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 - t Firm 2's profit function (used for the constraint) is: π₂ = 12q + t - q² The constraint is: π₂ = k

To 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 - t subject to the constraint C = 100 - 10Q - t. The constraint C must be held constant at a value of 20. To solve this, the consultant first uses the substitution method to express the objective function P in terms of the single variable Q. The resulting single-variable objective function is P = ____.

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 constraint 5x + y = 100. The economist performs the following steps:

1. Rearranges the utility function to solve for y: y = (U - 10x) / 2.
2. Substitutes this expression for y into the constraint equation, resulting in: 5x + (U - 10x) / 2 = 100.
3. Prepares to solve this final equation for x to find the optimal quantity.

What is the fundamental flaw in the economist's approach?