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7/26/2019 Samuelson (1948)
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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"].