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ajaz_204_2013_2014_HW_5

ajaz_204_2013_2014_HW_5 - University of Toronto Department...

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University of Toronto, Department of Economics, ECO 204, 2013 - 2014 1 ECO 204 2013 2014 HW 5 For feedback, comments and suggestions, please e-mail [email protected] Fixed typos are shaded in yellow __________________________________________________________________________________________________ Question 1 Consider a (rational) consumer with an arbitrary utility function ( ) defined over the consumption set {( ) } (a) Write down the Lagrangian equation for the following (constrained optimization) general utility maximization problem: ( ) (b) Show that if one of the goods (say, good 1) is a “good” good then the consumer will spend her entire income (or the budget allocated to these two goods). (c) As you’ve shown in part (b), assuming that at least one good (say good 1) the consumer’s UMP becomes: ( ) Use the envelope theorem to interpret the Lagrange multipliers (i.e. after you solve the UMP, what will the Lagrange multipliers tell you?). (d) Suppose good 1 is a “good” good and that good 2 is a “neutral” good. True or false: a small incre ase in the minimum amount of good 2 that the consumer must consume (i.e. ) will definitely lower the consumer’s optimal utility?
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University of Toronto, Department of Economics, ECO 204, 2013 - 2014 2 Question 3 Consider a (rational) consumer with an arbitrary utility function ( ) defined over the consumption set {( ) } Assume the consumer like s at least one good (so that she’ll spend her entire income). Solve the general (two good) UMP (i.e. derive all possible scenarios and their respective conditions): ( ) Question 4 Consider a (rational) consumer with a Cobb-Douglas utility function defined over the consumption set {( ) } Assume that Solve the Cobb-Douglas UMP by taking a log positive monotonic transformation of the Cobb-Douglas utility function. You should compare your answers to the regular Cobb-Douglas UMP. Hint: Exploit the “rules” derived from the general UMP. Question 5 Consider a (rational) consumer with the utility function ( ) defined over the consumption set {( ) } Assume . Solve this UMP. Hint: Exploit the “rules” derived from the general UMP.
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University of Toronto, Department of Economics, ECO 204, 2013 - 2014 3 SOLUTIONS Question 1 Consider a (rational) consumer with an arbitrary utility function ( ) defined over the consumption set {( ) } (a) Write down the Lagrangian equation for the following (constrained optimization) general utility maximization problem: ( ) Answer First re-arrange the inequality constraints: ( ) ( ) Then write down the Lagrangian equation: ( ) [ ] [ ] [ ] ( ) [ ] (b) Show that if one of the goods (say, good 1) is a “good” good then the consumer will spend her entire income (or the budget allocated to these two goods).
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