How-to: Solve the Cournot duopoly model (part 2)

In part 1 the model was setup we will now continue with solving it.

Solving the model

We will solve the model in three steps.

Step 1: finding the profit function $\pi$

As the companies want to maximize profits we first need to find the profit function. As you probably know profit for a company is their Revenue minus their Cost. With revenue being the product of quantity and price, and cost being given by the cost function, or for company 1:

\[ \pi_1(q_1,q_2) = q_1 \left( P(Q) - c_1 \right) \]

filling in $P(Q)$ and $Q$:

\[ \pi_1(q_1,q_2) = q_1 \left(\alpha -q_1 - q_2  - c_1 \right) \]

Doing the same for company 2 would result in:

\[ \pi_2(q_1,q_2) = q_2 \left(\alpha -q_1 - q_2  - c_2 \right) \]

Note: we can already see the strategic interaction, all else constant (ceteris paribus) when company 2 will start to produce more it will lower the profit of company 1 and vice versa.

Step 2: finding the best response function

The next step is to determine what the best amount of quantity to produce is, given the quantity of their rival. Or as it is called in game theory, what is the best response for company 1 and 2? The most simple way to do this is by optimizing the profit function with respect to $q_1$, i.e. setting the derivative of the profit function with respect to $q_1$ equal to zero and solve for $q_1$. This will give us the quantity for which company 1 maximized his profits, given everything else. Just as the model assumes. We can distinguish two situations:

     1. $q_2 \le \alpha - c_1$, in this situation there is still demand of consumers. 

So we have profit function: $\pi_1(q_1,q_2) = q_1 \left(\alpha -q_1 - q_2  - c_1 \right) $, working out the brackets:

\[\pi_1(q_1,q_2) = q_1\alpha -q_i^2 -q_1q_2 - q_1c_1 \]

taking the derivative to $q_1$:

\[\frac{d\pi_1}{dq_1} = \alpha - 2q_1 -q_2-c_1 = 0\]

isolating $q_1$

\[2q_1 = \alpha - c_1 - q_2 \]

\[q_1 = \frac{1}{2} \left( \alpha - c_1 - q_2 \right) \]

for company 2:

\[q_2 = \frac{1}{2} \left( \alpha - c_2 - q_1 \right) \]

     2. $q_2 > \alpha - c_1$

Looking at the profit function above you can if this is the case the optimal $q_1$ would be negative. Hence in this situation it is best for a company to produce nothing.

We can summarize our finding in a best response function.

\[R_1(q_2) \left\{\begin{array}{11} \frac{1}{2} \left( \alpha - c_1 - q_2 \right) \qquad \mbox { if $q_2 \le \alpha -c_1$}\\ 0 \qquad \qquad \qquad \qquad \qquad \quad \mbox {if $q_2 > \alpha - c_1$} \end{array} \right. \]

for company 2:

\[R_2(q_1) \left\{\begin{array}{11} \frac{1}{2} \left( \alpha - c_2 - q_1 \right) \qquad \mbox { if $q_1 \le \alpha -c_2$}\\ 0 \qquad \qquad \qquad \qquad \qquad \quad \mbox {if $q_1 > \alpha - c_2$} \end{array} \right. \]

Step 3: solving the model

In equilibrium the companies both will play their best response. Thus the outputs chosen in equilibrium $q_1^*$ and $q_2^*$ are equal to:

\[q_1^* = R_1(q_2^*)\]

\[q_2^* = R_2(q_1^*)\]

We can solve this by substituting $q_2^*$, which is equal to the best response function of company 2, into the best response function of company 1: $R_1(q_2^*)$. Hence $q_1^*$ is:

\[q_1^* = \frac{1}{2} \left(\alpha - c_1 - \frac{1}{2} \left(\alpha - c_2 -q_1 \right) \right) \]

working out the inner brackets (note $-- = +$)

\[q_1^* = \frac{1}{2} \left(\frac{1}{2}\alpha - c_1 + \frac{1}{2}c_2 + \frac{1}{2}q_1\right) \]

working out the remaining brackets

\[q_1^* = \frac{1}{4}\alpha -\frac{1}{2}c_1 + \frac{1}{4}c_2 + \frac{1}{4}q_1\]   

 isolation $q_1$

\[\frac{3}{4}q_1^* = \frac{1}{4}\alpha -\frac{1}{2}c_1 + \frac{1}{4}c_2 \]

\[q_1^* = \frac{1}{3} \left(\alpha - 2c_1+c_2 \right) \]

for company 2 this would result in:

\[q_2^* = \frac{1}{3} \left(\alpha - 2c_2+c_1 \right) \]

These two functions show what quantity is best chosen by the companies in equilibrium, this is a Nash equilibrium as nobody can improve his situation by deviating from this strategy. 

Substituting this into the other functions we can report also that, the total output of the market is $Q = \frac{2}{3}\alpha - \frac{1}{3}c_1 - \frac{1}{3}c_2$; market price is: $P(Q) = \frac{1}{3}\left( \alpha +c_1+c_2 \right)$; and the profit functions are give by:

$\pi_1(q_1,q_2) = \frac{1}{9} \left(\alpha -2c_1 +c_2 \right)^2$ and $\pi_2(q_1,q_2) = \frac{1}{9} \left(\alpha -2c_2 +c_1 \right)^2$  

Implications

the optimal amount to produce can change via three ways:

1. Change of $\alpha$.  An increase (decrease) in $\alpha$ means that consumer are willing to pay more (less) for the product. Hence the optimal output increase (decreases).
2. Change in $c$ of own company. If it increases (decreases) it will be more (less) costly to produce, hence the optimal output decreases (increases).
3. Change of $c$ of competitors company. If it increases (decreases) the optimal output of your competitor decreases (increases), hence yours increases (decreases). 

This follows also from the total output, demand and profit functions..

In part 3 we will generalize this model to include more than two firms, we will also compare the findings to a monopoly.