Answer to Question #120773 in Microeconomics for christopher

Question

A monopolist faces two totally separated markets with inverse demand p=100 – qA and p=160−2qB respectively. The monopolist has no fixed costs and a marginal cost given by mc= 2 /3 q Find the profit maximizing total output and how much of it that is sold on market A and market B respectively if the monopoly uses third degree price discrimination.Calculate the price elasticity of demand in each market and explain the intuition behind the relationship between the prices and elasticities in these two separate markets.

Answer 1

A monopolist faces two totally separated markets with inverse demand p=100 – qA and p=160−2qB respectively. The monopolist has no fixed costs and a marginal cost given by mc= 2 /3q

Find the profit maximizing total output and how much of it that is sold on market A and market B respectively if the monopoly uses third degree price discrimination.

For the first market, the marginal revenue is:

$$MR_A = 100 - 2q_A$$

The marginal cost is:

$$MC_A = \dfrac{2}{3}q_A$$

Setting MR_A = MR_A, we get:

$$100 - 2q_A = \dfrac{2}{3}q_A\\[0.3cm] \dfrac{8}{3}q_A = 100\\[0.3cm] \color{red}{q_A^* = 37.5}$$

The price in this market is:

$$P = 100 - q_A\\[0.3cm] \color{blue}{P^* = \$62.5}$$

For the second market, the maginal revenue is:

$$MR_B = 160 - 4q_B$$

And the marginal cost is:

$$MC_B = \dfrac{2}{3}q_B$$

Setting MR = MC and solving for q:

$$160 - 4q_B = \dfrac{2}{3}q_B\\[0.3cm] \dfrac{14}{3}q_B = 160\\[0.3cm] \color{red}{q_B^* = 34.3}$$

The price on this market is:

$$P = 160 - 2(34.3)\\[0.3cm] \color{blue}{P^* = \$91.4}$$

Calculate the price elasticity of demand in each market and explain the intuition behind the relationship between the prices and elasticities in these two separate markets.

In the first market:

$$MC_A = P_A\left(1 - \dfrac{1}{e_A}\right)\\[0.4cm] MC_A = \dfrac{2}{3}(37.5) =\$25 \\[0.4cm] P_A = \$62.5$$

Therefore:

$$25= 62.5\left(1 - \dfrac{1}{e_A}\right)\\[0.3cm]$$

Solving for the elasticity, we get:

$$\color{red}{e_A = 1.67}$$

In the second market, we have:

$$MC_B = P_B\left(1 - \dfrac{1}{e_B}\right)\\[0.4cm] MC_B = \dfrac{2}{3}(37.5) =\$22.9\\[0.4cm] P_B = \$91.4$$

Therefore:

$$22.9= 91.4\left(1 - \dfrac{1}{e_B}\right)$$

Solving for the elasticity, we get:

$$\color{red}{e_B = 1.33}$$