Home > Duopoly and strategic behaviour, Micro concepts > Stackelberg competition

## Stackelberg competition

September 29, 2011

In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: $D^r (P) = D (P) - S^o (P)$.

In the simple model I’m using for these examples, the market demand is Q = 500 – P and the firm (both firms in this duopoly case) have no fixed costs and a constant marginal cost of 150. So if the firms are firm A and firm B, then the residual demand curves for each firm are: $Q_A = 500 - P - Q_B$ and $Q_B = 500 - P - Q_A$.

This means the inverse residual demand curves for each firm are $P= 500 - Q_A - Q_B$.

We established that the best response curve for firm A is $Q_A = 175 - 0.5 Q_B$ and the best response curve for firm B is $Q_B = 175 - 0.5 Q_A$.

When the firms had no knowledge of each other’s output making decision, and made their decisions simultaneously, they reached a Cournot equilibrium where each firm produced 116.667.

The total market output was 233.333, and the market price was 266.667. Each firm made profits of 13611.11, and had a Lerner index of 0.438.

Now we can consider the situation where the firms did not make their decisions simultaneously, but one firm effectively ‘moved first’ – the other firm knew what output the first firm was producing, when it came to make its own decision. This gives the first firm an advantage, because now if the second firm is going to respond by choosing the appropriate point on its best response curve, it has no choice but to choose the output that corresponds to the output chosen by the first firm.

This form of competition is called Stackelberg competition and the firm moving first is called the Stackelberg leader. Again we assume that the firms are producing identical goods and are competing with each other based on their quantity decisions. We will continue to use the case where the firms are identical to allow us to compare this more easily with the Cournot example, so both firm A and firm B have marginal costs of 150. But now we will assume firm A is the first mover, so firm B is entering the market but has to respond to the output level firm A is producing.

If A knows B is entering then it can take advantage of being first mover by choosing to produced the Stackelberg leader output. To do this we find a profit function for firm A (the leader firm) in terms of the quantity it produces and the quantity firm B produces, and we optimise it.

The inverse market demand function is $P= 500 - Q$ and as $Q = Q_A + Q_B$ we can write it $P = 500 - (Q_A + Q_B)$.

Because firm A, the Stackelberg leader, chooses its output first, then firm B has to respond by choosing its output based on the decision of A. The best response function for firm B is $Q_B = 175 - 0.5 Q_A$ so we know this is how B’s output will relate to A’s output. Subbing this into the inverse market demand function gives $P = 500 - (Q_A + 175 - 0.5 Q_A) \Rightarrow P = 325 - 0.5Q_A$.

Now we think about the profit function for firm A. This will be $\pi_A = TR_A - TC_A = PQ_A - (MC_A)Q_A = (P-MC_A)Q_A$. As we have an expression for the market price, and we know the marginal cost is 150, the profit function for firm A becomes $\pi_A = (325 - 0.5Q_A - 150)Q_A \Rightarrow \pi_A = 175Q_A-0.5{Q_A}^2$.

Now to optimise the profits of firm A we differentiate the function with respect to the output of firm A and set that equal to 0, $\frac{d\pi_A}{dQ_A} = 175-Q_A$ so when $\frac{d\pi_A}{dQ_A} = 0, Q_A = 175$. This means the Stackelberg leader output for firm A is 175.

If A produces 175, then B will produce $Q_B = 175 - 0.5 (175) = 87.5$.

So with A producing 175 and B producing 87.5, market output is 262.5 and market price is 500-262.5=237.5. Firm A will get profits of 237.5(175) – 150(175) = 15312.5, and firm B will have profits of 237.5(87.5) – 150(87.5) = 7656.25. Both firms have a Lerner index of 0.368.

Again we can look at the case of what would happen if the firms were not identical and had different marginal costs. First consider the case where the new entrant, firm B, was more efficient than the incumbent firm, firm A, and whereas firm A had a marginal cost of 150, firm B had a marginal cost of 120. On the Cournot section I found the best response curve for firm B here to be $Q_B = 190 - 0.5 Q_A$.

Subbing this into the inverse market demand function gives $P = 500 - (Q_A + 190 - 0.5 Q_A) \Rightarrow P = 310 - 0.5Q_A$.

The profit function for firm A will be $\pi_A = (P-MC_A)Q_A = (310 - 0.5Q_A - 150)Q_A =160 Q_A- 0.5{Q_A}^2$. Differentiating this we get $\frac{d\pi_A}{dQ_A} = 160-Q_A$ so when $\frac{d\pi_A}{dQ_A} = 0, Q_A = 160$. The Stackelberg leader output for firm A this time is 160. If A produces 160, then B will produce $Q_B = 190 - 0.5 (160) = 110$. The total market output will be 270 so the market price will be 230.

Alternatively we could consider the case where it was firm A that was more efficient than firm B as well as being the Stackelberg leader. What if A had the marginal cost of 120 and B had the marginal cost of 150. This time firm B’s best response curve would be $Q_B = 175 - 0.5 Q_A$ as before, because the best response curve depends on its own marginal cost. Subbing this into the inverse market demand function gives $P = 500 - (Q_A + 175 - 0.5 Q_A) \Rightarrow P = 325 - 0.5Q_A$.

Again the profit function for firm A will be $\pi_A = (P-MC_A)Q_A$, but this time the marginal cost for A is 120, so the profit function for firm A becomes $\pi_A = (325 - 0.5Q_A - 120)Q_A \Rightarrow \pi_A = 205Q_A-0.5{Q_A}^2$.

Differentiating this we get $\frac{d\pi_A}{dQ_A} = 205-Q_A$ so when $\frac{d\pi_A}{dQ_A} = 0, Q_A = 205$. The Stackelberg leader output for firm A this time is 205. If A produces 160, then B will produce $Q_B = 175 - 0.5 (205) = 72.5$. The total market output will be 277.5 so the market price will be 222.5.

The Stackelberg leader is able to be more aggressive (produce a greater share of the market) when it has a lower marginal cost.