ISI 2015 PEB Sample Question 4

[Zen]

[onionknight]

a.) I think your answer for this part is correct.

b.)I think you've misinterpreted this part. When you look at the best response function of the two firms, there is a chance that the equilibrium itself might have one firm producing zero and the other producing a non zero quantity. You need to state the condition for which such a situation can be avoided altogether. I've explained it in the picture on this link: http://s13.postimg.org/nw3r9z3br/20150405_173747_1.jpg

[Zen]

For (b). : What I get is this: The reaction curves need to intersect at a point with positive coordinates. This cannot happen in 2 cases: Firm 1's whole reaction curve lies below Firm 2's and vice-versa. Those are exactly the conditions I wrote. I am sorry if I am doing some mistake, please point it out.

What does the question ask for part (c)? Can you help me here.

[onionknight]

Please have a look at the image I've attached. I think there is a possibility of nash equilibria occuring at one or both of the axes. In such cases, one of the firms wouldn't produce anything. So we need to avoid all these cases.

About c, I'm not entirely sure either so I'm afraid I can't help you

[onionknight]

In the standard case, the two firms produce homogeneous and indistinguishable goods. Which would mean the inverse demand functions should be the same. A1=A2, b11=b21 and b12=b22. Not sure about this one.

[Vidit Chaudhary]

What is Standard Cournot Eq? Does that mean that response curves will become symmetric.
forgive me if it sounds stupid( I am an engineer).

[onionknight]

Haha,  I'm also an engineer and I'm not entirely sure about this part either. I actually read this on a webpage: http://en.wikipedia.org/wiki/Cournot_competition . It says that the goods must be indistinguishable unlike what's given in the question.

[Zen]

Please help me in this: http://discussion-forum.2150183.n2.nabble.com/ISI-2015-PEB-Sample-Question-5-td7596064.html

[L]

I think your graphs are not correct. they should not be discontinuous.
I think the best to specify the condition is that second reaction curve should not be steeper than the first's otherwise it would lead to unstable equilibrium, i.e., |2 b11 /b12 |>|b21/ 2 b22|.

[onionknight]

Yes and thanks for pointing that. However, the condition for attaining equilibrium with either firms producing positive quantities would still remain the same. If you look at the graph at the bottom, the only error is that the discontinuity, which shouldn't ideally be there and at Q2=A1/B12, the graph of Q1 should touch the Y axis (and likewise for Q2, it should touch the x axis at Q1=A1/2B11) . If by the "second curve", you mean the best response curve of firm 2, then as illustrated in the graph on the top (even after making the correction you pointed out), the second curve could be steeper than the first and there could still be a nash equilibrium on one of the axis(which would mean one firm produces 0)

[onionknight]

So the required condition can be derived by imposing a condition wherein the best response function of firm 1 touches the y axis(i.e. A1/b12) at a point above the point at which the best response of firm 2 starts from the Y-axis (i.e. A2/2b22) and a similar condition for the two points on the X-axis.

This would give:
1.) A1/B12 > A2/2(B22)
2.) A2/B21> A1/2(B11)

[L]

Yes second reaction curve means the reaction curve for the second firm. If its slope (absolute) is higher than that of first firm's reaction curve, that would mean that the curves don't intersect in the first quadrant.

I didn't get this point of yours :

there could still be a nash equilibrium on one of the axis(which would mean one firm produces 0)

How can you say that equilibrium is on the axis? And how it is an equilibrium at all?

[onionknight]

Equilibria exist wherever the best response functions of the two firms intersect and they do so on the axes if the aforementioned conditions are not satisfied.

[onionknight]

In the graph on the top (when drawn with the corrections you mentioned), there will be 3 equilibria points, one on either axes and one at the intersection point in the 1st quadrant.

[L]

I searched more about this and I find you are correct.
I also found that the graphs will have bumps, as you have shown, when average cost function is U shaped.
Thanks to you I learned something new.

[onionknight]

Happy to help :) !

[kk]

can we do something like this..part b) is like stakelberg equilibrium..where reaction curve of one is tangent to the isoprofit curve of another..and part c)that is cournot..i thnk we can get that by equating the reaction curves..as its is a stable equilibrium

[saumya singha]

hey, you have solved 4 , you must have looked at question 3 , please do share the answer incase you have solved it. I am stuck in the middle of it,cant seem to get the final answer