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Exercise Two firms compete for market shares producing similar goods under imperfect competition. The market demand curve is p = 24-y, where y = y₁ + y2 is the summation of the firms' outputs. Total costs are TC₁ = 4y₁ for the first company and TC₂ = 2y2 for the second company. The second company is therefore more productive than the first. Now answer the following questions: (a) Compute the best response functions of each company. Then compute the Nash equilibrium in quantities. The Nash equilibrium in quantities is also called the Cournot equilibrium. Note that in this case the payoffs are given by the profit functions rather than by the levels of utility. (b) Plot the best response functions and the Nash equilibrium. (c) Show that the Nash equilibrium is stable. (d) Assume that the first company is a first mover (even though it is less productive) and chooses its output level before the second company. Find the (Stackelberg) equilibrium in quantities. (e) Plot the Stackelberg equilibrium in the same plot from part (b). (f) Is the Stackelberg equilibrium stable or unstable? (g) Assume that an impartial spectator (which could be a cartel) maximizes joint profits. This is equivalent to an impartial spectator that maximizes a social welfare function that aggregates utility functions, where the payoffs in this case are given by the profit functions. Find the equilibrium level of joint output, and also the equilibrium levels of output of each company. Is there a conflict of interest between the companies within the cartel?

Exercise Two firms compete for market shares producing similar goods under imperfect competition. The market demand curve is p = 24-y, where y = y₁ + y2 is the summation of the firms' outputs. Total costs are TC₁ = 4y₁ for the first company and TC₂ = 2y2 for the second company. The second company is therefore more productive than the first. Now answer the following questions: (a) Compute the best response functions of each company. Then compute the Nash equilibrium in quantities. The Nash equilibrium in quantities is also called the Cournot equilibrium. Note that in this case the payoffs are given by the profit functions rather than by the levels of utility. (b) Plot the best response functions and the Nash equilibrium. (c) Show that the Nash equilibrium is stable. (d) Assume that the first company is a first mover (even though it is less productive) and chooses its output level before the second company. Find the (Stackelberg) equilibrium in quantities. (e) Plot the Stackelberg equilibrium in the same plot from part (b). (f) Is the Stackelberg equilibrium stable or unstable? (g) Assume that an impartial spectator (which could be a cartel) maximizes joint profits. This is equivalent to an impartial spectator that maximizes a social welfare function that aggregates utility functions, where the payoffs in this case are given by the profit functions. Find the equilibrium level of joint output, and also the equilibrium levels of output of each company. Is there a conflict of interest between the companies within the cartel?

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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Part G H I 

Exercise Two firms compete for market shares producing similar goods under imperfect competition. The market demand curve is p = 24 – y, where y = y, + y2 is the summation of the firms' outputs. Total costs are TC, = 4y1 for the first company and TC2 = 2y2 for the second company. The second company is therefore more productive than the first. Now answer the following questions: (a) Compute the best response functions of each company. Then compute the Nash equilibrium in quantities. The Nash equilibrium in quantities is also called the Cournot equilibrium. Note that in this case the payoffs are given by the profit functions rather than by the levels of utility. (b) Plot the best response functions and the Nash equilibrium. (c) Show that the Nash equilibrium is stable. (d) Assume that the first company is a first mover (even though it is less productive) and chooses its output level before the second company. Find the (Stackelberg) equilibrium in quantities. (e) Plot the Stackelberg equilibrium in the same plot from part (b). (f) Is the Stackelberg equilibrium stable or unstable? (g) Assume that an impartial spectator (which could be a cartel) maximizes joint profits. This is equivalent to an impartial spectator that maximizes a social welfare function that aggregates utility functions, where the payoffs in this case are given by the profit functions. Find the equilibrium level of joint output, and also the equilibrium levels of output of each company. Is there a conflict of interest between the companies within the cartel?
Transcribed Image Text:Exercise Two firms compete for market shares producing similar goods under imperfect competition. The market demand curve is p = 24 – y, where y = y, + y2 is the summation of the firms' outputs. Total costs are TC, = 4y1 for the first company and TC2 = 2y2 for the second company. The second company is therefore more productive than the first. Now answer the following questions: (a) Compute the best response functions of each company. Then compute the Nash equilibrium in quantities. The Nash equilibrium in quantities is also called the Cournot equilibrium. Note that in this case the payoffs are given by the profit functions rather than by the levels of utility. (b) Plot the best response functions and the Nash equilibrium. (c) Show that the Nash equilibrium is stable. (d) Assume that the first company is a first mover (even though it is less productive) and chooses its output level before the second company. Find the (Stackelberg) equilibrium in quantities. (e) Plot the Stackelberg equilibrium in the same plot from part (b). (f) Is the Stackelberg equilibrium stable or unstable? (g) Assume that an impartial spectator (which could be a cartel) maximizes joint profits. This is equivalent to an impartial spectator that maximizes a social welfare function that aggregates utility functions, where the payoffs in this case are given by the profit functions. Find the equilibrium level of joint output, and also the equilibrium levels of output of each company. Is there a conflict of interest between the companies within the cartel?
(h) Is the equilibrium given by the impartial spectator (cartel) stable or unstable? Why? (i) Plot the equilibrium given by the impartial spectator (cartel) in the same graph from part (e). (j) Find the equations of the iso-profit curves for each company. Remember that the iso-profit curves are similar to the iso-utility curves (indifference curves) for consumers. Plot the iso-profit curves of each company in the same graph from part (h). (k) Find the social optimum and the output levels of each company at the social optimum. Remember that the Pareto-efficient equilibrium must equalize the firms' marginal rates of substitution (mrs). (I) Compute the marginal rates of substitution of each firm at the Nash equilibrium. Is the Nash equilibrium Pareto optimal? Why? (m) Explain briefly why the Nash equilibrium is not Pareto optimal. Where is the source of the negative externalities?
Transcribed Image Text:(h) Is the equilibrium given by the impartial spectator (cartel) stable or unstable? Why? (i) Plot the equilibrium given by the impartial spectator (cartel) in the same graph from part (e). (j) Find the equations of the iso-profit curves for each company. Remember that the iso-profit curves are similar to the iso-utility curves (indifference curves) for consumers. Plot the iso-profit curves of each company in the same graph from part (h). (k) Find the social optimum and the output levels of each company at the social optimum. Remember that the Pareto-efficient equilibrium must equalize the firms' marginal rates of substitution (mrs). (I) Compute the marginal rates of substitution of each firm at the Nash equilibrium. Is the Nash equilibrium Pareto optimal? Why? (m) Explain briefly why the Nash equilibrium is not Pareto optimal. Where is the source of the negative externalities?
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