Stackelberg Model

A duopoly is a market situation where there are two sellers, for each set of actions and reaction of the duopolist, there corresponds a solution. Thus there is no unique solution for duopoly equilibrium. The solution depends on the actions of one duopolist and the counter of another duopolist.^duopoly-definition

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Tldr

In a Cournot Model situation, if one of them turns out to be a leader, to independently make the first move ⇒ Stackelberg Model

Let ${\Pi }_1={\Pi }_1(q_1,q_2)$, ${\Pi }_2={\Pi }_2(q_1,q_2)$ be the profit of the duopolist

Let $q_1={\Psi }_1(q_2)$ be the reaction function of $I$

$q_2={\Psi }_2(q_1)$ be the reaction function of $II$

Suppose $I$ wants to be leader then, he assumes that $II$ will be following leadership of $I$.

$\therefore$ $I$’s $\Pi$ function will be given

${\Pi }_1={\Pi }_1(q_1,{\Psi }_2(q_1))$ which is a function of $q_1$ only

$I$ will maximize this w.r.t. $q_1$ and get this leadership profit and leadership output. Similarly, $II$ will compete his leadership output and profit by assuming that $II$ is following through leadership function of $I$.

Suppose $I$ wants to be follower then he will accept $II$’s leadership output and his followership profit.

Similarly $I$ will compete with his followership profit.

By comparing the leadership profit and followership profit, each duopolist will decide the role to be a leader or to be a follower, depending on whichever is advantageous.

There are four possibilities given

1. $I$ → L, $II$ → F ✅
2. $I$ → F, $II$ → L ✅
3. $I$ → L, $II$ → L ❌
4. $I$ → F, $II$ → F ❌

From the four possibilities, we can observe that there cannot be Static Equilibrium in (3) and (4) cases, there can be equilibrium only if $I$ accepts the leadership of the other as in case of (1) and (2)

Example

$P=100-0.5[q_1+q_2]$ be the demand function

Let $c_1=5q_1$ ;$c_2=0.5q_2^2$ be the cost function

Reaction fn. are $q_1=95-0.5q_2$ for $I$

$q_2=50-0.25q_1$ for $II$

$I$’s profit is ${\Pi }_1=100q_1-0.5q_1^2-0.5q_1{q}_2-5q_1$

If $I$ wants to be leader then $I$’s $\Pi$ will be given by

$$\begin{array}{rl}{\Pi }_1& =100q_1-0.5q_1^2-0.5q_1(0.5-0.25q_1)-5q_1\\ & =70q_1-0.375q_1^2\end{array}$$

For maximizing ${\Pi }_1$, $\frac{\partial {\pi }_1}{\partial q_1}=70-0.75q_1=0$

$$\therefore q_1=\frac{70}{0.75}=93\frac{1}{3}$$

Thus, $q_1^L=93\frac{1}{3}$

$$\therefore {\Pi }_1^L=70\left(93\frac{1}{3}\right)-0.375\left(93\frac{1}{3}\right)=3266\frac{2}{3}$$

Similarly we get,

$$\begin{array}{rl}{q}_2^L& =35\\ {\Pi }_2^L& =918.75\\ q_1^L& =93\frac{1}{3}\\ {\Pi }_1^L& =3266\frac{2}{3}\end{array}$$

Suppose $I$ is follower then he accepts $II$’s leadership output $35$ and hence his followership output will be¹ $q_1^F=95-0.5(35)=77.5$. If this is substituted in $I$’s profit function then ${\Pi }_1^F=70(77.5)-0.375(77.5{)}^2=3003.125$

q_1 = 95 - 0.5*35
q_2 = 35
Profit_1 = 100q_1 - 0.5*q_1^2 - 0.5*q_1*q_2 - 5q_1

Now, if $II$ is a follower then he accepts $I$’s leadership output $93\frac{1}{3}$ and hence his followership output will be ¹ $q_2^F=50-0.25(93\frac{1}{3})=26\frac{2}{3}$.

The profit of $II$ is given by

$${\Pi }_2=100q_2-0.5q_1{q}_2-q_2^2$$

So, the followership profit of $II$ is given by ${\Pi }_2^F=711.11$

q_1 = 93+1/3
q_2 = 50 - 0.25*93.333
Profit_2 = 100*q_2 - 0.5*q_1*q_2 - q_2^2

Footnotes

1. By substituting in the reaction function of the other firm ↩ ↩²