ma253
,Spr
ing
2007
— Problem Set
1
Solutions
1.
Problems from the textbook:
a. Section
1
.
1
, problems
5
,
7
,
9
,*
20
28
.
Problems 5, 7, and 9 should be straightforward; the answers are in the back of the
book. Let’s do the other two.
20.
The total demand for the product of A is
1000
(from consumers) plus
0.1b
(from B). So we must have
a
=
1000
+
0.1b
. Similarly,
b
=
780
+
0.2b
.Th
isleads
to the system
a

0.1b
=
1000

0.2a
+
b
=
780
The unique solution is
a
=
1100
,
b
=
1000
.
28.
Thermal equilibrium requires that
T
1
=
T
2
+
200
+
0
+
0
4
T
2
=
T
1
+
T
3
+
200
+
0
4
T
3
=
T
2
+
400
+
0
+
0
4
Rewriting as a linear system and solving gives
(
T
1
,T
2
3
)=(
75, 100, 125
)
.
b. Section
1
.
2
, problems
5
,
11
,
13
38
44
61
.
Again, 5, 11, and 13 should be routine; the answers are in the back of the book.
You should be able to do 5 by hand, but getting 11 right by hand is challenging.
38.
This is in the same genre as problem 20 in section 1.1, but is a little closer to
what would happen in real economic modeling.
For (a), the vectors are
~
v
1
=
⎡
⎣
0
0.1
0.2
⎤
⎦
~
v
2
=
⎡
⎣
0.2
0
0.5
⎤
⎦
~
v
3
=
⎡
⎣
0.3
0.4
0
⎤
⎦
~
b
=
⎡
⎣
320
90
150
⎤
⎦
.
For (b), notice ±rst
x
j
is the output of
I
j
and the
i
th entry
a
ij
in
~
v
j
is the perunit
demand of
I
j
for the goods made by
I
i
.S
o
x
j
a
is the actual demand of
I
j
for
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