Chapter 8: Exchange
8.1: Introduction
In many ways this chapter is the most important in the book. If you have time to study just one, this
is the one that you should study (even though it might be a bit difficult studying it on its own). It
has a brilliant idea in it (the Edgeworth Box) and it generates an exciting number of important
results. It is used over and over again in the rest of the book.
It is exciting and insightful because it portrays in a very simple way the benefits people get out of
exchanging things. It shows what exchanges (if any) are possible, what exchanges are efficient and
which are not, and what exchanges may take place under different institutions. It shows that what
economists call perfect competition has some nice properties, while monopoly and monopsony have
some less-than-nice properties.
8.2: Exchange
We consider a really simple story. We have an economy in which there are just two individuals and
just two goods. Call the individuals A and B and the goods 1 and 2. The economy is a simple pureexchange economy – nothing is produced – the individuals wake up in the morning to find that they
each have an endowment of each of the two goods. They could, if they want, simply consume their
initial endowments – or they may find it mutually convenient to do some kind of trade or exchange
between the two of them. The purpose of this chapter is to see whether it might be possible to have
some mutually advantageous trade – and, if so, whether we can offer some advice as to how it
might be carried out.
Obviously the answers to these questions depend upon the preferences of the individuals and their
initial endowments. We shall start with a particular example and then we shall try and generalise the
results we have obtained. Later we shall present several more specific examples.
8.3: Individual A’s Preferences and Endowments
Although we work with a particular example you should pay attention to the general principles we
are using. We start with a statement of Individual A’s preferences and endowments, though perhaps
that should be preceded by a statement of the space in which we will be representing these
preferences and endowments. This will be the same as we have been using for the last 3 chapters:
with the quantity of good 1, denoted by q1, along the horizontal axis and the quantity of good 2,
denoted by q2, along the vertical axis.
We assume in this example that A has an initial endowment of 22 units of good 1 and 92 units of
good 2. Further we assume that A’s preferences over these two goods are Cobb-Douglas with
parameter a = 0.7. This enables us to draw A’s indifference curves and endowment point in the
following figure. Note that the endowment point is indicated with the letter E – it is at the point (22,
92).
You will note that we have drawn the indifference curve passing through the initial endowment
point. This enables us to answer the question: to where in this space would individual A voluntarily
move? The answer is simple: anywhere above and to the right of the indifference curve passing
through the point E.
8.4: Individual B’s Preferences and Endowments
We now do the same for Individual B. Here we assume that he or she starts with an initial
endowment of 128 units of good 1 and 8 units of good 2. We assume that B also has Cobb-Douglas
preferences but here with a weight of just 0.6 on good 1. So B absolutely prefers good 1 but relative
to A prefers good 2.
We draw B’s indifference curves and endowment point:
Note that B starts at (128, 8). In the figure is the indifference curve passing through the initial point.
B would be happy to move to any point above and to the right of this indifference curve.
8.5: The Edgeworth Box
We now do one of the clever things that Edgeworth did – he first turned Individual B upside down!
This gives us:
Let us carefully consider what this means. For B his or her origin – the zero point – is at the top
right hand corner of this figure. The quantity of good 1 that B has is measured from this top right
hand corner leftwards – so the horizontal distance from the top right hand corner to the endowment
point E is 128 units – the endowment that B has of good 1. Moreover, the quantity of good 2 that B
has is measured from this top right hand corner downwards – so the vertical distance from the top
right hand corner to the endowment point E is 8 units – the endowment that B has of good 2.
Obviously the further to the left and the further down the happier is Individual B – so his or
happiness increases as we move from the top right hand corner – his origin – down and to the left.
In this figure lower indifference curves mean more happiness for individual B. It follows that B
would be happy to move to any point in this space to the left of and below the indifference curve
passing through the initial endowment point E.
We note that A starts with 22 of good 1 and 92 of good 2, while B starts with 128 of good 1 and 8
of good 2. Between the two of them they have a grand total of 150 of good 1 and 100 of good 2.
The problem that we are going to discuss is the division of these 150 units of good 1 and 100 units
of good 2 between the two of them – given, of course, their initial allocation. To help us in this
discussion we shall use a brilliant device – named the Edgeworth Box after its originator. We have
already described one of the clever things that Edgeworth did – namely turn B upside down – we
now describe the second clever thing that he did.
He superimposed figure 8.3 on top of figure 8.1 in such a way that the initial endowment points
coincided.
What do we note: the width of this box is the sum of the horizontal distance from A’s origin to the
endowment point (A’s endowment of good 1 - 22 units) plus the horizontal distance from B’s origin
to the endowment point (B’s endowment of good 1 - 128 units). That is the width of the box is the
total amount of good 1 that the two individuals possess between them. That is, 150 units. We also
note that the height of this box is the sum of the vertical distance from A’s origin to the endowment
point (A’s endowment of good 2 - 92 units) plus the vertical distance from B’s origin to the
endowment point (B’s endowment of good 2 - 8 units), That is the height of the box is the total
amount of good 2 that the two individuals possess between them. That is, 100 units. So the
dimensions of the box are determined by the total amounts of the two goods that society (the two
individuals together) have of the two goods. The width is the total amount of good 1; the height the
total amount of good 2.
Now note that every point in the box is an allocation of the goods between the two members of
society. For example, point E is the initial allocation. The bottom origin (0, 0) represents an
allocation in which A gets nothing and B gets everything. The top origin (150, 100) represents an
allocation in which A gets everything and B gets nothing. The mid-point (75, 50) represents an
allocation in which A ends up with 75 units of good 1 and 50 of good 2 – and so does B – they split
the total up evenly. And so on.
The question is: what happens? Are they happy to stay where they are – at point E? Or might they
be better off moving to some other point in the space?
We have already seen that A would be happy moving to any point to the right and above his or her
indifference curve passing though E; similarly B would be happy moving to any point to the left
and below his or her indifference curve passing though E. A glance at the figure shows that there is
quite a large region to which both of them would be happy to move. Can we narrow things down a
bit?
8.6: The Contract Curve
If you look at the figure you will see that there are points of tangency between the indifference
curves of A and those of B. If we join them up we get a very important curve – which is known as
the contract curve. As its name suggests, it indicates the points where some kind of contract, some
kind of deal, some kind of exchange, between A and B might be made. Why?
First of all, let us understand its properties. Take any indifference curve of individual A and ask
yourself – “where on this indifference curve for A is B happiest?”
What is your answer? The point where there is an indifference curve of B tangential to that of A –
and this point, by definition, is on the contract curve. Now do the converse: take any indifference
curve of individual B and ask yourself – “where on this indifference curve for B is A happiest?”
The answer? The point where there is an indifference curve of A tangential to that of B – and this
point, by definition, is on the contract curve1. So the contract curve is the locus of points efficient in
the sense that, for any given level of utility for individual A the utility of B is maximised, and for
any given level of utility for individual B the utility of A is maximised.
It follows that points off the contract curve are inefficient. What does this mean? Simply that, from
any point off the contract curve, there is always some direction in which we can move and increase
the utility of at least one of the individuals without decreasing the utility of the other2. To show this,
take any point off the contract curve and then move towards the contract curve – by moving
between an indifference curve of A and an indifference curve of B. Starting at point E provides an
obvious example: what happens if we move from E towards the contract curve by moving between
A’s original indifference curve and B’s original indifference curve? The utility of both increase.
This is true no matter where we start from – as long as it is off the contract curve.
However, once we are on the contract curve, we can no longer increase the utility of one individual
without decreasing the utility of the other. Try it.
1
The contract curve is not necessarily the locus of tangency points. What it is the locus of points efficient in the sense
that will very shortly be defined.
2
Actually we can usually say something stronger: that there is always some direction in which we can move and
increase the utility of both of the individuals.
So the contract curve is the locus of efficient points. Points off the contract curve are inefficient in
the sense that we can usually make both individuals better off by moving away and towards the
contract curve. It seems reasonable then to conclude that any contract made between the two
individuals should be on the contract curve.
We have narrowed things down a lot: it seems that we can conclude that any contract should be on
the contract curve, between the point where A’s original indifference curve intersects it and the
point where B’s original indifference curve intersects it. (For neither would accept a deal that made
them worse off than at the initial point.) Can we narrow things down any further?
One way we can narrow things down further is by imposing a particular trading institution and
looking at the implications. An obvious choice – particularly in view of chapter 2 – is that
institution which economists call perfect competition. Essentially this is the imposition of a price,
which both individuals take as given, which is such that demand and supply are equal. In the
context of this chapter, it is the imposition of a price for which both individuals are happy to move
to the same point in the box. If such a price exists, then both are happy to move to the same point
and this point is the chosen allocation. Does such a price exist?
8.7: Price-Offer Curves
Let us first consider what individual A would want to do at different prices. A pair of prices (a price
for good 1 and a price for good 2) determines a budget constraint for A (and indeed for B) which
has the usual properties: it passes through the initial endowment point E and has a slope equal to –
p1/p2. Like in the attached figure.
For the budget line illustrated the optimal point for individual A lies outside the box, but as we
rotate the line around the point E then the optimal point eventually comes inside the box and then
steadily moves leftward as the price of good 1 rises relative to the price of good 2. If we take the
locus of optimal points – as the relative price p1/p2 varies – we get what is called the price-offer
curve for the individual. This is the curve passing through E drawn in figure 8.6. Note that the
price-offer curve must pass through E as the individual can always choose not to move (and in fact
so chooses when the relative price of good 1 equals the magnitude of the slope of the original
indifference curve at the point E).
We can do the same for B.
Here a particular budget constraint is shown (the straight line through E) and it is clear that the
optimal point is on the price-offer curve for B (the curve through E). Once again the price-offer
curve passes through E as the individual can always choose not to move.
8.8: Competitive Equilibrium
Let us now put the two price-offer curves together and ask whether there is a point where they
intersect. If there is, then this is the competitive equilibrium that we have been looking for.
There is such a point – and it is on the contract curve! That is interesting (and good news) – but is it
surprising? Let us call the competitive equilibrium point C. It is the intersection of the two priceoffer curves. We know that C is on the price-offer curve of A. This means that the budget constraint
through the initial point pictured in the above figure must be tangential to A’s indifference curve at
that point. Similarly, we know that C is on the price-offer curve of B. This means that the budget
constraint through the initial point pictured in the above figure must be tangential to B’s
indifference curve at that point. If the budget line is tangential to A’s indifference curve and to B’s
it must therefore follow that A’s indifference curve is tangential to B’s at the point C. It follows that
C must be on the contract curve. It is not surprising.
We thus have a really nice result: the competitive equilibrium must be on the contract curve and
must therefore be efficient.
Let us note the implications. The initial point has A with 22 units of good 1 and 92 of good 2, and B
with 128 of good 1 and 8 of good 2. As you will see from the figure above, the competitive
equilibrium is at the point (70, 36) as measured from the bottom left origin and is at the point (80,
64) as measured from the top right origin. In this competitive equilibrium A has 70 of good 1 and
36 of good 2, and B has 80 of good 1 and 64 of good 2. It might be easiest to see all of this in a
tabular form.
Initial allocation
Good 1
Good 2
Individual A
22
92
Individual B
128
8
Society
150
100
Competitive
equilibrium allocation
Individual A
Individual B
Society
Good 1
Good 2
70
36
80
64
150
100
Changes between the
two allocations
Good 1
Good 2
Individual A
Individual B
Society
+48
-56
-48
+56
0
0
What happens is that Individual B gives 48 units of good 1 to Individual A in exchange for 56 units
of good 2 – which Individual A gives to Individual B. The rate of exchange is 48 units of good 1
for 56 units of good 2. The slope of the line joining E and C determines this exchange rate – this
slope is – 56/48 – this is of course –p1/p2. So p1/p2 = 56/48 = 1.16666. Good 1 is more expensive
than good 2 in the sense that for each unit of good 1 exchanged the return is more than one unit of
good 2.
You might like to ask why the exchange is the way it is – why A gives good 2 to B and B gives
good 1 to A. You could simply say that A starts out with lots more of good 2 and B starts out with
lots more of good 1. But it is more to do with where the initial point is in relation to the contract
curve. Indeed you might like to ask why the contract curve is where it is – why is to the right and
below the line joining the two origins. This is a consequence of the fact that the two individuals
have different preferences – with B relatively (to A) preferring good 2. (Note that both individuals
absolutely prefer good 1 – their values of a are greater than 0.5 – but A’s is 0.7 while B’s is 0.6, so
relative to A¸ individual B prefers good 2. That is the reason the contract curve is where it is.)
8.9: Price-Setting Equilibria
We have investigated above one market institution – that of competitive equilibrium. Each agent
takes the price as given and we ask whether there is a price at which both individuals want to move
to the same point. If so, we have found a competitive equilibrium.
Now we explore other trading institutions. In particular we explore what happens if we give to one
of the individuals the ability to choose the price (the exchange rate). The other individual simply
takes the price as given and chooses the point to which they will move. This binds the first mover
(the price setter) to accept that point.
What happens? Suppose we give to Individual A the right to choose the price. He or she knows that
then Individual B will choose the point to which he or she wants to move – and this binds them
both. What does A do?
We can argue in this fashion. Given any choice of the price by A, individual B will respond by
choosing the point on his or her price-offer curve. So, in essence, A is choosing a point on B’s
price-offer curve. What point does he or she choose? Consider the figure:
B’s price-offer curve is the curve going through point E. If A can choose any point on this, which
point does he or she choose? Obviously the best point on it relative to A’s indifference curves.
Which is the highest point from that point of view? Point A – it is on the highest possible A
indifference curve.
So if A can choose the price, he or she chooses the budget line going from E through A. Given this
budget line, B’s best response is to choose point A. Point A is approximately (64, 72) as measured
from the bottom left origin and (86, 28) as measured from the top right origin. In tabular form:
Initial allocation
Good 1
Good 2
Individual A
22
92
Individual B
128
8
Society
150
100
Allocation determined
by A setting the price
Good 1
Good 2
Individual A
Individual B
Society
64
72
86
28
150
100
Changes between the
two allocations
Good 1
Good 2
Individual A
Individual B
Society
+42
-20
-42
+20
0
0
Note what happens – in this allocation A gives 20 units of good 2 to B in exchange for 42 units of
good 1. Compared to the competitive equilibrium it is obviously a much better deal for individual
A. This is hardly surprising – as it was chosen by A.
So it is better for A and worse for B. But there is something else. What do we see from the figure
above? That the point chosen by A setting the price – point A – is off the contract curve. It is
inefficient! This means that there is some direction from point A in which the individuals can move
which makes both of them better off. Why do they not do that? Simply because A is choosing the
price – not the point. If he or she could choose a point then A would choose the point on the
contract curve just below where it intersects B’s original indifference curve. But choosing a price is
not the same as choosing a point: choosing a price is choosing a direction to move from point E.
For completeness we also present the case when B sets the price and A responds by choosing the
point – but you can probably anticipate the result. In this case, B is effectively choosing a point on
A’s price-offer curve. Which point does he or she choose?
B chooses the point on A’s price-offer curve (the curve through the point E) which is best from B’s
point of view. B therefore chooses point B – by asking a budget line given by the line going from E
through B. What do we notice about point B? First, it implies the following trades:
Initial allocation
Good 1
Good 2
Individual A
22
92
Individual B
128
8
Society
150
100
Allocation determined
by B setting the price
Good 1
Good 2
Individual A
Individual B
Society
42
45
108
55
150
100
Changes between the
two allocations
Good 1
Good 2
Individual A
Individual B
Society
+20
-47
-47
+20
0
0
In this exchange, A gives 47 units of good 2 to B – in exchange for 20 units of good 1. Compared to
the competitive equilibrium it is obviously a much better deal for individual B. This is hardly
surprising – as it was chosen by B.
Like the point chosen by A however point B is off the contract curve – it is inefficient – and for the
same reasons.
So price-setting (monopoly or monopsony) by one of the two agents is inefficient - whereas perfect
competition is efficient. That is why governments like competition.
8.10: Two Theorems of Welfare Economics
At this stage we can show two important theorems of what is called ‘welfare economics’. Welfare
economics is the study of the welfare of the society and how it can be changed. For simplicity, in
stating these theorems, I assume that the indifference curves of the agents are everywhere smoothly
convex – as in the example that we have above. If the indifference curves are not everywhere
smoothly convex then the theorems may not be true. (You might like to think about this and
perhaps provide some examples – but this is not necessary for the course.)
These theorems are an immediate and important implication of the result that we have derived
above – that the competitive equilibrium is necessarily on the contract curve. This result first
implies the First Theorem of Welfare Economics: “whatever the initial allocation, price-taking trade
takes us to a point on the contract curve - and hence is Pareto Efficient.". This is an important and
interesting proposition and one that is not immediately obvious to those who have not done
economics. It implies that if we have the market institution of competitive trading then the resulting
outcome will be efficient. It is for this reason that governments like competition and encourage it.
The Second Theorem of Welfare Economics is a sort of converse: “any point on the contract curve
can be reached by competitive trading from some (in fact many) initial allocations." This too is
important as it implies that a government can achieve any desired efficient point (one on the
contract curve) by taxing people appropriately and then leaving the rest to the competitive system.
Precise planning is not required.
8.11: Alternative Scenarios
We have worked throughout with a particular example but it is hoped that you are convinced that
the results must be true in general. One thing which may not be obvious is how the competitive
equilibrium allocation depends upon the initial endowment point and on the preferences of the two
individuals. This can be shown mathematically (a mathematical derivation is provided in the
Mathematical Appendix to this chapter) but I would prefer to give you some further examples. This
section, therefore, gives a number of further examples (I call then scenarios) in which endowments
and/or preferences differ. From these you will be able to see how endowments and preferences
affect the competitive equilibrium allocation. The results are not surprising: if one good becomes
more abundant then its price falls; if one individual changes his or her preferences in such a way
that they like a good more than before then the price of that good rises.
Let us turn to the second scenario. This differs from the first in that the preferences of Individual B
are changed: in particular the Cobb-Douglas parameter is changed from 0.6 to 0.3. So individual B
now absolutely prefers good 2 to good 1, and relative to A, who has a parameter a equal to 0.7,
even more prefers good 2 to good 1 than in the first scenario. As a consequence, the contract curve
is even more convex and further away from the line that joins the two origins of the box. You will
see in the figure where the competitive equilibrium is. It is denoted it by C, as in scenario 1, and
note that it remains at the intersection of the contract curve and the two price-offer curves.
In this competitive equilibrium A ends up with more of good 1 than in scenario 1 – and we note that
the budget line (joining E and C) is flatter – so that the equilibrium price of good 1 is lower. This
follows because B likes good 1 less than before.
In scenario 3 we give the two individuals identical tastes (but the same endowments as in scenario
1). It follows that the contract curve is the straight line joining the two origins. With Cobb-Douglas
preferences this is always the case3. We thus have:
3
Indeed it is always true with identical homothetic preferences. (Though you do not need to know what homothetic
preferences are.) They are defined in the Mathematical Appendix.
The competitive equilibrium is on this line joining the two origins – so that A gets some 40% of the
total quantity of good 1 and 40% of the total quantity of good 2, while B gets around 60%. In this
scenario A does relatively badly as he or she starts with rather little of good 1 – the good that both
of the individuals prefer.
Scenario 4 is the same as scenario 1 in terms of preferences but the endowments are rearranged so
that A starts out with most of good 1 and B starts out with most of good 2.
You will see that the competitive equilibrium sort of corrects this.
Scenario 5 has the same endowments as Scenario 4 but gives the two individuals the same
preferences so that the contract curve is the straight line joining the two origins.
Once again the competitive equilibrium is on this line.
The next 3 scenarios are interesting in that we return to the same totals as in Scenario 1 (150 units
of good 1 and 100 units of good 2) but we assume that individual A starts out with all 100 units of
good 2 while B starts out with all 150 units of good 1. This is an interesting case in that the priceoffer curve of A is horizontal while that for B is vertical. Why?
Consider individual A. Suppose his or her Cobb-Douglas parameter is a. Suppose he or she has an
initial endowment of zero of good 1 and e2 of good 2. Then the value of his income is p2e2 and we
know that he or she wants to spend a fraction a of this on good 1 and a fraction (1-a) on good 2. We
thus have his demands:
from which we get
q1 = ap2e2/p1
q1 = ap2e2/p1
and
and
q2 = (1-a)p2e2/p2
q2 = (1-a)e2
Note what the second of these says: that the demand for good 2 is constant, independent of the
prices, and is always equal to a fixed fraction of the initial endowment. In Scenario 6 A’s parameter
a is 0.7 and therefore A always wants to spend a fraction 30% of his initial endowment of good 2.
His initial endowment of good 2 is 100 and therefore he or she always wants 30 units of good 2. He
or she sells the rest (70 units) and buys as many units of good 1 as he or she can with this. Hence
the horizontal price-offer curve for individual A.
A similar argument applies for B. If his or her value of the parameter is a and if he or she starts with
an endowment of e1 units of good 1 and zero units of good 2, then the value of his or her income is
p1e1. His or her demands are therefore:
q1 = ap1e1/p1 and q2 = (1-a)p1e1/p2
from which we get
q1 = ae1
and
q2 = (1-a)p1e1/p2
from which we note that the demand for good 1 is constant, independent of the prices. In Scenario 6
a for B is 0.6 and so he or she always spends 60% of his or her endowment of good 1 on good 1.
The endowment is 150 units – therefore his demand for good 1 is constant at 90 units. He sells the
remaining 60 units and buys as many units of good 2 as possible with this. Thus his or her priceoffer curve is vertical at the value 60 (= 150 – 90). (Recall that we measure B from the top right
origin.) Scenario 6 looks as follows:
We should note that the preferences – and hence the contract curve – and the total endowments of
the two goods are the same as in Scenario 1. What differs is how the endowments are initially
allocated. We have a different allocation – so we have a different competitive equilibrium.
Scenario 7 is to Scenario 2 as Scenario 6 is to Scenario 1: just the initial distribution differs.
Scenario 7 is:
Similarly, Scenario 8 is to Scenario 3 as Scenario 6 is to Scenario 1: just the initial allocation
differs. We have:
Note that the two individuals have identical tastes – so that the contract curve is the straight line
joining the two origins.
8.12: Comments
It should be clear that there are almost always possibilities for exchange. Only when the initial
allocation point lies on the contract curve are such possibilities absent. Even when the preferences
are identical (so that the contract curve joins the two origins of the box) there will usually be the
possibility of trade – unless the initial point is on the contract curve (for example when the
endowments are identical so that we start at the middle point of the box). Also even when the
endowments are identical (so that the initial point is in the centre of the box) there will usually be
the possibility of trade – unless the contract curve also goes through the centre of the box (for
example, when preferences are identical). So, as long as people are different we will generally have
mutually advantageous trade.
This chapter paid particular attention to the competitive trading mechanism – showing that it is
efficient and leads to trade on the contract curve. We also saw that price-setting (monopoly or
monopsony) behaviour is inefficient. There are obviously other trading mechanisms – and you
might like to consider what their properties are.
8.13: Summary
We have done a lot in this chapter. We have considered the general problem of exchange between
two individuals and have used the clever device of the Edgeworth Box to examine this.
We have discovered the contract curve.
The contract curve is the locus of points efficient in the sense that, once on the contract curve, it is
impossible to make one person better off without making the other worse off.
Points off the contract curve are inefficient in the sense that there is always some movement which
makes at least one person better off without making the other worse off.
We have shown that the competitive equilibrium is on the contract curve.
The two price-offer curves must intersect on the contract curve.
The competitive equilibrium is on the contract curve and hence is efficient in the above sense.
Price-setting equilibria (in which one agent sets the price and the other chooses the point) is
inefficient.
Moreover and very importantly:
The competitive equilibrium depends upon the preferences and the endowments.
8.14: Is equality good? Is planning good?
There are two propositions that might be considered self-evident. The first is that equality is a good
thing - and hence that inequality is a bad thing. The second is that planning an economy is good for
the people in the society - and hence that leaving people open to market forces is bad. While these
propositions are certainly true in some instances, we explore here whether there are situations in
which they are not true. We use the apparatus of this chapter and consider a very simple pureexchange economy in which there are just two individuals. The individuals, crucially, are different.
There are two goods in this society, Good 1 and Good 2, and two individuals, Individual A and
Individual B. We assume that there is available in the society a quantity of 100 units of Good 1 and
a quantity of 100 units of Good 2 to allocate between the two individuals. We ask whether the
allocation of 50 units of each good to each individual – and the enforced consumption of these
quantities – is a good thing or not. That is, is planning plus equality necessarily a Good Thing?
We assume that the individuals are different in their preferences. Specifically we assume that
Individual A has Cobb-Douglas preferences with parameter 0.7 (that is, weight 0.7 on Good 1 and
weight 0.3 on Good 2) while Individual B has Cobb-Douglas preferences with parameter 0.3 ((that
is, weight 0.3 on Good 1 and weight 0.7 on Good 2). So Individual A relatively prefers Good 1
while Individual B relatively prefers Good 2. Having a difference in the preferences is important
and drives what follows. If the preferences were identical then the two propositions that we are
looking at are rather self-evident. But we know that in the real world people are different.
We use the apparatus of this chapter to investigate the allocation of the 100 units of Good 1 and the
100 units of Good 2 between the two individuals. We use an Edgeworth Box of size 100 by 100, in
which we measure Individual A’s consumption from the bottom left-hand origin and Individual B’s
consumption from the top right-hand corner of the box. Every point within the box is an allocation
of the 100 units of Good 1 and the 100 units of Good 2 between the two individuals. Consider a
planned allocation in which both individuals are given and consume 50 units of each good. This
allocation is at the centre of the box – point E in the figure below.
The question is: are the two individuals happy to stay at this point – or would they prefer to be
“exposed to market forces”. The answer depends on what these ‘market forces’ are. If we have a
competitive market, in which each individual takes the price as given and we seek for an
equilibrium price (at which both individuals would agree to a particular exchange), we can see that
the answer must be ‘yes’. The competitive equilibrium of this allocation problem is at the point
labelled C in this figure: it is on the contract curve and is at the intersection of the two price-offer
curves. Note that at C, Individual A consumes 70 of Good 1 and 30 of Good 2, while Individual
consumes 30 of Good 1 and 70 of Good 2. Note that in equilibrium the relative price of the two
goods is 1 – the slope of the line joining E with C is –1. You might like to ask where these numbers
(70 and 30, 30 and 70) come from (recall the preferences of the two individuals). We end up at an
unequal allocation – having started at an equal planned allocation – and both individuals prefer the
unequal allocation to the equal allocation. Inequality is not necessarily bad. Market forces are not
necessarily bad.
Indeed, you might like to argue that point C is in a sense the best allocation in the box – we started
with an equal allocation and we ended up (after trading at the competitive price) at a point they both
prefer to the initial allocation. It is interesting to note that we could end up at this ‘best’ point even
if we have an initial allocation that is clearly not the same for the two individuals. For example,
suppose we start with A having all 100 units of the good he or she relatively dislikes (Good 2) and
with B having all 100 units of the good that he or she relatively dislikes (Good 1). See the following
figure. Notice also that in both equilibria the implicit prices of the two goods are equal – their
relative price (minus the slope of the line joining the endowment point and the equilibrium point) is
1.
It is further clear that it does not matter where we start, as long as it is one some point on the
equilibrium budget constraint – the line joining the top left-hand corner of the box with the bottom
right-hand corner. It is interesting to note that at each point on this line, the values of the
endowments of the two individuals are equal (at the equilibrium price ratio). So we could argue
that this is truly a fair – if not necessarily equal (in terms of consumption) – situation: start at any
point on this equilibrium budget constraint and let the competitive market do the rest.
A planner obviously could also find the point C if it knew the preferences of the two individuals.
Clearly the point C depends on the preferences, and if the planner miscalculates the preferences
then it will miscalculate the point C. One advantage of the market solution is that the government
does not really need to know the preferences of the two individuals – it just needs to start them off
on the equilibrium budget constraint. In a sense this is cheating since the equilibrium budget
constraint is also dependent on the preferences (here we have chosen an example in which the
equilibrium price ratio just happens to be 1, but this does not need to be the case). Nevertheless the
government can try and start the two individuals at the centre of the box, and leave the competitive
market to do the rest. We end up (if the individuals have different preferences) at a necessarily
unequal consumption point – but one we might argue is ‘fair’.
There are other qualifications we should make, particularly to what we mean by market forces. If
one of the two individuals can set the price, then it may well be the case that we end up at an
unsatisfactory equilibrium. Perhaps you would like to think about this case.
8.15: Mathematical Appendix
Here we provide a solution to the general problem of competitive exchange between two
individuals with Cobb-Douglas preferences. Let us denote the endowments (e1,e2) for individual A
and (f1,f2) for individual B. Let us suppose that A has Cobb-Douglas preferences with parameter a
and that B has Cobb-Douglas preferences with parameter b. Do recall that the parameter indicates
the relative weight that the individual places on good 1 – the relative weight put on good 2 is one
minus this parameter. From the material in Chapter 6, we know that A’s gross demands for the two
goods are:
q1 = a(p1e1 + p2e2/p1 and q2 = (1-a)(p1e1 + p2e2)/p2
while B’s gross demands are
q1 = b(p1f1 + p2f2)/p1 and q2 = (1-b)(p1f1 + p2f2)/p2
From these we can calculate the aggregate gross demand for good 1 and impose the market-clearing
equilibrium condition that the aggregate gross demand should equal the aggregate supply of good 1,
which is e1 + f1. This gives us the equilibrium condition:
a(p1e1 + p2e2)/p1 + b(p1f1 + p2f2)/p1 = e1 + f1
This is an equation which can be solved for the price ratio p1/p2 which gives equilibrium in the
market for good 1. Solving it yields:
p2/p1 = [(1-a)e1 + (1-b)f1]/(ae2 + bf2)
(8.1)
Before discussing the implications of this, let us also derive the market-clearing condition for good
2. Imposing the condition that the aggregate gross demand equals the supply of good 2, we have:
(1-a)(p1e1 + p2e2)/p2 + (1-b)(p1f1 + p2f2)/p2 = e2 + f2
If we solve this for the implied equilibrium price ratio p2/p1 we get….equation (8.1)! Is this a
surprise? Clearly not – as if the market for good 1 is in equilibrium then so must be the market for
good 2. (You should check this out. Suppose the equilibrium in the market for good 1 has A giving
x units to A and B receiving x units from A, then it must be the case that A is getting in exchange y
units of good 2 and B is giving in exchange y units of good 2, where p1x = p2y.)
Now let us examine the equilibrium condition, after noticing that it is a condition on the relative
prices of good 1 and good 2 (which determines the slope of the equilibrium budget constraint).
From (8.1) above we see that
p2/p1 increases if either of e1, f1 increase
and
p2/p1 decreases if any of e2, f2, a or b increases
The first of these says that if good 1 becomes more plentiful then its relative price decreases. The
second says: (1) that if good 2 becomes more plentiful then its relative prices decreases; and (2) that
if good 1 becomes more preferred by either individual (either a or b increases) then the relative
price of good 1 increases; (3) that if good 2 becomes more preferred by either individual (either a or
b decreases) then the relative price of good 2 increases. All of these accord with intuition.