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Answers to Selected Questions in Practice Problem Set 2 ECON 11 Summer Session C 2010 Microeconomic Theory Yong Yang 1. [Utility maximization] (a) A line conneting (0,30) and (45,0). A ﬁgure is omitted. (b) No because the MRS is not the same with the price ratio. (c) The optimal consumption is X = 30 and Y = 10. (d) X = 20 and Y = 10. (e) A curve passing through points (c) and (d). A ﬁgure is omitted. 2. [Marginal rate of substitution] (a) MU X = Y 2 and MU Y = 2 XY . (b) MU X does not depend on X and is constant given Y . So it is not decreasing in X . MU Y is increasing in X , but not decreasing. (c) The MRS is MU X MU Y = Y 2 X . It gets smaller as X increases, so the MRS is decreasing in X . (d) False. In the above example, the MRS is diminishing although the marginal utilies are not decreasing. 3. [Cardinality of utility function] (a) X = 50 and Y = 30. (b) X = 50 and Y = 30, which is the same as above. (c) X = 50 and Y = 30, which is the same again. (d) Note that Ted, Emily, and Alex have the same MRS= Y X , so the condition for optimal consumption is the same when the price ratio is the same. If even the income is the same, they would choose the same consumption. The statement is true. (e) Jessica’s marginal utilities are MU X = f 0 ( XY ) Y MU Y = f 0 ( XY ) X where f 0 ( · ) denotes the ﬁrst order derivative of f with respect to its argument. Then her MRS is given by MRS = f 0 ( XY ) Y f 0 ( XY ) X = Y X which is the same as Ted’s. So she would choose the same consumption given the same price ratio and the same income as Ted faces. 4. [Derivation of demand curve] The MRS is

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