경제학의 볼록성

경제학의 볼록성은 JEL 분류 코드에 JEL: C65로 포함되어 있다.

다음에 대한 시리즈 일부
경제학
역사 개요 색인
분기와 분류 경제학부 주류 경제학 헤테로독스 경제학 경제 방법론 경제이론 정치 경제, 경제학 미시경제학 거시경제학 국제경제학 응용경제학 수리경제학 계량학 JEL 분류 코드
개념, 이론 및 기술 경제 시스템 경제 성장 시장 국가회계 실험경제학 계산경제학 게임 이론 운영연구 중간소득트랩 산업단지
적용별 농업 행동 비즈니스 문화 인구통계학 개발 디지털화 생태학 경제 지리 경제사 경제기획 경제 정책 경제사회학 경제통계 교육 공학 환경 진화론 탐험가 페미니스트 금융 행복경제학 건강 인문경제학 산업조직 정보 기관 지식 노동 법 관리 통화 천연자원 조직의 인원 공공경제학 공개/사회적 선택 지역 농촌 서비스 사회경제학 연대경제 어반 복지 복지경제학
저명한 경제학자 프랑수아 퀘스나이 애덤 스미스 데이비드 리카르도 토머스 로버트 맬서스 존 스튜어트 밀 윌리엄 스탠리 제본스 레온 왈라스 알프레드 마셜 어빙 피셔 존 메이너드 케인스 아서 세실 피구 존 힉스 바실리 레온티프 폴 새뮤얼슨 더 많은
경제평론 정치경제 비판 캔 따개기 가정
경제에 대한 저명한 비평가들 카를 마르크스 존 러스킨 장 보드리야르 더 많은
목록 용어집 이코노미스트 간행물(저널)
비즈니스 및 경제 포털 머니포털
v t

볼록함은 경제학에서 중요한 주제다.^[1] 일반적 경제 평형화의 화살표-데브루 모델에서 대리인은 볼록한 예산 집합과 볼록한 선호도를 가진다. 평형가격에서 예산초과기는 달성 가능한 최고의 무관심 곡선을 지원한다.^[2] 이익 함수는 비용 함수의 볼록 결합이다.^[1][2] 볼록스 분석은 교과서 경제학을 분석하는 표준 도구다.^[1] 경제학에서 비-콘벡스 현상은 볼록한 분석을 일반화하는 비원활한 분석으로 연구되어 왔다.^[3]

Preliminaries

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The economics depends upon the following definitions and results from convex geometry.

Real vector spaces

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

(x₁, y₁) + (x₂, y₂) = (x₁+x₂, y₁+y₂);

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 { (v₁, v₂, . . . , 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

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 v₀ and v₁ in Q and for every real number λ in the unit interval [0,1], the point

(1 − λ) v₀ + λv₁

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 {v₀, v₁, . . . , v_D} of a vector space is any weighted average λ₀v₀ + λ₁v₁ + . . . + λ_Dv_D, for some indexed set of non‑negative real numbers {λ_d} satisfying the equation λ₀ + λ₁ + . . . + 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

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

Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set ${\displaystyle S}$ in the real n-space ${\displaystyle \mathbb {R} ^{n}}$ if it meets both of the following:

• ${\displaystyle S}$ is entirely contained in one of the two closed half-spaces determined by the hyperplane
• ${\displaystyle S}$ has at least one point on the hyperplane.

Here, a closed half-space is the half-space that includes the hyperplane.

Supporting hyperplane theorem

This theorem states that if ${\displaystyle S}$ is a closed convex set in ${\displaystyle \mathbb {R} ^{n},}$ and ${\displaystyle x}$ is a point on the boundary of ${\displaystyle S,}$ then there exists a supporting hyperplane containing ${\displaystyle x.}$

The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set ${\displaystyle S}$ is not convex, the statement of the theorem is not true at all points on the boundary of ${\displaystyle S,}$ as illustrated in the third picture on the right.

Economics

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

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,^[4] of market failures,^[5] and of public economics.^[6] These results are described in graduate-level textbooks in microeconomics,^[7] general equilibrium theory,^[8] game theory,^[9] mathematical economics,^[10] and applied mathematics (for economists).^[11] 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.^[12]

In "oligopolies" (markets dominated by a few producers), especially in "monopolies" (markets dominated by one producer), non‑convexities remain important.^[13] 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,^[14] after which Harold Hotelling wrote about marginal cost pricing in 1938.^[15] Both Sraffa and Hotelling illuminated the market power of producers without competitors, clearly stimulating a literature on the supply-side of the economy.^[16] Non‑convex sets arise also with environmental goods (and other externalities),^[17][18] with information economics,^[19] and with stock markets^[13] (and other incomplete markets).^[20][21] Such applications continued to motivate economists to study non‑convex sets.^[22]

Nonsmooth analysis

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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)^[23] and Mordukhovich (2006),^[24] according to Khan (2008).^[3] Brown (1995, pp. 1967–1968) harvtxt error: no target: CITEREFBrown1995 (help) 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 (1995, p. 1966) harvtxt error: no target: CITEREFBrown1995 (help), "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..^[25] Economists have also used algebraic topology.^[26]

See also

• Convex duality

Notes

References

• 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:10.1057/9780230226203.0411. ISBN 978-0-333-78676-5.
• Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E (eds.). The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 815–816. doi:10.1057/9780230226203.1375. ISBN 978-0-333-78676-5.
• 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. 1. Amsterdam: North-Holland Publishing Co. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
• Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D (eds.). Handbook of mathematical economics, Volume I. Handbooks in economics. 1. Amsterdam: North-Holland Publishing Co. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 978-0-444-86126-9. MR 0634800.
• Luenberger, David G. Microeconomic Theory, McGraw-Hill, Inc., New York, 1995.
• 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.
• Newman, Peter (1987c). "Convexity". In Eatwell, John; Milgate, Murray; Newman, Peter (eds.). The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. p. 1. doi:10.1057/9780230226203.2282. ISBN 9780333786765.
• Newman, Peter (1987d). "Duality". In Eatwell, John; Milgate, Murray; Newman, Peter (eds.). The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. p. 1. doi:10.1057/9780230226203.2412. ISBN 9780333786765.
• Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton landmarks in mathematics (Reprint of the 1979 Princeton mathematical series 28 ed.). Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6. MR 0274683..
• Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of mathematics and its applications. 44. Cambridge: Cambridge University Press. pp. xiv+490. doi:10.1017/CBO9780511526282. ISBN 978-0-521-35220-8. MR 1216521.