 3.21.

(c) Remember

The distribution of tax for linear functions is given as

\text{Consumer pays}\,\frac{|m_{\mathrm{d}}|}{|m_{\mathrm{d}}|+|m_{\mathrm{s}}|}\times\text{tax}     \text{Producer pays}\,{\frac{|m_{s}|}{|m_{s}|+|m_{d}|}}\times\text{tax}

See Appendix to chapter. The distribution of tax is graphically illustrated in Figures 3.20 and 3.21.