Consumption Theory in Terms of Revealed Preference

The Suntory and Toyota International Centres for Economics and Related Disciplines

Consumption Theory in Terms of Revealed PreferenceAuthor(s): Paul A. SamuelsonSource: Economica, New Series, Vol. 15, No. 60 (Nov., 1948), pp. 243-253Published by: Wiley on behalf of The London School of Economics and Political Science and TheSuntory and Toyota International Centres for Economics and Related DisciplinesStable URL: http://www.jstor.org/stable/2549561 .

Accessed: 14/07/2014 23:12

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Wiley, The London School of Economics and Political Science, The Suntory and Toyota International Centresfor Economics and Related Disciplines are collaborating with JSTOR to digitize, preserve and extend access toEconomica.

http://www.jstor.org

This content downloaded from 68.43.199.197 on Mon, 14 Jul 2014 23:12:55 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=black

http://www.jstor.org/action/showPublisher?publisherCode=lonschool

http://www.jstor.org/action/showPublisher?publisherCode=suntoy

http://www.jstor.org/action/showPublisher?publisherCode=suntoy

http://www.jstor.org/stable/2549561?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


I948]

Consumption Theory in Terms of Revealed Preference

By PAUL A. SAMUELSON

I. 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 preferred to 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 price- quantity 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 (xX y) 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 Economic Papers, New Series, No. i, January, I949; P. A. Samuelson: Foundations of 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,

353-354- 2 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 relations have the "simple concavity" property: f (af/ay) - (af/ax)> o.

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 Gestalt psychology that the eye tends to discern smooth contour lines from such a repre- sentation, 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.




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 line- segment 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

better approximation, y= I92. In Figure 2, the broken line from A 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.

10 15

30 A30

25 2.5

20 z" 20

15 Z5

10 15 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




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 Cauchy- Lipschitz 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 that point vertically above A which lies on the new contour line above A'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 preference field does not have simple concavity-and why should it ?-we may observe cases where A is preferred to B at some times, and B to A at others. If this is a pattern of consistency and not of chaos, we could choose to regard A and B as " indifferent " under those circumstances. If the preference field has simple concavity, " 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.

10 15

YI 30 A 30

25 25

20 Z 20

IS _ 15

10 15 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.




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 com- puter 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 nmay be 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$/pv-

dY 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 may verify that using the points x I0 IIo 12) 13, 14, i5 brings us to within i ot y = 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.



NUMERICAL APPENDIX

In the Cauchy-Lipschitz process, the straight line going from (x0, Yo) to (x,, Yl) is defined by the explicit equation

(a) Y=Y0-f(x0, Y0)(X-X0)=yY -90(X-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 APPROXIMATION

dy x Y = f (X, Y) d dx x

First Approximation initial point

30 - 3 (15 - 10)=

Second Approximation initial point 30-3(I2 - IO)

24 - 2 (15 - I2)=

Third Approximation initial point 30- 3 (II -IO)=

27 __ 270 27 - 1 II

II II

270 270 270 0I3 - Iz)= II (II)(I-) (-

270 _ 270 270 ('4 - 13)=- Iz (Iz)(I3) I3

270 _ 270 270

'3 (13)(14) I4

30

15

30

24 i8

30

27

24-A

22,

1 3

2091 I9$

- 301IO= - 3

- 30/IO=

- 24/12=

-3 -2

- 30/IO= -3 - 27/11= -ZII

270

(I I)(I2)

270

(Iz2)(13) =- 2

270 = -

(I3)(I4) 9-

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

(b) y,= yo -f(xI, y)(x - x0) In the case where f (x, y) = y/x, we have

Yo y zx1o- (xj x x or

XI Y1= Yo-

2xl-- X()

IO

'5

IO

12

'5

IO

12 I I

'3

I4

'5

[~NOVEMBER 25z ECONOMICA



1948] CONSUMPTION THEORY IN TERMS OF REVEALED PREFERENCI 253

Our numerical approximations are given in the following table:

TABLE 2. NEW APPROXIMATING PROCESS

Yi= x (Y) 2X1- xo

X1

IO

I5

10

I5

I0

II

12

13

I4

'5

First Approximation initial point

z(15)- (30) 2 (I5) --

Second Approximation initial point

12 (30)= 2 (i2)- I

15 80o

2 (5)- I2 7

Third Approximation initial point

II (30) =

2 (II) - I

I2 330 I2 330 2 (2)- II 12 13 12

13 330 13 330 2 (13)- 2 13 14 I3

I4 330 I4 330 2 (4)- I3 I4 I5 I4

I5 330 I5 330 2 (15)- 4 I5 I6 15

30

=22i

30 i8o

= -25T 7

I50 J - = 2I-

7 v7

-^

30

330 - 271

330 - =25T-7

330 225 I3

I4 330

I5 330 -6- 2o

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"].

I



Documents

Consumption Theory in Terms of Revealed Preference