Analyzing Preference Invariance
A microeconomist claims that the utility functions U(x, y) = x^0.3 * y^0.7 and V(x, y) = 3ln(x) + 7ln(y) represent identical preferences for a consumer choosing between positive quantities of goods x and y. Analyze this claim. Your response should first explain whether the claim is correct and why, based on the principle of ordinal utility. Then, provide a mathematical demonstration to support your conclusion, focusing on a key measure of consumer trade-offs that must be the same for both functions.
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CORE Econ
Economics
Social Science
Empirical Science
Science
Economy
Introduction to Microeconomics Course
The Economy 2.0 Microeconomics @ CORE Econ
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Consider two different utility functions for a consumer who consumes two goods, x and y: U₁(x, y) = x²y and U₂(x, y) = 2ln(x) + ln(y). Assuming the quantities of both goods are positive, which of the following statements accurately describes the relationship between the preferences represented by these two functions?
Comparing Consumer Choices
Demonstrating Preference Equivalence
A consumer's preferences for two goods (x and y, where quantities are positive) can be represented by the utility function U₁(x, y) = xy. It follows that this consumer prefers the bundle (4, 3) over the bundle (2, 5).
True or False: If this same consumer's preferences were instead represented by the utility function U₂(x, y) = 10 - (1/xy), they would prefer the bundle (2, 5) over the bundle (4, 3).
A consumer's preferences for two goods, X and Y (where quantities are positive), are represented by a utility function in the left column. Match this original function to the function in the right column that represents the exact same set of preferences (i.e., would result in the same indifference map and the same ranking of any two consumption bundles).
Analyzing Preference Invariance
A consumer's preferences for goods X and Y (with quantities X>0, Y>0) are described by the utility function U(X, Y) = X²Y. For the consumption bundle (10, 5), the marginal rate of substitution (the rate at which the consumer is willing to trade Y for X) is 1.
Now, suppose the same consumer's preferences are instead represented by a different function, V(X, Y) = 0.5 * ln(X²Y) + 20. The marginal rate of substitution for this consumer at the same bundle (10, 5) must be _____.
Comparing Consumer Choice Models
A consumer's preferences for two goods, Apples (A) and Bananas (B), where quantities are always positive, are represented by the utility function U(A, B) = AB. An economist proposes several alternative functions to model this consumer's behavior. Which of the following functions would represent a different set of underlying preferences, meaning it would not result in the same ranking of all possible consumption bundles?
Evaluating Competing Economic Models