# How to calculate the quantity of an incumbent that sets the profit of a challenger equal to 0?

I have the inverted market demand of $P=4Z-(q_1+q_2)$ and cost function of both firms is equal to $C=cq_i+f$ for $i=1,2$

So now, the existing firm wants to discourage a possible challenger to entry and sets an output that would make the profits of the challenger equal to 0.

My approach would be, to set up the profit function for the challenger:

Step 1: $(4Z-q_1-q_2)*q_2 - cq_2 = 0$ and then solve for $q_1$ in terms of $q_2$.

Step2: Then I can substitute $q_1$ in the profit function of firm 1 and take the derivative of the profit function with regard to $q_2$

Step3: The result of step 2 I would insert in step 1 in order to receive the quantity that firm one needs to produce in order to set the profit function of the challenger equal to zero.

Is this approach correct? I am a bit confused because of the strange algebra, but I could not come up with a different solution.

• I wanted to vote to close but retracted : your question is close to meeting the minimum standards for homework-like questions, but not quite yet. Please have a look at meta.economics.stackexchange.com/questions/1465/…. Apr 20 '16 at 15:40
• I totally understand your concern and I also do not want that stackexchange becomes a platform for homework questions, it is just that I saw the problem in a game theory textbook and while the verbal description seemed quite easy the proposed problem in the book gave me some headaches. I will be more careful in the future. Have a nice day/weekend. Apr 22 '16 at 9:20
• +1 for your comment, thanks for your understanding and efforts. I think the community here is ok with homework and self-study questions as long as they satisfy the standards for this kind of questions discussed in the above link. Apr 22 '16 at 14:16

Some steps are a bit unclear to me, but I would say that they are wrong. For example step 1 is wrong because the meaning of this equality is: I will set the q1 that makes the challenger do zero profits once (after) it enters, but the point of this problem is that we have to block it before it enters.

so here my version.

The first thing to do is to calculate the challenger best response function, which you obtain by taking the derivative to respect to q2 (challenger)with given q1(incumbent). You will then obtain the optimal q2 which you will substitute in the challenger profit function, and only now you choose the q1 by setting the challenger profit equal to zero.

intuition: once the challenger enters, he will face a price and a quantity given by the incumbent, by doing the derivatives, he will also choose a quantity that best help him maximize his profit. So we substitute the optimal q2 in his profit function

The incumbent knowing this, and knowing the challenger best response function, will choose a q1 that set the opponent profit function to zero.

Now you may wonder what is the difference with step 1 of your solution. Notice that in your solution the incumbent doesn't know what quantity the challenger will set ex-ante, but only ex-post, while in my analysis the incumbent knows that the challenger is rational and will set an optimal q2 which depends on the quantity q1, so in this case the incumbent knows ex-ante.

• Solving quantity-competition models necessitates finding best response functions. Profits of firm 1: (Price - Cost)*quantity $\to$ $(4Z-(q_1+q_2) - cq_1)*q_1$ The firm seeks to maximize this profit function. Take first order condition w.r.t. $q_1$ : FOC $q_1 \to$ $4Z-2q_1-q_2-c = 0$ Solving gives $q_1^{BR}(q_2) = \frac{4Z-q_2-c}{2}$ similarly: $q_2^{BR}(q_1) = \frac{4Z-q_1-c}{2}$