Provide solutions to the following attachments
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: Pm (e) = 26 82 Where 6 is the level of effort by the average player. 6 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, OOOPw(e). For simplicity assume that the only cost for the owner is to pay the salary of 11 players. Consider the following wage arrangements: 1. w = $150 (Regardless of the outcome of the game) 2. w : 1203) 3. w = R(e) 900 The utility of the average player is given by 15(5) 2 w 2505 (a) (10 points) Find the optimal level of effort for each of the proposed wage arrangements. (b) (5 points) Find the prot level for each different contract. (c) (5 points) What's the optimal contract for the owner? 1. Tami starts her new job and wants to build a new wardrobe. Her utility for pants and shirts is estimated by the equation U = P0.7.50.3, where U is her total utility from consuming pants and shirts, P is the quantity of pants she buys, and S is the number of shirts she buys. Suppose she has budgeted $1,000 for her wardrobe, and the price of pants is $70/each, and shirts $30 each. a. Putting pants on the x-axis and shirts on the y-axis, draw Tami's indifference curve (estimate it) and income constraint for pants and shirts. b. How many pairs of pants and how many shirts will she buy if maximizing her utility is her objective? c. What is her total utility? d. Suppose that the price of pants fell to $50. Show using the drawing you already created, show the change in Tami's income constraint and indifference curve. 2. A poor college students spends $100 per week on food and all other goods (a composite good consisting of everything else you'd consume other than food). The price of food per unit is $8, and the per unit price of all other goods is $5, and the student's utility for the two goods is U = FO.8.A0.2, where F is units of food and A is units of all other goods. a. Draw this student's indifference curve and income constraint for food and all other goods, with food on the x-axis. b. Next, show what happens when the government subsidizes food purchased by students by giving them vouchers for 12 off the price of all food items.2) Consider the following demand and supply functions: Qi =24-4P, demand Q;=-6+2E,_,P, supply and expectations generator E, _P. = P, a) Calculate the long run price and quantity in this market. Justify your answer. b) Is this system a stable or unstable Cobweb model? Justify your answer. c) Beginning at P, =4 and Q, =8, calculate three years of prices and quantities in this market. Suppose instead that the demand and supply functions are: Qi =24-2P, demand Q;=-6+4E,_P, supply d) Repeat questions a) through c) where in c) part you begin with P, =4 and Q =16
