Question 9 Two firms: Q1=75-P1+.5P2 P1=75+.5P2-Q1 >>>> 75+.5P2-2Q1=30 Q2=75-P2+.

Question

Question 9
Two firms:
Q1=75-P1+.5P2 P1=75+.5P2-Q1 >>>> 75+.5P2-2Q1=30
Q2=75-P2+.5P1

AC=MC=30

a) Confirm firm 1's optimal price depends on P2 according to P1=52.5+.25P2 (Set up profit expression of profit firm 1=(p1-30)Q1
=(P1-30)(75-P1+.5P2) and set Marginal Profit = 0 to solve for P1 in terms of P2. Alternatively, set MR1=MC and solve for Q1 and then
P1 in terms of P2

ANSWER: firm 1, P1=75+.5P2-Q1. Setting MR1=MC >>>> 75+.5P2-2Q1=30 >>>> Q1=22.5+.25P2>>> substituting this solution for
Q1 into the price equation, therefore P1=52.5+.25P2

Solving P1=52.5+.25P2 >>>>> P1=P2=70 >>> demand Q1=Q2=40 : Each firms profit is 1600 2800 1200 1600

Question 10

Instead, the firms compete by setting quantities rather than prices. Rewriting the demand equations P1=[150-(2/3)Q2]-(4/3)Q1
and P2=[150-(2/3)Q1]-(4/3)Q2. Increases in the competitor's output lowers the intercept of the firm's demand curve

a) MR1=MC to confirm that firm 1's optimal quantity depends on Q2 according to Q1=45-.25Q2



b) Equilibrium Q1=Q2

Explanation / Answer

I'm ignoring question 9 since you say you only need question 10 a and b. This is a Cournot equilibrium. a. You are given the price equation P1=[150-(2/3)Q2]-(4/3)Q1 By definition, revenue is: R1 = P1*Q1 Substituting your price equation: R1 = {[150-(2/3)Q2]-(4/3)Q1}*Q1 Take the derivative with respect to quantity to find the marginal revenue: MR1 = [150-(2/3)Q2] - (8/3)Q1 You are given the marginal cost: MC = 30 Set MR1=MC [150-(2/3)Q2] - (8/3)Q1 = 30 Solve for Q1 and simplify: Q1 = (3/8)*{[150-(2/3)Q2] - 30} Q1 = 56.25 - 0.25Q2 - 11.25 Q1=45-.25Q2 b. Analogously: Q2 = 45-.25Q1 You can get this by repeating the steps in part a for firm 2. Now, we can substitute Q1=45-.25Q2 Q1=45-.25*[45-.25Q1] Q1 = 45 - 11.25 + 0.0625Q1 Q1(1-0.0625) = 45 - 11.25 Q1 = (45 - 11.25)/((1-0.0625) Q1 = 36 You can similarly substitute: Q2=45-.25Q1 Q2=45-.25*[45-.25Q2] Q2 = 45 - 11.25 + 0.0625Q2 Q2(1-0.0625) = 45 - 11.25 Q2 = (45 - 11.25)/((1-0.0625) Q2 = 36 Q1 = Q2 = 36

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