Question 1 PLEASE

Part 1: Consider a principal-agent relationship between Coach and Paulo. Paulo is an elementary school kid who plays soccer; he can choose to practice either 2 times a week, the mandatory minimum, or 4 times a week. Since he is a key player on the team, when he practices 4 times a week, compared to the mandatory 2 , he has a big influence on his team's winning probability: If the team wins, they receive a prize of $400, and 0 if the team loses. After every game Coach takes the players to a restaurant and buys them a dinner, spending f (on food). Coach is maximizing team's expected income Y=bf where b is the prize. Paulo enjoys the post-game dinner and is risk-averse, his utility function is U=fp where p is the number of practices. Paulo can also quit the team and learn economics instead, with guaranteed utility of U=3. 1) Suppose first that Coach can call Paulo's mom and ask her about the number of times Paulo practiced. He will then spend on food f2 if Paulo practices 2 times and f4 if he practices 4 times. a) If Coach wants Paulo to practice twice, how much does he need to spend on food f2 ? Write the participation constraint for Paulo for p=2. What will be the team's expected income Y2 ? b) If Coach wants Paulo to practice four times a week, how much does he need to spend on food f4 ? Write the participation constraint for Paulo for p=4. What is team's expected 2) Now suppose effort isn't observable, but only the outcome. The Coach will decide whether to link spending on food to the team's result. a) If Coach does not link spending on food to the team result, will Paulo's effort depend on spending? How much will Coach spend on food fa ? Write the participation constraint. Explain why the incentive compatibility constraint is not relevant here. What will be team's expected income Y2a ? b) If Coach decides to link spending to the team result to incentivize Paulo to practice four times a week, how much will Coach spend if the team wins fW and if the team loses fL ? Write the participation constraint and the incentive compatibility constraint. What will be team's expected income Y2b ? **Note: if Paulo is indifferent between 2 and 4 practices, then he will choose 4 to help his team win c) What will Coach decide? What will be Paulo's equilibrium utility U, given Coach's optimal choice? d) Draw the game tree for this sequential game. Coach moves first, deciding whether to link spending on food to team's result. Paulo moves second, choosing his level of effort. Indicate payoffs to players for each feasible outcome of the game. Use the check method to mark the Nash equilibrium. 3) Paulo's parents become concerned that their son is not getting enough food in case of a loss. They demand that no matter what the result of the game is, Coach spends at least fmin= $25 on food. a) Will Coach choose again to link spending on food to team's result, given the minimum spending constraint? What will be Paulo's equilibrium utility U, given Coach's optimal choice? b) Draw the new game tree, indicating payoffs to players for each feasible outcome of the game. Use the check method to mark the Nash equilibrium. c) Is the minimum spending rule beneficial for Paulo? Does it help the team win more matches? Part 1: Consider a principal-agent relationship between Coach and Paulo. Paulo is an elementary school kid who plays soccer; he can choose to practice either 2 times a week, the mandatory minimum, or 4 times a week. Since he is a key player on the team, when he practices 4 times a week, compared to the mandatory 2 , he has a big influence on his team's winning probability: If the team wins, they receive a prize of $400, and 0 if the team loses. After every game Coach takes the players to a restaurant and buys them a dinner, spending f (on food). Coach is maximizing team's expected income Y=bf where b is the prize. Paulo enjoys the post-game dinner and is risk-averse, his utility function is U=fp where p is the number of practices. Paulo can also quit the team and learn economics instead, with guaranteed utility of U=3. 1) Suppose first that Coach can call Paulo's mom and ask her about the number of times Paulo practiced. He will then spend on food f2 if Paulo practices 2 times and f4 if he practices 4 times. a) If Coach wants Paulo to practice twice, how much does he need to spend on food f2 ? Write the participation constraint for Paulo for p=2. What will be the team's expected income Y2 ? b) If Coach wants Paulo to practice four times a week, how much does he need to spend on food f4 ? Write the participation constraint for Paulo for p=4. What is team's expected 2) Now suppose effort isn't observable, but only the outcome. The Coach will decide whether to link spending on food to the team's result. a) If Coach does not link spending on food to the team result, will Paulo's effort depend on spending? How much will Coach spend on food fa ? Write the participation constraint. Explain why the incentive compatibility constraint is not relevant here. What will be team's expected income Y2a ? b) If Coach decides to link spending to the team result to incentivize Paulo to practice four times a week, how much will Coach spend if the team wins fW and if the team loses fL ? Write the participation constraint and the incentive compatibility constraint. What will be team's expected income Y2b ? **Note: if Paulo is indifferent between 2 and 4 practices, then he will choose 4 to help his team win c) What will Coach decide? What will be Paulo's equilibrium utility U, given Coach's optimal choice? d) Draw the game tree for this sequential game. Coach moves first, deciding whether to link spending on food to team's result. Paulo moves second, choosing his level of effort. Indicate payoffs to players for each feasible outcome of the game. Use the check method to mark the Nash equilibrium. 3) Paulo's parents become concerned that their son is not getting enough food in case of a loss. They demand that no matter what the result of the game is, Coach spends at least fmin= $25 on food. a) Will Coach choose again to link spending on food to team's result, given the minimum spending constraint? What will be Paulo's equilibrium utility U, given Coach's optimal choice? b) Draw the new game tree, indicating payoffs to players for each feasible outcome of the game. Use the check method to mark the Nash equilibrium. c) Is the minimum spending rule beneficial for Paulo? Does it help the team win more matches