4 Consider the following system of production functions, G(gi,I2) : 91(r, s, t) = 2r² + 3s2 + 4t? – rs +5rt 1 92(r, 8, t) = t° /5 – r You are also provided with the following additional information: r(v, w) = v + w s(v, w) = v²w t(v, w) = Jw 4.1 First compute the Jacobian derivative matrix for G(g1,92) using the Chain Rule and then evaluate the matrix where v = 3 and w = 1. 4.2 Using information from the Jacobian in 4.1, in which proportions should the firm increase or decrease v and w to ensure the most rapid change in the production of good 1 (i.e. what is the direction of most rapid change for gi with respect to v and w)?
4 Consider the following system of production functions, G(gi,I2) : 91(r, s, t) = 2r² + 3s2 + 4t? – rs +5rt 1 92(r, 8, t) = t° /5 – r You are also provided with the following additional information: r(v, w) = v + w s(v, w) = v²w t(v, w) = Jw 4.1 First compute the Jacobian derivative matrix for G(g1,92) using the Chain Rule and then evaluate the matrix where v = 3 and w = 1. 4.2 Using information from the Jacobian in 4.1, in which proportions should the firm increase or decrease v and w to ensure the most rapid change in the production of good 1 (i.e. what is the direction of most rapid change for gi with respect to v and w)?
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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Transcribed Image Text:4 Consider the following system of production functions, G(g1, 92) :
91(r, s, t) = 2r2 + 3s? + 4t2 – rs + 5rt
1
You are also provided with the following additional information:
r(v, w) = v + w
s(v, w) = v²w
t(v, w)
4.1 First compute the Jacobian derivative matrix for G(91, 92) using the Chain Rule and then
evaluate the matrix where v = 3 and w = 1.
4.2 Using information from the Jacobian in 4.1, in which proportions should the firm increase
or decrease v and w to ensure the most rapid change in the production of good 1 (i.e. what
is the direction of most rapid change for g1 with respect to v and w)?
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