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    Consumption Theory in Terms of Revealed Preference

    Paul A. Samuelson

    Economica, New Series, Vol. 15, No. 60. (Nov., 1948), pp. 243-253.

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    Consumption Theory

    in

    Terms

    of

    Revealed Preference

    A DECADE ago

    I

    suggested th a t th e economic theo ry of consumer's

    beh aviou r can be largely built up on th e notio n of revealed.preference .

    B y com paring th e costs of different com binatio ns of goo ds at different

    relative price situations, we can infer whether a given batch of goods

    is preferred to anothe r batc h

    ;

    the individual guinea-pig, by his market

    beh aviou r, reveals his preference pattern-if the re is such a consistent

    pa t te rn .

    Recently, Mr. Ian

    M.

    D.

    L ittl e of Oxford U nive rsity has made a n

    imp ortant co ntr ibution to th is f ie ld ,l In addit ion t o showing the

    changes in viewpoint that this theory may lead to, he has presented

    a n ingenious proof th a t if enough judiciously selected price -qu an tity

    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 sam e result. W hile th e proof is a direct one, it requires a little

    more mathematical reasoning than does his.

    If we confine ourselves to th e case of two comm odities, x and y ,

    we could conceptually observe for any individual a number of price-

    qu an tity situa tions . Since only relative prices are assbmed to m att er,

    each obse rvation consists of the trip let of num bers, (p,/p,, x, y). B y

    manipulating prices and income, we could cause the individual to

    come in to equilibrium a t an y (x, y) point, at least within a given area.

    We may also make the simplifying assumption tha-t 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 p / p ; or

    (1) pz lp ,= f (2,

    Y

    wheref is an observable function, assumed to be continuous and with

    continuous partial derivative^.^

    I. M. D. Li t t l e

    :

    " A Refo rmula tion of the Theo ry of Consumers' Behaviour", Oxford

    Economic Papers, Neur Series, No.

    I ,

    January , 1949

    ; P.

    A. Sam uels on Foundatioizs of

    E co no mic A n a l ~ ~ s i s1947), Ch. V and VI

    ;

    P. A. Samuelson

    :

    A Note on the Pure Theory of

    Consumer 's Behav iour ; and an Adden dum Economics (1938 ), Vol.

    V

    (New Series), pp. 61-71,

    353-354.

    Mathem atical ly, the abo ve cont inui ty assumptions are over-st r ict . Also, ure shall mak e

    the unnecessari ly s t rong assumption th at in th e region under discussion the p rice-quant i ty

    relat ions hav e the simple concavi ty prop erty J(af iay) (af/ax)>

    O.

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    Th e central notion underlying the theo ry of revealed preference,

    a n d indeed th e whole modern economic theo ry of index

    numbers,

    is very simple. Thro ugh an y observed equilibrium point,

    A

    draw the

    budget-equation straight line with arithmetical slope given by the

    observed price ratio. T he n all com binations of goods on or with in

    the budget line could have been bought in preference to what was

    actually bought. B u t th ey weren t. Hence, the y are all revealed

    to be inferior to A.

    No other line of reasoning is needed.

    As ye t we ha ve no right to speak of ndifference

    ,

    and certainly

    no righ t t o speak of ndifference slopes

    .

    But nobody can object

    to our summarising our observable information graphically by drawing

    a little negative slope elemen t a t each

    x

    and y point, with numerical

    gradient equal to the price ratio in question.

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    194.81

    C O K S U h Z P T I O N

    T H E O R Y

    I N TERMS

    OF RFVFALED PREFERENCE 245

    This is shown in Figure I by the numerous li t t le arrows.

    These

    little slopes are all th a t we choose to d raw i n of th e bud get lines which

    go through each point and the directional arrows are only drawn in

    to guide the eye. I t is a well known obse rvation of

    Gestalt

    psychology

    that the eye tends to discern smooth contour l ines from such a repre-

    sen tation , altho ugh stric tly speaking, only a finite num ber of little

    line segm ents are depicted, an d the y do n ot .for the most pa rt r un into

    each other.1 (I n th e present illustration the contour lines hav e been

    taken to be the familiar rectangular hyperbola: or unitary-elasticity

    curves an d (x, y takes the simple form p,/p,:y/x.)

    There is an exact mathematical counterpart of this phenomenon of

    estalt

    psychology. L et us identify a littl e slope, dyldx, w ith each

    price ratio, p,/p,. Th en , from (I), we have the simplest differential

    equat ion

    It is known mathematically that this defines a unique curve through

    an y given poin t, an d a (one-parameter) family of c urves throu gho ut

    the surrounding

    x ,

    y) plane. These solutio n curve s (or integra l

    solutions as th ey are often called) are such th a t when a ny one of

    them is substituted into the above differential equation, it will be

    found to sat isfy th a t equat ion. Later we shall verify t h a t these solution

    curv es are th e con ven tional indifference curve s of m odern economic

    theo ry. Also, an d this is the novel pa rt of th e present paper,

    I

    shall

    show th a t these solution curves are in fact th e lim iting loci of revealed

    preference-or in Mr. Little's termin olog y th e y are th e behaviour

    curv es defined for specified ini tial po int s. Th is is our excuse for

    arbitrarily associating the differential equation system (2) with our

    observable pattern of prices and quantities summarised in I).

    Mathematicians are able to establish rigorously the existence of

    solutions to the differential equations without having to rely upon

    th e mind's eye as a prim itive differential-analyser or int eg rato r .2

    Also, mathematicians have devised rigorous methods for numerical

    solution of such eq ua tion s to an y desire d (and recognisable) degree of

    accuracy.

    I t so happens th at one of th e simples t methods for proving the

    existence of, and numerically approximating, a solution is that called

    the Cauchy-Lipschitz method afte r the men who first ma de it

    Ev ery st ud en t of elem entary physics has du sted i ron f il ings on a piece of p ape r suspended

    on a perm ane nt magnet . The l i t t le f il ings become magnet ised an d orient themselves in a s imple

    pat te rn .

    T o th e mind 's eye these ap pe ar as lin es of for ce of the mag netic field.

    The usual proof found in such intermediate texts as F. R Moulton,

    Differential Equations

    Ch.

    X I I - X I I I

    is th a t of Picard's m et ho d of successive appro xima tions

    .

    But the earl ier

    rigor& proofs are by the Cauchy-Lipschitz m ethod , which is very closely related to the economic

    the ory of index num bers and revealed preference. See also, R. G. D. Allen,

    Ma~hematical nalysis

    for Economists

    1938 Ch. XVI.

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    246

    ECONOMIC [NOVEMBER

    rigorous, even thou gh i t really goes back t o a t least the tim e of Eule r.

    In this method we approx ima te to our tru e solution .curve by a

    connected series of stra igh t line-segments, each line having t h e slope

    dictated by the differential equation for

    the beginning point

    of the

    straig ht line-segment in question. This mea ns th a t our differential

    equation is not perfectly satisfied at all other points

    ;

    but if we make

    our line-segments num erous a nd sh ort 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 (10,30) and going from

    x =

    10

    to the vertical l ine x=

    15 .

    The top smooth curve is the t rue

    unitary-elasticity curve th at we hope to approxim ate. Th e three

    lower broken-line curves are successive approximations, improving

    in accuracy as we move to higher curves.

    Our crudest Cauchy-Lipschitz approximation is to use one line-

    segment for the whole interval. We pass a straig ht line throu gh A

    with a s lope equal to the l i t t le arrow at A, or equal t o 3. This is

    nothing but the familiar budget l ine through the ini t ia l point A ; i t

    intersects the vertical line

    x= 15 , a t the va lue y =

    1 5 or at the point

    marked Z'.'

    (Actually, from the economic theory of index numbers and con-

    sumer's choice, we know that this first crude approximation Z' : (x, y)

    (15, 15) clearly revealed itself t o be worse th a n (x,

    y)

    (10,30)

    -since th e former was actually chosen over the lat ter even thou gh

    both cost the same amou nt. This suggests tha t the Cauchy-Lipschitz

    process will always approa ch th e tru e solution curv e, or indifference

    curve , from

    below

    This is in fact a general truth, as we are about

    to see.) Can we not get a better approx ima tion to the correct solution

    th a n this crude st ra ig ht line, AZ' Yes, if we use tw o line-segments

    instead of one. As b e f ~ r eet us first proceed on a straig ht line throu gh

    A with slope equ al to A's little arrow. B u t let us tra ve l on this line

    only two-fifths as far as before : t o x = 12 r a the r than x= 15. This

    gives us a new point B' (12, 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; an d our second, better , approx ima tion to

    t h e t r u e v a l u e a t x = 15 is given by th e new intersection, Z , w ith

    th e ve rtical l ine, a t th e level y = 18. (The tr u e value is obviously

    at Z on the smooth curve where

    y

    must equal 20 if we are to be on

    the hyperbola with the proper ty xy=

    10

    x

    30=

    15

    x

    20 ; and our

    second app roxim ation has only th e error of our first.)

    T he gen eral procedure of th e Cauchy-L ipschitz process is now clear.

    Suppose we divide the interval between x =

    10

    a nd x = 15 into 5 equal

    segments ; suppose we follow each straight line with slope equal to

    its initial arrow until we reach th e end of t he in terva l, an d then begin a

    new straight line. T he n as our numerical table shows, we get the still

    A Numerical Appendix gives the exact arithmetic underlying this and the following

    figure.

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    better approximation, y 199. In Figure 2, the broken line from

    A

    to Z ' shows our third approxim ation..

    I n the limit as we take e nough sub-intervals so t h a t th e size of eac h

    line-segment becomes indefinitely small, we approach the true value

    of y

    20

    an d the same is t rue for the tru e value at an y other

    x

    point.

    Ho w do we know this Because the pure ma them atician assures us

    that this can be rigorously proved.

    In economic terms, the individual is definitely going downhill along

    any one

    Cauchy-lips chit^,

    curve.

    For just as

    A

    was revealed to be

    better than Z', so also was it revealed to be better than

    B'.

    Note too

    tha t

    Z

    is on the bud get line of B' a nd is hence revealed t o be inferior

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    248

    ECONOMIC [NOVEMBER

    t o B', which already has been revealed to be worse th an A. I t follows

    t h a t Z is worse than A.

    By the same reasoning

    2 '

    on the third approximation curve is

    shown to be inferior to A, although it now takes four intermediate

    points to m ake this certain. I t follows as a general rule

    :

    an y 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

    t o reveal th e ne ut ral case of indifference along th e tru e solution

    curve to the differential equation.

    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 t o be worse th a n A.

    We have not rigorously proved that points falling on the solution

    contour curve are really equal to A. Indeed in terms of th e strict

    algebra of revealed preference we ha ve as ye t no definition of

    wh at is m ean t by equ ality or indifference

    .

    Still it would be a gr ea t ste p forward if we could definitely prove

    th e following: all points

    above

    the true mathematical solut ion are

    definitely revealed t o be bette r th an A.

    T he ne xt following section gives a direct proof of thi s fact b y defining

    a new process which is similar to the Cauchy-Lipschitz process and

    which definitely approximates to the true integral solution

    from above

    Bu t it m ay be as well to digress in this section an d show tha t b y indirect

    reasoning like th a t of Mr. Little , we m ay establish t he proposition

    that all points above the solution-contour are clearly better than A.

    I shall only ske tch th e reasoning. Suppose we tak e an y point just

    vertically above the point

    Z

    and regard it as our new initial point.

    Th e m athem atician assures us th a t a new higher solution-contour

    goes throu gh such a point. Le t us con struc t a Cauchy-Lipschitz process

    leftward, or backwards. T he n by using small enough line-segments we

    m ay app roac h indefinitely close to

    that poin t vertically above

    A

    which

    lzes

    on the new contour line above

    A's

    contour will

    the n have to lie below

    the leftward-mov ing Cauchy-Lipschitz curve, an d is thu s revealed t o be

    worse th a n a n y new initial point lying above the old contour line.

    Q E D

    W e m ay follow Mr. Little 's terminology an d give the nam e behaviour

    li n e to the unique curve which lies between the points definitely

    shown to be better than A, and those definitely shown to be worse

    th an A. This happens to coincide with the mathem atical solution to

    the differential equation, and we may care to give this contour line,

    by cou rtesy, th e title of an indifference cu rve.l

    If

    our preference field does no t hav e simple concavity-and wh y should it ?-we m ay observe

    cases where is preferred to B at some t imes , and B t o A a t others. If this is a patte rn of

    consistenc y and n ot of ch aos, we could choose to regard

    A

    and

    B

    as indiffere nt und er those

    circumstances. If th e preference field has simple con cav ity, indifference will never explicitly

    reveal itself to us except as the rcs~lltsof an infinite limiting process.

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    Le t me retur n now to the problem of defining a new app roxim ating

    process, like the conventional Cauchy-Lipschitz process, but which

    I )

    approaches the mathematical solut ion from above rather than

    below, and which 2) definitely reveals the economic preference of

    the individual at every point .

    Our new process will consist of broke n stra igh t lines a n d in th e

    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. Inste ad, the slope

    will be determined by t h e j n a l point of th e sub-interval s line-segment.

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    After the reader ponders over this for some time and considers its

    geom etrical significance, he m ay feel th a t he is being swindled. H ow

    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 ha s alr ea dy been determ ined Clearly, we

    are a t something of a circular impasse. T o determine the slope, we

    seem already to require the slope.

    The way out of this dilemma is perfectly straightforward to anyone

    who has grasped th e ma them atica l solution of a simu ltaneous equ ation .

    Th e logical circle is a virtu ous rath er th an a vicious one. B y solving th e

    implied sim ultaneous equ ation , we cu t throug h t he problem of circular

    interdependence . And in this case we do not need a n electronic com-

    puter t o solve the implied equation . Our h um an guinea-pig, s imply by

    following his ow n bent, inad verte ntly helps t o solve our problem for us.

    In F igure

    3

    we again begin with the initial point

    A

    Again we

    wish to find the true solution for y at x

    15.

    Our first an d crudest

    ap pro xim ati on will consist of one stra igh t line. B u t its slope will

    be determined a t the en d of th e interv al an d is initially unknow n.

    Let us, therefore, through A swing a straight line through all possible

    angles. One an d on ly one of these slopes will give us a line t h a t is

    exa ctly tangen t t o one of the little arrows a t the en d of our interval.

    Le t

    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 ro tat e a straig ht budget li ne

    arou nd a n initial point A, an d let the individual pick the best combi-pa-

    tion of goods in each situa tion , we trac e ou t a so-called offer curve .

    This curve is not drawn in on the figure, but the point Z' is the inter-

    section of th e offer curv e w ith th e vertical line. I t should be obvious

    from our earlier reasoning that

    2'

    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 m uch for our crude first approximation. Le t us tr y dividing the

    interval between x=

    1

    a nd x =

    15

    up into two sub-intervals so that

    two connected straight lines may be used.

    If

    we wish the first line

    to end a t x = 12, we ro tat e our line throug h A u ntil its final slope is

    just equ al to th e indica ted little arrow (or price ratio) along th e vertical

    l ine x=

    12.

    For the simple hyperbole in question, where

    p /p =

    y

    y/x , our straight l ine will be found t o end at the point

    B ,

    whose

    dx-

    x ,

    y) coordinates are (12, 25$) an d whose a rro w ha s a slope of just

    less than (- 2).

    W e now begin a t B as a new initial point an d repeat th e process

    by finding a new straight l ine over the interval from

    x 1 2

    t o x =

    15 .

    Pivoting a line through all possible angles, we find tangency only at

    the point

    Z ,

    where y = 21?, which is

    a

    stil l better approximation to

    the true value,

    y 20.

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    Th e interested reader m ay easily verify th a t using more sub-intervals

    and intermediate points will bring us indefinitely close to the true

    solution-contour.1 I t is clear therefore t h a t our new process brings

    us to the true solution in the limit, but unlike the Cauchy-Lipschitz

    process, it now approache s th e solution from abov e. And we can use

    th e word above in more th an a geometrical sense.

    Along the

    new process lines, the individual is revealing himself to be getting

    be tter off. For just as

    A

    is inferior to Z', it is by the same reasoning

    inferior to B , which is likewise inferior t o Z from which it follows

    th a t A is inferior to Z .

    I t should be clear , therefore, tha t no m atte r how ma ny intermediate

    points there are in the new process, the consumer none the less reveals

    himself to be travelling uphill. I t follows th a t every point a bove th e

    mathematical contour line can reveal itself to be better than A.

    This essentially completes the present demonstration.

    The mathe-

    matical contour lines defined by our differential equation have been

    proved to be the frontier between points revealed to be inferior to

    A

    an d points revealed to be superior. T he points lying literally on a

    (concave) frontier locus can never themselves be revealed to be better

    or worse th a n A. If we wish, the n, we m a y speak of the m as being

    indifferent to A.

    T he whole the ory of consumer's behaviour can th us be based upon

    operat ionally meaningful foundations in terms of revealed p r e f e r e n ~ e . ~

    He m a y ver i fy tha t using the points 1 11 12 13, 14, 5 brings us to within ot

    y = zo

    as shown in the second table of the Numerical Appendix.

    T h e above remarks apply without qualification to tw o dimensional problems where the

    problem of int egr ab ilit y cannot appear. In the multidimensional case there still remain

    some problems, awaiting a solution for more than a decade now.

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    252 ECONOMIC

    [ N O V E M B E R

    N U M E R I C A L P P E N D I X

    I n t he Cauchy-Lipschitz process, th e s tra ig ht line going from (xo, yo)

    to (x,, y,) is defined by the explicit equation

    (0) y = y o - f ( ~ 0 , o ) ( x - x o ) = ~ o - - ( x - x o )

    X

    where dy/dx=

    f

    (x, y) is the differential equation requiring solution

    -in th is case being

    - y / x .

    The three approximations given in

    Figure 2 are derived numerically in the following table.

    First Approximation

    1

    initial point 30

    30/1o= 3

    15

    30 3 (15 lo )=

    15

    Second Approximation

    1

    initial point 30

    30 /10=

    I 2

    3 0 - 3 ( 1 2 - 1 0 ) - 24 24/12=

    2

    15

    2 4 - 2 (15 IZ)=

    I 8

    Third Approximation

    initial point

    30 - 3 (11 IO)=

    I

    In th e new process which approaches th e true solution, y = 3oo/x,

    from above, the straight lines hdve their slopes determined

    by

    t h e

    final poin t of e ac h inte rva l, or by the implicit equation

    4 Y l = ~ o - f ( ~ l ~ ~ l ) ( ~ l - ~ o )

    In the case where

    f

    (x, y)= ylx, we have

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    Our numerical approximations are given in the following table

    First Appvoximation

    1

    initial point

    30

    Second Approximation

    1

    initial ~ o i n t

    hird

    Approximation

    1 initial point

    t m ay be mentioned t h a t the third Cauchy-Lipschitz approximation

    satisfies th e equation 2 7 0 1 ~ hich is less th an the true solution,

    ~ O O / X

    an d the th ird appro xim ation of th e new upper process satisfies th e

    equation 330/x, which happens to be equally in excess of the true

    solution.

    [ In F igure

    3

    the poin t be tween and Z" should be labelled B"].