45
6.11 Graphing complements
a,b. The figure shows that the loss in
1
x
can be compensated for by an additional
j
of
3
x
or
k
of
2.x
c. The new indifference curve is given by
2.U
0
1
x
U(
0
1
x
)
j
k
x3
x2
U2
U0
j
k
x3
x2
Chapter 6: Demand Relationships among Goods
46
Chapter 6: Demand Relationships among Goods
46
33
22
32
2 3 2 3
2
3
33
2 2 2
3 2 2 2 3 3 2 3 3
23 32 33
2 22
3 2 2 3 3
1 1 1 1
.
cc
cc
cc
c
cc
c c c
c c c c c
c
c c c c c
xx
xx
xx
p p p p
x
xx
x x x
x x p x p x p x p
sss
xs
x x x x x










d. Hicks’ third law is
3
10
ij
je
, for
1,2,3.i
If we substitute for
23
e
and
33
e
in
the above equation, we get
23 21 22
e e e
and
33 31 32.e e e
Thus,
22 32
23
11
( ) 0.ee pp



We are almost done finding the sign of the last equation in part (d); we only need
to deal with the last term:
Chapter 6: Demand Relationships among Goods
46
31 21
3
.
ee
p
This term is likely to be small if we assume that goods 2 and 3 have similar
term
22 32
ee
should be negative (as explained above) and
22 32
23
11
( ) 0.ee pp



Moreover, we can assume that apples, whether of high or low quality,
have a similar relationship with the other commodities, thus
31
e
and
21
e
should have close values. Hence, transaction costs such as the
transportation costs are expected to increase the relative demand for the
more expensive fruits, making it harder to find such products.
b) In this example, expensive restaurant meals and cheap restaurant
implying
22 32
23
11
( ) 0.ee pp



c) We can assume that flying the Concorde falls in the category of expensive
flights, and these are close substitutes to cheaper flights. Thus, the
Concorde flights can be represented as good 2 (using the above notation)
and the cheaper flights as good 3. Since goods 2 and 3 come from the
same category, namely flights, we can expect them to have similar
relationships with the other categories of goods, represented by
1.x