Chapter 31
NAME
Exchange
Introduction.
The
Edgeworth box
is a thing of beauty. An amazing
amount of information is displayed with a few lines, points and curves.
In fact one can use an Edgeworth box to show just about everything
there is to say about the case of two traders dealing in two commodities.
Economists know that the real world has more than two people and more
than two commodities. But it turns out that the insights gained from this
model extend nicely to the case of many traders and many commodities.
So for the purpose of introducing the subject of exchange equilibrium, the
Edgeworth box is exactly the right tool. We will start you out with an
example of two gardeners engaged in trade. You will get most out of this
example if you Fll in the box as you read along.
Example:
Alice and Byron consume two goods, camelias and dahlias.
Alice has 16 camelias and 4 dahlias. Byron has 8 camelias and 8 dahlias.
They consume no other goods, and they trade only with each other. To
describe the possible allocations of ﬂowers, we Frst draw a box whose
width is the total number of camelias and whose height is the total number
of dahlias that Alice and Byron have between them. The width of the
box is therefore 16 + 8 = 24 and the height of the box is 4 + 8 = 12.
Dahlias
Byron
12
6
0
6
1
21
82
4
Alice
Camelias
Any feasible allocation of ﬂowers between Alice and Byron is fully
described by a single point in the box. Consider, for example, the alloca
tion where Alice gets the bundle (15
,
9) and Byron gets the bundle (9
,
3).
This allocation is represented by the point
A
=(15
,
9) in the Edgeworth
box, which you should draw in. The distance 15 from
A
to the left side of
the box is the number of camelias for Alice and the distance 9 from
A
to
the bottom of the box is the number of dahlias for Alice. This point also
determines Byron’s consumption of camelias and dahlias. The distance
9f
rom
A
to the right side of the box is the total number of camelias
consumed by Byron, and the distance from
A
to the top of the box is the
number of dahlias consumed by Byron. Since the width of the box is the
total supply of camelias and the height of the box is the total supply of
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EXCHANGE
(Ch. 31)
dahlias, these conventions ensure that any point in the box represents a
feasible allocation of the total supply of camelias and dahlias.
It is also useful to mark the initial allocation in the Edgeworth box,
which, in this case, is the point
E
=(16
,
4). Now suppose that Alice’s
utility function is
U
(
c, d
)=
c
+2
d
and Byron’s utility funtion is
U
(
c, d
cd
. Alice’s indiFerence curves will be straight lines with slope
−
1
/
2. The
indiFerence curve that passes through her initial endowment, for example,
will be a line that runs from the point (24
,
0) to the point (0
,
12). Since
Byron has CobbDouglas utility, his indiFerence curves will be rectangular
hyperbolas, but since quantities for Byron are measured from the upper
right corner of the box, these indiFerence curves will be ﬂipped over as in
the Edgeworth box diagrams in your textbook.
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