Pareto and the Wicksell–Cobb–Douglas Functional Form

Journal of the History of Economic Thought, Volume 20, Number 2, 1998
203
PARETO AND THE
WICKSELL-COBB-DOUGLAS
FUNCTIONAL FORM
BY
CHRISTIAN E. WEBER
I. INTRODUCTION
It is understood that Charles Cobb and Paul Douglas (1928) were not the
first to use the production function named after them. Joseph Schumpeter
(1954, p. 1042), Carl-Axel Olsson (1971), and Henry Spiegel (1991, p. 816)
all note that the production function Y = AK*\J ~x had been used by Knut
Wicksell (1901, 1906) more than twenty years before Cobb and Douglas
published their study.1 While it is quite possible that Wicksell was the first
to use the Cobb-Douglas functional form to study production, he was not
the first to apply it to economic analysis in general. Vilfredo Pareto had
worked out several implications of a specific version of the Cobb-Douglas
utility function as early as 1892. Later, he repeated and extended this
analysis in the mathematical appendix to the French translation of his
Manual of Political Economy (1909).
This note discusses Pareto's use of the Cobb-Douglas functional form. Its
purpose is to explain his rationale for adopting it and the results he derived
from it, and to point out some interesting omissions and minor errors in his
later analysis.
II. FUNCTIONAL FORM IN PARETO'S "CONSIDERAZIONI"
Pareto (1892a) criticized Alfred Marshall's (1890) assumption that the
marginal utility of income2 remains constant when the price of a good
Seattle University, USA.
I would like to thank Dean Peterson for helpful comments on an earlier draft of this paper and John
Chipman for providing me with an English translation of Pareto, 1892a, b, 1893 and 1900a, b. All
errors are my own responsibility.
1
Douglas (1976) provides an informative survey on the development of the Cobb-Douglas production
function between 1928 and 1976. He claims, without providing a reference, that Wicksteed had also
discussed the functional form later named after Cobb and Douglas.
2
Pareto (1892a, b) referred to the marginal utility of income as the "final degree of utility of capital
goods." He referred to it as "the elementary index of the good whose price is one, that is, of money"
(Pareto, 1909, p. 421).
1042-7716/98/020203-08 © 1998 The History of Economics Society
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204
JOURNAL OF THE HISTORY OF ECONOMIC THOUGHT
changes.3 He sought to prove that in general, a change in the price of any good
must change the marginal utility of income.4 He reasoned as follows:5 Suppose
that the household consumes goods b, c, etc. in the quantities, rb, rc, etc. and that
the marginal utility of good i, denoted </>,, depends only on the level of
consumption of good i: (j)b = <j>b{rb), <j)c = <t>c{rc), etc.6 Goods b, c, etc. are
purchased at prices, pb, pc, etc. Finally, assume that the marginal utility of
income, m, does not vary with pb, pc, etc. In this case, the first order conditions
for maximizing utility,
m = (j)b{rb)lpb = <t>c{rc)lpc = ...
(1)
7
can be solved for r^ rc, etc. This yields
rb = i//b(mpb), rc = ^c(mpc), ...
(2)
where i/f,(m/?,) = (f>r '(>;)• If qa is the constant income available to the individual
to spend buying market goods, then substituting the solutions for n, rc, etc. in
equation (2) into the budget constraint yields
qa = Pb^bimpb) + Pc^cimpc) + ...
(3)
Since m is constant by assumption and the pi's are all independent of each other,
qa can only be constant if for the given value of qa each term of the form
Piif/iimpi) is constant. Pareto denoted the constant value of p,i/',(mp,) by Aj.
Substituting this into equation (3) yields
qa = Ah + Ac+...
(4)
Equation (4) implies that total spending on each good must be a constant
fraction of income if the marginal utility of income is to be constant. Since
Cobb-Douglas utility functions have become popular pedagogical tools, modern
economists immediately recognize that Pareto has shown that utility must be a
monotone increasing function of the Cobb-Douglas functional form,
3
A number of authors have argued that Marshall only assumed that the marginal utility of income
is nearly constant following a change in income. For example, George Stigler (1950) claims that it
"seems beyond doubt that Marshall treated the marginal utility of income as approximately, and not
rigorously, constant, and fairly clear that it is constant with respect to variations in the price of a
commodity whose total cost is not too large a part of the budget."
In his discussion of the constancy of the marginal utility of income, Paul Samuelson (1942, p. 79)
discussed two different meanings which have been attached to the phrase. One refers to the constancy
of the marginal utility of income with respect to a change in any price. The other refers to constancy
with respect to income. By showing that the marginal utility of income must be homogeneous of
degree minus one in income and all prices, Samuelson, showed that the marginal utility of income
cannot be simultaneously constant with respect to income and all prices.
4
Peter Dooley (1983b) presents an interesting account of the debate over the possible constancy of
the marginal utility of income following the publication of the first edition of Marshall's Principles
in 1890.
5
The notation used here follows Pareto (1892a) closely.
6
That is, Pareto assumed here, as he occasionally did elsewhere, that the total utility function is
additively separable.
7
According to P. Dooley (1983b), Marshall (1890) was the first to show how to solve for demand
functions from marginal utility functions when the total utility function is additive.
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PARETO AND THE COBB-DOUGLAS FUNCTION
205
U = rj,ryc... , if the marginal utility of income is to remain constant when prices
change. Although he did not actually write down the particular function of
U = r\ryc... which must apply, Pareto did point the reader in the right direction.
He noted that since equation (4) implies /?, = A/r,-, the first order conditions in
equation (1) can be rewritten
(j>b{rb) - mpb = mAb/rb, (j)c(rc) = mpc = mAJrc, ...
(5)
so that simple integration (which Pareto did carry out) yields the separate utility
functions for rb, rc, etc., which are denoted &b, <PC, etc.
&b(rb) = mAb\nrb,
<Pc(rc) = mAc\nrc, ...
(6)
The results in equation (6) can then be combined to yield the additive log-linear
version of the Cobb-Douglas total utility function
U(rb,rc, ...) = 0b(rb) + <Pc(rc) + ...= mAb\nrb + mAc\nrc + ...
(7)
Thus, by considering marginal utility only (no expression similar to equation
(7) appears in Pareto's analysis), Pareto arrived at the fact that Cobb-Douglas
utility implies constant budget shares for all goods. Based on this analysis,
Pareto rejected the possibility that the marginal utility of income might remain
constant when the price of a good changes on the grounds that marginal utilities
such as those in equation (5) are too special to hold in the real world:8 "Hence,
it would be necessary for the final degree of utility of all the commodities to
have that extremely singular expression for the final degree of utility of a capital
good to remain constant. Which amounts to saying that it will never be so"
(emphasis in the original).
Pareto also derived part of another property of Cobb-Douglas utility functions. He noticed that the price elasticity of demand must be constant if marginal
utilities take the form in equation (5). However, he did not go so far as to say
explicitly that the price elasticity must be unity, nor did he discuss income
elasticities at all.
III. FUNCTIONAL FORM IN THE MANUAL
Pareto returned to the question of the constancy of the marginal utility of income
in the Manual (1909, sections 56-59, pp. 426-29). He used essentially the same
method of analysis and produced the same results as he had earlier. However,
there are several small changes in the appendix to the Manual compared to the
earlier analysis, as well as several small flaws in the later analysis which bear
mention.
In the Manual, as in much of his other analyses, Pareto considered "general
equilibrium" demand curves, where the household is endowed not with money
income, but with non-negative quantities of some of the goods it consumes. It
sells a portion of its endowment to obtain income which it uses to purchase those
It is interesting to note that after he had gone to some lengths to show that the marginal utility of
income had to change in response to a change in price, several pages later, Pareto discussed cases
in which the marginal utility of income would be nearly constant following a change in price.
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206
JOURNAL OF THE HISTORY OF ECONOMIC THOUGHT
goods which it wants to consume in excess of its endowment. This leads to
budget constraints of the form found in Pareto's equation (51) (Pareto, 1909,
p. 414):
+ ...
(8)
0 = x- x0 + py(y-yo)+pz(z-zo)
where xo, yo, Zo, etc. are the household's initial endowments of the goods, and py,
pz, etc. are the relative prices of goods y and z in terms of the numeraire, good
x. Using this budget constraint, Pareto derived the following comparative static
results for additive utility functions using methods which he had pioneered in
Pareto (1892b, 1893):9
T = l/*« + pfeyy + p\l$a +...,
= (y-yo
+ $y/<Pyy)/T,
$a,
(9a)
(9b)
(9c)
where <P,, is the second derivative of the utility function with respect to good i,
and the second-order cross partial derivatives, <P,y with / ±j are all zero, so that
the total utility function is once again assumed to be additively separable.
Pareto's discussion of the implications of equations (9a)-(9c) was incomplete.
Several additional points related to constancy of the marginal utility of income
had either already been made by Arthur Berry (1891) and F. Y. Edgeworth
(1891) or were added later by Paul A. Samuelson (1942). Two of these points
made by other authors merit brief mention.
First, Pareto understood by 1900 at the latest that economic theory requires
only ordinal utility indices (see Pareto, 1900a, b; Schumpeter, 1954, p. 1062; and
Weber, 1997b). Economic theory requires only that consumers be able to rank
different consumption bundles according to whether they are indifferent to each
other or one is strictly preferred to the other. Pareto reiterated this point in the
mathematical appendix to the Manual (sections 4—18, pp. 392-403). What he did
not realize was that this methodological improvement was the last nail in the
coffin of constant marginal utility of income. Any restriction on the marginal
utility of income, including the possibility that it might be constant, clearly
requires that utility be cardinally measurable; otherwise such a restriction is
meaningless. Once economics had discarded cardinal utility functions, cardinal
restrictions on the marginal utility of income had no place in the analysis. If
Pareto understood this, he did not discuss it in the Manual. It appears that
Samuelson (1942) was the first to make this point.
Second, despite his careful discussion of T in equations (9b) and (9c) Pareto
did not recognize a second case in which T is infinite. That is the case of the
quasi-linear utility function: U(x, y, z, ...) =Ax + u(y, z, ...). where A is a
constant, so that utility is linear in exactly one good (Varian, 1992). Since
^.a = 0, T= oo. Here, one of the first-order conditions is: m = Alpx, so that m
depends only on px. dm/dp* = 0 for i i1 x. This point had been made by Arthur
9
The reader will notice a slight difference in notation between Pareto's "Considerazioni ..." and the
appendix to the Manual.
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PARETO AND THE COBB-DOUGLAS FUNCTION
207
Berry (1891) and Edgeworth (1891) and was later reiterated by Samuelson
(1942). No doubt, Pareto neglected this case since he assumed diminishing
marginal utility for all goods throughout his analysis, both in the appendix to the
Manual and elsewhere.
In addition to these lacunae, Pareto's analysis is subject to the following
criticisms:
First, although he showed that if the marginal utility of income remains
constant when prices change, then uncompensated cross price effects must be
zero, his proof did not use marginal utility functions of the form discussed
above, and he apparently failed to recognize that Cobb-Douglas preferences
imply zero uncompensated cross price effects. At the very least, he did not state
this result explicitly. Instead, he argued using equations (9a)-(9c) above. He
recognized from equation (9c) that if the marginal utility of income, m, did not
change in response to a change in py, then what we would today call the
uncompensated cross price effect, dz/dpy, would necessarily be zero.
However, he discussed two conditions under which the marginal utility of
income would be constant in the face of a change in py. One is the case in which
T in equations (9b) and (9c) is "very large." Pareto stated that constant marginal
utility of income implies zero cross price effects following his discussion of the
case where T is very large (section 58). He recognized that this provides an
argument against the constancy of the marginal utility of income in addition to
that quoted above from the "Considerazioni sui principii ..." of 1892. Referring
to the hypothesis that the marginal utility of income is constant, he observed:
"Hence, the hypothesis is tantamount to assuming that when py varies, only the
quantity of y varies, while z, u,... remain constant. This assumption may be
admissible in certain cases, but in general it is inadmissible" (Pareto, 1909,
p. 427).
The other case in which the marginal utility of income is constant is that
where the numerator of equation (9b) is zero. Following his discussion of the
case where T is very large, Pareto considered this second case. The numerator
of equation (9b) will be zero so that the marginal utility of income does not vary
with the price of good y if:
dm/dpy = 0 if: y - y0 + $>/<£yy = 0,
or:
0y = Bliy - y0)
where B is an arbitrary constant. Integrating equation (10) yields a logarithmic
total utility function in net trades of y, y — y0. Again, this shows that marginal
utility must take the reciprocal form of equations (5), so that utility had to be
logarithmic. However, Pareto did not mention that in this case the cross price
effects would be zero. At least in print, Pareto did not make the connection
between Cobb-Douglas preferences and zero cross price effects.
Finally, in the Manual, as in the rest of his work, Pareto neglected income
effects in his comparative statics analysis of demand. One crucial result of this
neglect was that although Pareto pioneered the method of analysis which E. E.
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208
JOURNAL OF THE HISTORY OF ECONOMIC THOUGHT
Slutsky (1915) later used to derive separate income and substitution effects of a
change in price, Pareto did not discover these effects himself.10
For present purposes, however, a more interesting result of this neglect occurs
in the Manual (Pareto, 1909, p. 428). From equation (10), Pareto reasoned that
constancy of the marginal utility of income required that the marginal utility of
good y depend on both the level of consumption of _y, and on yo, the initial
endowment of good y, rather than on the actual level of consumption only. Since
he found this implication unacceptable, he believed that he had found yet a third
argument against a constant marginal utility of money. What he apparently failed
to recognize was that the household's consumption of all goods, including good
y, depends in general on the market prices of the goods and on the market value
of the household's initial endowment of the various goods. In other words, the
general form of the demand function for y is:
y = y(Py>P* •••;x0,y0,zo,
...).
(11)
Moreover, defining the market value of the household's initial endowment
(measured in terms of the numeraire) as:
s = xo + pyyo + pzzo + . . . ,
(12)
the Cobb-Douglas functional form for the utility function implies:
y = Bslpy = B(x0 + pyyo + pzzo + ••• )/py,
(13)
where B is a constant. Equation (13) shows that a change in yo alters y. Thus,
at constant prices, the first-order conditions in equation (1) imply that a change
in y0 does affect the marginal utility of income. Pareto's attempt to argue that the
marginal utility of income should not depend on initial endowments is fundamentally flawed and only two of his three arguments against a constant marginal utility
of money stand: A constant marginal utility of money does imply that total utility
must be log linear, which is highly restrictive. Also, a constant marginal utility
of money implies that uncompensated cross price effects must be zero, which can
be rejected empirically. However, one cannot argue that a constant marginal utility
of income should be rejected since it implies that marginal utilities depend on
endowment levels.
IV. CONCLUSION
On purely aesthetic grounds, there is no need to rename the Cobb-Douglas utility
function the Pareto-Wicksell-Cobb-Douglas utility function." However, it is
worth noting that in addition to his other achievements, Pareto did present an early,
10
On the relationship between the comparative statics analyses of Pareto and Slutsky, see Dooley,
1983a. On Pareto's neglect of income effects, see Weber, 1997a.
'' It appears that neither Wicksell nor Cobb and Douglas ever used this functional form in connection
with consumer preferences. Two of the earliest authors to use Cobb-Douglas utility functions were
R. G. D. Allen (1938) and Paul Samuelson (1942).
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PARETO AND THE COBB-DOUGLAS FUNCTION
209
if incomplete, discussion of the Cobb-Douglas utility function.12 His conclusions
were generally sound, and anticipated closely results which Samuelson (1942)
would use half a century later to put to rest forever economists' concerns
over the possible constancy of the marginal utility of money. The logic and
clarity with which Pareto stated his results and their implications make it all the
more disappointing that, as Peter Dooley (1983b, p. 36) notes, Marshall and
his defenders largely dominated thinking on the marginal utility of income for
four decades until more careful analysis was applied to the question in the 1930s
and '40s.
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12
Pareto (1900b) also wrote down the functional form for the linear expenditure system (LES) utility
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