Question 4 Consider a consumer who consumes three goods, 1, x2 and r3. The consumer's consumption of the three goods is dependent on four variables, p, s, y and w, in the following way: a1(p, s, w, y) p*yw - sy = 0 (5) %3D x2(p, s, w, y) 3yw 1 --w = 2 (6) x3(p, s, w, y) 4ps + 3w + 2y = 9 (7) 4 Currently, p = , s = }, y = 1, and w = 2, so that the consumer is consuming 0 units of a1, 2 units of r2 and 9 units of a3. 4.1 Making use of the Implicit Function Theorem, show that the system is defined in the neighbourhood of p = }, s = }, y = 1, and w = 2 if p, s, and y are treated as endogenous variables in the model and w is treated as an exogenous variable. 4.2 Now, assuming the consumption is at the levels set out in 4.1, use calculus to approximate the new value of p if w increases by one unit. [NOTE: you may make use of Cramer's rule or use linearisation and invert the coefficient matrix]. 4.3 Assume that the following additional information becomes available about how p, y and w vary with time (t): 3 p(t) 2t s(t) 18 y(t) 2t – 5 ( – 21) + 1] w(t) %3D Using the Chain Rule, first form the Jacobian derivative matrix of the change in r1, x2 and r3 as t changes. Then evaluate the matrix at the point where t 3. I| ||
Question 4 Consider a consumer who consumes three goods, 1, x2 and r3. The consumer's consumption of the three goods is dependent on four variables, p, s, y and w, in the following way: a1(p, s, w, y) p*yw - sy = 0 (5) %3D x2(p, s, w, y) 3yw 1 --w = 2 (6) x3(p, s, w, y) 4ps + 3w + 2y = 9 (7) 4 Currently, p = , s = }, y = 1, and w = 2, so that the consumer is consuming 0 units of a1, 2 units of r2 and 9 units of a3. 4.1 Making use of the Implicit Function Theorem, show that the system is defined in the neighbourhood of p = }, s = }, y = 1, and w = 2 if p, s, and y are treated as endogenous variables in the model and w is treated as an exogenous variable. 4.2 Now, assuming the consumption is at the levels set out in 4.1, use calculus to approximate the new value of p if w increases by one unit. [NOTE: you may make use of Cramer's rule or use linearisation and invert the coefficient matrix]. 4.3 Assume that the following additional information becomes available about how p, y and w vary with time (t): 3 p(t) 2t s(t) 18 y(t) 2t – 5 ( – 21) + 1] w(t) %3D Using the Chain Rule, first form the Jacobian derivative matrix of the change in r1, x2 and r3 as t changes. Then evaluate the matrix at the point where t 3. I| ||
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Question 4
Consider a consumer who consumes three goods, x1, x2 and x3. The consumer's consumption
of the three goods is dependent on four variables, p, s, y and w, in the following way:
a1(p, s, w, y)
p*yw – sy = 0
(5)
x2(p, s, w, y)
1
3yw – -w = 2
(6)
x3(p, s, w, y)
4ps + 3w + 2y = 9
(7)
Currently, p = ;, s = ;, y = 1, and w = 2, so that the consumer is consuming 0 units of x1, 2
units of x2 and 9 units of x3.
4.1 Making use of the Implicit Function Theorem, show that the system is defined in the
neighbourhood of p = }, s = }, y = 1, and w = 2 if p, s, and y are treated as endogenous
variables in the model and w is treated as an exogenous variable.
4.2 Now, assuming the consumption is at the levels set out in 4.1, use calculus to approximate
the new value of p if w increases by one unit. [NOTE: you may make use of Cramer's rule
or use linearisation and invert the coefficient matrix].
4.3 Assume that the following additional information becomes available about how p, y and
w vary with time (t):
3
p(t)
2t
s(t)
18
y(t)
2t – 5
%3D
( - 20) + 1]
w(t)
Using the Chain Rule, first form the Jacobian derivative matrix of the change in x1, x2
and x3 as t changes. Then evaluate the matrix at the point where t = 3.
I|||](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87b53351-2067-4a4d-a81c-a5cf0ffaa5a8%2F2161e067-8ef6-46b8-b809-1712b89250a9%2Fjmh6y4i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 4
Consider a consumer who consumes three goods, x1, x2 and x3. The consumer's consumption
of the three goods is dependent on four variables, p, s, y and w, in the following way:
a1(p, s, w, y)
p*yw – sy = 0
(5)
x2(p, s, w, y)
1
3yw – -w = 2
(6)
x3(p, s, w, y)
4ps + 3w + 2y = 9
(7)
Currently, p = ;, s = ;, y = 1, and w = 2, so that the consumer is consuming 0 units of x1, 2
units of x2 and 9 units of x3.
4.1 Making use of the Implicit Function Theorem, show that the system is defined in the
neighbourhood of p = }, s = }, y = 1, and w = 2 if p, s, and y are treated as endogenous
variables in the model and w is treated as an exogenous variable.
4.2 Now, assuming the consumption is at the levels set out in 4.1, use calculus to approximate
the new value of p if w increases by one unit. [NOTE: you may make use of Cramer's rule
or use linearisation and invert the coefficient matrix].
4.3 Assume that the following additional information becomes available about how p, y and
w vary with time (t):
3
p(t)
2t
s(t)
18
y(t)
2t – 5
%3D
( - 20) + 1]
w(t)
Using the Chain Rule, first form the Jacobian derivative matrix of the change in x1, x2
and x3 as t changes. Then evaluate the matrix at the point where t = 3.
I|||
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