4 Duopolies: Collusion, Cournot, Stackelberg, and Bertrand

This is a worked out problem in which I solve for firm quantity and market price in 4 different industry competitive structures: collusion, Cournot, Stackelberg, and Bertrand competition. This is appropriate for an Intermediate Microeconomics course without calculus, as well as for an Industrial Organization course.

[@mayaframe]

Matt Birch you single handedly are gonna be the only reason I pass my Intermediate Econ Class.

[@MattBirch]

Haha, glad to help!

[@ruiren]

@@MattBirch Nice video!!! Really helpful!!! I got 90/100 on my midterm. Could you please post a video about game theory? It will be on my final! Thanks a lot!

[@MattBirch]

@@ruiren Congrats on your midterm! This is all I have for Game Theory. I hope they help! kzbin.info/aero/PLWd1brOYtkZWFq22nZVxi3y_rOdEitAS9

[@ruiren]

@@MattBirch Thanks so much! You are my superhero!

[@MattBirch]

Heck yeah! Spread the word.

[@sabeernarula3735]

This is the most helpful video I have watched for these duopoly models - thank you so much!

[@MattBirch]

Glad to hear it. Spread the word! And good luck.

[@TheXozin]

Wow thank you! Exam in 3 days this cleared it all up, best explanation I've found on KZbin!

[@MattBirch]

Glad to hear it. Kick butt on that exam for me. (And spread the word to other students who may need help).

[@justinomartinez5447]

Great video. Currently I'm Industrial Organization and needed a refresher.

[@MattBirch]

Glad it is helpful. This stuff gets pretty crazy when you forget which is which!

[@jake______]

Great work explaining these four methods so succinctly. Just watched an MIT professor spend 20 minutes on Cournot with less clarity than you spending the same time on four models. Good stuff mate.

[@MattBirch]

Well I bet he captured some detail that I did not, but I thought this approach would be helpful too. Glad you liked it so much!

[@jake______]

@@MattBirch I mean it was a whole intro micro course so it's fair enough, but you knocked it outta the park either way.

[@MattBirch]

@@jake______ Thank you much! Feel free to spread the word to any students who may benefit. And also let me know if you need more content. No promises, but I will do what I can.

[@TY1998able]

Great, concise and clearly understandable explanations with proper links to core elements of each market version. thanks!

[@MattBirch]

Glad you enjoyed it!

[@Mo-ut4xb]

my final exam is in 20 minutes thank you so much!!

[@MattBirch]

Hope you did well. Good luck!

[@Summer-zb8rw]

this is an amazing video, awesome explanation!

[@MattBirch]

Glad it helps! Happy econ-ing!

[@gabrielasoler218]

Very helpful! Great explanation. 100% recommend

[@MattBirch]

Well thank you! If you know any students who could benefit, please recommend it :) And good luck!

[@dropgamer6524]

You explained everything very well. Great teacher imo if your looking into that profession

[@MattBirch]

Well Joshua that is good to hear! That is my profession, so thank you much!

[@yanruzhu6993]

You save my micro!! Thx

[@MattBirch]

Glad to hear it! Congrats!

[@RaymondLiswag]

Great video, thanks Matt!

[@MattBirch]

Thanks! And you're welcome!

[@muradpashazade6566]

Perfectly simple. Thanks a lot

[@MattBirch]

My pleasure!

[@rodreckdellainemindisi997]

wow I'm very grateful for your explanations. keep up the good work... thank you

[@MattBirch]

My pleasure! Good luck in micro! (and if you know any other students who need help, spread the word)

[@federicocampi7964]

Thanks dude! This video is insane !

[@MattBirch]

Haha glad to hear it!

[@misslola3112]

Thank you for all your videos! They have been helping all of us so so much. I have a question concerning the MR in Cournot (and also at the Collusion): Why does P=500-2Q1-2Q2 become 500-4Q1-2Q2...how do I know that with the MR, the 2 becomes a 4?

[@MattBirch]

First: I am glad that they help! Second, that is a very good question. This slope doubling is true if the demand curve is linear, and it comes from the calculus. Firm 1 will choose q1, and its total revenue =TR1=(P)(q1)=(500-2q1-2q2)q1=500q1-2q1^2-2q1q2. When you take the derivative of that, with respect to q1, you get MR=500-4q1-2q2. With linear demand curves, MR will always be the price equation but with double the slope of the q you are choosing.

[@Espiritu_Santu]

Perfect explanation. Cournot, forth row. You say - here I’m gonna double the the slope and you write down 4Q instead of 2 and continue. Why do you do that, why double, why not triple, you just move on. I wonder what we should think here…

[@MattBirch]

Great question. Short answer: it comes from the calculus. If the inverse demand curve is straight, MR has same intercept but double the slope.

[@thomokgetse9802]

Definitely useful💐💐💯

[@MattBirch]

Haha, glad to hear it!

[@shadreckkazhinga5871]

Amazing stuff! Thanks!

[@MattBirch]

You are quite welcome! Good luck.

[@avestaabdulrahman6549]

Very helpful, thank you!

[@humaek]

great video, thank you!

[@MattBirch]

My pleasure! And good luck.

[@teresap4193]

Thanks so much fo the video!! It’s been so useful for me 😊

[@MattBirch]

Well good! Glad to hear it.

[@veronikahassova6024]

Seriously! Thank you!

[@MattBirch]

Heck yes! I am glad it helps. (This one is usually the best)

[@johnchung5299]

Mr Birch, I have computed different values of the deadweight loss under different market structures. For Bertrand(same as perfectly competitive market where P=MC), deadweight loss is zero. As for Cournot .model, the D.L. is 1/2(180-20)(240-60)=$6400. As for the Stackelberg model, the D.L. is 1/2(140-20)(240-180)=$3,600. Finally, as for the Collusion(Monopoly)model, the D.L. is 1/2(260-20)(240-120)=$14400. I hope you or somebody can check the answers for me. If the se values are found correct, they may provide some additional information to this video. If not, I hope you can correct them. Thank you

[@johnchung5299]

Some corrections: D.L of Cournot= 1/2(180-20)(240-160)=$6,400

[@MattBirch]

I just checked and got all of the same answers. You got it. And thanks!

[@cknbeatz]

you're a legend

[@MattBirch]

Haha! #econswag

[@abdurezakmeka]

in the case of collusive shouldnt the marginal cost 1 must be added with marginal cost 2 to get total MC. and become MC=40

[@MattBirch]

Good question! The answer is no though. The next unit produced, in this case, will always be at a cost of 20. It doesn't matter if firm 1 pays the 20 or if firm 2 pays the 20, the cost of the next unit is still 20. To add them together would be to make one product for the cost of two and would lead you to under-produce the good.

[@rashmishetty3811]

Amazing

[@MattBirch]

Haha! Glad you like it. Spread the word :)

[@ElsaAbiKhalilPrettyCapricieuse]

Amazing, thanks !

[@MattBirch]

Glad it was useful!

[@elyesgabsi1998]

I am wondering please I need series (exercises) with correction I have an exam!:)

[@MattBirch]

@@elyesgabsi1998 I don't understand what you mean.

[@elyesgabsi1998]

I cant believe you replied so fast ! I need exercises for practice with correction :) I need to practice more Another thing : i study in french a chapter after oligopoly called « la différenciation des produits « ( product diffirenciation) if you have any idea about it

[@MattBirch]

@@elyesgabsi1998 some of these may be helpful: kzbin.info/aero/PLWd1brOYtkZWHvPN624Tx4kA8oxhcrrvy

[@farkhundasaidmansoor4761]

Thank you sooooo much!!!

[@MattBirch]

Heck yes. Good luck.

[@parvezbinkamal8570]

What will happen in collusion when marginal cost is different? Then how will marginal cost be calculated

[@MattBirch]

If they are both constant marginal costs, it is just a weighted average of the two costs. If one firm has a cost of $10 and the other has a cost of $20, and each produces 50% of the total quantity, then the average MC for choosing collusive output it 0.5(10)+0.5(20)=15.

[@TheAbelma]

Very good video

[@MattBirch]

Well thank you much! And good luck.

[@amosmoono4838]

Thanks for this been struggling just to differentiate the four...

[@olivernummi4022]

In the Stackelberg model how did the 2Q(1) come when you made MR=MC. Meaning when the function after that is 260-2Q=20?

[@MattBirch]

Good question. That comes out of the calculus of profit maximizing. If your demand curve is a straight line, then your MR curve is equal to the inverse demand curve with double the slope. So if P=a-bQ, then MR=a-2bQ.

[@BLINKBOXHD]

excellent video, i have on question. How do you know Q1 will equal Q2 for the Betrand problem.

[@MattBirch]

That is because this is the simplest case of Bertrand: with identical products and identical costs. No firm can gain any advantage so they sell at cost and split he market. I go more into detail on how different costs can change things in this video: kzbin.info/www/bejne/l3zGiX-on514ia8 I also show how it changes if they can differentiate their products in this video: kzbin.info/www/bejne/mWSygYCMf7KBbtE

[@ramanathansundaram5811]

Hi Matt, Thanks for explaining all 4 models in a single example, could please advice how to calculate the producer surplus in Cournot and stackelberg Model? Thanks

[@mohdip]

You probably figured it out by now but in case some else is wondering, here is how.. [P1-MC1(set Q=0)] multiplied by 1/2(Q1). So, if P1=100, MC(Q=0)=80, and Q1 =40: answer is (1/2)*(100-80)*(40)

[@mohdip]

For consumer surplus, its [ intercept of price curve: P(Q=0) minus P* ] multiplied by [ (1/2)Q* ]

[@MattBirch]

Hi there, I am sure it is too late, and apologize for the late response. I think M's has it mostly right, but I want to make a change. Since my marginal cost lines are flat, the area or producer surplus will be a rectangle (not a triangle), so the multiplied by 1/2 part is not correct. Producer surplus is the area between the MC curve and the price. In this Cournot example, the price is 180, MC=20, and Q=160. PS=(180-20)*160. Another way of approaching it if the geometry is not working out in your favor and you still don't have calculus: PS=Q(p-AVC).

[@cbd8692]

Unreal value! Btw, can collusion be replaced with a monopoly? If so, is the calculation process for the price and total output of monopoly gonna be the same as that of the collusion?

[@MattBirch]

Oh it's real! Glad you like it. Total collusion quantity is the same as monopoly quantity. Price is the same in each also.

[@jenicabunderson2451]

Thanks for making this! Question: why do you double the slope of the inverse demand function to get the marginal revenue? Could we use calculus to get the same answer?

[@MattBirch]

You can. It comes from the calculus, and is true whenever the demand curve is linear. If P=a-bQ, then total revenue is TR=QP=Q(a-bQ)=aQ-bQ^2. dTR/dQ=a-2bQ.

[@pheromoneblink-1827]

He does use it, but he skips showing it. I'd say we should go step by step unless we're arithmetic champions like him.

[@MattBirch]

@@pheromoneblink-1827 Hah, nothing so fancy as being a champion. Just lazy. If the demand curve is straight, MR is same intercept and double the slope. The class I was teaching when I made this video did not have any calculus preparation, so I just demanded that they remembered that. Then I changed the prerequisites so that they need calculus.

[@liyongpeng1173]

I am confused if each firm has a different cost function like firm 1 is 20Q1, firm 2 is 30Q2, how I can solve the collusion question?

[@MattBirch]

You would have some sort of an average marginal cost. If they split 50 50, then .5(20q)+.5(30q)=25q. That would be the cartel cost. Hope that helps.

[@liyongpeng1173]

@@MattBirch Thanks for replying. It is really helpful.

[@MattBirch]

@@liyongpeng1173 glad to hear it. Good luck!

[@oskarmoore4525]

Hi Matt. Thanks for that. In cournot why are you ignoring TC1 = 20Q. Because it is a fixed cost (assuming that TC is your fixed cost - however, usually a fixed cost is not in terms of how many q you produce)? When doing the more calculus way with differentiation, I would include that TC and it would then be q1=115 - 0.5q1. What are your thoughts behind ignoring this? Thanks

[@MattBirch]

Hi Oskar, you are correct that fixed costs are not linked to quantity. TC here stands for total cost. To simplify this already long video, I set fixed cost equal to 0. There is no fixed cost here. TC=20Q. I am not sure where you got your cost function, but you have definitely over-complicated it. Hope that helps!

[@farcecrafter]

Thanks for the vid

[@MattBirch]

Heck yes. Hope it was helpful.

[@muhammadirfanislami818]

2:08 is normal to assume Qtotal splitted to 60:40?

[@MattBirch]

It is less common. I sometimes mess with that ratio just to drive home the point that you choose the monopoly quantity first, and then the firms divide it. But the majority of examples I see just assume a 50-50 split.

[@muhammadirfanislami818]

@@MattBirch i still dont understand for collusion (monopoly) behavior if the cost structure is different. Have you any suggested source about it?

[@MattBirch]

@@muhammadirfanislami818 it is relatively easy if they both have constant marginal cost. If MC1=10 and MC2=8, then a 50-50 split would have half costing 10 and half costing 8, for an average of (.5×10)+(.5×8)= 9. If they split production 60-40, then 60 percent is made at a cost of 10 and 40 percent at a cost of 8, or (.6×10)+(.4×8)=6+3.2=9.2. The collusive mc with a 60-40 would be 9.2.

[@kevall3651]

for collusion: why price = 500 - 2Q not 500-4Q ?

[@kevall3651]

ignore please

[@MattBirch]

No worries. Glad you got it.

[@farkhundasaidmansoor4761]

I didn't get it???

[@MattBirch]

@@farkhundasaidmansoor4761 The inverse demand curve was represented by p=500-2q. When you choose q to maximize profit with a linear demand curve, the MR curve will be the same as the inverse demand curve but with double the slope: 500-(2)2q=500-4q.

[@jeelaravari]

For the cournot example, so what value is the cournot equilibrium?

[@MattBirch]

Well I give q1, q2, and p. What exactly are you looking for?

[@jeelaravari]

@@MattBirch never mind I understand it now, thank you!

[@MattBirch]

@@jeelaravari awesome!

[@wergraphy4220]

Dont think that I'll get an answere today, but why is p = 500-2Q(1)-2Q(2) and then MR(1) = 500 - 4Q(1) - 2Q(2) ? Where is the 4 coming from in the MR(1) function?

[@MattBirch]

It comes out of the calculus of profit maximization. If your inverse demand curve is linear, then the MR from the optimal q will always be the inverse demand curve, but with double the slope on the q you are choosing.

[@aprildeath1015]

In Europe we call It derivation. We multiply P.Q (which equals revenue, yes?). Then we derivate to convert revenue into marginal revenue.

[@alfiegreenwood2208]

At 16:50, why do we use 500-2Q as opposed to 500-2q1-2q2?

[@alfiegreenwood2208]

PS this video is a real life saver

[@MattBirch]

@@alfiegreenwood2208 That would give the same thing, given that q1+q2=Q. In Bertrand, we choose price and then we use prices to choose quantities, and since the demand curve is a function of Q, I prefer to leave Q as Q when choosing price. You could do it either way, but you would have to remember the intuition of them splitting the Q when they have the same MC. I hope that helps. PS Glad to hear it!

[@danhanlon3516]

For the collusion one what what the outcomes be if the two firms have different cost functions?

[@MattBirch]

That depends on the cost functions. You would need to aggregate them into a cartel cost function, and then do MR=MC to maximize profit. An example of aggregating cost functions: If neither firm has a fixed cost, firm 1 produces 60% of cartel output, and MC1=20 and MC2=30, then the cartel MC would effectively be 0.6*20+0.4*30=24.

[@danhanlon3516]

@@MattBirch Thank you!!!

[@MattBirch]

Heck yes. Good luck!

[@amadeus242]

Excuse me how you find Q1=72 and Q2=48 for Collusion ?

[@mohdip]

0.6Q=72, 0.4Q=48

[@MattBirch]

M's math is right. Often in collusion examples we just assume a 50-50 split between two firms, in which case both firms would have produced 60. In this case, I assumed that one firm produced 60 percent and the other firm produced 40 percent.

[@SimplyMartin]

Explained in a great way, approachable to the student. Like & Subscribe from me.

[@MattBirch]

I am glad it helped. Thank you much and good luck!

[@indahmawarni1986]

Hi, how do we get the 3/4Q1 at 8:45?

[@MattBirch]

I subtracted 1/4 Q1 from both sides of the equation.