How-to: Solve the Cournot duopoly model (part 3)

In part 1 we had the model setup, in part 2 we have solved the model. Now in part 3 we will generalize the model.

Generalizing to N firms

The model can be generalised to a situation where there are n firms active in the market, i.e. the amount of companies goes from 1 to n. For simplicity we will now assume that the cost are identical between firms or:

\[\ C_i(q_i)  = cq_i \mbox { for each} \quad i  \]

Demand is still given by:

\[ P(Q) \left\{\begin{array}{11} \alpha - Q \qquad \mbox {if $Q \le \alpha$}\\ 0 \qquad \qquad \quad \mbox {if $Q > \alpha$} \end{array} \right. \]

with $Q = \sum q_i$ or $Q = nq_i$ and $Q_{-i} = Q-q_i$ or $Q = Q_{-i} +q_i$

I do not define $Q$ in so many ways just for fun. We will need this to easier solve the model.

The above says that $Q$ is the summations of all $q_i$'s, which is the same as multiplying the output of one firm with the total amount of firms. I also define $Q_{-i}$, which we need to solve the model. Although it might seem complicated its is actually quite simple. If we take for example $ n = 10$ and $i = 1$, it says that the output of nine firms $(Q_{-1})$ is equal to the output of ten firms $(Q)$ minus the output of one firm $(q_1)$.

Step 1: finding the profit function

For company $i$ the profit function is:

\[\pi_i(q_i,Q_{-i}) = q_i \left(P(Q) - c \right) \]

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

\[\pi_i(q_i,Q_{-i}) = q_i \left(\alpha - q_i-Q_{-i} - c \right) \]

Step 2: best response

As before we need to take the derivative with respect to $q_i$ and set it equal to zero:

\[\frac{d\pi_i}{dq_i} = \alpha - 2q_i - Q_{-i} -c = 0 \]

isolating $q_i$:

\[2q_i = \alpha - Q_{-i} -c\]

filling in $Q_{-i}$

\[2q_i = \alpha - (Q-q_i) -c\]

gives the best response:

\[q_i = \alpha - c - Q\]

Step 3: solving the model

From here on we can solve the model again. However we do this a little bit different than before. As the best response functions are identical for each company we could also sum up both the right and left side of the equation, or multiplying both sides with n (  $Q = \sum q_i$ or $Q = nq_i$ ). Thus,

\[nq_i = n\alpha - nc - nQ \]

isolating $Q$ (remember $Q = nq_i$)

\[Q+nQ = n\alpha - nc \]

rewriting gives:

\[(1+n)Q = n(\alpha - c) \]

\[Q = \frac{n}{1 + n} (\alpha - c)\]

it follows that:

\[q_i = \frac{1}{1 + n} (\alpha - c)\]

\[P(Q) = \frac{\alpha + nc}{ 1 + n }\]

\[\pi_i = \left( \frac{1}{1 + n } (\alpha - c) \right)^2 \]

Implications

If the total amount of firms increases ($n$), the optimal output of each individual company decreases, the total output of the market increases, resulting in a lower price, and lower profits for the companies. But consumers are better of because the price is lower!

If we take the limit of n to indifinite (n is equal to a really big number) we can conclude that:

\[P(Q) = c \quad \mbox {and}\quad \pi_i = 0\] which is basically the outcome under perfect competition.

Comparing to a monopolist

Having seen the extreme of many companies, what about if there is only one company active. The monopolist would simply maximize his profits:

\[\pi = Q (P(Q) - c)) \]

\[\pi = Q (\alpha -Q - c) \]

taking again the derivative to maximize profits:

\[\frac {d\pi}{dQ} = \alpha - 2Q - c \]

\[ Q= \frac{1}{2} (\alpha - c)\]

leading to:

\[ P = \frac {1}{2} (\alpha + c) \]

\[\pi = \frac{1}{4} (\alpha - c)^2 \]

(note: filling in $n=1$ in the generalized model gives the same outcome..)

If we compare this with the outcomes of the duopoly we see that under a monopoly consumers are even worse off. The total output of the market is lower, prices are higher and profits for the company is higher.

We can see from this example and the generalised model that firms have an incentive to collude. As if the firm would reduce the total output of the market, they would have higher profits. That’s why many governments have competition regulators keeping an eye on the markets.

Thus 

• Total output is greater with cournot duopoly than in a monopoly. However lower than under perfect competition 
• Prices are lower in duopoly than in monopoly, but again not as low as under perfect competition 
• If the amount of companies increases the outcome will become closer to the situation of perfect competition. 
• Firms will have an incentive to collude. 

So now you hopefully know how to solve a basic Cournot model and have an understanding of the implications of this model. The only thing you are missing is a numerical example, although you should be able to work out any numerical example for yourself now, and how to graphically find the solution. Maybe that will be subject of a post in the future, although I would not hold your breath.

References

• An introduction to Game Theory by Martin J. Osborne, on amazon .   
• Microeconomics and Behavior by Robert H. Frank, on amazon. This one is less technical than the first one
• Some lecture notes..