Significant topic in economics
Convexity
is a geometric property with a variety of applications in
economics
[1]
. Informally, an economic phenomenon is convex when "intermediates (or combinations) are better than extremes". For example, an economic agent with
convex preferences
prefers
combinations
of goods over having a lot of any
one
sort of good; this represents a kind of
diminishing marginal utility
of having more of the same good.
Convexity is a key simplifying assumption in many economic models, as it leads to market behavior that is easy to understand and which has desirable properties. For example, the
Arrow?Debreu model
of
general economic equilibrium
posits that if preferences are convex and there is perfect competition, then
aggregate supplies
will equal
aggregate demands
for every commodity in the economy.
In contrast,
non-convexity
is associated with
market failures
, where
supply and demand
differ or where
market equilibria
can be
inefficient
.
The branch of mathematics which supplies the tools for convex functions and their properties is called
convex analysis
; non-convex phenomena are studied under
nonsmooth analysis
.
Preliminaries
[
edit
]
The economics depends upon the following definitions and results from
convex geometry
.
Real vector spaces
[
edit
]
A non?convex set fails to
cover
a point in some
line segment
joining two of its points.
A
real
vector space
of two
dimensions
may be given a
Cartesian coordinate system
in which every point is identified by a list of two real numbers, called "coordinates", which are conventionally denoted by
x
and
y
. Two points in the Cartesian plane can be
added
coordinate-wise
- (
x
1
,
y
1
) + (
x
2
,
y
2
) = (
x
1
+
x
2
,
y
1
+
y
2
);
further, a point can be
multiplied
by each real number
λ
coordinate-wise
- λ
(
x
,
y
) = (
λx
,
λy
).
More generally, any real vector space of (finite) dimension
D
can be viewed as the
set
of all possible lists of
D
real numbers
{ (
v
1
,
v
2
, . . . ,
v
D
)
} together with two
operations
:
vector addition
and
multiplication by a real number
. For finite-dimensional vector spaces, the operations of vector addition and real-number multiplication can each be defined coordinate-wise, following the example of the Cartesian plane.
Convex sets
[
edit
]
In a real vector space, a set is defined to be
convex
if, for each pair of its points, every point on the
line segment
that joins them is
covered
by the set. For example, a solid
cube
is convex; however, anything that is hollow or dented, for example, a
crescent
shape, is non?convex.
Trivially
, the
empty set
is convex.
More formally, a set
Q
is convex if, for all points
v
0
and
v
1
in
Q
and for every real number
λ
in the
unit interval
[0,1]
, the point
- (1 −
λ
)
v
0
+
λv
1
is a
member
of
Q
.
By
mathematical induction
, a set
Q
is convex if and only if every
convex combination
of members of
Q
also belongs to
Q
. By definition, a
convex combination
of an indexed subset {
v
0
,
v
1
, . . . ,
v
D
} of a vector space is any
weighted average
λ
0
v
0
+
λ
1
v
1
+ . . . +
λ
D
v
D
,
for some indexed set of non?negative real numbers {
λ
d
} satisfying the
equation
λ
0
+
λ
1
+ . . . +
λ
D
= 1.
The definition of a convex set implies that the
intersection
of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set.
Convex hull
[
edit
]
For every subset
Q
of a real vector space, its
convex hull
Conv(
Q
)
is the
minimal
convex set that contains
Q
. Thus Conv(
Q
) is the intersection of all the convex sets that
cover
Q
. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in
Q
.
Duality: Intersecting half-spaces
[
edit
]
Supporting hyperplane
is a concept in
geometry
. A
hyperplane
divides a space into two
half-spaces
. A hyperplane is said to
support
a
set
in the
real
n
-space
if it meets both of the following:
- is entirely contained in one of the two
closed
half-spaces determined by the hyperplane
- has at least one point on the hyperplane.
Here, a closed half-space is the half-space that includes the hyperplane.
Supporting hyperplane theorem
[
edit
]
This
theorem
states that if
is a closed
convex set
in
and
is a point on the
boundary
of
then there exists a supporting hyperplane containing
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set
is not convex, the statement of the theorem is not true at all points on the boundary of
as illustrated in the third picture on the right.
Economics
[
edit
]
An optimal basket of goods occurs where the consumer's convex
preference set
is
supported
by the budget constraint, as shown in the diagram. If the preference set is convex, then the consumer's set of optimal decisions is a convex set, for example, a unique optimal basket (or even a line segment of optimal baskets).
For simplicity, we shall assume that the preferences of a consumer can be described by a
utility function
that is a
continuous function
, which implies that the
preference sets
are
closed
. (The meanings of "closed set" is explained below, in the subsection on optimization applications.)
Non-convexity
[
edit
]
If a preference set is non?convex, then some prices produce a budget supporting two different optimal consumption decisions. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeper does not want to purchase a half an eagle and a
half a lion
(or a
griffin
)! Thus, the contemporary zoo-keeper's preferences are non?convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.
Non?convex sets have been incorporated in the theories of general economic equilibria,
[2]
of
market failures
,
[3]
and of
public economics
.
[4]
These results are described in graduate-level textbooks in
microeconomics
,
[5]
general equilibrium theory,
[6]
game theory
,
[7]
mathematical economics
,
[8]
and applied mathematics (for economists).
[9]
The
Shapley?Folkman lemma
results establish that non?convexities are compatible with approximate equilibria in markets with many consumers; these results also apply to
production economies
with many small
firms
.
[10]
In "
oligopolies
" (markets dominated by a few producers), especially in "
monopolies
" (markets dominated by one producer), non?convexities remain important.
[11]
Concerns with large producers exploiting market power in fact initiated the literature on non?convex sets, when
Piero Sraffa
wrote about on firms with increasing
returns to scale
in 1926,
[12]
after which
Harold Hotelling
wrote about
marginal cost pricing
in 1938.
[13]
Both Sraffa and Hotelling illuminated the
market power
of producers without competitors, clearly stimulating a literature on the supply-side of the economy.
[14]
Non?convex sets arise also with
environmental goods
(and other
externalities
),
[15]
[16]
with
information economics
,
[17]
and with
stock markets
[11]
(and other
incomplete markets
).
[18]
[19]
Such applications continued to motivate economists to study non?convex sets.
[20]
Nonsmooth analysis
[
edit
]
Economists have increasingly studied non?convex sets with
nonsmooth analysis
, which generalizes
convex analysis
. "Non?convexities in [both] production and consumption ... required mathematical tools that went beyond convexity, and further development had to await the invention of non?smooth calculus" (for example, Francis Clarke's
locally Lipschitz
calculus), as described by
Rockafellar & Wets (1998)
[21]
and
Mordukhovich (2006)
,
[22]
according to
Khan (2008)
.
[23]
Brown (1991
, pp. 1967?1968) wrote that the "major methodological innovation in the general equilibrium analysis of firms with pricing rules" was "the introduction of the methods of non?smooth analysis, as a [synthesis] of global analysis (differential topology) and [of] convex analysis." According to
Brown (1991
, p. 1966), "Non?smooth analysis extends the local approximation of manifolds by tangent planes [and extends] the analogous approximation of convex sets by tangent cones to sets" that can be non?smooth or non?convex.
[24]
Economists have also used
algebraic topology
.
[25]
See also
[
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]
Notes
[
edit
]
- ^
Newman (1987c)
- ^
Pages 392?399 and page 188:
Arrow, Kenneth J.
;
Hahn, Frank H.
(1971).
"Appendix B: Convex and related sets"
.
General competitive analysis
. Mathematical economics texts [Advanced textbooks in economics]. San Francisco: Holden-Day, Inc. [North-Holland]. pp.
375?401
.
ISBN
978-0-444-85497-1
.
MR
0439057
.
Pages 52?55 with applications on pages 145?146, 152?153, and 274?275:
Mas-Colell, Andreu
(1985). "1.L Averages of sets".
The Theory of General Economic Equilibrium: A
Differentiable
Approach
. Econometric Society Monographs. Cambridge University Press.
ISBN
978-0-521-26514-0
.
MR
1113262
.
Theorem C(6) on page 37 and applications on pages 115?116, 122, and 168:
Hildenbrand, Werner
(1974).
Core and equilibria of a large economy
. Princeton studies in mathematical economics. Princeton University Press. pp. viii+251.
ISBN
978-0-691-04189-6
.
MR
0389160
.
- ^
Pages 112?113 in Section 7.2 "Convexification by numbers" (and more generally pp. 107?115):
Salanie, Bernard (2000). "7 Nonconvexities".
Microeconomics of market failures
(English translation of the (1998) French
Microeconomie: Les defaillances du marche
(Economica, Paris) ed.). MIT Press. pp. 107?125.
ISBN
978-0-262-19443-3
.
- ^
Pages 63?65:
Laffont, Jean-Jacques
(1988).
"3 Nonconvexities"
.
Fundamentals of public economics
. MIT.
ISBN
978-0-262-12127-9
.
- ^
Varian, Hal R.
(1992).
"21.2 Convexity and size"
.
Microeconomic Analysis
(3rd ed.). W. W. Norton & Company. pp.
393?394
.
ISBN
978-0-393-95735-8
.
MR
1036734
.
Page 628:
Mas?Colell, Andreu
; Whinston, Michael D.; Green, Jerry R. (1995). "17.1 Large economies and nonconvexities".
Microeconomic theory
. Oxford University Press. pp. 627?630.
ISBN
978-0-19-507340-9
.
- ^
Page 169 in the first edition:
Starr, Ross M. (2011). "8 Convex sets, separation theorems, and non?convex sets in
R
N
".
General equilibrium theory: An introduction
(Second ed.). Cambridge: Cambridge University Press.
doi
:
10.1017/CBO9781139174749
.
ISBN
978-0-521-53386-7
.
MR
1462618
.
In Ellickson, page xviii, and especially Chapter 7 "Walras meets Nash" (especially section 7.4 "Nonconvexity" pages 306?310 and 312, and also 328?329) and Chapter 8 "What is Competition?" (pages 347 and 352):
Ellickson, Bryan (1994).
Competitive equilibrium: Theory and applications
. Cambridge University Press. p. 420.
ISBN
978-0-521-31988-1
.
- ^
Theorem 1.6.5 on pages 24?25:
Ichiishi, Tatsuro (1983).
Game theory for economic analysis
. Economic theory, econometrics, and mathematical economics. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. x+164.
ISBN
978-0-12-370180-0
.
MR
0700688
.
- ^
Pages 127 and 33?34:
Cassels, J. W. S.
(1981). "Appendix A Convex sets".
Economics for mathematicians
. London Mathematical Society lecture note series. Vol. 62. Cambridge, New York: Cambridge University Press. pp. xi+145.
ISBN
978-0-521-28614-5
.
MR
0657578
.
- ^
Pages 93?94 (especially example 1.92), 143, 318?319, 375?377, and 416:
Carter, Michael (2001).
Foundations of mathematical economics
. MIT Press. pp. xx+649.
ISBN
978-0-262-53192-4
.
MR
1865841
.
Page 309:
Moore, James C. (1999).
Mathematical methods for economic theory: Volume
I
. Studies in economic theory. Vol. 9. Berlin: Springer-Verlag. pp. xii+414.
doi
:
10.1007/978-3-662-08544-8
.
ISBN
978-3-540-66235-8
.
MR
1727000
.
Pages 47?48:
Florenzano, Monique; Le Van, Cuong (2001).
Finite dimensional convexity and optimization
. Studies in economic theory. Vol. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. pp. xii+154.
doi
:
10.1007/978-3-642-56522-9
.
ISBN
978-3-540-41516-9
.
MR
1878374
.
S2CID
117240618
.
- ^
Economists have studied non?convex sets using advanced mathematics, particularly
differential geometry
and
topology
,
Baire category
,
measure
and
integration theory
, and
ergodic theory
:
Trockel, Walter (1984).
Market demand: An analysis of large economies with nonconvex preferences
. Lecture Notes in Economics and Mathematical Systems. Vol. 223. Berlin: Springer-Verlag. pp. viii+205.
doi
:
10.1007/978-3-642-46488-1
.
ISBN
978-3-540-12881-6
.
MR
0737006
.
- ^
a
b
Page 1:
Guesnerie, Roger
(1975). "Pareto optimality in non?convex economies".
Econometrica
.
43
(1): 1?29.
doi
:
10.2307/1913410
.
JSTOR
1913410
.
MR
0443877
.
(
Guesnerie, Roger (1975). "Errata".
Econometrica
.
43
(5?6): 1010.
doi
:
10.2307/1911353
.
JSTOR
1911353
.
MR
0443878
.
)
- ^
Sraffa, Piero
(1926). "The Laws of returns under competitive conditions".
Economic Journal
.
36
(144): 535?550.
doi
:
10.2307/2959866
.
JSTOR
2959866
.
S2CID
6458099
.
- ^
Hotelling, Harold
(July 1938). "The General welfare in relation to problems of taxation and of railway and utility rates".
Econometrica
.
6
(3): 242?269.
doi
:
10.2307/1907054
.
JSTOR
1907054
.
- ^
Pages 5?7:
Quinzii, Martine
(1992).
Increasing returns and efficiency
(Revised translation of (1988)
Rendements croissants et efficacite economique
. Paris: Editions du Centre National de la Recherche Scientifique ed.). New York: Oxford University Press. pp. viii+165.
ISBN
978-0-19-506553-4
.
- ^
Pages 106, 110?137, 172, and 248:
Baumol, William J.
; Oates, Wallace E. (1988). "8 Detrimental externalities and nonconvexities in the production set".
The Theory of environmental policy
. with contributions by V. S. Bawa and David F. Bradford (Second ed.). Cambridge: Cambridge University Press. pp. x+299.
ISBN
978-0-521-31112-0
.
- ^
Starrett, David A. (1972). "Fundamental nonconvexities in the theory of externalities".
Journal of Economic Theory
.
4
(2): 180?199.
doi
:
10.1016/0022-0531(72)90148-2
.
MR
0449575
.
Starrett discusses non?convexities in his textbook on
public economics
(pages 33, 43, 48, 56, 70?72, 82, 147, and 234?236):
Starrett, David A. (1988).
Foundations of public economics
. Cambridge economic handbooks. Cambridge: Cambridge University Press.
ISBN
9780521348010
.
nonconvex OR nonconvexities.
- ^
Radner, Roy
(1968). "Competitive equilibrium under uncertainty".
Econometrica
.
36
(1): 31?53.
doi
:
10.2307/1909602
.
JSTOR
1909602
.
- ^
Page 270:
Dreze, Jacques H.
(1987). "14 Investment under private ownership: Optimality, equilibrium and stability". In Dreze, J. H. (ed.).
Essays on economic decisions under uncertainty
. Cambridge: Cambridge University Press. pp. 261?297.
doi
:
10.1017/CBO9780511559464
.
ISBN
978-0-521-26484-6
.
MR
0926685
.
(Originally published as
Dreze, Jacques H.
(1974). "Investment under private ownership: Optimality, equilibrium and stability". In Dreze, J. H. (ed.).
Allocation under Uncertainty: Equilibrium and Optimality
. New York: Wiley. pp. 129?165.
)
- ^
Page 371:
Magill, Michael;
Quinzii, Martine
(1996). "6 Production in a finance economy, Section 31 Partnerships".
The Theory of incomplete markets
. Cambridge, Massachusetts: MIT Press. pp. 329?425.
- ^
Mas-Colell, A.
(1987).
"Non?convexity"
(PDF)
. In Eatwell, John; Milgate, Murray; Newman, Peter (eds.).
The New Palgrave: A Dictionary of Economics
(first ed.). Palgrave Macmillan. pp. 653?661.
doi
:
10.1057/9780230226203.3173
.
ISBN
9780333786765
.
- ^
Rockafellar, R. Tyrrell
;
Wets, Roger J-B
(1998).
Variational analysis
. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 317. Berlin: Springer-Verlag. pp. xiv+733.
doi
:
10.1007/978-3-642-02431-3
.
ISBN
978-3-540-62772-2
.
MR
1491362
.
S2CID
198120391
.
- ^
Chapter 8 "Applications to economics", especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495:
Mordukhovich, Boris S.
(2006).
Variational analysis and generalized differentiation
II
: Applications
. Grundlehren Series (Fundamental Principles of Mathematical Sciences). Vol. 331. Springer. pp. i?xxii and 1?610.
MR
2191745
.
- ^
Khan, M. Ali (2008).
"Perfect competition"
. In Durlauf, Steven N.; Blume, Lawrence E. (eds.).
The New Palgrave Dictionary of Economics
(Second ed.). Palgrave Macmillan. pp. 354?365.
doi
:
10.1057/9780230226203.1267
.
ISBN
978-0-333-78676-5
.
- ^
Brown, Donald J. (1991). "36 Equilibrium analysis with non?convex technologies". In
Hildenbrand, Werner
;
Sonnenschein, Hugo
(eds.).
Handbook of mathematical economics, Volume
IV
. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 1963?1995 [1966].
doi
:
10.1016/S1573-4382(05)80011-6
.
ISBN
0-444-87461-5
.
MR
1207195
.
- ^
Chichilnisky, G.
(1993).
"Intersecting families of sets and the topology of cones in economics"
(PDF)
.
Bulletin of the American Mathematical Society
. New Series.
29
(2): 189?207.
arXiv
:
math/9310228
.
Bibcode
:
1993math.....10228C
.
CiteSeerX
10.1.1.234.3909
.
doi
:
10.1090/S0273-0979-1993-00439-7
.
MR
1218037
.
References
[
edit
]
- Blume, Lawrence E.
(2008a).
"Convexity"
. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
The New Palgrave Dictionary of Economics
(Second ed.). Palgrave Macmillan. pp. 225?226.
doi
:
10.1057/9780230226203.0315
.
ISBN
978-0-333-78676-5
.
- Blume, Lawrence E. (2008b).
"Convex programming"
. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
The New Palgrave Dictionary of Economics
(Second ed.). Palgrave Macmillan. pp. 220?225.
doi
:
10.1057/9780230226203.0314
.
ISBN
978-0-333-78676-5
.
- Blume, Lawrence E. (2008c).
"Duality"
. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
The New Palgrave Dictionary of Economics
(Second ed.). Palgrave Macmillan. pp. 551?555.
doi
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10.1057/9780230226203.0411
.
ISBN
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.
- Crouzeix, J.-P. (2008).
"Quasi-concavity"
. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
The New Palgrave Dictionary of Economics
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.
- Diewert, W. E. (1982). "12 Duality approaches to microeconomic theory". In
Arrow, Kenneth Joseph
; Intriligator, Michael D (eds.).
Handbook of mathematical economics, Volume
II
. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 535?599.
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.
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MR
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.
- Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In
Arrow, Kenneth Joseph
; Intriligator, Michael D (eds.).
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I
. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 15?52.
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.
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MR
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.
- Luenberger, David G.
Microeconomic Theory
, McGraw-Hill, Inc., New York, 1995.
- Mas-Colell, A.
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"Non?convexity"
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. In Eatwell, John; Milgate, Murray;
Newman, Peter
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(first ed.). Palgrave Macmillan. pp. 653?661.
doi
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.
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.
- Newman, Peter
(1987c).
"Convexity"
. In Eatwell, John; Milgate, Murray;
Newman, Peter
(eds.).
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(first ed.). Palgrave Macmillan. p. 1.
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.
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.
- Newman, Peter
(1987d).
"Duality"
. In Eatwell, John; Milgate, Murray;
Newman, Peter
(eds.).
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(first ed.). Palgrave Macmillan. p. 1.
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.
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.
- Rockafellar, R. Tyrrell
(1997).
Convex analysis
. Princeton landmarks in mathematics (Reprint of the 1979 Princeton mathematical series
28
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.
MR
0274683
.
.
- Schneider, Rolf (1993).
Convex bodies: The Brunn?Minkowski theory
. Encyclopedia of mathematics and its applications. Vol. 44. Cambridge: Cambridge University Press. pp. xiv+490.
doi
:
10.1017/CBO9780511526282
.
ISBN
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.
MR
1216521
.
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