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