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1. John is a student at university who spends most of the stipend he receives from his parents on meals m ,books b andonmov iet ickets t . His preferences are given by the utility function U ( m, b, t )=2log m +log b +log t ( i ) If the price of a meal is 5\$, the price of a book is 25\$ and the price of a ticket is 10\$, and John’s stipend is 400\$ per month, how will he allocate his stipend each month? ( ii ) John’s parents are concerned that over all the movies that he watches, he does not read enough books. His parents consider reducing his stipend by 100\$. By how much will such a change in the stipend reduce the number of movies John watches? How will his consumption of meals and books be a f ected? Do you think this is an e f ective measure to reduce the time John spends at the movies? ( iii ) Ultimately, John’s parents are most concerned that John eats well and reads enough books for his exams, and do not care about his movie consumption. Their preferences for John can be expressed by the utility function V ( m, b )=log m +log b If they could dictate how John should allocate his stipend, how would they like him to spend his money? ( iv ) The parents realize they cannot monitor how their son spends his stipend, but they nevertheless want to in f uence his behavior. They decide to reduce his stipend to some new amount X , and they decide to subsidize each purchase of a book by an additional Y dollars. Rewrite John’s budget line for his parent’s new policy, and solve the new optimization problem for a given allowance X and book subsidy Y . ( v ) How do John’s optimal purchases of m , b ,and t depend on X and Y ? What are his purchases, if X =200 and Y =20 . What is the total cost to his parents? Show that this new agreement reduces John’s utility, but increases his parent’s utility. ( vi ) Suppose that John’s parents want to spend a total of 400\$ per month for the stipend and the book subsidy. Use John’s optimal choice of b to F nd the parental budget constraint between the allowance X and the subsidy Y . Sketch the parental budget constraint. Explain, why this budget constraint is no longer a straight line. ( vii ) Use John’s optimal choices of m , b and t ,g iventheva lueso f X and Y ,toder ive the utility that his parents obtain from a stipend X and a subsidy Y . ( viii ) Set up the parents’ maximization problem and solve for the optimal stipend X and book subsidy Y . ( ix ) Brie f y comment on your results. When trying to in f

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