ps1_fall2011ECONW3211 - Department of Economics Columbia...

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Unformatted text preview: Department of Economics Columbia University W3211 Fall 2011 P r o b l e m Se t 1 I n t e r m e d i a t e M i c r o e co n om i cs P r of . Se y h a n E A r k o n a c Se c t i o n 3 1. Suppose a consumer’s utility derived from consuming bananas is described by the function U 10X 3X 2 (1/3)X 3. Compute marginal utility. a. Make a table showing total and marginal utility for X from 0 to 7 units. b. Would this individual ever choose to consume more than 7 units? Explain. 2. For each of the utility functions below, draw a set of indifference curves showing utility levels U 12, U 16, and U 24. Find MRS for each. a. U XY b. U X Y c. U X Y d. U lnX lnY e. U min{2X, Y} f. What is true about the commodities in (b)? g. What is true about commodities in (e)? h. What about the commodities in (c)? i. What can you say about the utility function in (a) and (d)? 3. Suppose a consumer has an income of $500 and faces prices pX 5 and pZ 10. a. Write the equation for the budget constraint. b. Draw the budget constraint, placing good X on the horizontal axis. Label it BC. c. What is the slope of BC? d. Suppose income decreases to $300. Draw the new budget constraint and label it RS. 4. Larry and Teri allocate their consumption between two goods: hats and bats. The price of hats is $4 each and the price of bats is $8 each. For Larry, the marginal utility of the last hat consumed was 8 and the marginal utility of the last bat was 24. For Teri the marginal utility of the last hat was 6 and the marginal utility of the last bat was 12. Which consumer is not maximizing his/her utility? How can you tell? How should he/she change their allocation? 5. Suppose a consumer has income of $120 per period, and faces prices pX 2 and pZ 3. Her goal is to maximize her utility, described by the function U 10X 0.5Z 0.5. Calculate the utility maximizing bundle ( X *, Z *) using the Lagrangian method. 6. Howie consumes only beer (B) and donuts (D) each week with his $100 income. Beer costs $1, while donuts cost only 50¢. Howie has Cobb-Douglas preferences given by: U(B,D) = B(D-d) where d is the quantity of donuts that his neighbor Nord consumes. Assume throughout this question that Howie always consumes more donuts than Nord. a . How does Nord's donut consumption influence Howie's utility function? Specifically, compute and determine the sign of ∂U/∂d. Intuitively, what does this tell you about Howie's happiness and Nord's consumption of donuts? b . Holding d fixed, compute Howie's optimal consumption bundle of Beer and Donuts as functions of Nord's consumption of donuts, d. c . How does Nord's consumption of donuts affect Howie's optimal bundle? Specifically calculate and determine the sign of ∂B*/∂d and ∂D*/∂d where (B*,D*) is Howie's optimal consumption bundle. F ol l ow i ng q u est i ons w i l l not b e g r a d e d , t h e y a r e f o r you to p r a c t i c e a n d w i l l b e d i sc usse d a t t h e r e c i t a t ion : 1. 2. 3. 4. 5. Ch. 3 question 7 Ch. 3 question 10 Ch. 3 problem 27 parts (a) and (b) only Ch. 3 problem 26 Ch. 3 problem 30 ...
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