2.7 (Page 154)
1.
We let x represent A’s wages, y represent B’s wages, and z represent C’s wages, which we are told
equal $30,000. The amount paid out by each A, B, and C equals the amount each receives, so
⎡1
⎢2
⎡x ⎤ ⎢
⎢y⎥ = ⎢ 1
⎢ ⎥ ⎢4
⎢⎣ z ⎥⎦ ⎢
⎢1
⎣⎢ 4
1
3
1
3
1
3
1⎤
4⎥
⎥
1⎥
4⎥
⎥
1⎥
2 ⎥⎦
⎧
⎪x =
⎡x ⎤
⎪
⎢ y ⎥ giving the system ⎪ y =
⎨
⎢ ⎥
⎪
⎢⎣ z ⎥⎦
⎪
⎪z =
⎩
Solving the equations for x, y, and z, we find
⎡ 1
⎢ 2
⎢
⎢− 1
⎢ 4
⎢
⎢− 1
⎢⎣ 4
1
1
x+ y+
2
3
1
1
x+ y+
4
3
1
1
x+ y+
4
3
1
1
1
⎧ 1
z
x− y− z =0
⎪
4
2
3
4
⎪
1
1
2
1
⎪
z or ⎨− x + y − z = 0
4
3
4
⎪ 4
1
1
1
⎪ 1
z
− x− y+ z =0
⎪
2
3
2
⎩ 4
1
1
2
1
⎤
⎡
⎤
0⎥
1 −
0⎥
−
−
⎢
⎡ 1 0 − 1 0⎤
3
4
3
2
⎥
⎢
⎥
⎢
⎥
2
1
1
3
⎢0
⎥ ⎯⎯⎯⎯→ ⎢ 0 1 − 3 0⎥
0⎥ ⎯⎯⎯⎯
0
−
→
−
⎥ R1 = 21r1
⎢
⎥ R2 = 22r2
3
4
2
8
4
⎢
⎥
⎥ R2 = 4 r1 + r2 ⎢
⎥ R1 = 3 r2 + r1 ⎢
⎥
0
0
0
0
1
1
1
3
⎦
0⎥ R3 = 1 r1 + r3 ⎢ 0 −
0⎥ R3 = 1 r2 + r3 ⎣
−
4
2
⎥⎦
⎢⎣
⎥⎦
3
2
2
8
−
3
z where z is the parameter. Since we are told C’s wages are $30,000, we know
4
3
A’s wages are also $30,000, and B’s wages are (30,000) = $22,500.
4
We let x represent A’s wages, y represent B’s wages, and z represent C’s wages, which we are told
equal $30,000. Since the amount paid out by each A, B, and C equals the amount each receives, we
⎡ x ⎤ ⎡ 0.2 0.3 0.1⎤ ⎡ x ⎤
get ⎢ y ⎥ = ⎢ 0.6 0.4 0.2⎥ ⎢ y ⎥ which gives the system
⎢ ⎥ ⎢
⎥⎢ ⎥
⎢⎣ z ⎥⎦ ⎢⎣ 0.2 0.3 0.7⎥⎦ ⎢⎣ z ⎥⎦
or x = z and y =
3.
⎧ x = 0.2 x + 0.3 y + 0.1z
⎧ 0.8 x − 0.3 y − 0.1z = 0
⎧ 8 x − 3y − 1z = 0
⎪
⎪
⎪
⎨ y = 0.6 x + 0.4 y + 0.2z or ⎨ −0.6 x + 0.6 y − 0.2z = 0 or ⎨ − 6x + 6 y − 2z = 0
⎪
⎪ − 2 x − 3 y + 3z = 0
⎪ z = 0.2 x + 0.3 y + 0.7z
⎩ −0.2 x − 0.3 y + 0.3z = 0
⎩
⎩
Solving the equations for x, y, and z, we find
3
1
2
⎡
⎤
⎡
⎤
⎢ 1 − 8 − 8 0⎥
⎢ 1 0 − 5 0⎥
⎡ 8 −3 − 1 0 ⎤
⎢
⎥
⎢
⎥
15
11 ⎥
11 ⎥
⎢ −6 6 −2 0⎥ ⎯⎯⎯⎯→ ⎢ 0
⎢
−
⎯
→ 0 1 −
0 ⎯⎯⎯⎯
0
⎢
⎥ R1 = 1 r1
⎢
⎥ R2 = 4 r2
⎢
4
4
15 ⎥
8
15
⎢⎣ −2 −3 3 0⎥⎦ R2 = 6 r1 + r2 ⎢
⎥
⎢
⎥
11 ⎥ R1 = 3 r2 + r1 ⎣⎢ 0 0
0 0⎦⎥
R 3 = 2 r1 + r3
⎢ 0 − 15
8
0
⎢⎣
⎥⎦ R3 = 15 r2 + r3
4
4
4
2
11
z and y =
z where z is the parameter. Since we are told C’s wages are $30,000, we find
5
15
2
11
(30,000) = $22,000.
A’s wages are (30,000) = $12,000, and B’s wages are
5
15
or x =
5.
The total output vector X for an open Leontief model is from the system X = [I3 − A]−1 · D where D
⎡ 80 ⎤
represents future demand for the goods produced in the system. If D2 = ⎢⎢90 ⎥⎥ , then using the
⎢⎣ 60 ⎥⎦
information from Table 5, we get
⎡1.6048 0.3568 0.7131⎤ ⎡80 ⎤ ⎡ 203.28 ⎤
X = ⎢⎢0.2946 1.3363 0.3857 ⎥⎥ ⎢⎢90 ⎥⎥ = ⎢⎢166.98 ⎥⎥
⎢⎣0.3660 0.2721 1.4013 ⎥⎦ ⎢⎣60 ⎥⎦ ⎢⎣137.85 ⎥⎦
The total output of R, S, and T required for the forecast demand D2 is to produce 203.28 units of
product R, 166.98 units of product S, and 137.85 units of product T.
6.
The total output vector X for an open Leontief model is from the system X = [I3 − A]−1 · D where D
⎡100 ⎤
represents future demand for the goods produced in the system. If D4 = ⎢⎢ 80 ⎥⎥ , then using the
⎢⎣ 60 ⎥⎦
information from Table 5, we get
⎡1.6048 0.3568 0.7131⎤ ⎡100 ⎤ ⎡ 231.81⎤
X = ⎢⎢0.2946 1.3363 0.3857 ⎥⎥ ⎢⎢ 80 ⎥⎥ = ⎢⎢159.51⎥⎥
⎢⎣0.3660 0.2721 1.4013 ⎥⎦ ⎢⎣ 60 ⎥⎦ ⎢⎣142.45 ⎥⎦
The total output of R, S, and T required for the forecast demand D4 is to produce 231.81 units of
product R, 159.51 units of product S, and 142.45 units of product T.