Question

Two firms compete by choosing price. Their demand functions are

Q1 = 20 - P1 + P2

and

Q2 = 20 + P1 - P2

where P1 and P2 are the prices charged by each firm, respectively, and Q1 and Q2 are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero.

1. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.)
2. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be?
3. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

Short answer

1. Each firm will produce 20 units at $20. The profit for each firm will be $400.
2. Firm 1 will charge $30 and sell 15 units; firm 2 will charge $25 and sell 25 units. The profit from firm 1 will be $450, and for firm 2 will be $625.
3. Option (iii) will be the best choice as it gives a higher profit.

Step-by-step solution

Explanation for part (a)

The Nash equilibrium will be obtained at the interaction of both the firm’s reaction curves.

The reaction curve of firm 1 is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1={\mathrm{P}}_1{\mathrm{Q}}_1-{\mathrm{C}}_1 \\ =20{\mathrm{P}}_1-{\mathrm{P}}_1^2-{\mathrm{P}}_1{\mathrm{P}}_2-0 \\ \frac{{\mathrm{d\pi }}_1}{{\mathrm{dP}}_1}=20-2{\mathrm{P}}_1+{\mathrm{P}}_2=0 \\ 20-2{\mathrm{P}}_1+{\mathrm{P}}_2=0 \\ 2{\mathrm{P}}_1=20+{\mathrm{P}}_2 \\ {\mathrm{P}}_1=10+0.5{\mathrm{P}}_2 \end{gathered}$$

The reaction curve of firm 2 is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_2={\mathrm{P}}_2{\mathrm{Q}}_2-{\mathrm{C}}_2 \\ =20{\mathrm{P}}_2-{\mathrm{P}}_2^2+{\mathrm{P}}_1{\mathrm{P}}_2-0 \\ \frac{{\mathrm{d\pi }}_2}{{\mathrm{dP}}_2}=20-2{\mathrm{P}}_2+{\mathrm{P}}_1=0 \\ 20-2{\mathrm{P}}_2+{\mathrm{P}}_1=0 \\ 2{\mathrm{P}}_2=20+{\mathrm{P}}_1 \\ {\mathrm{P}}_2=10+0.5{\mathrm{P}}_1 \end{gathered}$$

From both the reaction curve,

$$\begin{gathered} {\mathrm{P}}_1=10+0.5\left(10+0.5{\mathrm{P}}_1\right) \\ =10+5+0.25{\mathrm{P}}_1 \\ 0.75{\mathrm{P}}_1=15 \\ {\mathrm{P}}_1=\$20 \\ {\mathrm{P}}_2=10+0.5\left(20\right) \\ =10+10 \\ =\$20 \end{gathered}$$

Each firm will charge $20.

The output of the firm is calculated below:

$$\begin{gathered} Q_1=20-20+20=20 \\ {Q}_2=20+20-20=20 \end{gathered}$$

Each firm will produce 20 units.

The profit of each firm is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1=\left(20\times 20\right)-0=\$400 \\ {\mathrm{\pi }}_2=\left(20\times 20\right)-0=\$400 \end{gathered}$$

Each firm profit will be $400.

Explanation for part (b)

Firm 1 is the leading firm and sets the price; thus, it takes the reaction curve of firm 2 into account while selecting the price. Therefore, firm 1 price will be:

$$\begin{gathered} {\mathrm{\pi }}_1=20{\mathrm{P}}_1-{\mathrm{P}}_1^2+{\mathrm{P}}_1\left(10+0.5{\mathrm{P}}_1\right)-0 \\ =30{\mathrm{P}}_1-0.5{\mathrm{P}}_1^2 \\ \frac{{\mathrm{d\pi }}_1}{{\mathrm{dP}}_1}=30-{\mathrm{P}}_1=0 \\ {\mathrm{P}}_1=\$30 \end{gathered}$$

Firm 2 price will be:

$P_2=10+0.5\left(30\right)=\$25$

The price charged by firm 1 will be $30, and by firm 2 will be $25.

The quantity of both firms is calculated below:

$$\begin{gathered} {\text{Q}}_{\text{1}}\text{= 20 - 30 + 25} \\ \text{= 15} \\ {\text{Q}}_{\text{2}}\text{= 20 + 30 - 25} \\ \text{= 25} \end{gathered}$$

Firm 1 will produce 15 units, and firm 2 will be 25 units.

The profit of both the firm will be:

$$\begin{gathered} {\mathrm{\pi }}_1=\left(30\times 15\right)-0=\$450 \\ {\mathrm{\pi }}_2=\left(25\times 25\right)-0=\$625 \end{gathered}$$

The profit of firm 1 will be $450, and firm 2 will be $625.

Explanation for part (c)

The nash equilibrium profit will be $400. The profit of the leader firm will be $450, and for the follower firm, it will be $625. Thus, option (iii) will be preferred as it gives higher profit.