The following three equations F' = (x, y, w; z) = 0 F2 = (x, y, w; z) = 0 F3 = (x, y, w; z) = 0 xy – w = 0 y – w3 – 3z = 0 w3 + z3 – 2 zw = 0 are satisfied at point P: (x, y, w; 2) = (, 4, 1, 1). The F' functions obviously possess con- tinuous derivatives. Thus, if the Jacobian | || is nonzero at point P, we can use the implicit- function theorem to find the comparative-static derivative (ax/a2). To do this, we can first take the total differential of the system: y dx + x dy – dw = 0 dy – 3w? dw – 3 dz=0 (3w² – 22) dw + (3z² – 2w) dz=0 Moving the exogenous differential (and its coefficients) to the right-hand side and writing in matrix form, we get -1 dx 0 1 0 0 (3w² – 2z) -3w2 dy dw 3 dz 2w – 3z2
The following three equations F' = (x, y, w; z) = 0 F2 = (x, y, w; z) = 0 F3 = (x, y, w; z) = 0 xy – w = 0 y – w3 – 3z = 0 w3 + z3 – 2 zw = 0 are satisfied at point P: (x, y, w; 2) = (, 4, 1, 1). The F' functions obviously possess con- tinuous derivatives. Thus, if the Jacobian | || is nonzero at point P, we can use the implicit- function theorem to find the comparative-static derivative (ax/a2). To do this, we can first take the total differential of the system: y dx + x dy – dw = 0 dy – 3w? dw – 3 dz=0 (3w² – 22) dw + (3z² – 2w) dz=0 Moving the exogenous differential (and its coefficients) to the right-hand side and writing in matrix form, we get -1 dx 0 1 0 0 (3w² – 2z) -3w2 dy dw 3 dz 2w – 3z2
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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Hi,
I have been trying to crack a mathematical economics book before my master's program but stuck with a question. Could you help me solve the b part in 2.6? I uploaded both the model and question.
Best Regards,
Transcribed Image Text:Example 5
The following three equations
PD
Exc
F' = (x, y, w; z) = 0
F2 = (x, y, w; z) = 0
F3 = (x, y, w; z) = 0
xy – w = 0
y - w3 – 3z = 0
w3 + z3 – 2 zw = 0
are satisfied at point P: (x, y, w; 2) = (, 4, 1, 1). The F' functions obviously possess con-
tinuous derivatives. Thus, if the Jacobian | || is nonzero at point P, we can use the implicit-
function theorem to find the comparative-static derivative (ax/a2).
To do this, we can first take the total differential of the system:
y dx + x dy – dw = 0
dy – 3w? dw – 3 dz= 0
(3w? – 22) dw + (3z² – 2w) dz=0
E
Moving the exogenous differential (and its coefficients) to the right-hand side and writing
in matrix form, we get
y x
0 1
0 0 (3w2 – 2z)
-1
-3w2
dx
dy
dw
3
dz
2w – 3z2
Transcribed Image Text:2.6 Consider the three dimensional vector function F(z)" = (w(z), x(z), y(z)) implicitly defined
by the equations in Example 5 on page 202.
dF d
a) Find the inverse for computing
dz
dz
y
b) Find the formula for
d
x and its value at (w, x, y; z) = (1, ¼, 4; 1).
dz
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