9. Solve the Leontief production equation for an economy with three sectors, given that .2 .0 .2 C = .3 .1 40 1.3 and d = 60 02 brin 80
9. Solve the Leontief production equation for an economy with three sectors, given that .2 .0 .2 C = .3 .1 40 1.3 and d = 60 02 brin 80
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
9
![ere is a final
f input-outpu
impu
final de-
riculture,
conomy
n level
51
[30].
b. Use an inverse matrix to determine the production level
necessary to satisfy a final demand of
c. Use the fact that
0] - [50] + [6]
30
51
30
and why the answers to parts (a) and (b) and to Exercise
5 are related.
8. Let C be an nxn consumption matrix whose column sums
are less than 1. Let x be the production vector that satisfies
a final demand d, and let Ax be a production vector that
satisfies a different final demand Ad.
a. Show that if the final demand changes from d to d + Ad,
then the new production level must be x + Ax. Thus Ax
gives the amounts by which production must change in
order to accommodate the change Ad in demand.
1
b. Let Ad be the vector in R" with 1 as the first entry and
O's elsewhere. Explain why the corresponding production
Ax is the first column of (I - C). This shows that the
first column of (IC)-¹ gives the amounts the various
sectors must produce to satisfy an increase of 1 unit in the
final demand for output from sector 1.
9. Solve the Leontief production equation for an economy with
three sectors, given that
C =
.2
.3
.1
.2
.1
.0
.0
.3
.2
to explain how
and d =
[40
60
ni brs RA
da ih ist
the
s
10. The consumption matrix C for the U.S. economy in 1972
has the property that every entry in the matrix (1-C)-¹ is
nonzero (and positive).¹ What does that say about the effect
of raising the demand for the output of just one sector of the
economy? nem of bozu zosmodism bied or
80
80
12. Let C be a consumption matr
m → ∞, and for m = : 1, 2,
Cm. Find a difference equation th
thereby obtain an iterative proce
(8) for (I-C)-¹.
11. The Leontief production equation, x = Cx+d, is usually
accompanied by a dual price equation,
bovito.mail.asvius ban 20mil odf
p = C¹p+v
elle
Sug
rgil
where p is a price vector whose entries list the price per unit
for each sector's output, and v is a value added vector whose
boxcentries list the value added per unit of output. (Value added
includes wages, profit, depreciation, etc.) An important fact
in economics is that the gross domestic product (GDP) can
be expressed in two ways: ob i Ha hugt
{gross domestic product} = p'd = v'x
Wassily W. Leontief, "The World Economy of the Year 2000,"
Scientific American, September 1980, pp. 206-231.
Verify the second equality. [Hint: Compute p¹x in two
ways.]
13. [M] The consumption matrix
output data for the U.S. econo
sectors grouped into 7 larger se
and personal products, (2) fina
vehicles), (3) basic metal pr
nonmetal products and agric
and (7) entertainment and mi
production levels needed to s
are in millions of dollars.)
..
.1588 .0064 .0025 .0
.0057 2645 .0436
.0264 1506 .3557.
.3299 .0565 .0495
.0089 .0081 .0333
.1190 .0901 .0996
.0063 .0126 .0196
d=
74,000
56,000
10,500
25,000
17,500
196,000
5,000
14. [M] The demand vect
1958 data, but Leontie
reference cited there
data:
d = (99640, 75548,
Find the production
15. [M] Use equation
cise 13. Set x(0)
x(k)= d + Cx(k-
the answer in Exerc
2 Wassily W. Leonti
Scientific American](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6bb2273-ae71-4436-8787-039b32bc3b2f%2Fe9f85398-39aa-483d-8cf4-16c2bc0d8c00%2Fptta1ek_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ere is a final
f input-outpu
impu
final de-
riculture,
conomy
n level
51
[30].
b. Use an inverse matrix to determine the production level
necessary to satisfy a final demand of
c. Use the fact that
0] - [50] + [6]
30
51
30
and why the answers to parts (a) and (b) and to Exercise
5 are related.
8. Let C be an nxn consumption matrix whose column sums
are less than 1. Let x be the production vector that satisfies
a final demand d, and let Ax be a production vector that
satisfies a different final demand Ad.
a. Show that if the final demand changes from d to d + Ad,
then the new production level must be x + Ax. Thus Ax
gives the amounts by which production must change in
order to accommodate the change Ad in demand.
1
b. Let Ad be the vector in R" with 1 as the first entry and
O's elsewhere. Explain why the corresponding production
Ax is the first column of (I - C). This shows that the
first column of (IC)-¹ gives the amounts the various
sectors must produce to satisfy an increase of 1 unit in the
final demand for output from sector 1.
9. Solve the Leontief production equation for an economy with
three sectors, given that
C =
.2
.3
.1
.2
.1
.0
.0
.3
.2
to explain how
and d =
[40
60
ni brs RA
da ih ist
the
s
10. The consumption matrix C for the U.S. economy in 1972
has the property that every entry in the matrix (1-C)-¹ is
nonzero (and positive).¹ What does that say about the effect
of raising the demand for the output of just one sector of the
economy? nem of bozu zosmodism bied or
80
80
12. Let C be a consumption matr
m → ∞, and for m = : 1, 2,
Cm. Find a difference equation th
thereby obtain an iterative proce
(8) for (I-C)-¹.
11. The Leontief production equation, x = Cx+d, is usually
accompanied by a dual price equation,
bovito.mail.asvius ban 20mil odf
p = C¹p+v
elle
Sug
rgil
where p is a price vector whose entries list the price per unit
for each sector's output, and v is a value added vector whose
boxcentries list the value added per unit of output. (Value added
includes wages, profit, depreciation, etc.) An important fact
in economics is that the gross domestic product (GDP) can
be expressed in two ways: ob i Ha hugt
{gross domestic product} = p'd = v'x
Wassily W. Leontief, "The World Economy of the Year 2000,"
Scientific American, September 1980, pp. 206-231.
Verify the second equality. [Hint: Compute p¹x in two
ways.]
13. [M] The consumption matrix
output data for the U.S. econo
sectors grouped into 7 larger se
and personal products, (2) fina
vehicles), (3) basic metal pr
nonmetal products and agric
and (7) entertainment and mi
production levels needed to s
are in millions of dollars.)
..
.1588 .0064 .0025 .0
.0057 2645 .0436
.0264 1506 .3557.
.3299 .0565 .0495
.0089 .0081 .0333
.1190 .0901 .0996
.0063 .0126 .0196
d=
74,000
56,000
10,500
25,000
17,500
196,000
5,000
14. [M] The demand vect
1958 data, but Leontie
reference cited there
data:
d = (99640, 75548,
Find the production
15. [M] Use equation
cise 13. Set x(0)
x(k)= d + Cx(k-
the answer in Exerc
2 Wassily W. Leonti
Scientific American
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