Bertrand Competition in a Product Differentiated Market

[@grantbaker240]

This video explained it very clearly! Thanks!

[@MattBirch]

My pleasure!

[@lauracastro110]

How did you get the 5/6*P1 when calculating the nash equilibrium?

[@MattBirch]

Can you give me a time stamp of when I lost you?

[@danielpainter8767]

Question - if you're differentiating TR from a function including p1 & p2, how can you simply set that equal to a static mc when finding the reaction functions? The intuition of mr=mc comes from differentiating the profit function which is tr-tc. Wouldn't you need to solve the differentiated profit function since cost in that setting would be a function of mc, p1, and p2? I'm brushing up for a class I'm working on and remember this being pretty crucial, but maybe I'm missing something.

[@MattBirch]

I'm not quite sure I follow you, but let me say a few things and see how it goes. TR and MR are now much more complicated than in a monopoly model because there are two firms, and 2 different quantities that matter. A firm has control over its own quantity, but not the other. It will choose its own quantity (choice means use calculus), but not the other. The firm still wants to increase production until MR=MC, and that hasn't changed. It is just that now that quantity that balances MR and MC also depends on the other quantity, so it is a BRF instead of a specific q.

[@ryanfeuz9956]

@@MattBirch I believe Daniel is correct. In the video you state that the MR is 400-2p1+2p2. However, this can not be marginal revenue as you took the partial derivative with respect to p1. MR is the partial derivative of TR with respect to Q. I think Daniel is correct in stating that you would need to differentiate the profit function and set equal to 0. The solution I come up with is P1=460, Q1=360, Profit1=129600, P2=210, Q2=330 and Profit 2=59400.

[@MattBirch]

​@@ryanfeuz9956 Here is the math. Hope it helps. q1 = 400 - p1 + 2p2 TR1 = p1*q1 TR1 = p1(400-p1+2p2) TR1 = 400p1-p1^2+2p1p2 MR1 = dTR1/dp1 MR1 = 400-2p1+2p2

[@ryanfeuz9956]

@@MattBirch Thanks for the reply. I understand the math. However, when you take the partial derivative of TR with respect to P1 this is not Marginal Revenue. Marginal revenue is partial derivative with respect to quantity. Therefore, you can not simply equate the partial derivative of total revenue with respect to price and the MC. I believe the correct answer can only be found if you maximize the entire profit function and then solve for P1 or P2 respectively to get the reaction functions. The profit function for firm 1 would be Profit1=(P1)(400-P1+2P2)-((100)(400-P1+2P2)). This is what I believe must be maximized to find the correct reaction functions.

[@MattBirch]

​@@ryanfeuz9956 So are we just quibbling about the definition of "Marginal Revenue"? We are still fine here. Where does MR=MC come from? Profit = TR-TC. Take the derivative of profit: MR-MC Set that derivative = 0: MR=MC Setting MR=MC is the same as taking the derivative of profit and setting it equal to 0. So the question is about MR. In a Cournot model, the firm chooses Q first, so they take the derivative (for MR and MC) with respect to Q. That is the choice that needs optimizing. It is the choice that gets calculus. Calculus always goes with the choice. In this model, the firm is choosing P first, so they will take derivative with respect to P. If you want to call it something other than MR and MC you can, but you will still get dTR/dp = dTC/dp. What you said about setting the derivative of profit equal to 0 IS right. It is literally the same thing. If you do your math right, you will get the exact same thing.

[@shyruti]

What about if one firm moves first? How do we proceed from there?

[@MattBirch]

It would be like Stackelberg, only choosing prices instead of quantities. I don't have a video on this yet, but if you use the steps from my Stackelberg video, and the demand curve and best response function from this framework, that should get you where you want to be. kzbin.info/www/bejne/lZqmfpWefad0isk

[@shubhamkumar-ig8ko]

I'm in Need for more numericals of micro and macro economics to prepare for M.A entrance exam of economics,india. Any suggestions or study resources?

[@MattBirch]

I am not exactly sure. If you can find any practice tests or old tests, that is where I would start. There are a lot more videos than mine here on KZbin. Also, I could not have survived grad school without wikipedia. It has some really great stuff on economic theory.

[@kaushaalk8043]

How is ATC equal to 100?

[@MattBirch]

Around 1:28ish I show that both firms have a constant MC=100 and no fixed costs. If MC is constant and FC=0, then ATC=MC.