3. Three firms compete for hiring workers in the labour market. The equilibrium wage is w=...

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3. Three firms compete for hiring workers in the labour market. The equilibrium wage is w= a + 3 (H₂ + H₂ + H3), where a, 3> 0 are parameters, and H, is the number of workers hired by firm i = 1,2,3. Firm i's profit is (y-w) H₁, where y is the output per worker. Assume y> a. Firm i chooses H, to maximize its profit. Let us only consider the symmetric equilibrium whenever there are simultaneous decisions. (a) [10 points] Suppose the firms decide on H, simultaneously. Derive the symmetric Nash equilibrium. Write down the outcome of the game. (b) [15 points] Suppose the game is sequential. Firms 1 and 2 first simultaneously decide on the respective H₁ and H₂, then firm 3 decide on H3. Derive the subgame perfect equilibrium. Write down the outcome of the game. (c) [15 points] Suppose the game is sequential. Firm 1 first decides on H₁, then firms 2 and 3 make decisions simultaneously. Derive the subgame perfect equilibrium. Write down the outcome of the game. Find which firm is the biggest winner across all these three games by comparing firms' market shares in each game. [A firm's market share is given by H₁/ (H₁+ H₂ + H3), where i = 1, 2, 3.] 3. Three firms compete for hiring workers in the labour market. The equilibrium wage is w= a + 3 (H₂ + H₂ + H3), where a, 3> 0 are parameters, and H, is the number of workers hired by firm i = 1,2,3. Firm i's profit is (y-w) H₁, where y is the output per worker. Assume y> a. Firm i chooses H, to maximize its profit. Let us only consider the symmetric equilibrium whenever there are simultaneous decisions. (a) [10 points] Suppose the firms decide on H, simultaneously. Derive the symmetric Nash equilibrium. Write down the outcome of the game. (b) [15 points] Suppose the game is sequential. Firms 1 and 2 first simultaneously decide on the respective H₁ and H₂, then firm 3 decide on H3. Derive the subgame perfect equilibrium. Write down the outcome of the game. (c) [15 points] Suppose the game is sequential. Firm 1 first decides on H₁, then firms 2 and 3 make decisions simultaneously. Derive the subgame perfect equilibrium. Write down the outcome of the game. Find which firm is the biggest winner across all these three games by comparing firms' market shares in each game. [A firm's market share is given by H₁/ (H₁+ H₂ + H3), where i = 1, 2, 3.]

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a Simultaneous Decision Given the equilibrium wage equation w alpha betaH1 H2 H3 Each firms profit function Profiti y w Hi To find the symmetric Nash ... View the full answer