Deriving the First-Order Condition for General Preferences Using the Chain Rule

Formulating a Constrained Choice Problem for Externalities

An economist is modeling the interaction between a chemical plant and a downstream fishery to find a Pareto-efficient level of pollution, Q. The fishery's utility is given by the function U(m, Q) = m * (100 - Q), where 'm' is its income and 'Q' is the units of pollution. The fishery's initial income is m_f^0. The chemical plant's profit from production is π(Q) = 40Q - 2Q^2. A transfer, τ, can be paid from the fishery to the plant. To find an efficient outcome, the economist sets up a problem to maximize the fishery's utility, subject to the constraint that the plant's final payoff is held constant at a level y_0. Which of the following mathematical statements correctly represents this constrained choice problem?

Modeling Pareto Efficiency with Externalities

An economist is modeling the interaction between a beekeeper and an adjacent apple orchard to find a set of Pareto-efficient outcomes. The economist formulates a constrained choice problem to maximize the beekeeper's utility by choosing the number of bee colonies (Q) and a monetary transfer (τ), subject to the constraint that the orchard owner's payoff is held constant at a level y_0. For this formulation to correctly identify the full set of all possible Pareto-efficient allocations, the constant y_0 must be set equal to the profit the orchard owner would earn in the absence of any bees.

An economist is setting up a constrained choice problem to find a Pareto-efficient allocation between two parties. The general mathematical formulation is: max_{Q, τ} u(m_1^0 - τ, Q) subject to τ + π_2(Q) = y_0. Match each component of this formulation to its correct economic interpretation.

Evaluating the Constrained Choice Framework for Pareto Efficiency

An economist is modeling the interaction between a factory and a local fishery to find a Pareto-efficient level of production, Q. The factory's profit is π(Q) = 100Q - 2Q^2. The fishery's utility is given by u(m, Q) = m - 5Q, where m is its monetary income. The fishery has an initial income of m_f^0, and a monetary transfer, τ, can be paid from the fishery to the factory. The economist sets up the following constrained choice problem: max_{Q, τ} (100Q - 2Q^2) subject to m_f^0 - τ - 5Q = y_0. Which statement best identifies the conceptual error in this formulation?

Constructing the Constraint for a Pareto Efficiency Problem

An economist is analyzing the interaction between a chemical factory and a nearby residential community to find a Pareto-efficient outcome. The factory's profit from producing quantity Q is given by π(Q) = 50Q - Q^2. The community's utility is affected by its net monetary wealth (m) and the pollution (Q), represented by the function u(m, Q) = m - 3Q^2. The community's initial wealth is m_c^0, and a monetary transfer, τ, can be paid from the community to the factory. The economist's specific goal is to formulate a problem that maximizes the factory's final payoff, subject to the constraint that the community's final utility is held constant at a level U_bar. Which of the following mathematical statements correctly represents this specific constrained choice problem?

An economist sets up a problem to find a Pareto-efficient allocation by maximizing Party A's utility, u_A(m_A^0 - τ, Q), subject to the constraint that Party B's payoff is constant, π_B(Q) + τ = y_0. To solve this, the economist first substitutes the constraint into the utility function to express it solely in terms of the variable Q. The resulting objective function to be maximized is u_A(______, Q).

An economist is modeling the interaction between a beekeeper and an adjacent apple orchard to find a set of Pareto-efficient outcomes. The economist formulates a constrained choice problem to maximize the beekeeper's utility by choosing the number of bee colonies (Q) and a monetary transfer (τ), subject to the constraint that the orchard owner's payoff is held constant at a level y_0. For this formulation to correctly identify the full set of all possible Pareto-efficient allocations, the constant y_0 must be set equal to the profit the orchard owner would earn in the absence of any bees.