Finding an Optimum for a Single-Variable Function using First and Second-Order Conditions

An individual's preferences are represented by the utility function U(t, c) = 4√t + c, where t is hours of free time and c is units of consumption. The individual's consumption is limited by a feasible frontier described by the equation c = 50 - t^2. To find the optimal choice of free time and consumption, the problem can be simplified by substituting the constraint into the utility function. Which of the following expressions correctly represents the individual's utility as a function of only free time, t?

Interpreting a Simplified Choice Problem

Error Analysis in Problem Simplification

An economist wants to find the combination of consumption (c) and free time (t) that maximizes an individual's utility, given that their choices are limited by a feasible production frontier. The problem is initially expressed with a utility function of two variables, u(t, c), and a constraint equation that links c and t. To simplify this into a problem of only one variable, the economist uses the substitution method. Arrange the following steps in the correct logical sequence for this method.

Critique of the Substitution Method in Constrained Choice

True or False: In a constrained choice problem where an individual maximizes utility u(t, c) subject to a feasible frontier c = f(t), the process of substituting the constraint into the utility function and finding the value of t that maximizes the resulting single-variable function is mathematically equivalent to finding the point on the feasible frontier where the marginal rate of substitution equals the marginal rate of transformation.

An economist is solving a constrained choice problem to find an optimal allocation. The problem involves maximizing a utility function of two variables, u(t, c), subject to a constraint c = f(t). The economist simplifies this by substituting the constraint into the utility function to create a new function of a single variable, H(t). Match each mathematical component from the original and simplified problems to its correct economic interpretation.

Formulating a Simplified Optimization Problem

The Role of the Constraint after Substitution

Interpreting a Simplified Utility Maximization Problem