Question

This exercise is a continuation of Exercise 3. We return to two firms with the same constant average and marginal cost, AC = MC = 5, facing the market demand curve Q1 + Q2 = 53 - P. Now we will use the Stackelberg model to analyze what will happen if one of the firms makes its output decision before the other.

1. Suppose Firm 1 is the Stackelberg leader (i.e., makes its output decisions before Firm 2). Find the reaction curves that tell each firm how much to produce in terms of the output of its competitor.
2. How much will each firm produce, and what will its profit be?

Short answer

1. There cannot be any reaction curve for firm1 as it is leading the market. The reaction curve for firm 2 will be Q₂ = 24 – 1/2Q₁.
2. Output for firm 1 will be 24 units, and for firm 2 will be 12 units. The profit for firm 1 will be $288, and for firm 2 will be $144.

Step-by-step solution

Explanation for part (a)

The reaction curve of firm 2 will be $Q_2=24-\frac{1}{2}{Q}_1$. There is no reaction curve for firm 1 as it sets the output decision earlier than firm 2; thus, there is nothing to react to.

Explanation for part (b)

Firm 1 is the leading firm, and firm 2 is the follower; thus, the firm 2 reaction curve will put in the profit function of firm 1.

The output of firm 1, and firm 2 are calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1=\left(53-{\mathrm{Q}}_1-\left(24-\frac{{\mathrm{Q}}_1}{2}\right)\right){\mathrm{Q}}_1-5{\mathrm{Q}}_1 \\ \frac{{\mathrm{d\pi }}_1}{{\mathrm{dQ}}_1}=53-2{\mathrm{Q}}_1-24+{\mathrm{Q}}_1-5=0 \\ 24-{\mathrm{Q}}_1=0 \\ {\mathrm{Q}}_1=24 \\ {\mathrm{Q}}_2=24-\frac{24}{2} \\ =24-12 \\ =12 \end{gathered}$$

Firm 1 will produce 24 units, and firm 2 will produce 12 units.

The profit of firm 1 and firm 2 are calculated below:

role="math" localid="1644230814271" $$\begin{gathered} P=53-24-12 \\ =\$17 \\ {\mathrm{\pi }}_1=\left(17\times 24\right)-\left(5\times 24\right) \\ =408-120 \\ =\$288 \\ {\mathrm{\pi }}_1=\left(17\times 12\right)-\left(5\times 12\right) \\ =204-60 \\ =\$144 \end{gathered}$$

The profit of firm 1 will be $288, and for firm 2 will be $144.