Solving the Constrained Choice Problem via the Substitution Method
To solve the constrained choice problem for Pareto efficiency, the substitution method is employed. First, the constraint equation, , is rearranged to express the monetary transfer () as a function of the production quantity (Q). This expression for is then substituted into the utility function of the party whose payoff is being maximized—in this case, the fishermen. This transforms their utility into the objective function, now dependent solely on the variable Q, which can then be maximized to find the optimal quantity.
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In the mathematical problem to find a Pareto-efficient allocation, a planner seeks to maximize the fisherman's payoff,
m_f^0 - τ - C_e(Q), by choosing the production quantity (Q) and a monetary transfer (τ), subject to the constraint that the plantation owner's payoff is held constant:τ + P^W Q - C_p(Q) = y_0. What is the primary economic reason for structuring the problem in this specific way?Formulating a Constrained Choice Problem for Externalities
A social planner is analyzing an externality between a steel mill and a laundry. The mill's profit is , and the laundry's profit is . Here, is the quantity of steel produced, is a monetary transfer, is the price of steel, is the mill's production cost, is the laundry's baseline income, and is the damage cost to the laundry from the mill's pollution. To find a Pareto-efficient allocation, the planner decides to maximize the steel mill's profit while holding the laundry's profit constant at a level . Which of the following correctly states this constrained choice problem?
A planner is setting up a problem to find a Pareto-efficient outcome between a chemical plant and a downstream fishery. The problem is stated as: Maximize the fishery's payoff,
Profit_F = R - D(Q) - τ, by choosing the plant's output level (Q) and a monetary transfer (τ), subject to the constraint that the plant's payoff is held constant at a levelk, whereProfit_P = P*Q - C(Q) + τ = k. Match each mathematical component to its role in this optimization problem.A student attempts to set up the constrained choice problem to find a single Pareto-efficient allocation between a fisherman and a plantation owner. Their formulation is as follows:
Objective: Maximize the fisherman's payoff,
m_f^0 - τ - C_e(Q), by choosing the production quantity (Q) and a monetary transfer (τ). Constraint: The plantation owner's payoff must satisfyτ + P^W Q - C_p(Q) ≥ y_0.What is the fundamental conceptual error in this formulation for the stated goal?
Consider the problem of finding a Pareto-efficient allocation between two parties by maximizing one party's payoff while holding the other's constant at a specific level. If we switch the roles—maximizing the second party's payoff while holding the first party's constant—the resulting efficient quantity of production (Q) will change.
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Learn After
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An analyst is trying to determine the optimal production level,
q, that maximizes Firm 1's profit, subject to the constraint that Firm 2's profit is held constant at a specific level,k.Firm 1's profit function (the objective function) is:
π₁ = 20q - tFirm 2's profit function (used for the constraint) is:π₂ = 12q + t - q²The constraint is:π₂ = kTo solve this, the analyst uses the substitution method to create a new objective function that depends only on the variable
q. Which of the following expressions represents the correct objective function for the analyst to maximize?A student is tasked with finding the optimal level of an activity, Q, that maximizes a payoff function, P(Q, t), subject to a constraint C(Q, t) = k, where 't' is a transfer payment. To solve this, they will use the substitution method. Arrange the following core steps of this method into the correct logical order.
A student aims to solve a constrained choice problem by maximizing an objective function, U = f(x, y), subject to a constraint, k = g(x, y), where 'k' is a constant. The student's first step is to rearrange the objective function to solve for 'x', yielding x = h(U, y). The student then substitutes this expression for 'x' into the constraint equation, resulting in k = g(h(U, y), y).
True or False: The student has correctly applied the substitution method to create a new, single-variable objective function ready for maximization.
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A consultant is solving a constrained choice problem. The goal is to maximize the objective function
P = 50Q - tsubject to the constraintC = 100 - 10Q - t. The constraintCmust be held constant at a value of 20. To solve this, the consultant first uses the substitution method to express the objective functionPin terms of the single variableQ. The resulting single-variable objective function isP = ____.An economist wants to maximize a firm's utility,
U = 10Q + 2t, subject to a profit constraint,π = 50Q - t = 1000, where Q is output and t is a transfer. They use the substitution method to solve this problem. Match each mathematical component from this process to its correct role or description.Critique of a Constrained Optimization Approach
An economist is attempting to solve a constrained choice problem. The goal is to maximize a utility function,
U(x, y) = 10x + 2y, subject to the constraint5x + y = 100. The economist performs the following steps:- Rearranges the utility function to solve for
y:y = (U - 10x) / 2. - Substitutes this expression for
yinto the constraint equation, resulting in:5x + (U - 10x) / 2 = 100. - Prepares to solve this final equation for
xto find the optimal quantity.
What is the fundamental flaw in the economist's approach?
- Rearranges the utility function to solve for