Non-convexity (economics)
Encyclopedia
In economics
, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences
(that do not prefer extremes to in-between values) and convex budget set
s and on producers with convex production set
s; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failure
s, where supply and demand
differ or where market equilibria can be inefficient
. Non-convex economies are studied with nonsmooth analysis
, which is a generalization of convex analysis
.
)! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.
When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connected
; A disconnected demand implies some discontinuous behavior by the consumer, as discussed by Harold Hotelling
:
The difficulties of studying non-convex preferences were emphasized by Herman Wold
and again by Paul Samuelson
, who wrote that non-convexities are "shrouded in eternal according to Diewert.
When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failure
s, where supply and demand
differ or where market equilibria can be inefficient
.
Non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in The Journal of Political Economy (JPE). The main contributors were Farrell, Bator, Koopmans
, and Rothenberg. In particular, Rothenberg's paper discussed the approximate convexity of sums of non-convex sets. These JPE-papers stimulated a paper by Lloyd Shapley
and Martin Shubik
, which considered convexified consumer-preferences and introduced the concept of an "approximate equilibrium". The JPE-papers and the Shapley–Shubik paper influenced another notion of "quasi-equilibria", due to Robert Aumann
.
Non-convex sets have been incorporated in the theories of general economic equilibria,. These results are described in graduate-level textbooks in microeconomics
, general equilibrium theory, game theory
, mathematical economics
,
and applied mathematics (for economists). The Shapley–Folkman lemma
establishes that non-convexities are compatible with approximate equilibria in markets with many consumers; these results also apply to production economies
with many small firm
s.
and especially monopolies
. Concerns with large producers exploiting market power initiated the literature on non-convex sets, when Piero Sraffa
wrote about on firms with increasing returns to scale
in 1926, after which Harold Hotelling
wrote about marginal cost
pricing in 1938. Both Sraffa and Hotelling illuminated the market power
of producers without competitors, clearly stimulating a literature on the supply-side of the economy.
s, where equilibria
need not be efficient
or where no equilibrium exists because supply and demand
differ. Non-convex sets arise also with environmental goods
(and other externalities
), and with market failures, and public economics.
Non-convexities occur also with information economics
, and with stock market
s (and other incomplete markets
). Such applications continued to motivate economists to study non-convex sets.
s, where points represent commodity bundles. However, economists also consider dynamic problems of optimization over time, using the theories of differential equation
s, dynamic systems, stochastic process
es, and functional analysis
: Economists use the following optimization methods:
In these theories, regular problems involve convex functions defined on convex domains, and this convexity allows simplifications of techniques and economic meaningful interpretations of the results. In economics, dynamic programing was used by Martin Beckmann and Richard F. Muth for work on inventory theory
and consumption theory. Robert C. Merton used dynamic programming in his 1973 article on the intertemporal capital asset pricing model. (See also Merton's portfolio problem
). In Merton's model, investors chose between income today and future income or capital gains, and their solution is found via dynamic programming. Stokey, Lucas & Prescott use dynamic programming to solve problems in economic theory, problems involving stochastic processes. Dynamic programming has been used in optimal economic growth
, resource extraction
, principal–agent problems, public finance
, business investment
, asset pricing, factor supply, and industrial organization
. Ljungqvist & Sargent apply dynamic programming to study a variety of theoretical questions in monetary policy
, fiscal policy
, taxation, economic growth, search theory
, and labor economics. Dixit & Pindyck used dynamic programming for capital budgeting
. For dynamic problems, non-convexities also are associated with market failures, just as they are for fixed-time problems.
, which generalizes convex analysis. Convex analysis centers on convex sets and convex functions, for which it provides powerful ideas and clear results, but it is not adequate for the analysis of non-convexities, such as increasing returns to scale. "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, Clarke's differential calculus
for Lipschitz continuous functions
, which uses Rademacher's theorem
and which is described by and , according to . 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 , "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.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences
Convex preferences
In economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes"...
(that do not prefer extremes to in-between values) and convex budget set
Budget set
A budget set or opportunity set includes all possible consumption bundles that someone can afford given the prices of goods and the person's income level...
s and on producers with convex production set
Production set
A production set is the set of all possible output bundles that a firm can produce given its vector of inputs. Used as part of profit maximization calculations....
s; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failure
Market failure
Market failure is a concept within economic theory wherein the allocation of goods and services by a free market is not efficient. That is, there exists another conceivable outcome where a market participant may be made better-off without making someone else worse-off...
s, where supply and demand
Supply and demand
Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...
differ or where market equilibria can be inefficient
Pareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
. Non-convex economies are studied with nonsmooth analysis
Subderivative
In mathematics, the concepts of subderivative, subgradient, and subdifferential arise in convex analysis, that is, in the study of convex functions, often in connection to convex optimization....
, which is a generalization of convex analysis
Convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....
.
Demand with many consumers
If a preference set is non-convex, then some prices determine a budget-line that supports two separate optimal-baskets. 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 half of an eagle and half of a lion (or a griffinGriffin
The griffin, griffon, or gryphon is a legendary creature with the body of a lion and the head and wings of an eagle...
)! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.
When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
; A disconnected demand implies some discontinuous behavior by the consumer, as discussed by Harold Hotelling
Harold Hotelling
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...
:
If indifference curves for purchases be thought of as possessing a wavy character, convex to the origin in some regions and concave in others, we are forced to the conclusion that it is only the portions convex to the origin that can be regarded as possessing any importance, since the others are essentially unobservable. They can be detected only by the discontinuities that may occur in demand with variation in price-ratios, leading to an abrupt jumping of a point of tangency across a chasm when the straight line is rotated. But, while such discontinuities may reveal the existence of chasms, they can never measure their depth. The concave portions of the indifference curves and their many-dimensional generalizations, if they exist, must forever remain in
unmeasurable obscurity.
The difficulties of studying non-convex preferences were emphasized by Herman Wold
Herman Wold
Herman Ole Andreas Wold was a Norwegian-born econometrician and statistician who had a long career in Sweden...
and again by Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...
, who wrote that non-convexities are "shrouded in eternal according to Diewert.
When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failure
Market failure
Market failure is a concept within economic theory wherein the allocation of goods and services by a free market is not efficient. That is, there exists another conceivable outcome where a market participant may be made better-off without making someone else worse-off...
s, where supply and demand
Supply and demand
Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...
differ or where market equilibria can be inefficient
Pareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
.
Non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in The Journal of Political Economy (JPE). The main contributors were Farrell, Bator, Koopmans
Tjalling Koopmans
Tjalling Charles Koopmans was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences....
, and Rothenberg. In particular, Rothenberg's paper discussed the approximate convexity of sums of non-convex sets. These JPE-papers stimulated a paper by Lloyd Shapley
Lloyd Shapley
Lloyd Stowell Shapley is a distinguished American mathematician and economist. He is a Professor Emeritus at University of California, Los Angeles, affiliated with departments of Mathematics and Economics...
and Martin Shubik
Martin Shubik
Martin Shubik is an American economist, who is Professor Emeritus of Mathematical Institutional Economics at Yale University. He was educated at the University of Toronto and Princeton University...
, which considered convexified consumer-preferences and introduced the concept of an "approximate equilibrium". The JPE-papers and the Shapley–Shubik paper influenced another notion of "quasi-equilibria", due to Robert Aumann
Robert Aumann
Robert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
.
Non-convex sets have been incorporated in the theories of general economic equilibria,. These results are described in graduate-level textbooks in microeconomics
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...
, general equilibrium theory, game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, mathematical economics
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...
,
and applied mathematics (for economists). The Shapley–Folkman lemma
Shapley–Folkman lemma
In geometry and economics, the Shapley–Folkman lemma describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of...
establishes that non-convexities are compatible with approximate equilibria in markets with many consumers; these results also apply to production economies
Production (economics)
In economics, production is the act of creating 'use' value or 'utility' that can satisfy a want or need. The act may or may not include factors of production other than labor...
with many small firm
Business
A business is an organization engaged in the trade of goods, services, or both to consumers. Businesses are predominant in capitalist economies, where most of them are privately owned and administered to earn profit to increase the wealth of their owners. Businesses may also be not-for-profit...
s.
Supply with few producers
Non-convexity is important under oligopoliesOligopoly
An oligopoly is a market form in which a market or industry is dominated by a small number of sellers . The word is derived, by analogy with "monopoly", from the Greek ὀλίγοι "few" + πόλειν "to sell". Because there are few sellers, each oligopolist is likely to be aware of the actions of the others...
and especially monopolies
Monopoly
A monopoly exists when a specific person or enterprise is the only supplier of a particular commodity...
. Concerns with large producers exploiting market power initiated the literature on non-convex sets, when Piero Sraffa
Piero Sraffa
Piero Sraffa was an influential Italian economist whose book Production of Commodities by Means of Commodities is taken as founding the Neo-Ricardian school of Economics.- Early life :...
wrote about on firms with increasing returns to scale
Returns to scale
In economics, returns to scale and economies of scale are related terms that describe what happens as the scale of production increases in the long run, when all input levels including physical capital usage are variable...
in 1926, after which Harold Hotelling
Harold Hotelling
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...
wrote about marginal cost
Marginal cost
In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. That is, it is the cost of producing one more unit of a good...
pricing in 1938. Both Sraffa and Hotelling illuminated the market power
Market power
In economics, market power is the ability of a firm to alter the market price of a good or service. In perfectly competitive markets, market participants have no market power. A firm with market power can raise prices without losing its customers to competitors...
of producers without competitors, clearly stimulating a literature on the supply-side of the economy.
Contemporary economics
Recent research in economics has recognized non-convexity in new areas of economics. In these areas, non-convexity is associated with market failureMarket failure
Market failure is a concept within economic theory wherein the allocation of goods and services by a free market is not efficient. That is, there exists another conceivable outcome where a market participant may be made better-off without making someone else worse-off...
s, where equilibria
Economic equilibrium
In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...
need not be efficient
Pareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
or where no equilibrium exists because supply and demand
Supply and demand
Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...
differ. Non-convex sets arise also with environmental goods
Environmental economics
Environmental economics is a subfield of economics concerned with environmental issues. Quoting from the National Bureau of Economic Research Environmental Economics program:...
(and other externalities
Externality
In economics, an externality is a cost or benefit, not transmitted through prices, incurred by a party who did not agree to the action causing the cost or benefit...
), and with market failures, and public economics.
Non-convexities occur also with information economics
Information economics
Information economics or the economics of informationis a branch of microeconomic theory that studies how information affects an economy and economic decisions. Information has special characteristics. It is easy to create but hard to trust. It is easy to spread but hard to control. It...
, and with stock market
Stock market
A stock market or equity market is a public entity for the trading of company stock and derivatives at an agreed price; these are securities listed on a stock exchange as well as those only traded privately.The size of the world stock market was estimated at about $36.6 trillion...
s (and other incomplete markets
Incomplete markets
In economics, incomplete markets refers to markets in which the number of Arrow–Debreu securities is less than the number of states of nature...
). Such applications continued to motivate economists to study non-convex sets.
Optimization over time
The previously mentioned applications concern non-convexities in finite-dimensional vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, where points represent commodity bundles. However, economists also consider dynamic problems of optimization over time, using the theories of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, dynamic systems, stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es, and functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
: Economists use the following optimization methods:
- calculus of variationsCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, following Frank P. RamseyFrank P. RamseyFrank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...
and Harold HotellingHarold HotellingHarold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...
; - dynamic programmingDynamic programmingIn mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure...
, following Richard BellmanRichard BellmanRichard Ernest Bellman was an American applied mathematician, celebrated for his invention of dynamic programming in 1953, and important contributions in other fields of mathematics.-Biography:...
and Ronald Howard; and - control theoryControl theoryControl theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
.
In these theories, regular problems involve convex functions defined on convex domains, and this convexity allows simplifications of techniques and economic meaningful interpretations of the results. In economics, dynamic programing was used by Martin Beckmann and Richard F. Muth for work on inventory theory
Inventory theory
Inventory theory is the sub-specialty within operations research that is concerned with the design of production/inventory systems to minimize costs...
and consumption theory. Robert C. Merton used dynamic programming in his 1973 article on the intertemporal capital asset pricing model. (See also Merton's portfolio problem
Merton's portfolio problem
Merton's Portfolio Problem is a well known problem in continuous-time finance. An investor with a finite lifetime must choose how much to consume and must allocate his wealth between stocks and a risk-free asset so as to maximize expected lifetime utility. The problem was formulated and solved by...
). In Merton's model, investors chose between income today and future income or capital gains, and their solution is found via dynamic programming. Stokey, Lucas & Prescott use dynamic programming to solve problems in economic theory, problems involving stochastic processes. Dynamic programming has been used in optimal economic growth
Economic growth
In economics, economic growth is defined as the increasing capacity of the economy to satisfy the wants of goods and services of the members of society. Economic growth is enabled by increases in productivity, which lowers the inputs for a given amount of output. Lowered costs increase demand...
, resource extraction
Resource extraction
The related terms natural resource extraction both refer to the practice of locating, acquiring and selling natural resources....
, principal–agent problems, public finance
Public finance
Public finance is the revenue and expenditure of public authoritiesThe purview of public finance is considered to be threefold: governmental effects on efficient allocation of resources, distribution of income, and macroeconomic stabilization.-Overview:The proper role of government provides a...
, business investment
Investment
Investment has different meanings in finance and economics. Finance investment is putting money into something with the expectation of gain, that upon thorough analysis, has a high degree of security for the principal amount, as well as security of return, within an expected period of time...
, asset pricing, factor supply, and industrial organization
Industrial organization
Industrial organization is the field of economics that builds on the theory of the firm in examining the structure of, and boundaries between, firms and markets....
. Ljungqvist & Sargent apply dynamic programming to study a variety of theoretical questions in monetary policy
Monetary policy
Monetary policy is the process by which the monetary authority of a country controls the supply of money, often targeting a rate of interest for the purpose of promoting economic growth and stability. The official goals usually include relatively stable prices and low unemployment...
, fiscal policy
Fiscal policy
In economics and political science, fiscal policy is the use of government expenditure and revenue collection to influence the economy....
, taxation, economic growth, search theory
Search theory
In microeconomics, search theory studies buyers or sellers who cannot instantly find a trading partner, and must therefore search for a partner prior to transacting....
, and labor economics. Dixit & Pindyck used dynamic programming for capital budgeting
Capital budgeting
Capital budgeting is the planning process used to determine whether an organization's long term investments such as new machinery, replacement machinery, new plants, new products, and research development projects are worth pursuing...
. For dynamic problems, non-convexities also are associated with market failures, just as they are for fixed-time problems.
Nonsmooth analysis
Economists have increasingly studied non-convex sets with nonsmooth analysisSubderivative
In mathematics, the concepts of subderivative, subgradient, and subdifferential arise in convex analysis, that is, in the study of convex functions, often in connection to convex optimization....
, which generalizes convex analysis. Convex analysis centers on convex sets and convex functions, for which it provides powerful ideas and clear results, but it is not adequate for the analysis of non-convexities, such as increasing returns to scale. "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, Clarke's differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
for Lipschitz continuous functions
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
, which uses Rademacher's theorem
Rademacher's theorem
In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn andis Lipschitz continuous, then f is Fréchet-differentiable almost everywhere in U In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the...
and which is described by and , according to . 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 , "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.