And now what were going to do since we have these two reaction curves we have a

And now what were going to do since we have these two

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And now what we're going to do since we have these two reaction curves, we have a reaction for Firm A and a reaction for Firm B, all we're going to do is we're going to plug-in for this qB, qB's reaction curve. And when we do that, when we plug-in 66.67 minus 0.33qA, we can solve through for just qA. Doing this we're going to find that Firm A is going to produce approximately 77 units. And then taking this 77 and plugging it in to Firm B's reaction function, we can solve for Firm B's production amount. And we're going to find that Firm B is going to produce approximately 41 units. Now to find the equilibrium price, we're just going to come back up here to our disaggregated demand function, and we're going to plug-in for qA and qB. And we can solve through for the price being equal to 82. Now what we've just done for this problem is we solved on our graph for the intersection of the two reaction functions. We found that qB is going to be equal to 41 and we found that qA is going to be equal to 77. So we just calculated the intersection point for the Cournot equilibrium. Now the last part of this problem asks us to calculate the profits for both the firms. The profits for Firm A are just going to be price times the quantity that A is producing minus the cost as a function of qA. So we're just going to take the total revenues minus the total costs. For Firm A, we're going to find that the total profits are going to be about $5,929. And doing the same process for Firm B, we can find that the profits for Firm B are going to be equal to approximately $2,521. Now part B of this problem is going to ask us instead of having this Cournot equilibrium where neither firm can go ahead and produce a higher quantity or move first in the market, we're
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going to look at something different than the Cournot equilibrium. We're going to look at the case where one of the firms gets to decide how much they're going to produce before the other firm. And if you get to decide first, you get to produce a higher quantity and get more of the profits. Part B says, now suppose that Firm A chooses how much to produce before firm B does. In this case, Firm A is a Stackelberg leader and B a follower. We're going to calculate the quantities, the market price, and the profit for each firm. Now coming over to this side of the board, we see that I'm going to keep Firm B's reaction function the same. So Firm B is going to be reacting in the same way to Firm A's decision. Only now, the only difference is when we calculate marginal revenue equal the marginal cost for Firm A, instead of just saying the qB is going to be random, we can't account for it. We're going to plug-in, we're going to take into account Firm B's reaction when we're maximizing or taking the derivative with respect to qA. So instead of having qB in here, I'm going to plug-in this reaction function.
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  • Summer '17
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