THE INTRODUCTION OF SATIATION INTO
UTILITY MAXIMIZATION PROBLEM
NIROTH CHHY∗
Graduate School of Economics, Kobe University
Allowing some goods to be satiated, in this paper, I will make a compromise between consumer theories with local nonsatiation and those allowing
for satiation. Particularly, I will argue that local nonsatiation is innocuous
as long as there exists nonsatiated good, while satiation allows one to deal
with change in preferences parsimoniously. To ensure a well-defined utility
function, I will allow consumers to choose both consumption and satiated
level, by assuming that utility, attained from satiated goods, is measured
by the distance from exogenously known satiation set. Then, under standard assumptions, I will prove the existence of Walrasian equilibrium and
first and second welfare theorem. Finally, I will provide an application of
endogenous change in preferences, by assuming that satiation depends on
product qualities, endogenously determined by firms.
Keywords: Satiation, Walrasian equilibrium, First and Second Welfare
Theorem.
1
Introduction
Following the works of Arrow and Debreu (1954) and Mckenzie (1959), utility function has been the most fundamental tool in modern economic theories. Economic models with utility maximization problem are said to be
equipped with microeconomic foundations and have been successful in explaining many phenomena in real world. However, utility function is said
to be based on one infamous assumption, i.e., local nonsatiation assumption. As pointed out by Mas-Colell (1992), for instance, satiation is not rare
in economic models because, occasionally, consumption sets are naturally
compact (and thus if one assumes continuous utility function, Weierstrass
theorem will guarantee the existence of satiation). Also, Allouch and Le Van
(2008, p. 907) stated that “the main critique to the insatiability assumption is that the human nature calls it into question. Namely, any moderately
∗
I am greatly indebted to Professor Tamotsu Nakamura (Kobe University) for many
valuable comments and suggestions. All errors are my responsibility. I gratefully acknowledge financial support from the Ministry of Education, Culture, Sports, Science,
and Technology of Japan.
1
greedy person will testify to their occasional satiation.” To tackle local nonsatiation assumption, some studies have tried to find weaker assumptions
(see Dana et al., 1999; Allouch and Le Van, 2008-9; Sato, 2010; Won and
Yannelis 2008, 2011). At the same time, other studies try to restore equilibrium by modifying the definition of Walrasian equilibrium (see, Drèze and
Müller, 1980; Makarov, 1981; Aumann and Drèze, 1986; Mas-Colell, 1992;
Cornet et al., 2003).
On the one hand, the problem of consumer theories with local nonsatiation is not due to local nonsatiation assumption per se, but due to the
fact that these theories often go further to assume that no good is satiated
(through the assumption of strong monotonicity). On the other hand, those
allowing for satiation only focus on the situation when all goods are satiated.
Accordingly, this paper will make a compromise by allowing some goods to
be satiated, and argue that local nonsatiation is innocuous as long as there
exists at least one nonsatiated good and that allowing for satiated goods
provides a parsimonious way to deal with change in preferences. In other
words, with at least one nonsatiated good, local nonsatiation still holds and
is compatible with change in preference when at least one good is satiated.
To achieve this goal, this paper will introduce satiation into utility function
exogenously.1 In particular, this paper will allow consumers to choose not
only consumption level but also satiated level by assuming that utility, attained from satiated goods, is measured by the distance from exogenously
known satiation set.2
As we shall see, the above assumption is used to ensure a well-defined
utility function, which will be convenient in applications. Also, even though
this assumption comes with a cost, i.e., it imposes a new constraint (regarding to satiation) on utility maximization problem, this paper will show that
fundamental results—e.g., the existence of Walrasian equilibrium, and first
and second welfare theorem—still hold under standard assumptions, used in
classical consumer theories, and the fundamental properties of norm. Furthermore, this paper will show that, with differentiability assumptions, consumers will never consume satiated goods at satiated levels. This, one the
one hand, suggests that one does not have to worry about the possibility of
satiated consumers in applications. On the other hand, it casts doubt on the
role played by satiation. Accordingly, this paper will provide an application,
regarding to endogenous qualities and preferences with Euclidean norm. In
particular, this paper will assume that satiation depends on product quali1
Throughout this paper, satiation is referred to a situation when some goods are satiated.
2
If all goods are satiated, this assumption can be viewed as the compromise programming, which is widely used in management science and operation research (see Yu, 1973;
Zeleny, 1974; Ballestero and Romero, 1994). Simply put, when utility function is unknown
(say, due to cognitive limitation) but satiation is known, one will choose the closest point
to satiation.
2
ties, endogenously determined by monopolistically competitive firms.
The organization of the rest of this paper is as follow. Section 2 provides
the setup and the discussion of demand function, the existence of Walrasian
equilibrium, and welfare theorem. Section 3 provides an application regarding to endogenous change in preferences and qualities. Section 4 provides
concluding remarks. Finally, appendix provides the proofs of Lemmas.
2
The Setup
First, consider why the assumption, that utility is measured by the distance
from satiation and, thus, consumers also choose satiated levels, is important.
In particular, consider a situation when there is one good with two satiated
levels, {1, 3}, and utility function is given by
{
−(1 − s)2 for s ≤ 2
u(s) =
.
−(3 − s)2 for s > 2
As shown in Figure 1, utility function is not differentiable at s = 2 and is
Figure 1: An Example of Unwell-Defined Utility Function with Satiation
u
1
2
3
s
non-concave. In such case, if dependent not only on consumption but also
on satiated level, i.e., u(s̄, s) = −(s̄ − s)2 , where s̄ is satiated level, utility
function will be both differentiable and concave. One problem is that this
comes with a cost, i.e., it imposes a new constraint regarding to satiation
(i.e., s̄ ∈ {1, 3}) on utility maximization problem. Thus, it is important to
explicitly characterize satiation set.
To begin, consider a consumer behavior problem, with ℓ ∈ N++ different goods, ℓs (0 ≤ ℓs ≤ ℓ) of which are satiated with exogenously known
3
satiation set. Particularly, let C, a close and convex subset of Rℓ+ , be conℓ−ℓs
sumption set, and let C s and C ns be the projection of C onto Rℓ+s and R+
,
respectively. Then, satiation set, S, is defined as
S ≡ {s̄ ∈ C s : f (s̄) ≥ 0} ,
where f is an exogenously known function, satisfying the following Assumption 1.
Assumption 1: f (·) is a continuous function from C s to Rk (1 ≤ k) and
there exists s̄¯ ∈ C s such that f (s̄¯) ≥ 0.
The continuity of f (·) ensures that satiation set is closed, while s̄¯ ensures
that satiation set is nonempty.
Utility is measured by the distance from satiation and consumption of
nonsatiated goods. More specifically, let s and c be consumption vector of
satiated and nonsatiated goods, respectively. Then, utility function, U :
C s × C → R, is defined as follow
U (s̄, s, c) = u (n(s̄ − s), c) ,
where n(s̄ − s) is the norm of vector s̄ − s and u(·) is composite utility
function. These functions are assumed to satisfy the following standard
assumptions.
Assumption 2:
1. n : Rℓs → R+ satisfies n(0) = 0 and n(tx) = |t|n(x), ∀t ∈ R, is
s
s
continuous, and is strictly increasing (decreasing) on Rℓ++
(Rℓ−−
),
2. u : R+ ×C ns → R is continuous, strictly decreasing in its first argument
n, and strictly increasing in the rest.
In Assumption 2.1, n(·) ≥ 0, n(0) = 0 and n(tx) = |t|n(x) simply reflect the
two fundamental properties of norm in general, i.e., properness and linearity
(see, for example, Appendix A.10 of Acemoglu 2009). Also, the strictly
decreasing and increasing assumption simply reflects the fact that norm
rises as the vector points further away from the origin. This, along with the
fact that u(·) is strictly decreasing in n(·) in Assumption 2.2, ensures that
utility is strictly increasing as consumptions of satiated goods get closer
to satiated levels. Furthermore, the rest of Assumption 2.2 ensures the
standard properties of utility function in classical consumer theory. Finally,
it is worth mentioning that local nonsatiation holds as long as nonsatiated
good exists (i.e., ℓs < ℓ).
4
Let p ∈ Rℓ++ and y ∈ R++ be price vector and income level, respectively.
Then, one can write utility maximization problem (UMP) as follow:
max
(s̄,s,c)∈C s ×C
U (s̄, s, c)
s.t.
(s, c) ∈ B(p, y) , s̄ ∈ S,
(U)
where B(p, y) ≡ {x ∈ C : p · x ≤ y} is budget set. It is noteworthy that
consumer will never consume any satiated good at a level strictly higher
than its satiated level (since he or she is strictly better off by consuming at
satiated level instead) as stated in Theorem 1 below:
Theorem 1: Under Assumption 2, the solution to Problem (U) satisfies
s ≤ s̄.
As usual, before solving Problem (U), it is important to check the existence of solution. Although satiation set needs not be bounded, similar to
conventional UMP, it is possible to use Weierstrass theorem to ensure the
existence of solution. This is trivial when there is no satiated good (ℓs = 0),
or when there is no nonsatiated good and satiation belongs to budget set
(i.e., S ∩ B(p, y) ̸= ∅). Thus, in what follows, assume that ℓs > 0 and
Assumption 3: S ∩ B(p, y) = ∅ when ℓs = ℓ.
Note first that if satiation set is a singleton (i.e., S = {s̄}), Problem (U)
becomes
max U (s̄, s, c)
s.t.
(s, c) ∈ B(p, y).
(s,c)∈C
Obviously, the above UMP has solution since objective function is continuous and the constraint set is compact and nonempty. Denoting by v(p, y, s̄)
the indirect utility function, one can show that v(·) is continuous.3 Then, it
immediately follows that, for any pair (p, y), one can restrict the domain of
satiation in Problem (U) to a subset of S given by
S(p, y) ≡ {s̄ ∈ C s : f (s̄) ≥ 0, v(p, y, s̄) ≥ v(p, y, s̄¯)} .
It is apparent that S(p, y) is compact and nonempty, since f (·) and v(·) are
continuous and s̄¯ ∈ S(p, y).4 Then, since budget set is also compact and
3
A simplified version of proof to Theorem 3 ensures that (s(p, y, s̄), c(p, y, s̄)) is upper
hemicontinuous. Then, the continuity of function v(·) follows immediately from the continuity of U (·); or, one can simply apply the theorem of the maximum (see Theorem 3.6 in
Stokey and Lucas, 1989) since B(·) is conventionally known to be a compact-valued and
continuous correspondence.
4
The boundedness follows from the homogeneity of norm. Particularly,
consider s̄)with
(
s̄ −s
sufficiently high s̄i for some i. For any (s, c) ∈ B(p, y), n(s̄−s) = s̄i n 1 − ss̄ii , −is̄i −i will
be strictly higher than n(s̄¯ − s) for sufficiently high s̄i , implying that U (s̄¯, s, c) > U (s̄, s, c),
∀(s, c) ∈ B(p, y).
5
nonempty and utility function is continuous, Weierstrass theorem ensures
the existence of solution to Problem (U), as stated in Theorem 2.
Theorem 2: Under Assumption 1-2, Problem (U) has at least one solution
for any pair (p, y) ∈ Rℓ++ × R++ .
With Theorem 2, it is now safe to analyze the properties of demand function.
As a benchmark, this paper will only consider three properties of demand
function—namely, Walras’s law, continuity, and uniqueness of solution to
Problem (U)—which will be useful for the existence of general equilibrium
allocation in the sequel. The first two and the last properties will be stated
in the following Theorem 3 and 4, respectively. Then, Theorem 5 will state
the existence of Walrasian equilibrium in pure exchange economy, while
Theorem 6-7 will state the first and second welfare theorem.
Theorem 3: Under Assumption 1-3, the solution to Problem (U) satisfies
Walras’ law and is upper hemicontinuous for all (p, y) ∈ Rℓ++ × R++ .
Proof : Walras’ law follows immediately from Assumption 2-3. To show
the upper hemicontinuity, it suffices to show that the solution has closed
graph and the images of compact sets are bounded (see definition M.H.3 in
mathematical appendix of Mas-Colell et al., 1995):
(i). Closed Graph: for any sequence (pm , y m ) → (p, y) ∈ Rℓ++ × R++ ,
choose (s̄m , sm , cm ) ∈ s̄ (pm , y m )×s (pm , y m )×c (pm , y m ) for all m. Then, we
must show that if (s̄m , sm , cm ) → (s̄, s, c), (s̄, s, c) ∈ s̄(p, y) × s(p, y) × c(p, y).
The proof is similar to that in chapter 3 (Appendix A) of Mas-Colell et al.,
(1995) and, therefore, will not be provided here.
(ii). Bounded Images of Compact Sets: for any compact set P × Y ⊂
Rℓ++ × R++ , we must show that (s̄ (P, Y) , s (P, Y) , c (P, Y)) is bounded.5
-s̄ (P, Y) is bounded: since s̄ (P, Y) ⊆ S (P, Y), it suffices to show that
the latter is bounded. Suppose that this is not the case; i.e., there is s̄ ∈
S (P, Y), with sufficiently high s̄i for some i. Then, there is some (p, y) ∈
P × Y such that v(p, y, s̄) ≥ v(p, y, s̄¯), which is a contraction (see footnote
6).
-(s (P, Y) , c (P, Y)) is bounded: let Pi be a projection of P onto the set
of pi . Then, there exists pi > 0 s.t. pi ≥ pi , ∀ pi ∈ Pi . Otherwise, there
exists a sequence {pni } in Pi converging to zero. Then, since P is compact,
the corresponding sequence {pn } in P has a convergence subsequence, whose
limit point p∗ , with p∗i = 0, belongs to P, which is a contradiction. Furthermore, since Y is bounded, there exists ȳ ∈ R++ such that y ≤ ȳ for all
y ∈ Y, implying that
( )
B (P, Y) ⊆ B p, ȳ .
To save notation, c(P, Y) is used to denote the set containing all c(p, y), for any
(p, y) ∈ P × Y.
5
6
Figure 2: The Uniqueness of Solution
f (s̄) = 0
f (s̄) = 0
S
S
B(p , y)
B(p , y)
Since B(p, ȳ) is bounded, B(P, Y) is also bounded. Then, since (s(p, y), c(p, y))
⊆ B(p, y) for all (p, y) ∈ P × Y, it follows that (s(P, Y), c(P, Y)) is a subset
of B(P, Y) and, thus, is bounded. ■
Now, before showing the uniqueness of solution to Problem (U), recall that
strict quasiconcavity of utility function is required. Note also that strict
quasiconvexity of norm, n(·), and strict quasiconcavity of composite function, u(·), are not sufficient for the strict quasiconcavity of utility function
(because n(·) depends only on s̄ − s). Thus, it is important to impose some
restriction on satiation set. As shown in Figure 2, which depicts the case
of ℓ = ℓs = 2 with Euclidean norm, an apparent candidate is the strict
quasiconcavity of function f (·) as stated in Assumption 4.
Assumption 4: n(·) is strictly quasiconvex, u(·) is strictly quasiconcave,
and f (·) is strictly quasiconcave.
Then, as stated in Theorem 4, Assumption 4 is indeed sufficient for the
uniqueness of solution.
Theorem 4: Under Assumption 1-4, solution to Problem (U) is unique.
Proof : Suppose that Problem (U) has two different solutions: (s̄, s, c)
and (s̄′ , s′ , (c′ ). With) strict quasiconcavity of f (·) and u(·), it is effortless to
show that s̄t , st , ct = t(s̄, s, c) + (1 − t) (s̄′ , s′ , c′ ) is feasible for all t ∈ (0, 1)
and U (s̄t , st , ct ) ≥ U (s̄, s, c). Since (s̄, s, c) is a solution, (s̄t , st , ct ) must also
be( a solution.
Then, if c ̸= c′ , strict quasiconcavity of u(·) implies that
)
U s̄t , st , ct > U (s̄, s, c), which is a contradiction. If c = c′ , we must have
s̄ ̸= s̄′ (see Lemma 1 in Appendix A) and, thus,
( )
{
}
f s̄t > min f (s̄), f (s̄′ ) ≥ 0,
because f (·) is strictly quasiconcave. By continuity of f (·), there exists s̄′′
7
arbitrarily close to s̄t from below, i.e., s̄′′ < s̄t , such that f (s̄′′ ) > 0.
-If st < s̄t , we can choose s̄′′ such that st < s̄′′ < s̄t and, thus, n(s̄′′ −st ) <
n(s̄t − st ). Since (s̄′′ , st , ct ) is feasible, we have U (s̄′′ , st , ct ) > U (s̄t , st , ct ),
giving a contradiction.
-If st = s̄t , consumer is strictly better off by lowering consumption of
satiated goods to s̄′′ and using the available resources to raise consumption
of nonsatiated goods, giving a contradiction.6 ■
With Theorem 3 and 4, we are now ready to take on the existence of Walrasian equilibrium. Provided that numerous approaches already exist, there
is no need to introduce a novel approach here.
It might sound natural to follow the works, allowing for satiation (e.g.,
Dana et al., 1999; Allouch and Le Van, 2008-9; Sato, 2010; Won and Yannelis, 2008, 2011). Unfortunately, these works are not compatible here because they allow for negative domain of prices. Accordingly, this paper
will instead follow older approach, which can be found in microeconomic
textbooks (say, chapter 5 of Jehle and Reny, 2011). Specifically, the sufficient conditions for the existence of Walrasian equilibrium in pure exchange
economy are given by:
(i) Walras’ law,
(ii) z(p) is continuous on Rℓ++ ,
(iii) and, if {pm } converges to p̄ ∈ Rℓ+ \{0} with p̄i = 0 for some i, then for
some good j with p̄j = 0, {zj (pm )} is unbounded above,
where z(p) denotes the aggregate excess demand function. Obviously, excess demand function deduced from Problem (U) already satisfies (i)-(ii), as
stated in Theorem 3-4. Thus, it remains to discuss (iii), which needs not
hold in general (because of Theorem 1). Carefully reading the proof of Theorem 5.3 in Jehle and Reny (2011), one will realize that (iii) is required to
ensures strictly positive equilibrium price. In particularly, in pure exchange
economy, since aggregate supply is fixed, there will exist excess demand (for
at least one good) as prices converge to zero. Therefore, even when {zj (pm )}
converges to a strictly positive amount, Walrasian equilibrium prices—i.e.,
p∗ ≫ 0 such that z(p∗ ) = 0—still exist.
Now, consider a pure exchange economy, E(H, ℓ, eh ), with H consumers
and ℓ goods. Each consumer h, whose behavior
∑is characterized by Problem
(U), is endowed with eh ∈ Rℓ+ \{0}, satisfying h∈H eh ≫ 0. Then, from the
above discussion and Theorem 1, it is appropriate to expect the following
Assumption 3’ (which is a stronger assumption that Assumption 3) to be
sufficient.
6
Note that if ℓs = ℓ, Assumption 3 ensures that st < s̄t .
8
Assumption 3’: For all h ∈ H and for all satiated good i, there exists
δ > 0 such that
s̄hi − ehi ≥ δ,
∀s̄hi ∈ Sih ,
where Sih is the projection of S h onto the set of good i.
Particularly, Assumption 3’ excludes the situation when consumer’s highest
demand is less than what he or she has, guaranteeing the existence of Walrasian equilibrium, as stated in Theorem 5.
Theorem 5: In a pure exchange economy, E(H, ℓ, eh ), Walrasian equilibrium exists under Assumption 1, 2, 3’, 4, and 5.
Proof : As already discussed, one should refer to the proof of Theorem
5.3 in Jehle and Reny (2011). This paper will only show that if pm →
p̄ ∈ Rℓ+ with p̄i = 0 for some i, for some good j with p̄j = 0, {zj (pm )} is
either unbounded above or converging to a strictly positive amount. To save
notation, let ℓ0 be the set of all good i with p̄i = 0.
(a) For any k, if p̄·ek > 0, Lemma
2 in
all goods
{
} Appendix A states that if
{ k m }
0
0
k
m
in ℓ are satiated, ∃j ∈ ℓ s.t. zj (p ) is unbounded above or zi (p ) ,
0
0
∀i
{ ∈ ℓ , }converges to a strictly positive amount. Otherwise, ∃j ∈ ℓ s.t.
zjk (pm ) is unbounded above.7
{∑ k m }
(b) Let
z (p ) be the excess demand function of all consumer
k
0
k with
{∑p̄ · e =} 0. Then, Lemma 3 in Appendix A states that ∃j ∈ ℓ
s.t.
zjk (pm ) is either unbounded above or converging to a nonnegative
amount.
(a)-(b) give the desired
result because there always exists k ∈ H such
∑
that p̄ · ek > 0 (since h∈H eh ≫ 0). ■
Note that Theorem 5 states a more general result than that in existing
literature because Theorem 5 is true whether there is no satiated good, there
are some satiated goods, or there is no nonsatiated good. Also, Theorem 5
does not require all consumers to consume the same satiated good basket.
Theorem 5 is stated under Assumption 3’ that no consumer is satiated. If
this is not the case, Walrasian equilibrium will fail to exist. In such case, one
can restore equilibrium by modifying the definition of Walrasian equilibrium,
e.g., dividend equilibrium (Drèze and Müller ,1980; Makarov, 1981; Aumann
and Drèze ,1986, Mas-Colell, 1992) or monetary equilibrium (Kajii, 1996).
Another possibility is to work with dynamic models (Ryder and Heal, 1973;
and Drugeon and Wigniolle, 2007) because satiated consumers will save all
unneeded resource. Doing all these is beyond the scope of this paper and,
thus, will be left to future works. This section will be ended with the first
Note that if eh ≫ 0 and, thus, p̄ · eh > 0, ∀h, Assumption 3’ can be relaxed to the all
goods that are satiated for all consumers.
7
9
and second welfare theorem, stated without proof—because the proof is
standard (see, for example, chapter 16 of Mas-Colell, et al., 1995; or chapter
5 of Jehle and Reny, 2011)—in Theorem 6-7, respectively:
Theorem 6 (First Welfare Theorem) : In a pure exchange economy, E(H, ℓ, eh ),
under Assumption 1-3, Walrasian equilibrium is Pareto efficient.
Theorem 7 (Second Welfare Theorem): In a pure exchange economy,
E(H, ℓ, eh ), under Assumption 1,2,3’,4, and 5, any Pareto-efficient allocation (s∗ , c∗ ) is Walrasian equilibrium of the market with initial endowment
(s∗ , c∗ ).
3
Endogenous Quality and Preference
Section 2 discussed the fundamental results without differentiability assumption. To go further, this section will make use of differentiability as stated
in Assumption 5 below:
Assumption 5: n(·), u(·), and f (·) are continuously differentiable and f (·)
has full rank Jacobian (ℓs × k) matrix, df ; i.e., rank(df ) = k ≤ ℓs .
The differentiability, along with the full rank Jacobian matrix, enables the
use of conventional Kuhn-Tucker conditions,8 from which follows a stronger
version of Theorem 1:
Theorem 1’: Under Assumption 1-3 and 5, if s̄ ≫ 0, ∀s̄ ∈ S, then the
solution to Problem (U) must satisfy s ≪ s̄.
Proof : Let λ ≥ 0 and (µs , µc ) ≥ 0 be Lagrange multiplier on budget
constraint and on (s, c) ≥ 0, respectively. Kuhn-Tucker conditions can be
written as
−un ∇n(s̄ − s) + µs = λps ,
uc + µc = λpc ,
s
c
where un = ∂u/∂n,
(
)p (p ) is price vector of satiated (non-satiated) goods,
un
and ∇n and
are gradient vectors of n(·) and u(·), respectively. If
uc
µsi > 0, si = 0 < s̄i . Thus, suppose that µs = 0. Then, when ℓs < ℓ (resp.
ℓs = ℓ), since λ > 0 because uc ≫ 0 (resp. because of Assumption 3), it
8
Specifically,
is because the (non-degenerated) constraint qualification is satisfied,
( t this )
p
0
i.e., rank
= k + 1, (see chapter 18 of Simon and Blume, 1994; or Appendix
0 df t
M.K of Mas-Colell et al., 1995).
10
follows that ∇n ≫ 0, implying that s ≪ s̄ (see Assumptoin 2). ■
On the one hand, Theorem 1’ is very useful in applications because one
does not have to worry about the possibility of satiated consumers. On the
other hand, it casts doubt on the role played by satiation. Accordingly,
this section will provide a stylized application, discussing the link between
satiation and change in preference, as discussed in the introduction.
To begin, consider an economy, populated by a continuum of identical
individuals with measure 1. There are one non-satiated numeraire and n
satiated goods. The numeraire is the only factor of production and each
individual is endowed with one unit of numeraire. Furthermore, each satiated good has a unique satiated level, which is strictly increasing in product
quality. Let c ∈ R+ be consumption of numeraire, s ∈ Rn+ be consumption
vector of satiated goods, and s̄ ∈ Rn++ be satiation vector. Utility function
is defined as
U = u (c) − ∥s̄ − s∥2 ,
where u(·) is strictly concave, is strictly increasing, and is continuously differentiable (i.e., u′ > 0, u′′ < 0). Without loss of generality, price of numeraire
is normalized to one. Then, letting p ∈ Rn++ be price vector of satiated
goods, one can write budget constraint of each individual as follow
c + p · s = y,
(1)
where y ∈ R++ is total income, equal to one (i.e., the endowment of numeraire) plus dividends from monopolistically competitive firms. Letting
µ ∈ R+ and λ ∈ Rn+1
be Lagrange multipliers on budget constraint and
+
nonnegative constraints, respectively, one can write Kuhn-Tucker conditions
as follows
u′ (c) + λ0 = µ,
2 (s̄i − si ) + λi = µpi ,
(2)
i = 1, ..., n.
(3)
Without loss of generality, suppose that consumption of each satiated good
i ≥ m (where m ≤ n) is strictly positive while that of the rest is zero. Then,
from equations (1) and (3), it is straightforward to show that
si = s̄i − αpi > 0,
∀i ≥ m,
(4)
s̄i − αpi ≤ 0,
∀i < m,
(5)
where α is given by
∑n
α=
p s̄ − y
i=m
∑ni i 2
i=m pi
+c
.
Note from equations (4)-(5) that α is a cutoff between firms, facing zero
and non-zero demand. Thus, similar to Melitz and Ottaviano (2008), one
11
can say that α determines the toughness of competition. Particularly, if α
rises (i.e., when competition gets tougher), less firms are likely to survive.
In contrast to Melitz and Ottaviano (2008), however, equations (4)-(5) suggest that price or productivity (i.e., the inverse of marginal cost) is not the
main determinant of survival, but s̄i /pi is. For instance, firms, with high
marginal costs and, thus, high prices, might have high demand because of
high satiated levels (say, due to high qualities). For convenience, since x̄i is
the maximum demand, highlighting desire toward good i, let us call s̄i /pi
market desirability of firm i. Then, one can conclude that, in free market,
only desirable firms (whose market desirability is strictly higher than α) can
survive. In other words, free market trims the least desirable (not the least
productive) firms off the markets.
Having discussed that the least desirable firms cannot survive, now, let
us analyze the behavior of the survivor, who face demand function, given
by equation (4). In the sense of monopolistic competition, assume that firm
takes α as a constant and, thus, there is no strategic interaction among firms.
As indicated by Dixit and Stiglitz (1993), this helps simplify the analysis
and can be justified by assuming that the number of firms is sufficiently
large. Alternatively, one can say that firms do not fully know individuals’
preferences and, thus, they use equation (4), which helps simplifying the
profit maximization problem and is easily estimated.
Following Dixit and Stiglitz (1977), assume that each firm i’s total cost
is given by mi si + fi , where mi is marginal cost and fi > 0 is fixed cost.
Assume that marginal cost is strictly increasing in product quality, Qi . In
other words, to produce an extra unit of good with higher quality requires
higher cost. Then, profit can be written as
πi = (pi − mi ) si − fi .
Before solving profit maximization problem, first, note from equation (4)
that a rise in s̄i raises the minimum market desirability, α. Then, demand for every good j ̸= i will falls while that of good i will rise, since
∂ (s̄i − αpi ) /∂s̄i > 0. In other words, preferences shift toward goods with
high quality, motivating firms to change their product qualities. Accordingly, in this paper, firm chooses both price and quality to maximize profit
subject to equation. Then, it is straightforward to show that price and
quality of each firm i must satisfy the following conditions:
[ ′
]
s̄i (Qi ) − αm′i (Qi ) [s̄i (Qi ) − αmi (Qi )] ≤ 0 , with equality if Qi > 0,(6)
s̄i (Qi ) + αmi (Qi )
.
(7)
pi =
2α
From condition (6), we know that firm chooses zero quality if and only if
s̄i (0) > αmi (0) and s̄′i (0) ≤ αm′i (0). The former states that demand is
non-zero at zero quality, while the latter states that, at zero quality, firm is
12
worse off when it raises quality. More specifically, when quality rises by one
unit, satiated level rises by s̄′i units and, hence, it immediately follows from
equation (4) that firm can raise price by s̄′i /α units, which must be higher
than the rise in cost, m′i , for raising quality to be beneficial. In the same
fashion, one can tell that when quality is non-zero, at optimum, one must
have
s̄′i (Qi ) = αm′i (Qi ),
(8)
which follows immediately from condition (6). Furthermore, since s̄i > αpi ,
it immediately follows from equation (7) that the markup is strictly higher
than unity, i.e., pi > mi . Also, one can tell from equation (7) that quality
have both cost-push and demand-pull effects on prices. In particular, a rise
in quality raises marginal cost (satiated level and, thus demand), pushing
(pulling) price up.
It is obvious that, for the existence of product qualities, it is important
to impose some restrictions on satiation function, s̄i (Qi ), and marginal cost
function, mi (Qi ). In particular, assume that the following conditions hold
(s̄′i − αm′i )2
,
2α
lim αm′i (Qi ).
s̄′′i (pi − mi ) − m′′i (s̄i − αpi ) < −
lim s̄′i (Qi ) <
Qi →∞
Qi →∞
The first condition is the sufficient condition for the strict concavity of profit
function, which ensures that the solution to profit maximization problem
must be unique. The second condition ensures that when s̄′i (0) > αm′i (0),
optimal quality always exist. More specifically, the locus of left hand side
(lhs) of equation (8) cuts that of the right hand side (rhs) only once from
above, i.e., s̄′′i < αm′′i . This, in turn, implies that optimal quality must
satisfy s̄i > αmi , which ensures that the markup is strictly higher than
one and demand is strictly positive. To see this more clearly, substitute
equations (7)-(8) into equation (9) and rearrange terms to obtain
( ′′
)
s̄i − αm′′i (s̄i − αmi ) < 0.
Then, since s̄′′i − αm′i < 0, we must have s̄i > αmi and, thus, pi > mi and
s̄i > αpi . Furthermore, the fact that s̄′′i − αm′i < 0 implies that optimal
quality price are decreasing in α. More specifically, from equations (7)-(8),
one can obtain the following conditions
Q′i (α) =
p′i (α) =
m′i (Qi )
< 0,
s̄′′i (Qi ) − αm′′i (Qi )
2αs̄′i (Qi )Q′i (α) − s̄i (Qi )
< 0.
2α2
Intuitively, a rise in α raises minimum market desirability and, thus, the
competition gets tougher. Then, firms will react by worsening product
13
qualities (and thus costs) in order to lower prices and to raise their market desirabilities (i.e., ∂(s̄i /pi )/∂α > 0).
Now, let us consider general equilibrium. Since firms, facing nonpositive
demands, exit the markets, we can focus on the surviving firms. Without loss
of generality, assume that all firms n survive. Then, the general equilibrium
is characterized by equations (4), (6)-(8), and the following equations
)
∑n (
fi
c = 1−
s i mi +
,
(9)
i=1
L
u′ (c) = 2α.
(10)
Equation (9) is simply the market clearing condition for numeraire,
∑ which
follows immediately from equation (1) and the fact that y = 1 + ni=1 πi /L.
In addition, equation (10) illustrates the trade-off between numeraire and
satiated goods, and follows immediately from equations (2)-(4). To solve the
this system of equations, note from equations (4), (6)-(8), and (9) that one
can write all variables as functions of α alone. Thus, it suffices to determine
α from equation (10). Given the continuity of u′ (·), to ensure the existence
of general equilibrium, it suffices to assume that the following conditions
hold:
(
)
lim u′ (c) − 2α > 0,
α→0
(
)
lim u′ (c) − 2α < 0.
α→∞
Furthermore, it is straightforward to check that c is strictly increasing in α.
Hence, since u′′ (·) < 0, α, satisfying equation (10), must be unique.
Note finally that if either the number of firms, n, or the ratio of fixed
cost to population, fi /L, rises, the locus of lhs of equation (10) shifts upward and that of rhs is unchanged. Then, equilibrium value of α will rise.
As a result, prices, qualities, and demands for all satiated goods will fall.
Intuitively, a new entry (i.e., a rise in n) makes competition tougher through
the business-stealing effects; that is, the new entrant lowers demand for the
incumbent firms. Then, as already discussed, incumbent firms will react
by lower qualities in order to lower prices. Similarly, a rise in fi /L lowers
demand through either a fall in fixed cost (and, thus, income) or a fall in
population.
4
Concluding Remarks
Despite being undeniable, it is surprising that satiation has received little attention. The knowledge regarding to satiation and how it affects the
economy is very limited. Accordingly, this paper has made the first attempt, aiming at igniting the interest in satiation. Particularly, this paper
has introduced exogenous satiation into conventional utility maximization
14
problem, in the sense that satiation set is exogenously given. Under standard assumptions, this paper is able to reproduce the fundamental results
in classical consumer theory, namely the existence of Walrasian equilibrium
and the first and second welfare theorem. Furthermore, since utility depends
on satiation, in turn, depending on prices, this paper suggests that the introduction of satiation might lead to some properties of demand function
beyond classical demand theory. Finally, this paper suggests that satiation
provides a parsimonious way to deal with change in preference, a problem
being ignored in applications. Thus, this paper calls for more studies, regarding to not only properties of demand function but also the impacts of
satiation in applications.
Appendix: Proofs of Lemmas
Lemma 1: In the proof of Theorem 4, when c = c′ , we must have s̄ ̸= s̄′ .
Proof : Since (s̄, s, c) is a solution to Problem (U), s must solve the
following minimization problem:
min n(s̄ − s)
s∈C s
s.t.
ps · s ≤ y − pc · c,
(U’)
where ps and pc are price vectors of satiated and nonsatiated consumption,
respectively. Given the strict quasiconvexity of n(·), the solution to Problem
(U’) must be unique (since C s is convex). Then, since (s̄′ , s′ , c) is a different
solution to Problem (U), if s̄′ = s̄, we must have s = s′ , which is a contradiction. ■
Lemma 2: In the proof
5, when p̄ · ek > 0, if all goods in ℓ0 are
{ of Theorem
}
{
}
satiated, ∃j ∈ ℓ0 s.t. zjk (pm ) is unbounded above or zik (pm ) , ∀i ∈ ℓ0 ,
{
}
converges to a strictly positive amount. Otherwise, ∃j ∈ ℓ0 s.t. zjk (pm )
is unbounded above.
Proof : The {(
proof follows
)}that of Theorem 5.4 in Jehle and Reny (2011).
m
m
Particularly, let
xℓ0 , x−ℓ0
be demand sequence corresponding to {pm }.
{ m}
If
{ mx}ℓ0 is unbounded above, the result is trivial. Therefore, assume that
xℓ0 is bounded and, thus, has convergence subsequence. The following
0
will consider the case that
{ mall
} goods in ℓ are satiated. Without loss of
generality, suppose that xℓ0 , itself, converges to x̄ℓ0 . Using contradiction,
suppose that ∃i ∈ ℓ0 and ε > 0 s.t. x̄i + ε < s̄i , ∀si ∈ Sik .
Under Assumption 2-3, Walras’ law holds (see Theorem 3):
(
)
m
m
k
pm · xm
ℓ0 , x−ℓ0 = p · e .
15
Taking} limit, we have p̄ · (x̄ℓ0 , x̄−ℓ0 ) = p̄ · ek > 0, where x̄−ℓ0 is the limit of
{
xm
−ℓ0 (which is continuos). Then, from Assumption 2, we know that
U (s̄, x̄i + ε, x̄−i , x̄−ℓ0 ) > U (s̄, x̄ℓ0 , x̄−ℓ0 ) , ∀s̄ ∈ S k ,
p̄ · (x̄i + ε, x̄−i , x̄−ℓ0 ) = p̄ · ek > 0.
By continuity of U (·), we have
U (s̄, tx̄i + tε, tx̄−i , tx̄−ℓ0 ) > U (s̄, x̄ℓ0 , x̄−ℓ0 ) ,
tp̄ · (x̄i + ε, x̄−i , x̄−ℓ0 ) < p̄ · ek ,
where t ∈ (0, 1) and is arbitrarily close to 1. This, in turn, implies that
(
)
m
k
U (s̄, tx̄i + tε, tx̄−i , tx̄−ℓ0 ) > U s̄, xm
ℓ0 , x−ℓ0 , ∀s̄ ∈ S ,
tpm · (x̄i + ε, x̄−i , x̄−ℓ0 ) < pm · ek ,
for m large enough, giving a contradiction. Thus, ∃s̄i ∈ Sik s.t. x̄i ≥ s̄i − ε,
∀i ∈ ℓ0 , ∀ε > 0. This, along with Assumption 5, gives desired result.
Note that when at least one good in ℓ0 is nonsatiated, a similar
{ pro}
cess (with nonsatiated good i) will contradict the assumption that xm
ℓ0 is
bounded. ■
k
0
Lemma
{∑ 3: In }the proof of Theorem 5, when p̄ · e = 0, then ∃j ∈ ℓ
s.t.
zjk (pm ) is either unbounded above or converging to a nonnegative
amount.
{(( ) (
)m )}
m
xkℓ0 , xk−ℓ0
Proof : Now, let
be demand sequence of conm
sumer k corresponding to {p }. Following
the)proof of Lemma 2, when
(
{( k )m }
k
k
x ℓ0
is unbounded, we have p̄ · x̄ℓ0 , x̄−ℓ0 = p̄ · ek = 0, implying
{(( ) (
)m )}
m
that
xkℓ0 , xk−ℓ0
→ (x̄kℓ0 , 0). Using contradiction, suppose that
)
∑ k
∑ k
∑( k
∑
x̄ℓ0 ≪ eℓ0 . Then, we can find ε ≫ 0 such that
x̄ℓ0 + ε ≪ ekℓ0 .
Without loss of generality, assume that there is no h such that x̄hℓ0 + ε ≥
0
k
k
ehℓ0 . Hence,
Assumption 3’ guarantees
( ∀k ∈, ∃i ∈ ℓ s.t.)x̄i + εi (< ei . Then,
)
k
k
k
k
k
k
k
that U s̄ , x̄i + εi , x̄ℓ0 −i , 0 > U s̄ , x̄ℓ0 , 0 , ∀s̄k ∈ S k . As a result, by
continuity of U k (·), we must have
(
)
( ( )m (
)m )
U k s̄k , x̄ki + εi , x̄kℓ0 −i , 0 > U k s̄k , xkℓ0
, xk−ℓ0
, ∀s̄k ∈ S k ,
)
∑( h
∑ h
m·
m
for
m
large
enough.
Since
p
x̄
eℓ0 , there exists k s.t.
0
0 + ε < p ℓ0 ·
ℓ
ℓ
( k
)
m
x̄ℓ0 + ε, 0 is feasible under price p , giving a contradiction. ■
16
References
Acemoglu, D. (2009): Introduction to Modern Economic Growth. Princeton: Princeton
University Press.
Allouch, N. and C. Le Van (2008): “Walras and Dividends Equilibrium with Possibly
Satiated Consumers,” Journal of Mathematical Economics, 44, 907-918.
—— (2009): Erratum to “Walras and Dividends Equilibrium with Possibly Satiated
Consumers,” Journal of Mathematical Economics, 45, 320-328.
Arrow, K.J. and G. Debreu (1954): “Existence of an Equilibrium for a Competitive
Economy,” Econometrica, 22 (3), 265-290.
Aumann, R.J. and J.H. Drèze (1986): “Values of Markets with Satiation or Fixed Prices,”
Econometrica, 54 (6), 1271-1318.
Ballestero, E. and C. Romero (1994): “Utility Optimization When the Utility Function
Is Virtually Unknown,” Theory and Decision, 37, 233-243.
Cornet, B., M. Topuzu and A. Yildiz (2003): “Equilibrium Theory with a Measure Space
of Possibly Satiated Consumers,” Journal of Mathematical Economics, 39, 175-196.
Dana, R.A., C. Le Van and F. Magnien (1999): “On the Different Notions of Arbitrage
and Existence of Equilibrium,” Journal of Economic Theory, 87, 169-193.
Dixit, A. and J.E. Stiglitz (1977): “Monopolistic Competition and Optimum Product
Diversity”, American Economic Review, 67, 297-308.
—— (1993): “Monopolistic Competition and Optimum Product Diversity: Reply”,
American Economic Review, 83, 302-304.
Drèze, J.H. and H. Müller (1980): “Optimality Properties of Rationing Schemes,” Journal of Economic Theory, 23, 131-149.
Drugeon, J.P. and B. Wigniolle (2007): “On Time Preference, Rational Addiction and
Utility Satiation,” Journal of Mathematical Economics, 43, 249-286.
Jehle, G.A. and P.J. Reny (2011): Advanced Microeconomic Theory. Harlow; Tokyo:
Financial Times/Prentice Hall.
Johnson, R. (2012) “Trade and Prices with Heterogeneous Firms,” Journal of International Economics, 86, 43-56.
Kajii, A. (1996): “How to Discard Nonsatiation and Free Disposal with Paper Money,”
Journal of Mathematical Economics, 25, 75-84.
Makarov, V.L. (1981): “Some Results on General Assumptions about the Existence of
Economic Equilibrium,” Journal of Mathematical Economics, 8, 87-99.
Mas-Colell, A. (1992): “Equilibrium Theory with Possibly Satiated Preferences,” In:
Majumdar, M. (Ed.) Equilibrium and Dynamics: Essays in Honor of David Gale.
Macmillan, 201-213.
——–, M.D. Whinston and J.R. Green (1995): Microeconomic Theory. New York:
Oxford University Press.
Mckenzie, L.W. (1959): “On the Existence of General Equilibrium for a Competitive
Market,” Econometrica, 27 (1), 54-71.
Melitz, M.J. and G.I.P. Ottaviano (2008): “Market Size, Trade, and Productivity,” Review of Economic Studies, 75, 295-316.
Ryder, H.E. and G.M. Heal (1973): “Optimal Growth with Intertemporally Dependent
Preferences,” Review of Economic Studies, 40 (1), 1-31.
Sato, N. (2010): “Satiation and Existence of Competitive Equilibrium,” Journal of Mathematical Economics, 46, 534-551.
17
Simon, C.P. and L. Blume (1994): Mathematics for Economists. New York: Norton.
Stokey, N.L. and R.E.Jr. Lucas with E.C. Prescott (1989): Recursive Methods in Economic Dynamics. Cambridge, Mass.: Harvard University Press.
Won, D.C. and N.C. Yannelis (2008): “Equilibrium Theory with Unbounded Consumption Sets and Non-order Preferences Part I. Non-satiation,” Journal of Mathematical Economics, 44, 1266-1283.
—— (2011): “Equilibrium Theory with Satiable and Non-Ordered Preferences,” Journal
of Mathematical Economics, 47, 245-250.
Yu, P.L. (1973): “A Class of Solutions for Group Decision Problems,” Management
Science, 19 (8), 936-946.
Zeleny, M. (1974): “A Concept of Compromise Solutions and the Method of the Displaced
Ideal,” Computers and Operations Research, 1, 479-496.
18