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 function v(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 for β and α would result in a function with these properties?

Modeling Consumer Preferences

Analyzing a Utility Function's Properties

A consumer's satisfaction from consuming a quantity t of a good is modeled by the function v(t) = βt^α. The values of the parameters β and α determine 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 t of a good is described by the function v(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 t of a particular good is modeled by the functions v_A(t) = 10t^0.2 and v_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)?