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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 30015 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

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 30015 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

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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(3h) How much does each supplier earn under the Bertrand-Nash Equilibrium in (2g) (accounting for the investment cost)? Are payoffs lower or higher in this equilibrium than in the other two Bertrand-Nash equilibria (considered above)? Consider the following two-stage game. 1. First, each supplier simultaneously and independently decides whether to invest in the new technology at a cost of $500. 2. Second, with each supplier having made his/her investment decision (revealed to each other), each supplier simultaneously and independently decides on the price to charge for his/her product. So the second stage of this game is the Bertrand game, conditional on each supplier's first stage investment decision. This two-stage game can be solved using backward induction ... first solve the second stage game for each possible outcome of the first stage and then solve the first stage. You've actually done the first step of the induction in the earlier parts of this question ... (3i) Fill-in the following payoff matrix for the first-stage “investment game" using your answers from (2d), (2f), (2h) Sarah Not Invest Invest Joe Not Invest ?,? ?, ? ?,? ?, ? Invest HINT: The payoffs from the Nash-Bertrand Equilibrium where Sarah invests but Joe does not are symmetric to the case where Joe invests but Sarah does not. (3j) Solve for the pure strategy Nash equilibrium for this "investment game." (3k) Explain why this "investment game" is a “Prisoner's Dilemma" game. Who benefits the most from the availability of this new low cost technology: Joe, Sarah, or consumers?
Transcribed Image Text:(3h) How much does each supplier earn under the Bertrand-Nash Equilibrium in (2g) (accounting for the investment cost)? Are payoffs lower or higher in this equilibrium than in the other two Bertrand-Nash equilibria (considered above)? Consider the following two-stage game. 1. First, each supplier simultaneously and independently decides whether to invest in the new technology at a cost of $500. 2. Second, with each supplier having made his/her investment decision (revealed to each other), each supplier simultaneously and independently decides on the price to charge for his/her product. So the second stage of this game is the Bertrand game, conditional on each supplier's first stage investment decision. This two-stage game can be solved using backward induction ... first solve the second stage game for each possible outcome of the first stage and then solve the first stage. You've actually done the first step of the induction in the earlier parts of this question ... (3i) Fill-in the following payoff matrix for the first-stage “investment game" using your answers from (2d), (2f), (2h) Sarah Not Invest Invest Joe Not Invest ?,? ?, ? ?,? ?, ? Invest HINT: The payoffs from the Nash-Bertrand Equilibrium where Sarah invests but Joe does not are symmetric to the case where Joe invests but Sarah does not. (3j) Solve for the pure strategy Nash equilibrium for this "investment game." (3k) Explain why this "investment game" is a “Prisoner's Dilemma" game. Who benefits the most from the availability of this new low cost technology: Joe, Sarah, or consumers?
(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 (раyoff)? (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?
Transcribed Image Text:(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 (раyoff)? (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?
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