The First-Order Condition for a Pareto-Efficient Allocation

Deriving the Profit-Maximizing Condition

An agent's net benefit from producing a quantity (Q) of a good is given by the objective function U(Q) = 80Q - 2Q^2 - 30. To find the quantity that maximizes this benefit, the first step is to derive the first-order condition. What is the correct first-order condition for this maximization problem?

A company's profit as a function of the quantity (Q) it produces is given by the equation Π(Q) = 100Q - 5Q^2. To find the quantity that maximizes profit, the first-order condition is derived by setting the profit function itself equal to zero (i.e., 100Q - 5Q^2 = 0).

Deriving the First-Order Condition for Utility Maximization

An agent's objective is to choose a quantity (Q) to maximize their net benefit, which is represented by the function B(Q). Arrange the following steps in the correct logical order to find the first-order condition that identifies the optimal quantity.

Critiquing the Derivation of an Optimal Choice Condition

An agent's objective is to choose a quantity (Q) to maximize their net benefit. Match each net benefit function with its corresponding first-order condition, which is used to find the optimal quantity.

An agent's net benefit from an activity is described by the function B(Q) = 150Q - 3Q^2. To find the quantity (Q) that maximizes this benefit, the first step is to find the derivative of the benefit function with respect to Q. This derivative, which is then set to zero to form the first-order condition, is ____.

Analyzing an Optimization Attempt

A social planner's objective is to choose a quantity (Q) of a good to maximize net social benefit. The objective function is represented as the difference between total benefits, B(Q), and total costs, C(Q):

W(Q) = B(Q) - C(Q) = (200Q - 2Q^2) - (20Q + 3Q^2)

To find the optimal quantity that maximizes W(Q), an analyst proposes finding the first-order condition by setting the derivative of the benefit function equal to the derivative of the cost function (i.e., B'(Q) = C'(Q)).

Evaluate the analyst's proposed method.