(3) Joe and Sarah's Investment Dilemma Suppose Joe and Sarah each have a patent on their respective product: no other supplier can provide their particular product. However, Joe and Sarah's products are imperfect substitutes for each other. Consequently, they face the following respective consumer demand Qjoe = 300 - 15 Pjoe + 10 Psarah Qsarah = 300 - 15 Psarah + 10 Pjoe They face the following costs characterized by constant marginal cost and no fixed costs C(Qjoe) = 8 Qjoe C(Qsarah) = 8 Qsarah (3a) Assuming that each supplier charges marginal cost Pjoe = Psarah = $8, calculate the own-price and cross-price elasticities for Joe. (Sarah's are the same due to symmetry) (3b) Solve for Joe's and Sarah's respective reaction curves, assuming a Bertrand game. (3c) Solve for the Bertrand-Nash Equilibrium. (3d) Under the Betrand-Nash Equilibrium, how much does each supplier earn (payoff)? (3e) Suppose Joe has the opportunity to invest and lower his costs as follows: C*(Qjoe) = 4 Qjoe If Joe invests in this new technology and Sarah is stuck with her current costs (constant marginal cost of $8), what would the new Bertrand-Nash Equilibrium be? (3f) How much does each supplier earn under the Bertrand-Nash Equilibrium in (2e) given that the investment cost for Joe is $500 ? Assuming that Sarah is stuck with her current costs, what is the most that Joe would have been willing to spend for the new technology? (3g) Suppose Sarah has the same opportunity to invest in the lower cost technology (at $500). If both Joe and Sarah make the investment and lower their marginal costs to $4, what is the new Bertrand-Nash Equilibrium