3. (20 points) Principal-Agent The probability of winning a soccer match is determined in part by...

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3. (20 points) Principal-Agent The probability of winning a soccer match is determined in part by the level of effort of the players. Assume that the probability of winning a match is given by the function: arrangements: Pw(e) Where e is the level of effort by the average player. e is continuous and bounded by the [0, 1] interval. If a team wins a match; the owner of the team will earn $1,000; if the match ends in a loss or a draw then the owner earns nothing. Then, the expected revenues are equal to R(e) = 1,000P(e). For simplicity assume that the only cost for the owner is to pay the salary of 11 players. Consider the following wage 2. w = = 1. w $ 150 (Regardless of the outcome of the game) R(e) R(e) - 900 3. w = 2ee² The utility of the average player is given by u(e) = w - 250e (a) (10 points) Find the optimal level of effort for each of the proposed wage arrangements. (b) (5 points) Find the profit level for each different contract. (c) (5 points) What's the optimal contract for the owner? (22) Consider a monopoly facing the demand curve P(Q) = 100 - 2Q. The monopolist's marginal cost is MC(Q) = 10+5Q, and its fixed cost is FC = 100. (a) Find the quantity the monopolist produces. (b) What is the market price? TWO (c) What is producer surplus? (d) what is profit? uvos tas What is the own-price elasticity of demand at the equilibrium price? 1. [Market Equilibrium I - Consumers and Producers Surplus, Price Floor] Find (a) equilibrium price, (b) equilibrium quantity, (c) consumers' surplus, (d) producers' surplus, and (e) total surplus for the following markets. Qd = 125-2P Qs =-45+8P. What is the surplus or shortage if the government imposes a price floor of P = 20 in this market. 2. [Market Equilibrium II - Consumers and Producers Surplus, Price Ceiling] Find (a) equilibrium price, (b) equilibrium quantity, (c) consumers' surplus, (d) producers' surplus, and (e) total surplus for the following markets. Qd 24+4P=0 13P - Q -27 = 0. What is the surplus or shortage if the government imposes a price ceiling of P = 2 in this market. 3. (20 points) Principal-Agent The probability of winning a soccer match is determined in part by the level of effort of the players. Assume that the probability of winning a match is given by the function: arrangements: Pw(e) Where e is the level of effort by the average player. e is continuous and bounded by the [0, 1] interval. If a team wins a match; the owner of the team will earn $1,000; if the match ends in a loss or a draw then the owner earns nothing. Then, the expected revenues are equal to R(e) = 1,000P(e). For simplicity assume that the only cost for the owner is to pay the salary of 11 players. Consider the following wage 2. w = = 1. w $ 150 (Regardless of the outcome of the game) R(e) R(e) - 900 3. w = 2ee² The utility of the average player is given by u(e) = w - 250e (a) (10 points) Find the optimal level of effort for each of the proposed wage arrangements. (b) (5 points) Find the profit level for each different contract. (c) (5 points) What's the optimal contract for the owner? (22) Consider a monopoly facing the demand curve P(Q) = 100 - 2Q. The monopolist's marginal cost is MC(Q) = 10+5Q, and its fixed cost is FC = 100. (a) Find the quantity the monopolist produces. (b) What is the market price? TWO (c) What is producer surplus? (d) what is profit? uvos tas What is the own-price elasticity of demand at the equilibrium price? 1. [Market Equilibrium I - Consumers and Producers Surplus, Price Floor] Find (a) equilibrium price, (b) equilibrium quantity, (c) consumers' surplus, (d) producers' surplus, and (e) total surplus for the following markets. Qd = 125-2P Qs =-45+8P. What is the surplus or shortage if the government imposes a price floor of P = 20 in this market. 2. [Market Equilibrium II - Consumers and Producers Surplus, Price Ceiling] Find (a) equilibrium price, (b) equilibrium quantity, (c) consumers' surplus, (d) producers' surplus, and (e) total surplus for the following markets. Qd 24+4P=0 13P - Q -27 = 0. What is the surplus or shortage if the government imposes a price ceiling of P = 2 in this market.