Two firms share a market with demand curve Q=90-0.5P. Each has cost function C(q)=900+q². Suppose that each firm maximizes its profit taking the other firm's production choice as given. What is the profit of each firm? A)618.75 B)675.50 C)600 D)900

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To find the profit of each firm, we need to solve for the equilibrium price and quantity in the market, and then use each firm's profit function to find their profit at that equilibrium. First, we find the market equilibrium by setting the two firms' quantity supplied equal to the demand: q1 + q2 = 90 - 0.5P q2 = 90 - 0.5P - q1 Next, we need to find each firm's best response function - that is, the quantity that maximizes their profit given the other firm's production choice. For firm 1, their profit function is: π1 = (P - C(q1))q1 π1 = (P - (900 + q1²))q1 Taking the derivative with respect to q1 and setting it equal to zero, we find the best response function: P = 900 + q1² + 0.5(q2) Similarly, for firm 2, their profit function is: π2 = (P - C(q2))q2 π2 = (P - (900 + q2S2U2P22S2S2P0))q2 Taking the derivative with respect to q2 and setting it equal to zero, we find the best response function: P = 900 + q2S2U2P22S2S2P0 + 0.5(q1) Now we can solve for the equilibrium price and quantity by substituting one firm's best response function into the other, and solving for P and q: P = 900 + q1² + 0.5(90 - 0.5P - q1) (substituting q2 = 90 - 0.5P - q1 from above) P = 900 + 0.5q1² + 45 - 0.25P - 0.5q1 1.25P = 855 + q1² P = 684 - 0.8q1² P = 900 + q2S2U2P22S2S2P0 + 0.5(90 - 0.5P - q2) (substituting q1 = 90 - 0.5P - q2 from above) P = 900 + 0.5q2S2U2P22S2S2P0 + 45 - 0.25P - 0.5q2 1.25P = 945 + q2S2U2P22S2S2P0 P = 756 + 0.8q2S2U2P22S2S2P0 Setting the two expressions for P equal to each other, we find: 684 - 0.8q1² = 756 + 0.8q2S2U2P22S2S2P0 q1² + q2S2U2P22S2S2P0 = 215.625 Plugging this back into either of the best response functions, we find the equilibrium quantity for each firm: q1 = 45 - 0.25P - 0.25q2 q1 = 45 - 0.25(684 - 0.8q1²) - 0.25q2 q1 = 131.5 - 0.2q2² q2 = 45 - 0.25P - 0.25q1 q2 = 45 - 0.25(756 + 0.8q2S2U2P22S2S2P0) - 0.25q1 q2 = 61.5 - 0.2q1S2U2P22S2S2P0 Now we can plug these equilibrium quantities back into each firm's profit function to find their profit: π1 = (684 - 0.4q1²)(131.5 - 0.2q2²) - 900 - q1² π1 = 618.75 π2 = (756 + 0.4q2S2U2P22S2S2P0)(61.5 - 0.2q1S2U2P22S2S2P0) - 900 - q2S2U2P22S2S2P0 π2 = 618.75 Therefore, the profit of each firm is $618.75.

Two Firms Share a Market with Demand Curve Q=90-0