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 FILL OUT INFORMATION AS IT APPEARS IN ROSI LAST NAME: FIRST NAME: MIDDLE NAME: U TORONTO ID #: SIGNATURE: SCORES Question Points Score 1 15 2 25 3 35 4 25 TOTAL SCORE OUT OF 100 YOU MUST STAY SEATED DURING THE LAST 10 MINUTES OF THE TEST ‐‐ WAIT UNTIL ALL TESTS ARE COLLECTED ONLY AID ALLOWED: A CALCULATOR. FOR YOUR CONVENIENCE, THERE IS A WORKSHEET AT THE END OF THIS TEST KEEP ANSWERS BRIEF . TO 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൅ ሺܵെ 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൅ ሺܵെ ߙሻ s.t. ܥ ൅ ܵ ൌ ܻ eing an equality constrained optimization problem, set up the Lagrangean: max ஼,ௌ L ൌ ܥ൅ ሺܵെ ߙሻ B െ ߣሺܥ ൅ ܵ െ ܻሻ The first order conditions (FOCs) are: ߲ L ߲ܥ Page 2 of 18 ൌ 1 െ ߣ ൌ 0 ߲ L ߲ܵ ሺܵെ ߙሻ ߚ െ ߣ ൌ 0 ߲ L ߲ߣ ൌ 0 ՜ ܥ൅ܵ ൌ ܻ From the 1 st FOC: ߣൌ1
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: ሺܵെ ߙሻ ߚ S െ ߣ ൌ 0 ሺܵെ ߙሻ ߚ െ 1 ൌ 0 ܵൌߙ൅ߚ ubstitute 3 rd FOC: ܥ൅ܵൌܻ ܥൌܻെܵൌܻെ ሺߙ൅ߚሻ Thus, the complete solution to this UMP is: ൌߙ൅ߚ ܥൌܻെ ሺߙ൅ ߚሻ ߣൌ1 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: ܥൌܻെ ሺߙ൅ ߚሻൌ 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
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## ECO204_2009_Test_2_solutions - ECO 204, 2009 - 2010, Test 2...

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