Question

A monopolist is deciding how to allocate output between two geographically separated markets (East Coast and Midwest). Demand and marginal revenue for the two markets are

P₁ = 15 – Q₁ MR₁ = 15 - 2Q₁

P₂ = 25 - 2Q₂ MR₂ = 25 - 4Q₂

The monopolist’s total cost is C = 5 + 3(Q₁ + Q₂). What are price, output, profits, marginal revenues, and deadweight loss (i) if the monopolist can price discriminate? (ii) if the law prohibits charging different prices in the two regions?

Short answer

1. The output in market 1 will be 6 units at $9, and in market 2 will be 5.5 units at $14. Marginal revenue in market 1 will be $3, and in-market 2 will be $3. The total profit will be $91.50, and the deadweight loss will be $48.25.
2. The output in market 1 will be 4.33 units, and inmarket 2 will be 7.17 units at $10.67. Marginal revenue in market 1 will be $6.34, and in market 2 will be -$3.68. The total profit will be $82.4, and the deadweight loss will be $44.10.

Step-by-step solution

Explanation for part (a)

The optimum level for each market will be when the marginal revenue is equal to Marginal cost.

The price and quantity in the East Coast (Q₁) are calculated below:

$$\begin{gathered} {\text{MR}}_{\text{1}}{\text{= 15 - 2Q}}_{\text{1}} \\ \text{MC = 3} \\ {\text{MR}}_{\text{1}}\text{= MC} \\ {\text{15 - 2Q}}_{\text{1}}\text{= 3} \\ {\text{2Q}}_{\text{1}}\text{= 12} \\ {\text{Q}}_{\text{1}}\text{= 6} \\ \text{P = 15 - 6} \\ \text{= \$ 9} \end{gathered}$$

In the East Coast, the quantity will be 6 units at $9.

The price and quantity in the Midwest (Q₂) are calculated below:

$$\begin{gathered} {\text{MR}}_{\text{2}}{\text{= 25 - 4Q}}_{\text{2}} \\ \text{MC = 3} \\ {\text{MR}}_{\text{2}}\text{= MC} \\ {\text{25 - 4Q}}_{\text{2}}\text{= 3} \\ {\text{4Q}}_{\text{2}}\text{= 22} \\ {\text{Q}}_{\text{2}}\text{= 5.5} \\ \text{P = 25 - 2}\left(\text{5.5}\right) \\ \text{= 25 - 11} \\ \text{= \$ 14} \end{gathered}$$

In the Midwest, the quantity will be 5.5 units at $14.

The marginal revenue in both the market is calculated below:

$$\begin{gathered} \text{M}{{\text{R}}_{\text{1}}}_{{\text{Q}}_{\text{1}}\text{= 6}}\text{= 15 - 2}\left(\text{6}\right) \\ \text{= 15 - 12} \\ \text{= \$ 3} \\ \text{M}{{\text{R}}_{\text{2}}}_{{\text{Q}}_{\text{2}}\text{= 5.5}}\text{= 25 - 4}\left(\text{5.5}\right) \\ \text{= 25 - 22} \\ \text{= \$ 3} \end{gathered}$$

The marginal revenue in both markets will be $3.

The total profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=\left(9\times 6\right)+\left(14\times 5.5\right)-\left[5+3\left(11.5\right)\right] \\ =54+77-5-34.5 \\ =\$91.5 \end{gathered}$$

The total profit will be $91.5.

The deadweight loss will be DWL = 0.5 (Q_C-Q_M)(P_M- P_C). The Q_C and P_C represent quantity and price of competitive market; Q_M and P_M represent quantity and price of monopoly market. In the competitive market, the P_C = MC is optimal; thus, the price will be $3 in a competitive market, and the quantity at this price in market 1 will be 12 units, and in market 2 will be 11 units.

The total deadweight loss is calculated below:

role="math" localid="1644382519200" $$\begin{gathered} DW{L}_1=0.5\left(12-6\right)\left(9-3\right) \\ =0.5\times 6\times 6 \\ =\$18 \\ DW{L}_2=0.5\left(11-5.5\right)\left(14-3\right) \\ =0.5\times 5.5\times 11 \\ =\$30.25 \\ TotalDWL=18+30.25 \\ =\$48.25 \end{gathered}$$

The total deadweight loss will be $48.25.

Explanation for part (b)

The total demand curve will be:

$$\begin{gathered} {\text{Q}}_{\text{1}}{\text{= 15 -P}}_{\text{1}} \\ {\text{Q}}_{\text{2}}{\text{= 12.5 - 0.5P}}_{\text{2}} \\ {\text{Q =Q}}_{\text{1}}{\text{+Q}}_{\text{2}} \\ {\text{= 15 -P}}_{\text{1}}{\text{+ 12.5 - 0.5P}}_{\text{2}} \\ \text{Q = 27.5 - 1.5P} \\ \text{P = 18.33 - 0.67Q} \end{gathered}$$

The quantity and price at the optimum level are calculated below:

$$\begin{gathered} {\text{TR = 18.33Q - 0.67Q}}^{\text{2}} \\ \text{MR = 18.33 - 1.33Q} \\ \text{MC = 3} \\ \text{MR = MC} \\ \text{18.33 - 1.33Q = 3} \\ \text{1.33Q = 15.33} \\ \text{Q = 11.5} \\ \text{P = 18.33 - 0.67}\left(\text{11.5}\right) \\ \text{= 18.33 - 7.7} \\ \text{= \$ 10.6} \end{gathered}$$

At $10.6, the quantity in each market will be:

$$\begin{gathered} {\text{Q}}_{\text{1}}\text{= 15 - 10.6} \\ \text{= 4.4} \\ {\text{Q}}_{\text{2}}\text{= 12.5 - 0.5}\left(\text{10.6}\right) \\ \text{= 12.5 - 5.3} \\ \text{= 7.2} \end{gathered}$$

Output in market 1 will be 4.4 units, and in market 2 will be 7.2 units.

The total profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=\left(10.6\times 11.5\right)-5-3\left(11.5\right) \\ =121.9-5-34.5 \\ =\$82.4 \end{gathered}$$

The total profit will be $82.4.

The total deadweight loss is calculated below:

$$\begin{gathered} DW{L}_1=0.5\left(10.6-3\right)\left(12-4.3\right) \\ =0.5\times 7.6\times 7.7 \\ =\$29.26 \\ DW{L}_2=0.5\left(10.6-3\right)\left(11-7.2\right) \\ =0.5\times 7.6\times 3.8 \\ =\$14.44 \\ TotalDWL=29.26+14.44 \\ =\$43.7 \end{gathered}$$

The total deadweight loss will be $43.7.