A consumer's preferences for free time (t) and consumption (c) are represented by a utility function u = t^α * c^β. The marginal utility of free time is found by taking the partial derivative with respect to t, which yields α * t^(α-1) * c^β. Given this information, how can this expression for the marginal utility of free time be simplified and correctly re-expressed in terms of total utility (u) and free time (t)?

Calculating Free Time from Utility Data

Explaining the Cobb-Douglas Simplification

For a utility function u(t, c) = t^α * c^β, where u is total utility, t is free time, and c is consumption, the marginal utility with respect to free time is correctly simplified as β * (u / t).

Consider a consumer whose preferences for free time (t) and consumption (c) are represented by the utility function u(t, c) = t^α * c^β. The marginal utility of free time (MU_t) for this function can be simplified and expressed as α * (u/t). By applying the same simplification logic, the marginal utility of consumption (MU_c) can be expressed as ____.

Calculating Marginal Utility with Incomplete Information

A student is simplifying the expression for the marginal utility of free time for the utility function u = t^α * c^β. Arrange the following steps in the correct logical sequence to show this simplification.

Evaluating a Simplification Method's Applicability

Analyzing a Flawed Derivation

For a consumer with a utility function represented as u(t, c) = t^α * c^β, where t is free time and c is consumption, match each economic concept with its correct mathematical expression.