M ath M e thods: EconomicsF ina l Exa m, May 12, 2021 Q uestion 4 (15 points)Suppose U (x, y) denotes the utility enjoyed by a perso n whe n havin g x hour s of le is u r e(hours not working) per day (24 hours) andy unit s per day of other goods. The perso ngets an hourly wage of w and pays a p ric e of p per unit of other goods. The individualspend s a ll of the mo ne y they earn and faces the follow ing budget constraint: py = w(24 x). Assume a utility function with the follow ing features Ux, Uy > 0; Uxy > 0; and Uxx, Uyy < 0. a)Set up the utility maximization problem usin g the Lagrangian method andstate the firs t order conditions. b)Let (x, y, ) denote the s o lut io n in the utility maximization problem. S o lvefo r x/w us in g C ra mer's rule and imp lic it functio n theore m. Be sure to state whether the conditions for the use of implicit function theorem hold. Show all your work. c)S ig n the comparative static solved fo r in part b.