Spring 2010 - Exam 1

Spring 2010 - Exam 1 - ECONOMICS 201 SPRING 2010 EXAM 1...

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ECONOMICS 201 SPRING 2010 EXAM 1 Instructions: Please mark your answers clearly. No class notes or other materials are allowed. 100 points total. Each question is worth 5 points with the exception of Question 6 (15 points), Question 7 (15 points), and Question 15 (10 points). 1. Dora likes hugs (good 1) and kisses (good 2). For the first 25 units of good 1 purchased, the price of good 1 is $2; for any units purchased beyond 25, the price ofgood 1 is $1. The price ofgood 2 is (always) $1. Her income is $100. Draw Dora's budget set. Label the intercepts and slopes. ~ooA ~ -------~ jod. \ 2. True or false: If we know a consumer's utility function but don't know prices, we are always able to determine whether two goods are perfect substitutes. Explain your answer with a mathematical expression and/or one or two sentences. 3. Let's say that Lady Gaga is indifferent between bundles (1,10) and (10,1). Which of the following leads us to conclude that her preferences are convex? Hint: The line connecting (1,10) and (10,1) is = -Xl + 11; her preferences satisfy monotonicity. (Please circle one response.) X 2 a. She strictly prefers (5,5) to (1,10). b. She strictly prefers (10,1) to (5,5). c. She strictly prefers (6,6) to (1,10). d. She strictly prefers (10,1) to (5,6). 4. True or false: While total utility derived from good 1 and good 2 may differ, marginal utility derived from good 1 and good 2 is equal at the optimal bundle (for an interior solution). Explain your answer with a mathematical expression and/or one or two sentences. 1
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] 5. Brad Pitt's utility function is u(XI'X 2 )=aJq +bX 2 ,where a,b>O. a. What is his MRS? b. Suppose that we transform Brad's utility function using v(u(X] ,X 2 )) = U(XI'X2)2. What are MU] and MU 2 based on the transformed function v? c. What is his MRS based on the transformed function v? How does it compare to your response in part a? 6. Benjamin Franklin likes red (good 1), white (good 2), and blue (good 3). Prices are (~, P 2 , P 3 ) , and income is m. His utility function is u(X p X 2 ,X 3 ) = X; X~ + X 3 , where a,b > 0 and a +b < 1. (15 points) a. What is his UMP (utility maximization problem)? b. What is the Lagrangian? What are the first-order conditions (FOCs) (with respect to XI ,X 2 ,X 3 , and A )? c. Use the FOCs to show that MRS I ,3 =Price Ratio l ,3' d. It turns out that XI (~, P2,P 3 ,m) = ( P 2 JB(~Jr, where B,y < O. Are red (good 1) and white bP 3 a~ (good 2) complements or substitutes? Take a derivative ofthe demand function. 2
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7. Al Capone consumes cigars (good I) and brandy (good 2). His preferences are U(XPX2) = min{XI'3X2}' Also, ~ = 1, P2 = 2, and m 20. (15 points) a. What is his UMP (utility maximization problem)? b. In this specific case, what are
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Spring 2010 - Exam 1 - ECONOMICS 201 SPRING 2010 EXAM 1...

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