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ECO204_2009_Test_2_solutions

ECO204_2009_Test_2_solutions - ECO 204 2009 2010 Test 2...

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ECO 204, 2009 - 2010, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission University of Toronto, Department of Economics , ECO 204 2009 2010 . Ajaz Hussain Test 2 Solutions F ILL O UT I NFORMATION AS IT APPEARS IN ROSI LAST NAME: FIRST NAME: MIDDLE NAME: U T ORONTO ID #: SIGNATURE: SCORES Question Points Score 1 15 2 25 3 35 4 25 TOTAL SCORE OUT OF 100 Y OU MUST STAY SEATED DURING THE LAST 10 MINUTES OF THE TEST ‐‐ WAIT UNTIL ALL TESTS ARE COLLECTED ONLY A ID ALLOWED : A C ALCULATOR . F OR YOUR CONVENIENCE , THERE IS A WORKSHEET AT THE END OF THIS TEST K EEP ANSWERS BRIEF . T O EARN CREDIT , SHOW CALCULATIONS AND GIVE ARGUMENTS TO SUPPORT YOUR ANSWERS Good luck! Page 1 of 18 S . Ajaz Hussain, Dept. of Economics, University of Toronto
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ECO 204, 2009 - 2010, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 1 [15 Points] In this 1 period problem, a consumer allocates her income ܻ between consumption ܥ and savings ܵ . Her utility function is: ܷ ൌ ܥ ൅ ሺܵ െ ߙሻ ߚ, ܻ ൐ 0 Assume ߙ, . (a) [10 Points] Solve the consumer’s UMP. Show all calculations and clearly state any assumptions. Answer: By definition, income must be allocated to consumption and savings. The consumer’s UMP is: max ஼,ௌ ܷ ൌ ܥ ൅ ሺܵ െ ߙሻ s.t. ܥ ൅ ܵ ൌ ܻ eing an equality constrained optimization problem, set up the Lagrangean: max ஼,ௌ L ൌ ܥ ൅ ሺܵ െ ߙሻ B െ ߣሺܥ ൅ ܵ െ ܻሻ The first order conditions (FOCs) are: ߲ L ߲ܥ Page 2 of 18 ൌ 1 െ ߣ ൌ 0 ߲ L ߲ܵ ሺܵ െ ߙሻ ߚ െ ߣ ൌ 0 ߲ L ߲ߣ ൌ 0 ՜ ܥ ൅ ܵ ൌ ܻ From the 1 st FOC: ߣ ൌ 1 S . Ajaz Hussain, Dept. of Economics, University of Toronto
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ECO 204, 2009 - 2010, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Page 3 of 18 ubstitute in second FOC: ሺܵ െ ߙሻ ߚ S െ ߣ ൌ 0 ሺܵ െ ߙሻ ߚ െ 1 ൌ 0 ܵ ൌ ߙ ൅ ߚ ubstitute 3 rd FOC: ܥ ൅ ܵ ൌ ܻ ܥ ൌ ܻ െ ܵ ൌ ܻ െ ሺߙ ൅ ߚሻ Thus, the complete solution to this UMP is: ൌ ߙ ൅ ߚ ܥ ൌ ܻ െ ሺߙ ൅ ߚሻ ߣ ൌ 1 Observe that her savings are always independent of income. This, of course, is due to her quasi near preferences (recall that ܷ ൌ ܳ ൅ ݂ሺܳ then ܳ is always independent of income). this consumer “g S in ܵ li (b) [5 Points] Is it possible that oes hungry” (i.e. consume nothing)? Answer: It is possible that this consumer consumes nothing. From above, note that she always saves a dollar amount $ሺߙ ൅ ߚሻ and consumes the remainder of her income. Thus, if her 0 ܻ ൌ ߙ ൅ ߚ constant income is less than ሺߙ ൅ ߚሻ she will consume nothing: ܥ ൌ ܻ െ ሺߙ ൅ ߚሻ ൌ S . Ajaz Hussain, Dept. of Economics, University of Toronto
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ECO 204, 2009 - 2010, Test 2 Solutions This test is copyright material and may not be used for commercial purposes without prior permission Page 4 of 18 Question 2 5 Points]
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