Consider a three-stage game: First, the incumbent may sign a quantity contract with consumers A contract...

 

 
 

Transcribed Image Text:

● ● Consider a three-stage game: First, the incumbent may sign a quantity contract with consumers A contract commits the incumbent to a quantity qfix (but not a price) - · Incumbent must pay a legal cost L = 3 to seal a contract Both firms observe the contract, including its quantity qfix Second, the entrant chooses Enter or Not The cost of entering is F = 4 Both firms observe the Entrant's choice Third, firms in the market choose quantities simultaneously - - The demand curve is p(q) = 10 — q; there are no variable costs - If Incumbent signed a quantity contract, it must produce exactly that much a) Diagram this game Observe that the subgame that starts after the Incumbent chooses not to sign a contract is identical to the "Cournot Entry" game considered in lecture. We found that in the unique sub- game perfect equilibrium in the Cournot Entry game, since F = 4 < 100, the Entrant chooses Enter, the Incumbent chooses quantity 5 if the Entrant does not Enter, and both firms choose quantity if the Entrant does Enter. This equilibrium yields a payoff of ¹00 ~ 11.1 for the Incum- bent and 3 9 100 9 4 7.1 for the Entrant. Consider a subgame that starts after the Incumbent chooses to sign a contract with quan- tity qfix. - b) Find a subgame perfect equilibrium in this subgame, as a function of qfix. c) What is the Incumbent's payoff in this subgame, as a function of qfix? (Work carefully, and notice that the Incumbent's payoff depends crucially on whether the Entrant chooses to Enter or Not given qfix.) Now consider a subgame perfect equilibrium of the entire game. d) What is the Incumbent's equilibrium choice in the first stage? (Work carefully, as the in- cumbent's payoff from signing a contract is discontinuous in qfix. You will need to consider several cases.) ● ● Consider a three-stage game: First, the incumbent may sign a quantity contract with consumers A contract commits the incumbent to a quantity qfix (but not a price) - · Incumbent must pay a legal cost L = 3 to seal a contract Both firms observe the contract, including its quantity qfix Second, the entrant chooses Enter or Not The cost of entering is F = 4 Both firms observe the Entrant's choice Third, firms in the market choose quantities simultaneously - - The demand curve is p(q) = 10 — q; there are no variable costs - If Incumbent signed a quantity contract, it must produce exactly that much a) Diagram this game Observe that the subgame that starts after the Incumbent chooses not to sign a contract is identical to the "Cournot Entry" game considered in lecture. We found that in the unique sub- game perfect equilibrium in the Cournot Entry game, since F = 4 < 100, the Entrant chooses Enter, the Incumbent chooses quantity 5 if the Entrant does not Enter, and both firms choose quantity if the Entrant does Enter. This equilibrium yields a payoff of ¹00 ~ 11.1 for the Incum- bent and 3 9 100 9 4 7.1 for the Entrant. Consider a subgame that starts after the Incumbent chooses to sign a contract with quan- tity qfix. - b) Find a subgame perfect equilibrium in this subgame, as a function of qfix. c) What is the Incumbent's payoff in this subgame, as a function of qfix? (Work carefully, and notice that the Incumbent's payoff depends crucially on whether the Entrant chooses to Enter or Not given qfix.) Now consider a subgame perfect equilibrium of the entire game. d) What is the Incumbent's equilibrium choice in the first stage? (Work carefully, as the in- cumbent's payoff from signing a contract is discontinuous in qfix. You will need to consider several cases.)