DECLINING MARGINAL UTILITY IS NOT ORDINALorion.math.iastate.edu/driessel/13Models/10.marginal_utility.pdfless. The marginal utility of an orange to you depends not only on the orange

DECLINING MARGINAL UTILITYIS NOT ORDINAL

KENNETH R. DRIESSEL

Mathematics Department

Iowa State University

[email protected]

Abstract

Economists should not confine their attention to only ordinal prop-

erties. For example, the principle of declining marginal utility is useful

and important but is not ordinal.

Table of Contents

• Introduction

• Ordinal Invariants

• Declining Marginal Utility

• Appendix: Mandelbrot Criteria

• References

Date: February 3, 2013.

Key words and phrases. Economics, utility, marginal utility, ordinal, group,

invariant.

1

2 KENNETH R. DRIESSEL

You are deciding how many oranges to consume. If the

question is whether to have one orange a week or none,

you would much prefer one. If the alternatives are 51

oranges a week or 50, you may still prefer the additional

orange, but the gain to you from one more orange is

less. The marginal utility of an orange to you depends

not only on the orange and you, but also on how many

oranges you are consuming. We would expect the util-

ity to you of a bundle of oranges to increase more and

more slowly with each additional orange. Total utility

increasing more and more slowly means marginal utility

decreasing ... Friedman(1990)

Introduction

Transformation groups are very important in mathematics. For ex-

ample, Felix Klein, in his Erlangen program, used them to classify

geometries. (See Klein(1893).) Groups are also very important in

physics. (See,for example, Sternberg(1995).) In this report we shall

discuss a number of groups that appear in economics.

In science an “invariant” is a property of a set of objects that remains

unchanged under a specified group of transformations. This notion is

important in mathematics and physics and, as a consequence, is even

discussed by philosophers. (See, for example, Nozick(2001).) In this

report, we consider utility functions and their invariants under groups

of transformations. We begin with a brief review of the use of utility

functions in economics.

Economists need to model individual choice behavior. In the tradi-

tional approach the decision maker’s tastes are modeled by by means

DECLINING MARGINAL UTILITY 3

of a “preference relation”. In particular, Mas-Colell, Whinston and

Green(1995) write: “The starting point for any individual decision

problem is a set of possible (mutually exclusive) alternatives from which

the individual must chose.” Let X denote this set of alternatives. A

preference relation on X is simply a complete pre-order on X. Recall

the definition of a complete preorder:

Definition: A preorder is a pair (X,≤) where X is a set and ≤ is

a binary relation on X that is reflexive (that is, for all x ∈ X, x ≤ x)

and transitive (that is, for all x, y, z in X, x ≤ y and y ≤ z implies

x ≤ z). A preorder is complete if every two elements are comparable

(that is, for all x, y in X, either x ≤ y or y ≤ x).

Economists usually represent a preorder by means of a “utility func-

tion”. Here is the formal definition:

Definition: Let u : X → R be a real-valued function defined on

the set X and let (X,≤) be a preorder. Then u is a utility function

representing the preorder if, for all x, y in X, x ≤ y iff u(x) ≤ u(y).

Note that if a preorder (X,≤) is represented by a utility function

then the preorder must be complete. Often complete preorders can

be represented by utility functions; see, for example, Debreu(1959) or

Mas-Colell, Whinston and Green(1995).

Let u : X → R and let (X,≤) be a preorder. Let f : R → R

be a strictly increasing function. It is easy to see that the function u

represents the preorder iff the composite function f ◦ u represents it.

In particular, f ◦ u is simply a rescaled version of u.

Definition: Properties of utility functions that are invariant under

rescaling by any strictly increasing function are called ordinal proper-

ties. Properties that are not preserved under all such transformations

are called cardinal properties.


This definition appears, for example, in Mas-Colell, Whinston and

Green(1995). We shall discuss several group-theoretic versions of this

definition in the section below on “Ordinal Invariants”. We need a

precise definition of “ordinal” in order to prove that a property is not

ordinal. Neoclassical economists are mostly interested in ordinal prop-

erties of utility functions.

Economists also need to model the preference behavior described in

the quote presented at the beginning of this introduction from Fried-

man(1990). Here is the standard model. (Friedman(1990) loosely de-

scribes this model.)

Definition: Let X be a closed, convex subset of Rn and let u :

X → R be a real-valued function defined on X. Then u is concave if

it satisfies the following condition: For all x and y in X and all t in the

interval [0, 1],

u(tx + (1− t)y) ≥ tu(x) + (1− t)u(y).

We say that a utility function u : X → R satisfies the condition of

declining marginal utility if u is concave.

This definition provides a precise mathematical version of the eco-

nomic principle.

Note that if X is an interval of real numbers and u : X → R is

differentiable then u is concave if u′ is a decreasing function. If the

second derivative u′′ exists and u′′(x) ≤ 0 for all x in X then u is

concave.

There is an analogous more general result. The following definition

is makes the result easier to state.

Definition: Let X ⊆ be closed, convex subset of Rn. Let u : X → R

have second derivatives D2u(x) for all x ∈ X. Then u is C2-concave if


it satisfies the following condition: For all x ∈ X the quadratic form

Q(x) : Rn → R defined by Q(x)v := D2u(x)vv is negative semi-definite.

Here is the more general result. (See the appendix on concave func-

tions for a proof.)

Proposition 1. If u is C2-concave the u is concave.

Here is the connection with the quote concerning oranges that ap-

pears at the beginning of this section: Regard x ∈ X as a quantity of

oranges and regard u(x) as the value of the quantity x to you. The

quote talks about change ∆u(x) in u at x relative to change ∆x in

x. The marginal utility is ∆u(x)/∆x. As ∆x approaches 0, this ratio

approaches u′(x).

Unfortunately, the principle of declining marginal utility is not or-

dinal - in other words, it is often not invariant under rescaling by a

strictly increasing function. This report is devoted to this matter.

Contents summary. In the section on “Ordinal Invariants” we define

several groups that can be used to rescale utility functions. We use

these groups to carefully define the notion of “ordinal invariance”. We

also recall the definition of the positive affine group of the line. We use

this group to define another notion of invariance for utility functions

called “positive affine invariance”.

In the section on “Declining Marginal Utility” we present examples

showing that this property is not an ordinal invariant. We also show

that the declining marginal property of utility functions is a positive

affine invariant.

In the appendix on “Concave functions” we present a proof of the

result concerning concave function given above.


Mandelbrot(1963) introduced several criteria that important eco-

nomic quantities should satisfy. In the appendix on “Mandebrot’s Cri-

teria” we show that declining marginal utility satisfies these criteria.

We sometimes repeat definitions for the reader’s convenience.

What’s new. As far as I know, the following items in this report are

new:

• Our careful discussion of the groups associated with the notion

of ordinal invariance.

• Our examples that show declining marginal utility is not ordi-

nal.

• Our result that shows declining marginal utility is a positive

affine invariant.

• Our demonstration that declining marginal utility satisfies Man-

delbrot’s criteria.

It easy to generalize the examples and results that appear in this

report; for simplicity, I have usually presented less general versions.

Prerequisites. In the main part of this report we shall repeatedly

calculate derivatives using standard formulas. (Here is a list of some

of the standard references: Lang(1969), Loomis and Sternberg(1968),

Dieudonne(1969).) In particular, we shall view the derivative of a func-

tion as a linear map: If f : X → Y is a function from an appropriate

subset X of Rm to a subset Y of Rn then, for p in X, the derivative

Df(p) : Rm → Rn of f at p is a linear map from Rm to Rn. We shall

repeatedly use the following chain rules for first and second derivatives:

D(g ◦ f)(p)x = Dg(f(p))(Df(p)x),

D2(g ◦ f)(p)xy = D2g(f(p))(Df(p)x)(Df(p)y) + Dg(f(p))(D2f(p)xy).


Ordinal Invriants

Let G be the set of invertible real-valued functions f : R→ R defined

on the real numbers. Since G is closed under composition and taking

inverses, we have a group. Let PC0 denote the set of continuous strictly

increasing functions in G. Note that this set forms a subgroup of G.

Let PC1 denote the set of continuously differential functions f in

G with positive derivative; in other words, f is in this set if, for all

p ∈ R, f ′(p) > 0. This set forms a subgroup of PC0. In particular, we

have the following result; this result and its proof are well-known; we

include the proof because it provides a model for a later proof.

Proposition 2. Let f : R→ R be a C1 function with positive deriva-

tive. Then f−1 is a C1 function with positive derivative. In particular,

for all p in R,

(f−1)′(f(p)) =1

f ′(p).

Proof. (sketch) Recall the chain rule:

D(g ◦ f)(p)x = Dg(f(p))(Df(p)x).

Note I = f−1 ◦ f where I is the identity function on R. Differentiating

this equation we get

t = DI(p)t = D(f−1 ◦ f)(p)t = Df−1(f(p))(Df(p)t)

= (f−1)′(f(p))f ′(p)t.

Hence 1 = (f−1)′(f(p))f ′(p). �

Let PC2 denote the set of twice continuously differential functions

f in PC1. This set forms a subgroup of PC1. In particular, we have

the following result.


Proposition 3. Let f : R → R be a twice continuously differentiable

function in PC1. Then f−1 is twice continuously differentiable. In

particular, for all p in R,

(f−1)′′(p) = − f ′′(p)

(f ′(p))3.

Proof. (sketch) Recall the chain rule for second derivatives:

D2(g ◦ f)(p)st = D2g(f(p))(Df(p)s)(Df(p)t) + Dg(f(p))(D2f(p)st).

Using this formula we get

0 = D2I(p)st = D2(f−1 ◦ f)(p)st

= D2(f−1)(f(p))(Df(p)s)(Df(p)t) + Df−1(f(p))(D2f(p)st)

= (f−1)′′(f(p))(f ′(p)s)(f ′(p)t) + (f−1)′(f(p))(f ′′(p)st).

Using the previous proposition concerning the first derivative of the

inverse of a function, we get

0 = (f−1)′′(f(p))(f ′(p))2 + (f−1)′(f(p))(f ′′(p))

= (f−1)′′(f(p))(f ′(p))2 +f ′′(p)

f ′(p)

�

Here is another group that we use below. Let a and b be real numbers

with 0 < a. Let the function f : R → R be defined by p 7→ ap + b.

Let PA (for “positive affine”) denote the set of all such functions.

This set forms a group. In particular, f−1 satisfies, for all q ∈ R,

f−1(q) = (q− b)/a. Note PA is a subgroup of PC2 since, for all p ∈ R,

f ′(p) = a and f ′′(p) = 0. (This group appears in Nash(1950).)

Recall that, in mathematics, an “invariant” is a property that re-

mains unchanged under the action of a group. For example, Euclidean


plane geometry is mainly the study of properties that are invariant

under the distance preserving transformation of the plane R2.

Let X ⊆ Rn be a closed, convex subset of real n-space. Let Ck(X →

R) denote the set of all real-valued functions u : X → R defined on

X that are k-times continuously differentiable. We have the following

group actions:

PCk × Ck(X → R)→ Ck(X → R) : (f, u) 7→ f ◦ u

and

PA× Ck(X → R)→ Ck(X → R) : (f, u) 7→ f ◦ u.

In other words, the action of f on u is given by f ∗ u := f ◦ u where

f ◦ u is the composition of the functions f and u.

Definition: A property of elements of Ck(X → R) is PCk-ordinal

if it is invariant with respect to the action of PCk on Ck(X → R).

A property of Ck(X → R) is a positive affine invariant or a PA-

invariant if it is invariant with respect to the action of the positive

affine group PA on Ck(X → R).

Example: Indifference sets.

Let u : X → R be a utility function. For r ∈ R the set u−1(r) :=

{x ∈ X : u(x) = r} is the “indifference” (or “level”) set of u determined

by r. The collection L(u) := {u−1(r) : r ∈ R} of all such indifference

sets is a PC0 ordinal invariant. In particular, if f : R → R is an

element of PC0 then L(f ◦ u) = L(u) since, for all r ∈ R we have

(f ◦ u)−1(r) = {x ∈ X : (f ◦ u)(x) = r}

= {x ∈ X : u(x) = f−1(r)} = u−1(f−1(r)).

We simply have a “relabeling” of the indifference sets.

Example: Gradient direction.


Let u : X → R be C1. We use the usual (Euclidean) inner product

on Rn which is defined by 〈x, y〉 =∑n

i=1 xiyi. Then, for p ∈ X, we

can represent the derivative Du(p) : Rn → R of u at p by the gradient

vector ∇u(p) of u at p; in particular, we can define this gradient vector

by the relation Du(p)x = 〈∇u(p), x〉. Assume that the gradient vector

∇u(p) is nonzero. Then we can define the “gradient direction” of u

at p to be the unit vector n(u)(p) := ∇u(p)/‖∇u(p)‖. This vector is

PC1-ordinal. In particular, if f : R→ R is an element of PC1 then,

n(f ◦ u)(p) =∇(f ◦ u)(p)

‖∇(f ◦ u)(p)‖=

f ′(u(p))∇u(p)

|f ′(u(p))|‖∇u(p)‖= n(u)(p).


Declining Marginal Utility

In economic analysis, we represent people’s preferences

by a utility function that allows them to rank various

alternatives. We place plausible restrictions on these

utility functions; for example, they are usually increas-

ing (more is better) and they are assumed to be concave

- an assumption that embodies the notion of diminishing

marginal utility. Acemoglu and Robinson(2005)

In this section we mainly discuss examples that show that declining

marginal utility is not ordinal. However, we also show that declining

marginal utility is invariant under the positive affine group.

In our first result we consider an increasing real-valued function de-

fined on an interval of real numbers. The graph of such a function is a

curve in R2. The resolute shows that we can straighten this graph to

a straight line by adjusting the scale of the range space.

Proposition 4. Straightening I. Let X ⊆ R be a closed interval. Let

u : X → R be a continuous, strictly increasing function. Then there is

a continuous, strictly increasing function f : R→ R such that f ◦ u is

the identity function on X.

Proof. (sketch) Note that the image J := u(X) is a closed interval.

Also note that u−1 : J → X is strictly increasing and continuous. Let

f : R → R be any continuous strictly increasing extension of u−1 to

the real line. Then f ◦ u is the identity function on X. �

Let v : X → R be any increasing function. It follows from the

straightening result that there is an increasing function g such that

v = g ◦ u. In particular, we can take g := v ◦ f . Note, for example,


that v may be convex even if u is concave. In this way we get a large

class of examples that show declining marginal utility is not an ordinal

invariant.

Recall that following notation: For p ∈ Rn and nonzero q ∈ Rn,

l(p, q) : R→ Rn is defined by l(p, q)(r) := p + rq.

In the next straightening result we consider a utility function u :

X → R defined on a closed compact subset X of Rn. In particular, we

consider the cross-section of the graph of u that lies above a line l(p, q).

The result says that we can rescale the range of u by an increasing

function f in such a way that the crossection of the graph of f ◦ u is

straight line.

Proposition 5. Straightening II. Let X be a closed, convex subset

of Rn and let u : X → R be a C1 function that satisfies the following

condition: For all x in X, the components of the gradient ∇u(x) are

strictly positive. Let p be an element of X and let q be a nonzero

element of Rn with non-negative components. Then there is a strictly

increasing function f : R→ R such that the graph of f ◦ u ◦ l(p, q) is a

straight line.

Remark: The condition on the gradient is a formal way of saying the

phrase “more is better” that appears in the quote at the beginning of

this section. This condition on the gradient of u appears, for example,

in Intriligator(1971).

Proof. (sketch) Let l := l(p, q) be the straight line in Rn defined by

p and q. Let J := {r ∈ R : l(r) ∈ X} = l−1(X). Note that J is a

non-empty closed interval of real numbers. Let v : J → R be defined

by v(r) = (u ◦ l)(r). Now, for all r and s in R, we have

Dv(r)s = D(u ◦ l)(r)s = Du(l(r))(Dl(r)s) = 〈∇u(l(r)), q〉s.


Hence v′(r) = 〈∇u(l(r)), q〉 and v′(r) > 0. Since v satisfies the hypothe-

ses of the first straightening proposition, there is a strictly increasing

function f : R→ R such that f ◦v is the identity on function on J . �

The next result says that declining marginal utility is a positive affine

invariant.

Proposition 6. Positive Affine Invariance. Let a and b be real

numbers with 0 < a. Let f : R → R be defined by f(r) := ar + b. Let

u : X → R be a C2 utility function where X is a closed, convex subset

of Rn. For p ∈ X, let Q(u)(p) : Rn → R be the quadratic form defined

by x 7→ D2u(p)xx. If Qu is negative definite then so is Q(f ◦ u).

Proof. (sketch) We have

D2(f ◦ u)(p)xy = D2f(u(p))(Du(p)x)(Du(p)y) + Df(u(p))(D2u(p)xy)

= f ′(u(p))(D2u(p)xy)

since D2f = 0. Hence, for any nonzero x ∈ Rn, D2(f ◦ u)(p)xx > 0

since f ′(u(p)) = a > 0. �

The following preservation result is well-known. It says that the

set of C2-concave utility functions u is preserved under rescaling by

increasing, C2-concave functions f . Unfortunately, the class of increas-

ing, C2-concave functions do not form a group; in particular, if f is

concave then usually f−1 is convex.

Proposition 7. Let X be a closed, convex subset of Rn. Let u : X → R

be C2. Let f : R→ R be increasing and C2-concave. If u is C2-concave

then so is f ◦ u.


Proof. Using the chain rule for second derivatives, we get

D2(f ◦ u)(p)xy = D2f(u(p))(Du(p)x)(Du(p)y) + Df(u(p))(D2u(p)xy)

= f ′′(u(p))(Du(p)x)(Du(p)y) + f ′(u(p))(D2u(p)xy).

Hence

D2(f ◦ u)(p)xx = f ′′(u(p))(Du(p)x)2 + f ′(u(p))(D2u(p)xx) ≤ 0

since f ′′(u(p)) ≤ 0, f ′(u(p)) ≥ 0 and D2u(p)xx ≤ 0. �


Appendix: Concave Functions

Recall that following definitions.

Definition: Let X be a closed, convex subset of Rn and let u :

X → R be a real-valued function defined on X. Then u is concave if

it satisfies the following condition: For all x and y in X and all t in the

interval [0, 1],

u(tx + (1− t)y) ≥ tu(x) + (1− t)u(y).

Definition: Let X be a closed, convex subset of Rn and let u : X →

R be a real-valued C2 function defined on X. Then u is C2-concave if

it satisfies the following condition: For all x in X, the quadratic form

Q(x) : Rn → R defined by Q(x)v := D2u(x)vv is negative semi-definite,

that is Q(x)v ≤ 0.

Proposition 8. If u : X → R is C2-concave then it is concave.

Proof. (sketch) Consider any x and y in X. We want to see that, for

all t in the interval [0, 1],

u(tx + (1− t)y) ≥ tu(x) + (1− t)u(y).

Note that this condition only involves the part of the graph of u that

lies above the line determined by x and y. (If x = y the condition holds

trivially.) In other words, we want to see that the part of the graph of

u lying above such a line is concave.

Let p and q be elements of Rn. Assume that q 6= 0. Define l :=

l(p, q) : R→ Rn by l(r) := p+ rq. This function is the line determined

by point p and vector q. Let v := u ◦ l. Note v : R → R. Clearly the

proposition follows from the following assertion.

Claim: The second derivative of v satisfies v′′(0) ≤ 0.


Using the chain rule for second derivatives, we get

v′′(0)st = D2v(0)st = D2(u ◦ l)(0)st

= D2u(p)(Dl(0)s)(Dl(0)t) + Du(p)D2l(0)st

= D2u(p)(sq)(tq) = stD2u(p)qq.

Hence v′′(0) = D2u(p)vv ≤ 0. �


Appendix: Mandelbrot Criteria

TODO: Finish writing this section.

The approach that I use to study the scaling distribu-

tion arose from physics. It occurred to me that before

attempting to explain an empirical regularity, it would

be a good idea to make sure that this empirical identity

is “robust” enough to be actually observed. ... [A] rela-

tionship will be discovered more rapidly, and established

with greater precision, if it “happens” to be invariant

with respect to certain observational transformations.

A relationship that is non-invariant will be discovered

later and remain less firmly established. Three transfor-

mations are fundamental to varying extents. Mandel-

brot(1963)

Here are the three transformations that Mandelbrot regards as fun-

damental (given in his words):

• “Linear aggregation, or simple addition of various quantities in

their natural scale.”

• “Weighted mixture.”

• “Maximizing choice, the selection of the largest or smallest

quantity in a set.”

In this appendix we want to see that the principle of declining marginal

utility satisfies these criteria.

Recall that we say that a real-valued utility function u : X → R

defined on a closed convex subset X of Rn satisfies the principle of

declining marginal utility if u is a concave function - that is, u satisfies

the following condition for all x and y in X and all t in the interval


[0, 1]:

u(tx + (1− t)y) ≥ tu(x) + (1− t)u(y).

This is our mathematical version of the economic principle of declining

marginal utility.

I believe that the following results show that the principle of declining

marginal utility satisfies Mandelbrot’s criteria. These results are well-

known. I include the (easy) proofs for the reader’s convenience.

Proposition 9. Addition. Let u : X → R and v : X → R be concave

functions. Then u + v is concave.

Proof. We have

(u + v)(tx + (1− t)y) = u(tx + (1− t)y) + v(tx + (1− t)y)

≥ tu(x) + (1− t)u(y) + tv(x) + (1− t)v(y)

= t(u + v)(x) + (1− t)(u + v)(y).

�

Proposition 10. Weighted mixture. Let u : X → R and v : X → R

be concave functions. And let 0 ≤ r and 0 ≤ s be real numbers. Then

ru + sv is concave.

Proof. We have

(ru + sv)(tx + (1− t)y) = ru(tx + (1− t)y) + sv(tx + (1− t)y)

≥ r(tu(x) + (1− t)u(y)) + s(tv(x) + (1− t)v(y))

= t(ru + sv)(x) + (1− t)(ru + sv)(y).

�


Proposition 11. Minimum choice. Let u : X → R and v : X → R

be concave functions. Then the function u ∧ v : X → R defined by

(u ∧ v)(x) := min{u(x), v(x)} is concave.

Proof. Case: u(tx + (1− t)y) ≤ v(tx + (1− t)y). We have

(u ∧ v)(tx + (1− t)y) = min{u(tx + (1− t)y), v(tx + (1− t)y)}

= u(tx + (1− t)y)

≥ tu(x) + (1− t)u(y)

≥ t(u ∧ v)(x) + (1− t)(u ∧ v)(y).

The other case is similar. �


References

• Acemoglu, D. and Robinson, J.A. (2005), Economic Origins of

Dictatorship and Democracy, Cambridge University Press

• Debreu, G. (1959) Theory of Value - An Axiomatic Analysis

of Economic Equilibrium, Cowles Foundation Monograph 17,

Yale University Press; originally published by Wiley. (Remark:

This monograph is freely available online; you can find it by

Googling: Debreu, Theory of Value.)

• Dieudonne, J. (1969) Foundations of Modern Analysis Aca-

demic Press

• Friedman, D.D. (1990) Price Theory, South-Western Publishing

• Intriligator, M.D. (1971), Mathematical Optimization and Eco-

nomic Theory, Prentice-Hall. Republished by SIAM in 2002.

• Klein, F. (1893) Verleichende Betrachtungen uber neuere ge-

ometrische Forschungen (A comparative review of recent re-

searches in geometry), Mathematische Annlen, 43, 63-100

• Lang, S. (1969) Real Analysis, Addison Wesley

• Loomis, L. and Sternberg, S. (1968) Advanced Calculus, Addison-

Wesley

• Mandelbrot, B. (1963) New methods in statistical economics,

The Journal of Political Economy 71, 421-440


• Mas-Colell, A., Whinston, M.D. and Green, J.R. (1995) Mi-

croeconomic Theory, Oxford University Press

• Nash, John F. (1950) The Bargaining Problem, Econometrica

18, 155-162.

• Nozick, R. (2001) Invariances: the structure of the objective

world, Harvard University Press

• Sternberg, S. (1995) Group Theory and Physics, Cambridge

University Press

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