Figure E5.4 - Angela's Optimal Choice as an Independent Farmer

Verifying the Maximality of Angela's Optimal Choice Using the Second-Order Condition

Calculating Optimal Resource Allocation

An individual chooses between hours of free time (t) and units of consumption (c). The marginal rate of substitution (the rate they are willing to trade) is MRS = c / t. The marginal rate of transformation (the rate they are able to trade) is MRT = 2. The boundary of their feasible set is defined by the equation c = 40 - 2t. What is the optimal combination of free time and consumption that maximizes their utility?

Evaluating an Agent's Production-Leisure Choice

An individual chooses between hours of free time (t) and units of a consumption good (c). The marginal rate at which they are willing to trade consumption for an extra hour of free time is given by the expression 2 / √t. The marginal rate at which they can technically transform free time into consumption is 2 / √(48 - 2t). Suppose this individual is currently choosing to have t = 8 hours of free time. Which of the following statements accurately analyzes their situation?

An individual chooses between hours of free time (t) and consumption. Their marginal rate of substitution (willingness to trade) is MRS = 10/t. Their marginal rate of transformation (ability to trade) is MRT = 2. True or False: The individual's optimal choice is to have 4 hours of free time.

Evaluating a Farmer's Time Allocation Advice

An individual seeks to maximize their satisfaction by choosing an optimal combination of free time and consumption. To find this optimal choice, they must follow a specific procedure. Arrange the following steps into the correct logical sequence.

An individual makes a choice between hours of free time (t) and units of consumption (c). Their willingness to trade consumption for an extra hour of free time is represented by the expression 8 / (t+1). The rate at which they can technologically transform an hour of free time into consumption is constant at 2 units. If this individual is currently choosing 5 hours of free time, which statement accurately analyzes their situation?

An individual is choosing their optimal combination of free time and consumption. Match each economic condition with its correct interpretation regarding the individual's choice.

An economics student is solving for an individual's optimal choice between consumption (c) and free time (t). The individual's willingness to trade is MRS = 40 / t and the technical possibility of trading is MRT = 4 / √(20 - t). The student's work is presented below:

Step 1: Set MRS = MRT → 40 / t = 4 / √(20 - t) Step 2: Simplify the equation → 10 / t = 1 / √(20 - t), which rearranges to 10 * √(20 - t) = t Step 3: Square both sides → 100 * (20 - t) = t^2 Step 4: Expand and rearrange into a quadratic equation → 2000 - 100t = t^2, which becomes t^2 + 100t - 2000 = 0 Step 5: Factor the quadratic equation → (t - 20)(t + 100) = 0 Step 6: Conclude that the solutions are t = 20 or t = -100. Since time cannot be negative or equal to 20 (which would make the MRT undefined), the student concludes there is no valid solution based on their work.

Analyze the student's work. Which statement best identifies the primary flaw in their reasoning?