1. ART AUCTION HOUSE PRICE RIVALRY Suppose Christie's and Sotheby's provide differentiated art auction house services. In this static duopolistic game, each profit- maximizing auction house simultaneously sets its price. Let us denote by subscript i = 1,2 the two services, so, for example, Firm F₁ maximizes its profit л₁ by choosing its price P₁, while taking its rival's price P2 as given. The demand function for Christie's (or F₁) is q₁ = 2 − 2P₁ + P2 and the demand function for Sotheby's (or F2) is q₂ = 1 − 2P₂ + P₁. Assume zero production costs for each firm so we can focus on the price competition. TIP: The demand intercepts are different. a) Derive the two firms' best response functions. b) Derive the heterogenous-good Bertrand Nash Equilibrium prices, quantities, and profits, (P1, P2, q1,9½‚ో˜²). HINT: You should find that π½ is “a cube divided by a square.” c) On a fully labeled diagram, illustrate the two firms' reaction curves and the Bertrand Nash Equilibrium. Put P₁ on the vertical (upward) axis and P2 on the horizontal (rightward) axis. d) Suppose the two firms collude to fix prices (i.e., they cooperatively maximize joint profits by picking both prices as if they were a monopoly). Find the collusive results, (P1, P2, 9,9Σ‚ÑƒÑ²)
1. ART AUCTION HOUSE PRICE RIVALRY Suppose Christie's and Sotheby's provide differentiated art auction house services. In this static duopolistic game, each profit- maximizing auction house simultaneously sets its price. Let us denote by subscript i = 1,2 the two services, so, for example, Firm F₁ maximizes its profit л₁ by choosing its price P₁, while taking its rival's price P2 as given. The demand function for Christie's (or F₁) is q₁ = 2 − 2P₁ + P2 and the demand function for Sotheby's (or F2) is q₂ = 1 − 2P₂ + P₁. Assume zero production costs for each firm so we can focus on the price competition. TIP: The demand intercepts are different. a) Derive the two firms' best response functions. b) Derive the heterogenous-good Bertrand Nash Equilibrium prices, quantities, and profits, (P1, P2, q1,9½‚ో˜²). HINT: You should find that π½ is “a cube divided by a square.” c) On a fully labeled diagram, illustrate the two firms' reaction curves and the Bertrand Nash Equilibrium. Put P₁ on the vertical (upward) axis and P2 on the horizontal (rightward) axis. d) Suppose the two firms collude to fix prices (i.e., they cooperatively maximize joint profits by picking both prices as if they were a monopoly). Find the collusive results, (P1, P2, 9,9Σ‚ÑƒÑ²)
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:1. ART AUCTION HOUSE PRICE RIVALRY
Suppose Christie's and Sotheby's provide differentiated art auction house services. In this static duopolistic game, each profit-
maximizing auction house simultaneously sets its price. Let us denote by subscript i = 1,2 the two services, so, for example,
Firm F₁ maximizes its profit л₁ by choosing its price P₁, while taking its rival's price P2 as given. The demand function for
Christie's (or F₁) is q₁ = 2 − 2P₁ + P2 and the demand function for Sotheby's (or F2) is q₂ = 1 − 2P₂ + P₁. Assume zero
production costs for each firm so we can focus on the price competition. TIP: The demand intercepts are different.
a)
Derive the two firms' best response functions.
b) Derive the heterogenous-good Bertrand Nash Equilibrium prices, quantities, and profits, (P1, P2, q1,9½‚ో˜²). HINT: You
should find that π½ is “a cube divided by a square.”
c) On a fully labeled diagram, illustrate the two firms' reaction curves and the Bertrand Nash Equilibrium. Put P₁ on the
vertical (upward) axis and P2 on the horizontal (rightward) axis.
d) Suppose the two firms collude to fix prices (i.e., they cooperatively maximize joint profits by picking both prices as if they
were a monopoly). Find the collusive results, (P1, P2, 9,9Σ‚ÑƒÑ²)
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