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Unformatted text preview: The University of Toronto Department of Economics ECO200Y5 Intermediate Microeconomics September 2009 1. preferences and indifference curves 1. What axioms are violated by indifference curves that cross? 2. Graph and describe indifference curves for the utility functions Cobb Douglas U = X 2/3 Y 1/3 , U = X 1/2 Y 1/2 and U = X 1/3 Y 2/3 . Linear Utility U = 3X + 2Y and Perfect Complements U = min(3X, 2Y) . 3. Explain, using the Axioms of Preferences why Indifference Curves for Good Y , representing yummy candy bars, an economic good, and Good X , representing mosquito bites, an economic bad, must slope upwards. No diagram is required. Which axioms are violated in this case? 4. A deadly poison Good X (an economic bad) and the antidote Good Y (an economic good) must be consumed together. For every two ml of poison the victim must consume at least three ml of antidote. More antidote is ¡ne, but any less is fatal. Describe the U = 6 and U = 12 indifference curves. Do these indifference curves cross? Does this matter? What Axioms do these crossing indifference curves violate? 5. Is diminishing marginal utility required to ensure convex indifference curves? Does the Cobb Douglas function U = X 2 Y 2 exhibit diminishing marginal utility? If not, why are the indifference curves convex? 6. Indifference Curves for Non-essential Goods intersect the axis. Indifference Curves for Essential Goods do not. Explain the logic behind this convention. 7. Axiom 3 requires that preferences are convex. Brie¢y restate the mathematical de¡nition of the axiom. Consider the U = 100. Show that indifference curve for U = XY will obey your de¡nition of convexity. What about U = min(Y, 2X) and U = X 2 + Y 2 ? _____________________________________________________________________________________________________________________ ECO200 Lee Bailey University of Toronto, Department of Economics September 2009 2. utility maximization 8. Mathematically derive the demand curve for Good X and calculate own price, cross price and income elasticities. Evaluate these elasticities at M = 12, P X = 3, P Y = 2 and comment on their meanings. Do this exercise for the utility functions: U = XY, U = min(Y, 2X), U = X + Y and U = X 2 + Y 2 9. Derive the demand curve for Good X associated with the the utility function U = aX + bY . How does this demand curve shift when P Y changes? How does this demand curve shift if the parameter b increases? Explain why. 10. Derive the income, cross price and own price elasticiticies for a general Cobb Douglas utility function U = X a Y b . What general feature of Cobb Douglas means that the demand curves must always be unit elastic? 11. Avery has preferences over goods X and Y deFned by U = X 2 + Y 2 . She has an income of M = $60 and faces competitively determined prices P X = $5 and P Y = $6. Solve for her optimal consumption bundle. Graph her demand curve. Comment on the results....
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