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A firm produces two products. Consider the function ƒ : Rf × Rf → R : (x1, x2) ↔ ƒ (x1, x2) = α₁ ln(x₁) + α₂ ln(x₂), with ₁ and ₂ the production levels of these products and a₁ and a2 non-negative constants. (a) Assume only for this part of the question that a₁ = a₂. Sketch some level curves of the function f. (b) The firm knows that production is optimal if the function ƒ is maximized. Denote the unit prices of both products by p₁ and p2. Assuming that the firm has a fixed budget B, determine appropriate expressions for the optimal production levels x₁ and 22.

A firm produces two products. Consider the function ƒ : Rf × Rf → R : (x1, x2) ↔ ƒ (x1, x2) = α₁ ln(x₁) + α₂ ln(x₂), with ₁ and ₂ the production levels of these products and a₁ and a2 non-negative constants. (a) Assume only for this part of the question that a₁ = a₂. Sketch some level curves of the function f. (b) The firm knows that production is optimal if the function ƒ is maximized. Denote the unit prices of both products by p₁ and p2. Assuming that the firm has a fixed budget B, determine appropriate expressions for the optimal production levels x₁ and 22.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. A firm produces two products. Consider the function ƒ : Rj × Rj → R : (x1, x2) → ƒ (x1, x2) = α₁ ln(x₁) + a₂ ln(x2), with ₁ and 2 the production levels of these products and a₁ and a2 non-negative constants. (a) Assume only for this part of the question that a₁ = a2. Sketch some level curves of the function f. (b) The firm knows that production is optimal if the function f is maximized. Denote the unit prices of both products by p₁ and p2. Assuming that the firm has a fixed budget B, determine appropriate expressions for the optimal production levels x₁ and £2.
Transcribed Image Text:3. A firm produces two products. Consider the function ƒ : Rj × Rj → R : (x1, x2) → ƒ (x1, x2) = α₁ ln(x₁) + a₂ ln(x2), with ₁ and 2 the production levels of these products and a₁ and a2 non-negative constants. (a) Assume only for this part of the question that a₁ = a2. Sketch some level curves of the function f. (b) The firm knows that production is optimal if the function f is maximized. Denote the unit prices of both products by p₁ and p2. Assuming that the firm has a fixed budget B, determine appropriate expressions for the optimal production levels x₁ and £2.
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