Positive Parameters and the Increasing Property of v(t) = βt^α
For the function to be a valid component of a quasi-linear utility function, it must be an increasing function of time, . This property is confirmed by examining its first derivative, . This derivative is positive, and thus the function is increasing, because the parameters and are defined as positive, and free time is also a positive quantity.
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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)?
Learn After
A function used to model the satisfaction gained from an amount of free time, t, is given by v(t) = βt^α, where t is always a positive value. For this model to be consistent with the principle that more free time is always better than less, the function must be strictly increasing. Which of the following parameter sets for β and α would violate this principle?
Conditions for an Increasing Value Function
Evaluating a Utility Model's Parameters
The Role of Parameters in Utility Function Properties
A function is given by the form v(t) = βt^α, where the variable t is always a positive quantity (t > 0). For this function to be strictly increasing for all possible positive values of t, it is a necessary and sufficient condition that the product of the parameters, αβ, is also positive.
A function is defined as v(t) = βt^α, where t represents a quantity of time (t > 0). The properties of this function depend on the values of the parameters α and β. Match each parameter condition with the resulting mathematical property of the function v(t).
A function is given by v(t) = βt^α, where the variable t and the parameters α and β are all defined as strictly positive quantities. The first derivative of this function, which represents its rate of change, is v'(t) = αβt^(α-1). Given that all components (α, β, t) are positive, the value of the derivative v'(t) must always be ______, confirming that the function v(t) is always increasing.
Analysis of a Proposed Utility Function
A function is defined as v(t) = βt^α, where the variable t and the parameters α and β are all strictly positive quantities. To mathematically demonstrate that this function is always increasing with respect to t, a series of logical steps must be followed. Arrange the following steps into the correct logical sequence that proves the function is increasing.
Critique of a Utility Function Model