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(a) Solve the following problem using Lagrange's method max xy subject to px + gy = m (b) The optimal choices x* and y of x and y both depend on p, q, and m. The value function is f* (p, q, m) = x*y*. Define the Lagrangian as L(x, y, p, q, m) = xy - A(px + gy - m), and let Q = (x*, y*,p, q, m). Verify the envelope theorem -). (i) a (ii) (iii) op

(a) Solve the following problem using Lagrange's method max xy subject to px + gy = m (b) The optimal choices x* and y of x and y both depend on p, q, and m. The value function is f* (p, q, m) = x*y*. Define the Lagrangian as L(x, y, p, q, m) = xy - A(px + gy - m), and let Q = (x*, y*,p, q, m). Verify the envelope theorem -). (i) a (ii) (iii) op

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(a) Solve the following problem using Lagrange's method max xy subject to px + qy = m (b) The optimal choices x* and y of x and y both depend on p, q, and m. The value function is f* (p, q, m) = x*y*. Define the Lagrangian as L(x, y, p, q, m) = xy - A(px + qy - m), and let Q. = (x*, y*, p, q, m). Verify the envelope theorem (i) op (i) Te (ii)
Transcribed Image Text:(a) Solve the following problem using Lagrange's method max xy subject to px + qy = m (b) The optimal choices x* and y of x and y both depend on p, q, and m. The value function is f* (p, q, m) = x*y*. Define the Lagrangian as L(x, y, p, q, m) = xy - A(px + qy - m), and let Q. = (x*, y*, p, q, m). Verify the envelope theorem (i) op (i) Te (ii)
(a) with A = m? () On the one hand, f* (p, q, m) = x* y* = 4pg (b) Then On the other hand, - at Q* (i) On the one hand, of * On the other hand, ÖL at Q+= (ii) On the one hand, af + am On the other hand, &L at Q+= We see that the envelope theorem is confirmed in all cases.
Transcribed Image Text:(a) with A = m? () On the one hand, f* (p, q, m) = x* y* = 4pg (b) Then On the other hand, - at Q* (i) On the one hand, of * On the other hand, ÖL at Q+= (ii) On the one hand, af + am On the other hand, &L at Q+= We see that the envelope theorem is confirmed in all cases.
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