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(a) A firm produces two product brands, the quantities of which are x1 and x2. The demand for brand x2 is three times that for brand x1. The profit function of the firm is T (x1, x2) = 30x1 – 3x² + 5x1x2 – 2x² + 10x2. %3D The firm wishes to maximize its profit. (i) (ii) (iii) Formulate the problem as a constrained optimization problem Write down the Lagrange function. Write down the first-order conditions.

(a) A firm produces two product brands, the quantities of which are x1 and x2. The demand for brand x2 is three times that for brand x1. The profit function of the firm is T (x1, x2) = 30x1 – 3x² + 5x1x2 – 2x² + 10x2. %3D The firm wishes to maximize its profit. (i) (ii) (iii) Formulate the problem as a constrained optimization problem Write down the Lagrange function. Write down the first-order conditions.

College Algebra
College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter4: Linear Functions
Section4.2: Modeling With Linear Functions
Problem 58SE: When hired at a new job selling electronics, you aregiven two pay options: • Option A: Base salary...
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(a) A firm produces two product brands, the quantities of which are x1 and x2. The demand for brand x2 is three times that for brand x1. The profit function of the firm is T (x1, x2) = 30x1 – 3x² + 5x1x2 – 2x² + 10x2. %3D The firm wishes to maximize its profit. (i) (ii) (iii) (iv) Formulate the problem as a constrained optimization problem Write down the Lagrange function. Write down the first-order conditions. Solve for the optimal values of x1, X2 and the Lagrange multiplier. (v) (vi) What is the maximum profit? Use the bordered Hessian to prove that the profit is indeed maximized.
Transcribed Image Text:(a) A firm produces two product brands, the quantities of which are x1 and x2. The demand for brand x2 is three times that for brand x1. The profit function of the firm is T (x1, x2) = 30x1 – 3x² + 5x1x2 – 2x² + 10x2. %3D The firm wishes to maximize its profit. (i) (ii) (iii) (iv) Formulate the problem as a constrained optimization problem Write down the Lagrange function. Write down the first-order conditions. Solve for the optimal values of x1, X2 and the Lagrange multiplier. (v) (vi) What is the maximum profit? Use the bordered Hessian to prove that the profit is indeed maximized.
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