Question

Firms X and Y operate very close to each other. Firms X has profits πX=240X− 1/2X²-1/3 XY These profits show a traditional structure (revenue minus quadratic costs) and also the fact that the quantities X and Y produce combine to create a negative effect on their profits due to pollution-related issues. Y 's profits are symmetrie: πY=240Y− 1/2Y²-1/3 XY

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XY. These profits show a traditional structure (revenue minus quadratic costs) and also the fact that the quantities X and Y produce combine to create a negative effect on their profits due to pollution-related issues.
(a) Which quantities would maximize thę sum of the profits? Which is the joint profit amount? You can use Wolfram Alpha to obtain all of these, they will be displayed on the "Global maximum" section.
(b) Show that Firm X 's best respotse function to the quantity level picked by Firm Y is X=240−1/3. Y (the derivative here is similar to the one we did in class, but if you are having trouble here let me know over e-mail) Note: Y′ s problem is symmetric, therefore we know that Y′ s best response function is Y=240−1/3,X
(c) Are the quantitien you found in item a) a Nash equilibrium? Why? Please answer this without relying on your answer for the next item (d)
(d) Which are the Nash equilibrium quantity lovels? Which are the profits for each company for those levels?
(e) Could aa agreement be good for both companios? Explonin (a qualitative auswer is enotagh).

Answer 1

To maximize the sum of the profits, we find the quantities that set the derivatives of the profit functions equal to zero. Firm X's best response function to the quantity level picked by Firm Y is X = 240 - 1/3Y. The joint profit amount is $9000.

Step-by-step explanation:

To find the quantities that would maximize the sum of the profits, we need to take the derivative of the profit functions and set them equal to zero. For firm X, the derivative is 240 - 2/3X - 1/3Y = 0. Solving for X gives X = 240 - 1/3Y. For firm Y, the derivative is 240 - 2/3Y - 1/3X = 0. Solving for Y gives Y = 240 - 1/3X.

Substituting the value of Y from firm Y's best response function into firm X's best response function, we get X = 240 - 1/3(240 - 1/3X), which simplifies to X = 240 - 80 + X/9. Solving for X gives X = 240 - 80 + X/9 simplifies to 8X/9 = 160, and X = 180.

The joint profit amount is obtained by plugging the values of X = 180 and Y = 240 - 1/3X into the profit function of either firm. Let's use firm X's profit function: πX = 240X - 1/2X² - 1/3XY. Plugging in X = 180 and Y = 240 - 1/3(180), we get πX = 240(180) - 1/2(180)² - 1/3(180)(240 - 1/3(180)). Simplifying gives πX = 32400 - 16200 - 7200 = 9000.