9. Solve the Leontief production equation for an economy with three sectors, given that .2 C = 3 .1 .2 .0 .07 1.3 .0 .2 and d = 40 60 80 TO 1070 .0063 7512 over 10 d=
9. Solve the Leontief production equation for an economy with three sectors, given that .2 C = 3 .1 .2 .0 .07 1.3 .0 .2 and d = 40 60 80 TO 1070 .0063 7512 over 10 d=
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
![c. Use the fact that
30
51
30
[50] [50] + [6]
30
and why the answers to parts (a) and (b) and to Exercise
5 are related.
8. Let C be an n x n 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 (1 − C)-¹. This shows that the
first column of (1-C) 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 =
to explain how
.2
.3
.I
.2
.0
.3
.1
.0 .2
and d =
1
40
60
80
A
21
10. The consumption matrix CR the U.S. economy in 1972
has the property that every entry in the matrix (I-C)-¹ is
nonzero (and positive). dces that say about the effect
of raising the demand for the suput of just one sector of the
economy?
11. The Leontief production equation, x = Cx+d, is usually
accompanied by a dual price equation,
p=CTp + v
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
entries 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:
{gross domestic product} = p'd = v^x
Verify the second equality. [Hint: Compute p'x in two
ways.]
Wassily W. Leontief, "The World Economy of the Year 2000,"
Scientific American, September 1980, pp. 206-231.
tar
a consumption matrix such that C-0 as
m→∞o, and for m = 1,2,..., let Dm = 1 + C ++
C". Find a difference equation that relates D and Dm+1 and
(8) for (I-C)-¹.
thereby obtain an iterative procedure for computing formula
m
13. [M] The consumption matrix C below is based on input-
output data for the U.S. economy in 1958, with data for 81
sectors grouped into 7 larger sectors: (1) nonmetal household
and personal products, (2) final metal products (such as motor
vehicles), (3) basic metal products and mining, (4) basic
nonmetal products and agriculture, (5) energy, (6) services,
and (7) entertainment and miscellaneous products. Find the
production levels needed to satisfy the final demand d. (Units
are in millions of dollars.)
.1588 .0064 .0025 .0304 .0014 .0083 .1594
.0057 .2645 .0436 .0099 .0083 0201 3413
0264
1506 .3557 .0139 0142 .0070 .0236
.3299 .0565 .0495 3636 .0204 .0483 0649
.0089 .0081 .0333 0295 .3412 0237 .0020
.1190 .0901 .0996 1260 1722 2368 .3369
.0063 .0126 .0196 .0098 .0064.0132 .0012
d=
74,000
56,000
10,500
25,000
17,500
196,000
5,000
14. [M] The demand vector in Exercise 13 is reasonable for
1958 data, but Leontief's discussion of the economy in the
reference cited there used a demand vector closer to 1964
data:
d = (99640, 75548, 14444, 33501, 23527, 263985, 6526)
Find the production levels needed to satisfy this demand.
15.
[M] Use equation (6) to solve the problem in Exer-
cise 13. Set x(0)=d, and for k = 1,2, ..., compute
x(k) = d + Cx(k-1). How many steps are needed to obtain
the answer in Exercise 13 to four significant figures?
2 Wassily W. Leontief, "The Structure of the U.S. Economy,"
Scientific American, April 1965, pp. 30-32.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6bb2273-ae71-4436-8787-039b32bc3b2f%2F0c1eb1f2-27b8-4276-8ab7-e762f09af5ab%2Fwlm7l8m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:c. Use the fact that
30
51
30
[50] [50] + [6]
30
and why the answers to parts (a) and (b) and to Exercise
5 are related.
8. Let C be an n x n 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 (1 − C)-¹. This shows that the
first column of (1-C) 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 =
to explain how
.2
.3
.I
.2
.0
.3
.1
.0 .2
and d =
1
40
60
80
A
21
10. The consumption matrix CR the U.S. economy in 1972
has the property that every entry in the matrix (I-C)-¹ is
nonzero (and positive). dces that say about the effect
of raising the demand for the suput of just one sector of the
economy?
11. The Leontief production equation, x = Cx+d, is usually
accompanied by a dual price equation,
p=CTp + v
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
entries 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:
{gross domestic product} = p'd = v^x
Verify the second equality. [Hint: Compute p'x in two
ways.]
Wassily W. Leontief, "The World Economy of the Year 2000,"
Scientific American, September 1980, pp. 206-231.
tar
a consumption matrix such that C-0 as
m→∞o, and for m = 1,2,..., let Dm = 1 + C ++
C". Find a difference equation that relates D and Dm+1 and
(8) for (I-C)-¹.
thereby obtain an iterative procedure for computing formula
m
13. [M] The consumption matrix C below is based on input-
output data for the U.S. economy in 1958, with data for 81
sectors grouped into 7 larger sectors: (1) nonmetal household
and personal products, (2) final metal products (such as motor
vehicles), (3) basic metal products and mining, (4) basic
nonmetal products and agriculture, (5) energy, (6) services,
and (7) entertainment and miscellaneous products. Find the
production levels needed to satisfy the final demand d. (Units
are in millions of dollars.)
.1588 .0064 .0025 .0304 .0014 .0083 .1594
.0057 .2645 .0436 .0099 .0083 0201 3413
0264
1506 .3557 .0139 0142 .0070 .0236
.3299 .0565 .0495 3636 .0204 .0483 0649
.0089 .0081 .0333 0295 .3412 0237 .0020
.1190 .0901 .0996 1260 1722 2368 .3369
.0063 .0126 .0196 .0098 .0064.0132 .0012
d=
74,000
56,000
10,500
25,000
17,500
196,000
5,000
14. [M] The demand vector in Exercise 13 is reasonable for
1958 data, but Leontief's discussion of the economy in the
reference cited there used a demand vector closer to 1964
data:
d = (99640, 75548, 14444, 33501, 23527, 263985, 6526)
Find the production levels needed to satisfy this demand.
15.
[M] Use equation (6) to solve the problem in Exer-
cise 13. Set x(0)=d, and for k = 1,2, ..., compute
x(k) = d + Cx(k-1). How many steps are needed to obtain
the answer in Exercise 13 to four significant figures?
2 Wassily W. Leontief, "The Structure of the U.S. Economy,"
Scientific American, April 1965, pp. 30-32.
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