The Suntory and Toyota International Centres for Economics and Related Disciplines
Consumption Theory in Terms of Revealed Preference
Author(s): Paul A. Samuelson
Source: Economica, New Series, Vol. 15, No. 60 (Nov., 1948), pp. 243-253
Published by: Blackwell Publishing on behalf of The London School of Economics and Political
Science and The Suntory and Toyota International Centres for Economics and Related
Disciplines
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I948]
ConsumptionTheory
in Terms of RevealedPreference
By
PAUL
I.
A.
SAMUELSON
INTRODUCTION
A DECADE ago I suggested that the economic theory of consumer's
behaviour can be largely built up on the notion of " revealed.preference".
By comparing the costs of different combinations of goods at different
relative price situations, we can infer whether a given batch of goods
is preferredto another batch; the individual guinea-pig, by his market
behaviour, reveals his preference pattern-if there is such a consistent
pattern.
Recently, Mr. Ian M. D. Little of Oxford University has made an
important contribution to this field.1 In addition to showing the
changes in viewpoint that this theory may lead to, he has presented
an ingenious proof that if enough judiciously selected price-quantity
situations are available for two goods, we may define a locus which
is the precise equivalent of the conventional indifference curve.
I should like, briefly, to present an alternative demonstration of
this same result. While the proof is a direct one, it requires a little
more mathematical reasoning than does his.
2.
OBSERVABLE
PRICE
RATIOS
AND A FUNDAMENTAL
DIFFERENTIAL
EQUATION
If we confine ourselves to the case of two commodities, x and y,
we could conceptually observe for any individual a number of pricequantity situations. Since only relative prices are asstimed to matter,
each observation consists of the triplet of numbers, (Px/Pi, x, y). By
manipulating prices and income, we could cause the individual to
come into equilibrium at any (x, y) point, at least within a given area.
We may also make the simplifying assumption that one and only one
price ratio can be associated with each combination of x and y.
Theoretically, therefore, we could for any point (x, y) determine a
unique pI/p,;
or
(I)
P$/Pv=--f (xXy)
wheref is an observable function, assumed to be continuous and with
continuous partial derivatives.2
1 I. M. D. Little:
"A Reformulation of the Theory of Consumers' Behaviour", Oxford
P. A. Samuelson: Foundations of
Economic Papers, New Series, No. i, January, I949;
Economic Analysis (I947), Ch. V and VI; P. A. Samuelson: "A Note on the Pure Theory of
Consumer's Behaviour; and an Addendum ", Economica (I938), Vol. V (New Series), pp. 6I-7I,
353-3542 Mathematically,
the above continuity assumptions are over-strict. Also, we shall make
the unnecessarily strong assumption that in the region under discussion the price-quantity
o.
relations have the "simple concavity" property: f (af/ay) - (af/ax)>
243
244
ECONOMICA
[NOVEMBER
The central notion underlying the theory of revealed preference,
and indeed the whole modern economic theory of index numbers,
is very simple. Through any observed equilibrium point, A, draw the
budget-equation straight line with arithmetical slope given by the
observed price ratio. Then all combinations of goods on or within
the budget line could have been bought in preference to what was
Y
30
25
20
15
10 _
5~~
0
5
10
15
20 o
FIG. I.
actually bought. But they weren't. Hence, they are all " revealed"
to be inferior to A. No other line of reasoning is needed.
As yet we have no right to speak of " indifference ", and certainly
no right to speak of " indifference slopes ". But nobody can object
to our summarising our observable information graphically by drawing
a little negative " slope element " at each x and y point, with numerical
gradient equal to the price ratio in question.
1948]
CONSUMPTION
THEORY IN TERMS OF REVEALED
PREFERENCE
245
This is shown in Figure i by the numerous little arrows. These
little slopes are all that we choose to draw in of the budget lines which
go through each point and the directional arrows are only drawn in
to guide the eye. It is a well known observation of Gestaltpsychology
that the eye tends to discern smooth contour lines from such a representation, although strictly speaking, only a finite number of little
line segments are depicted, and they do not for the most part run into
each other.' (In the present illustration the contour lines have been
taken to be the familiar rectangular hyperbolae or unitary-elasticity
curves and f (x, y) takes the simple form p yl,/px=y/x.)
There is an exact mathematical counterpart'of this phenomenon of
Gestalt psychology. Let us identify a little slope, dy/d:x, with each
price ratio, - P,/P,. Then, from (i), we have the simplest differential
equation
(2)
dy
(X)y).
It is known mathematically that this defines a unique curve through
any given point, and a (one-parameter) family of 'curves throughout
the surrounding (x, y) plane. These solution curves (or " integral
solutions " as they are often called) are such that when any one of
them is substituted into the above differential equation, it will be
found to satisfy that equation. Later we shall verify that these solution
curves are the conventional " indifference curves " of modern economic
theory. Also, and this is the novel part of the present paper, I shall
show that these solution curves are in fact the limiting loci of revealed
preference-or in Mr. Little's terminology they are the " behaviour
curves " defined for specified initial points. This is our excuse for
arbitrarily associating the differential equation system (2) with our
observable pattern of prices and quantities summarised in (i).
3.
THE
CAUCHY-LIPSCHITZ
PROCESS
OF APPROXIMATION
Mathematicians are able to establish rigorously the existence of
solutions to the differential equations without having to rely upon
the mind's eye as a primitive "differential-analyser " or " integrator "*2
Also, mathematicians have devised rigorous methods for numerical
solution of such equations to any desired (and recognisable) degree of
accuracy.
It so happens that one of the simplest methods for proving the
existence of, and numerically approximating, a solution is that called
the " Cauchy-Lipschitz " method after the men who first made it
1 Every student of elementary physics has dusted iron filings on a piece of paper suspended
on a permanent magnet. The little filings become magnetised and orient themselves in' a simple
pattern. To the mind's eye these appear as " lines of force " of the magnetic field.
2 The usual proof found in such intermediate texib as F. R. Moulton, Differential Equations,
Ch. XII-XIII, is that of Picard's " method of successive approximations ". But the earlier
rigorous proofs are by the Cauchy-Lipschitz method, which is very closely related to the economic
theory of index numbers and revealed preference. See also, R. G. D. Allen, Mathematical Analysis
for Economists, I938, Ch. XVI.
246
ECONOMICA
[NOVEMBER
rigorous, even though it really goes back to at least the time of Euler.
In this method we approximate to our true solution curve by a
connected series of straight line-segments, each line having the slope
dictated by the differential equation for the beginning point of the
straight line-segment in question. This means that our differential
equation is not perfectly satisfied at all other points ; but if we make
our line-segments numerous and short enough, the resulting error from
the true solution can be made- as small as we please.
Figure 2 illustrates the Cauchy-Lipschitz approximations to the
true solution passing through the point A (IO,30) and going from
X= IO, to the vertical line x= I5. The top smooth curve is the true
unitary-elasticity curve that we hope to approximate. The three
lower broken-line curves are successive approximations, improving
in accuracy as we n-ove to higher curves;
Our crudest Caichy-Lipschitz approximation is to use one linesegment for the whole interval. We pass a straight line through A
with a slope equal to the little arrow at A, or equal to - 3. This is
nothing but the familiar budget line through the initial point A; it
intersects the vertical line x= I5, at the value y= I5 or at the point
marked Z'.1
(Actually, from the economic theory of index numbers and consumer's choice, we know that this first crude approximation Z': (x, y)
= (I5, I5) clearly revealed itself to be "worse" than (x, y) = (IO,30)
-since the former was actually chosen over the latter even though
both cost the same amount. This suggests that the Cauchy-Lipschitz
process will always approach the true solution curve, or " indifference
curve ", from below. This is in fact a general truth, as we are about
to see.) Can we not get a better approximation to the correct solution
than this crude straight line, AZ' ? Yes, if we use two line-segments
instead of one. As before let us first proceed on a straight line through
A with slope equal to A's little arrow. But let us travel on this line
only two-fifths as far as before: to x= I2, rather than x_ I5. This
gives us a new point B' (I 2, 24), whose directional arrow is seen to
have the slope of - 2. Now, through B' we travel on a new straight
line with this new slope; and our second, better, approximation to
the true value at x= I5, is given by the new intersection, Z", with
the vertical line, at the level y_ I8. (The " true " value is obviously
at Z on the smooth curve where y must equal 20 if we are to be on
the hyperbola with the property xy= I0 X 30_ I5 X 20; and our
second approximation has only 3 the error of our first.)
The general procedure of the Cauchy-Lipschitz process is now clear.
Suppose we divide the interval between x= IO and x= I5 into 5 equal
segments ; suppose we follow each straight line with slope equal to
its initial arrow until we reach the end of the interval, and then begin a
new straight line. Then as our numerical table shows, we get the still
1 A Numerical Appendix gives the exact arithmetic underlying this and the following
figure.
I948]
CONSUMPTION
THEORY
IN TERMS
OF REVEALED
PREFERENCE
247
In Figure 2, the broken line from A
better approximation, y= I92.
to Z"' shows our third approximation.
In the limit as we take enough sub-intervals so that the size of each
line-segment becomes indefinitely small, we approach the true value
of y- 20, and the same is true for the true value at any other x point.
How do we know this ? Because the pure mathematician assures us
that this can be rigorously proved.
15
10
30
A30
2.5
25
z"
20
20
Z5
15
15
10
X
FIG4 2.
In economic terms, the individual is definitely going downhill along
any one Cauchy-Lipschitz curve. For just as A was revealed to be
better than Z', so also was it revealed to be better than B'. Note too
that Z" is on the budget line of B' and is hence revealed to be inferior
248
ECONOMICA
[NOVEMBER
to B', which already has been revealed to be worse than A. It follows
that Z" is worse than A.
By the same reasoning Z"' on the third approximation curve is
shown to be inferior to A, although it now takes four intermediate
points to make this certain. It follows as a general rule: any CauchyLipschitz path always leads to a final point worse than the initial.
And strictly speaking, it is only as an infinite limit that we can hope
to reveal the neutral case of " indifference" along the -true solution
curve to the differential equation.
4.
AN INDIRECT
PROOF OF
LIMITING
REVELATION
OF INDIFFERENCE"
We have really proved only one thing so far: all points below the
true mathematical solution passing through an initial point, A, are
definitely " revealed to be worse " than A.
We have not rigorously proved that points falling on the solution
contour curve are really " equal " to A. Indeed in terms of the strict
algebra of " revealed preference " we have as yet no definition of
what is meant by " equality " or " inciifference".
Still it would be a great step forward if we could definitely prove
the following: all points above the true mathematical solution are
definitely " revealed to be better " than A.
The next following section gives a direct proof of this fact by defining
a new process which is similar to the Cauchy-Lipschitz process and
which definitely approximates to the true integral solution from above.
But it may be as well to digress in this section and show that by indirect
reasoning like that of Mr. Little, we may establish the proposition
that all points above the solution-contour are clearly better than A.
I shall only sketch the reasoning. Suppose we take any point just
vertically above the point Z and regard it as our new initial point.
The mathematician assures us that a new " higher " solution-contour
goes through such a point. Let us-construct a Cauchy-Lipschitz process
leftward, or backwards. Then by using small enough line-segments we
may approach indefinitely close to thatpoint verticallyaboveA whichlies
on the new contourline aboveA's contour. A will then have to lie below
the leftward-moving Cauchy-Lipschitz curve, and is thus revealed to be
worse than any new initial point lying above the old contour line. Q.E.D.
We may follow Mr. Little's terminology and give the name " behaviour
line " to the unique curve which lies between the points definitely
shown to be better than A, and those definitely shown to be worse
than A. This happens to coincide with the mathematical solution to
the differential equation, and we may care to give this contour line,
by courtesy, the title of an indifference curve.1
1 If our preferencefield does not have simpleconcavity-and why shouldit ?-we may observe
cases where A is preferredto B at some times, and B to A at others. If this is a pattern of
consistencyand not of chaos, we could choose to regardA and B as " indifferent" underthose
circumstances. If the preferencefield has simpleconcavity, " indifference" will never explicitly
reveal itself to us except as the results of an infinite limiting process.
1948]
CONSUMPTION THEORY IN TERMS OF REVEALED
PREFERENCE
249
5. A NEW APPROXIMATION PROCESS FROM ABOVE
Let me return now to the problem of defining a new approximating
process, like the conventional Cauchy-Lipschitz process, but which:
(i)
approaches the mathematical solution from above rather than
below, and which (2) definitely reveals the economic preference of
the individual at every point.
15
10
YI
30
30
A
25
25
20
Z
20
15
IS _
15
10
X
FIG. 3.
Our new process will consist of broken straight lines; and in the
limit these will become numerous enough to approach a smooth curve.
But the slopes of the straight line-segments will not be given by their
initial points, as in the Cauchy-Lipschitz process. Instead, the slope
will be determined by the final point of the sub-interval's line-segment.
250
ECONOMICA
[NOVEMBER
After the reader ponders over this for some time and considers its
geometrical significance, he may feel that he is being swindled. How
can we determine the slope at the line's final point, without first
determining the final point ? But, how can we know the final point
of the line unless its slope has already been determined ? Clearly, we
are at something of a circular impasse. To determine the slope, we
seem already to require the slope.
The way out of this dilemma is perfectly straightforward to anyone
who has grasped the mathematical solution of a simultaneous equation.
The logical circle is a virtuous rather than a vicious one. By solving the
implied simultaneous equation, we cut through the problem of circular
interdependence. And in this case we do not need an electronic computer to solve the implied equation. Our human guinea-pig, simply by
following his own bent, inadvertently helps to solve our problem for us.
In Figure 3, we again begin with the initial point A. Again we
wish to find the true solution for y at x 15. Our first and crudest
approximation will consist of one straight line. But its slope will
be determined at the end of the interval and is initially unknown.
Let us, therefore, through A swing a straight line through all possible
angles. One and only one of these slopes will give us a line that is
exactly tangent to one of the little arrows at the end of our interval.
Let Z' be the point where our straight line is just tangent to an arrow
lying in the vertical line. It corresponds to a y value of 22, which is
above the true value of y = 20.
Economically speaking, when we rotate a straight " budget line"
around an initial point A, and let the individual pick the best combination of goods in each situation, we trace out a so-called " offer curve
This curve is not drawn in on the figure, but the point Z' is the intersection of the offer curve with the vertical line. It should be obvious
from our earlier reasoning that Z' and any other point on the offer
curve is revealed to be better than A, since any such equal-cost point
is chosen over A.
So much for our crude first approximation. Let us try dividing the
interval between x= io and x= I5 up into two sub-intervals so that
two connected straight lines nmaybe used. If we wish the first line
to end at X- I2, we rotate our line through A until its final slope is
just equal to the indicated little arrow (or price ratio) along the vertical
line x= I2. For the simple hyperbole in question, where - p$/pvdY
y/x, our straight line will be found to end at the point B", whose
(x, y) coordinates are (I 2, 25a) and whose arrow has a slope of just
less than (- 2).
We now begin at B" as a new initial point and repeat the process
by finding a new straight line over the interval from x I2 to x== I5.
Pivoting a line through all possible angles, we find tangency only at
the point Z", where y= 2I3, which is a still better approximation to
the true value, y = 20.
I948]
CONSUMPTION
THEORY IN TERMS OF REVEALED
PREFERENCE
25I
The interested reader may easily verify that using more sub-intervals
and intermediate points will bring us indefinitely close to the true
solution-contour.' It is clear therefore that our new process brings
us to the true solution in the limit, but unlike the Cauchy-Lipschitz
process, it now approaches the solution from above. And we can use
the word " above " in more than a geometrical sense. Along the
new process lines, the individual is revealing himself to be getting
better off. For just as A is inferior to Z', it is by the same reasoning
inferior to B", which is likewise inferior to Z" ; from which it follows
that A is inferior to Z".
It should be clear, therefore, that no matter how many intermediate
points there are in the new process, the consumer none the less reveals
himself to be travelling uphill. It follows that every point above the
mathematical contour line can reveal itself to be better than A.
6. CONCLUSION
This essentially completes the present demonstration. The mathematical contour lines defined by our differential equation have been
proved to be the frontier between points revealed to be inferior to
A, and points revealed to be superior. The points lying literally on a
(concave) frontier locus can never themselves be revealed to be better
or worse than A. If we wish, then, we may speak of them as being
indifferent to A.
The whole theory of consumer's behaviour can thus be based upon
operationally meaningful foundations in terms of revealed preference.2
1 He
y
=
may verify that using the points x I0 IIo
12) 13, 14, i5 brings us to within i ot
2o, as shown in the second table of the Numerical Appendix.
2 The above remarks apply without
qualification to two dimensional problems where the
problem of " integrability " cannot appear.
In the multidimensional case there still remain
some problems, awaiting a solution for more than a decade now.
ECONOMICA
25z
NUMERICAL
[~NOVEMBER
APPENDIX
In the Cauchy-Lipschitz process, the straight line going from (x0, Yo)
to (x,, Yl) is defined by the explicit equation
(a)
-90(X-X0)
Y0)(X-X0)=yY
Y=Y0-f(x0,
where dy/dx= -f (x, y) is the differential equation requiring solution
-in this case being = - y/x. The three approximations given in
Figure 2 are derived numerically in the following table.
TABLE
I.
CAUCHY-LIPSCHITZ
x
IO
'5
IO
12
'5
IO
II
12
'3
I4
'5
APPROXIMATION
dy
d
dx=
Y
First Approximation
initial point
-
30-3(I2
IO)
24 - 2 (15 - I2)=
II
270
_
270
Iz
270
__
(II)(I-)
(Iz)(I3)
_
- 24/12=
-2
- 30/IO=
- 27/11=
-ZII
24-A
II
- Iz)=
0I3
(-
('4 -
-3
270
270
22,
2
(Iz2)(13)
I3
270
(13)(14)
(I I)(I2)
270
270
270
270
13)=-
270
'3
-3
24
27
II
270
270
- 30/IO=
30
-IO)=
27
27 - II 1
30
i8
Third Approximation
initial point
3 (II
- 3
15
Second Approximation
initial point
30-
x
- 301IO=
30
30 - 3 (15 - 10)=
f (X, Y)
I4
13
(I3)(I4)
==
9-
-
2091
I9$
In ihe new process which approaches the true solution, y= 300/x,
from above, the straight lines hive their slopes determined by the
final point of each interval, or by the implicit equation
y,= yo -f(xI, y)(x - x0)
(b)
case
In the
where f (x, y) = y/x, we have
y
zx1o- (xj
Yo
XI
Yo-
Y1=
2xl--
X()
xx
or
1948]
CONSUMPTION
THEORY IN TERMS OF REVEALED
PREFERENCI
253
Our numerical approximations are given in the following table:
PROCESS
NEW APPROXIMATING
TABLE 2.
X1
IO
I5
10
Yi= 2X1-x
2 (5)-
(30)=
7
JI50-
I2
-^
7
7
=
-25T
= 2I- v7
Third Approximation
initial point
2 (II)
-
I2
2 (2)-
30
330
(30)=
- I
330
12
'5
30
i8o
80o
15
II
I4
=22i
Second Approximation
initial point
II
13
30
z(15)2 (I5)
-- (30)
2 (i2)- I
I0
(Y)
First Approximation
initial point
12
I5
xo
II
12
13
330
2 (13)- 2 13
330
I4
2 (4)- I3 I4
330
I5
2 (15)- 4 I5
I2 330
13
12
13 330
14 I3
I4 330
I5 I4
I5 330
I6 15
- 271
=25T-7
330
I3
330
I4
225
330
I5
330
-6- 2o
I
It may be mentioned that the third Cauchy-Lipschitz approximation
satisfies the equation 270/x which is less than the true solution, 300/x;
and the third approximation of the new upper process satisfies the
equation 33o/x, which happens to be equally in excess of the true
solution.
[In Figure 3, the point between A and Z" should be labelled B"].