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A General Form for v(t) in Quasi-Linear Utility (v(t) = βt^α)
The function is a common specification for the non-linear component of a quasi-linear utility function. To ensure this represents well-behaved, convex preferences, the function must be increasing and concave. These properties are achieved when its parameters satisfy the conditions and $0 < \alpha < 1$.
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Introduction to Microeconomics Course
The Economy 2.0 Microeconomics @ CORE Econ
Ch.5 The rules of the game: Who gets what and why - The Economy 2.0 Microeconomics @ CORE Econ
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A General Form for v(t) in Quasi-Linear Utility (v(t) = βt^α)
Independence of Marginal Utility from Income in Quasi-Linear Preferences
Marginal Utility of Income in a Quasi-Linear Function
A consumer's preferences are described as 'quasi-linear' if the utility function is linear with respect to one good (typically representing all other consumption) and non-linear with respect to another. A key implication of this form is that the marginal utility of the non-linear good does not depend on the quantity of the linear good. Given this information, which of the following utility functions,
u(x, m), represents quasi-linear preferences wherexis a specific good andmis money spent on all other goods?A consumer's preferences for a specific good
xand moneym(representing all other consumption) are described by the utility functionu(x, m) = 10√x + m. By analyzing the properties of this function, which statement accurately describes the consumer's behavior or preferences?Consider a consumer whose preferences for a specific good,
x, and money available for all other goods,m, can be represented by the utility functionu(x, m) = 20 * ln(x) + m. According to this model, if the consumer's income increases, their willingness to pay for an additional unit of goodxwill also increase.A utility function of the form u(x, m) = v(x) + m is said to represent 'well-behaved' quasi-linear preferences. For this to be true, the utility from good x must be increasing (meaning its first derivative, v'(x), is positive), and there must be diminishing marginal utility for good x (meaning its second derivative, v''(x), is negative), for all x > 0. Which of the following specifications for v(x) satisfies both of these conditions?
Analyzing Preferences with a Quasi-Linear Model
Modeling Consumer Preferences for Different Goods
An individual's preferences are modeled by a utility function
u(x, m), wherexis the quantity of a specific good andmis the amount of money available for all other goods. Match each utility function to the statement that correctly describes its marginal utility properties.A utility function of the form
u(x, m) = v(x) + mis referred to as 'quasi-linear' because while it is typically non-linear with respect to goodx, it is perfectly linear with respect to the variable ______, which represents an individual's income available for other goods.Evaluating Model Suitability for Different Goods
Critical Evaluation of the Quasi-Linear Utility Model
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Example of a Quasi-Linear Utility Function (u(t,c) = βt^α + c)
Positive Parameters and the Increasing Property of v(t) = βt^α
Parameter Constraints and the Concavity of v(t) = βt^α
A consumer's preference for a good,
t, is represented by the functionv(t) = βt^α. For this function to accurately model well-behaved preferences, it must be both increasing (meaning more of the good is always better) and concave (meaning the additional satisfaction from each extra unit of the good decreases). Which of the following parameter sets forβandαwould result in a function with these properties?Modeling Consumer Preferences
Analyzing a Utility Function's Properties
A consumer's satisfaction from consuming a quantity
tof a good is modeled by the functionv(t) = βt^α. The values of the parametersβandαdetermine the shape of this function and its economic interpretation. Match each set of parameter constraints to the corresponding description of the function's properties.In the function v(t) = βt^α, which represents a consumer's satisfaction from a quantity 't' of a good, setting the parameter α to a value greater than 1 (α > 1) implies that the consumer experiences increasing marginal satisfaction from each additional unit of the good.
Evaluating a Proposed Utility Function
A consumer's satisfaction from consuming a quantity
tof a good is described by the functionv(t) = 10t^0.5. The mathematical properties of this function (specifically, that it is increasing but concave) imply that the consumer experiences diminishing marginal ____ from consuming more of the good.Constructing a Valid Utility Component
Evaluating Economic Models for Consumer Preference
Consider two consumers, Alex and Ben, whose satisfaction from consuming a quantity
tof a particular good is modeled by the functionsv_A(t) = 10t^0.2andv_B(t) = 10t^0.8, respectively. Both functions represent valid, well-behaved preferences where more of the good is always preferred, but with diminishing added satisfaction. Which of the following statements accurately compares their preferences for quantities greater than one (t > 1)?