Deriving the First-Order Condition for Optimal Quantity in a Constrained Choice Problem
After using the substitution method to express the objective function in terms of a single variable, quantity (Q), the next step in solving the constrained choice problem is to apply calculus. By differentiating this function with respect to Q and setting the result to zero, one can derive the first-order condition used to find the optimal quantity.
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Deriving the First-Order Condition for Optimal Quantity in a Constrained Choice Problem
Applying the Substitution Method in a Consumer Choice Problem
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.
Evaluating a Constrained Choice Solution
Rationale and Mechanism of the Substitution Method in Constrained Choice
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
Learn After
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.