Please help me with this Economics of Sports problem set. I can send you the corresponding lecture slides if needed.
Economics 261 Economics 411 Problem Set #1 Tuesday, February 4, 2020 You may submit your completed problem sets during class or you may put them in Juan Margitic’s mail box by 5:00 pm. 1. A baseball team receives total profit of (Q) + aQ. represents profit from ticket sales and aQ represents profit from sales of apparel. The demand curve for tickets is P = 100 – (Q/1,000) and the team has marginal cost equal to zero. (i) Find the equilibrium quantity and profit from ticket sales when a = 20 and when a = 40. (ii) Explain why the profit from ticket sales changes the way it does when a rises. 2. A football team has two kinds of fans. Adults demand QA(p) = 1,000(100 – P) tickets when the price is P. Senior citizens demand QS(p) = 1,000(60 – P). Marginal cost is equal to zero, and the stadium has capacity K. Find the profit-maximizing ticket prices when K < 80,000="" and="" resale="" of="" tickets="" is="" impossible.="" 3.="" a="" soccer="" team="" faces="" demand="" uncertainty.="" in="" the="" high-demand="" state,="" the="" quantity="" demanded="" is="" qh(p)="2,000(50" –="" p);="" in="" the="" low-demand="" state,="" it="" is="" ql(p)="2,000(30" –="" p).="" the="" probability="" that="" demand="" is="" high="" is="" h="" ="" [0,="" ½].="" marginal="" cost="" is="" mc="0," and="" the="" stadium="" has="" capacity="" k="40,000." (i)="" find="" the="" profit-maximizing="" ticket="" price,="" assuming="" that="" the="" team="" sells="" tickets="" before="" demand="" uncertainty="" is="" resolved.="" (ii)="" suppose="" that="" the="" team="" sells="" tickets="" after="" demand="" uncertainty="" is="" resolved="" instead.="" find="" the="" profit-maximizing="" prices.="" (iii)="" in="" which="" of="" the="" two="" scenarios="" is="" the="" expected="" value="" of="" social="" surplus="" higher?="" 4.="" a="" team’s="" demand="" curve="" for="" tickets="" is="" =="" (200="" −="" 10,000="" )="" (="" 100="" ),="" where="" p="" is="" the="" ticket="" price,="" q="" is="" the="" annual="" attendance,="" and="" w="" is="" the="" team’s="" winning="" percentage.="" marginal="" cost="" equals="" zero="" and="" fixed="" costs="" are="" 8,000w2.="" (i)="" find="" the="" optimal="" ticket="" price="" and="" quantity="" when="" the="" team’s="" winning="" percentage="" is="" w.="" (ii)="" find="" the="" optimal="" winning="" percentage.="" 5.="" two="" teams="" play="" each="" other="" 100="" times.="" team="" alpha="" receives="" gate="" revenue="" equal="" to="" aw="" –="" 5w2="" if="" it="" wins="" w="" games;="" team="" beta="" team="" receives="" bw="" –="" 5w2="" if="" it="" wins="" w="" games.="" assume="" that="" a="" and="" b="" are="" known="" constants="" and="" 500="">< b="">< a="">< 500+b. (the inequalities simply guarantee that an interior solution exists.) the teams have no other sources of revenue. (i) find the equilibrium number of wins for each team and the marginal cost of a win. explain briefly why beta’s number of wins and marginal cost of a win change the way they do when b rises slightly. (ii) now suppose that there is revenue sharing. each team keeps 60% of its revenue and receives 40% of the other team’s revenue. solve for the equilibrium number of wins for each team and the marginal cost of a win, assuming that a = 1,000 and b = 800. (iii) finally, suppose that there is a salary cap but no revenue sharing. assume that the cap is binding for both teams. explain briefly how the equilibrium here differs from the equilibrium in (i). 6. two players on a basketball team share playing time. mark’s expected number of points scored is sm(x) = 30x – 5x 2 if he plays the fraction x [0,1] of the game. john’s expected number is sj(x) = 25x – 5x 2 points if he plays the fraction x of the game. (i) suppose that mark plays the entire game. how many points does he expect to score? how many points would john expect to score if he played the fraction x = of the game? find the average scoring rate, sj()/, as approaches zero. (ii) what fraction of the time should mark play in order to maximize the total points scored by mark and john? find the expected number of points scored by each player per unit time. 500+b.="" (the="" inequalities="" simply="" guarantee="" that="" an="" interior="" solution="" exists.)="" the="" teams="" have="" no="" other="" sources="" of="" revenue.="" (i)="" find="" the="" equilibrium="" number="" of="" wins="" for="" each="" team="" and="" the="" marginal="" cost="" of="" a="" win.="" explain="" briefly="" why="" beta’s="" number="" of="" wins="" and="" marginal="" cost="" of="" a="" win="" change="" the="" way="" they="" do="" when="" b="" rises="" slightly.="" (ii)="" now="" suppose="" that="" there="" is="" revenue="" sharing.="" each="" team="" keeps="" 60%="" of="" its="" revenue="" and="" receives="" 40%="" of="" the="" other="" team’s="" revenue.="" solve="" for="" the="" equilibrium="" number="" of="" wins="" for="" each="" team="" and="" the="" marginal="" cost="" of="" a="" win,="" assuming="" that="" a="1,000" and="" b="800." (iii)="" finally,="" suppose="" that="" there="" is="" a="" salary="" cap="" but="" no="" revenue="" sharing.="" assume="" that="" the="" cap="" is="" binding="" for="" both="" teams.="" explain="" briefly="" how="" the="" equilibrium="" here="" differs="" from="" the="" equilibrium="" in="" (i).="" 6.="" two="" players="" on="" a="" basketball="" team="" share="" playing="" time.="" mark’s="" expected="" number="" of="" points="" scored="" is="" sm(x)="30x" –="" 5x="" 2="" if="" he="" plays="" the="" fraction="" x="" ="" [0,1]="" of="" the="" game.="" john’s="" expected="" number="" is="" sj(x)="25x" –="" 5x="" 2="" points="" if="" he="" plays="" the="" fraction="" x="" of="" the="" game.="" (i)="" suppose="" that="" mark="" plays="" the="" entire="" game.="" how="" many="" points="" does="" he="" expect="" to="" score?="" how="" many="" points="" would="" john="" expect="" to="" score="" if="" he="" played="" the="" fraction="" x="" of="" the="" game?="" find="" the="" average="" scoring="" rate,="" sj()/,="" as="" ="" approaches="" zero.="" (ii)="" what="" fraction="" of="" the="" time="" should="" mark="" play="" in="" order="" to="" maximize="" the="" total="" points="" scored="" by="" mark="" and="" john?="" find="" the="" expected="" number="" of="" points="" scored="" by="" each="" player="" per="" unit="">