Opposing Slopes of Derivative Functions Guarantee a Unique Solution

An economist is analyzing a situation where an agent chooses an activity level, t, to maximize their net benefit. The optimal level is found where the marginal benefit of the activity equals its marginal cost. The analysis reveals that the marginal cost is a strictly increasing function of t. However, the economist finds that there are two distinct activity levels, t1 and t2, where the 'marginal benefit equals marginal cost' condition is met. What property of the marginal benefit function must be true to explain the existence of two solutions?

Consider an agent choosing a level of an activity, t, where the optimal choice is found by setting the marginal benefit equal to the marginal cost. If it is known that the marginal benefit is a strictly decreasing function of t and the marginal cost is also a strictly decreasing function of t, it is guaranteed that there will be at most one value of t that satisfies this optimality condition.

Conditions for a Unique Solution

Evaluating an Economic Model's Uniqueness Claim

The Role of Function Shape in Ensuring a Single Optimal Choice

In the context of a one-variable optimization problem where an agent chooses an activity level, t, to maximize net benefits (benefits minus costs), the optimal choice is found where marginal benefit equals marginal cost. Match each description of the model's components or outcomes to the corresponding mathematical or graphical property.

An economic model describes an agent's choice of an activity level, t. The net gain is defined by a benefit function minus a cost function. The optimal level of t is found where the marginal benefit equals the marginal cost. If the benefit function is strictly concave and the cost function is strictly convex, any solution that satisfies this optimality condition is guaranteed to be ________.

To demonstrate that a particular optimization problem has at most one solution, an economist constructs a logical argument. The problem involves an agent choosing an activity level, t, where the optimal choice occurs when marginal benefit equals marginal cost. Arrange the following statements to form a correct and logical proof for the uniqueness of the solution.

An economic model describes an agent's choice of an activity level, x > 0, to maximize their net benefit, defined as B(x) - C(x). The optimal choice, if an interior solution exists, is characterized by the condition where marginal benefit equals marginal cost. Which of the following specifications for the benefit function, B(x), and the cost function, C(x), is sufficient to guarantee that any solution satisfying this condition is a unique maximum?

In an optimization model, an agent's optimal choice of an activity level, x, is found where the marginal benefit of the activity equals its marginal cost. An analysis of the model reveals that for all positive levels of the activity, the marginal benefit is a strictly decreasing function. However, the marginal cost is found to be a U-shaped function (it first decreases and then increases as x increases). Based on this information, what can be concluded about the number of activity levels that satisfy the optimality condition?