Harold Hotelling
Encyclopedia
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 University of North Carolina at Chapel Hill
from 1946 until his death. A street in Chapel Hill
bears his name. In 1972 he received the North Carolina Award
for contributions to science.
and its use in statistical hypothesis testing
and confidence regions. He also introduced canonical correlation analysis.
At the beginning of his statistical career Hotelling came under the influence of R.A. Fisher
, whose Statistical Methods for Research Workers
had "revolutionary importance", according to Hotelling's review. Hotelling was able to maintain professional relations with Fisher, despite the latter's temper tantrums and polemics. Hotelling suggested that Fisher use the English word "cumulant
s" for the Thiele's Danish "semi-invariants". Fisher's emphasis on the sampling distribution of a statistic was extended by Jerzy Neyman
and Egon Pearson
with greater precision and wider applications, which Hotelling recognized. Hotelling sponsored refugees from European anti-semitism and Nazism, welcoming Henry Mann
and Abraham Wald
to his research group at Columbia. While at Hotelling's group, Wald developed sequential analysis
and statistical decision theory, which Hotelling described as "pragmatism in action".
In the United States, Harold Hotelling is known for his leadership of the statistics profession, in particular for his vision of a statistics department at a university, which convinced many universities to start statistics departments. Hotelling was known for his leadership of departments at Columbia University
and the University of North Carolina
.
. Later, at Columbia University
(where during 1933-34 he taught Milton Friedman
statistics) in the '40s, Hotelling in turn encouraged young Kenneth Arrow
to switch from mathematics and statistics applied to actuarial studies towards more general applications of mathematics in general economic theory. Hotelling is the eponym
of Hotelling's law
, Hotelling's lemma
, and Hotelling's rule
in economics
.
In economics
, non-convexity refers to violations of the convexity assumptions of elementary economics
. Basic economics textbooks concentrate on consumers with convex preferences
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
.
" (markets dominated by a few producers), especially in "monopolies
" (markets dominated by one producer), non-convexities remain important. Concerns with large producers exploiting market power initiated the literature on non-convex sets, when Piero Sraffa
wrote about firms with increasing returns to scale
in 1926, after which 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.
; A disconnected demand implies some discontinuous behavior by the consumer, as discussed by Hotelling:
according to Diewert.
Following Hotelling's pioneering research on non-convexities in economics, research in economics has recognized non-convexity in new areas of economics. In these areas, non-convexity is associated with market failure
s, where any equilibrium
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 failure
s, 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.
These entries have photographs. There is another at
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
statistician
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and an influential economic
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"...
theorist.
He was Associate Professor of Mathematics at Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
from 1927 until 1931, a member of the faculty of Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
from 1931 until 1946, and a Professor of Mathematical Statistics at the University of North Carolina at Chapel Hill
University of North Carolina at Chapel Hill
The University of North Carolina at Chapel Hill is a public research university located in Chapel Hill, North Carolina, United States...
from 1946 until his death. A street in Chapel Hill
Chapel Hill, North Carolina
Chapel Hill is a town in Orange County, North Carolina, United States and the home of the University of North Carolina at Chapel Hill and UNC Health Care...
bears his name. In 1972 he received the North Carolina Award
North Carolina Award
The North Carolina Award is the highest civilian award bestowed by the U.S. state of North Carolina. It is awarded in the four fields of science, literature, the fine arts, and public service....
for contributions to science.
Statistics
Hotelling is known to statisticians because of Hotelling's T-square distributionHotelling's T-square distribution
In statistics Hotelling's T-squared distribution is important because it arises as the distribution of a set of statistics which are natural generalisations of the statistics underlying Student's t distribution...
and its use in statistical hypothesis testing
Statistical hypothesis testing
A statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study . In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre-determined threshold...
and confidence regions. He also introduced canonical correlation analysis.
At the beginning of his statistical career Hotelling came under the influence of R.A. Fisher
Ronald Fisher
Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation...
, whose Statistical Methods for Research Workers
Statistical Methods for Research Workers
Statistical Methods for Research Workers is a classic 1925 book on statistics by the statistician R.A. Fisher. It is considered by some to be one of the 20th century's most influential books on statistical methods. According to ,...
had "revolutionary importance", according to Hotelling's review. Hotelling was able to maintain professional relations with Fisher, despite the latter's temper tantrums and polemics. Hotelling suggested that Fisher use the English word "cumulant
Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...
s" for the Thiele's Danish "semi-invariants". Fisher's emphasis on the sampling distribution of a statistic was extended by Jerzy Neyman
Jerzy Neyman
Jerzy Neyman , born Jerzy Spława-Neyman, was a Polish American mathematician and statistician who spent most of his professional career at the University of California, Berkeley.-Life and career:...
and Egon Pearson
Egon Pearson
Egon Sharpe Pearson, CBE FRS was the only son of Karl Pearson, and like his father, a leading British statistician....
with greater precision and wider applications, which Hotelling recognized. Hotelling sponsored refugees from European anti-semitism and Nazism, welcoming Henry Mann
Henry Mann
Henry Berthold Mann was a professor of mathematics and statistics at Ohio State University. Mann proved the Schnirelmann-Landau conjecture in number theory, and as a result earned the 1946 Cole Prize. He and his student developed the U-statistic of nonparametric statistics...
and Abraham Wald
Abraham Wald
- See also :* Sequential probability ratio test * Wald distribution* Wald–Wolfowitz runs test...
to his research group at Columbia. While at Hotelling's group, Wald developed sequential analysis
Sequential analysis
In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre-defined stopping rule as soon as significant results...
and statistical decision theory, which Hotelling described as "pragmatism in action".
In the United States, Harold Hotelling is known for his leadership of the statistics profession, in particular for his vision of a statistics department at a university, which convinced many universities to start statistics departments. Hotelling was known for his leadership of departments at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
and the University of North Carolina
University of North Carolina
Chartered in 1789, the University of North Carolina was one of the first public universities in the United States and the only one to graduate students in the eighteenth century...
.
Economics
Hotelling has a crucial place in the growth of mathematical economics; several areas of active research were influenced by his economics papers. While at the University of Washington, he was encouraged to switch from pure mathematics toward mathematical economics by the famous mathematician Eric Temple BellEric Temple Bell
Eric Temple Bell , was a mathematician and science fiction author born in Scotland who lived in the U.S. for most of his life...
. Later, at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
(where during 1933-34 he taught Milton Friedman
Milton Friedman
Milton Friedman was an American economist, statistician, academic, and author who taught at the University of Chicago for more than three decades...
statistics) in the '40s, Hotelling in turn encouraged young Kenneth Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
to switch from mathematics and statistics applied to actuarial studies towards more general applications of mathematics in general economic theory. Hotelling is the eponym
Eponym
An eponym is the name of a person or thing, whether real or fictitious, after which a particular place, tribe, era, discovery, or other item is named or thought to be named...
of Hotelling's law
Hotelling's law
Hotelling's law is an observation in economics that in many markets it is rational for producers to make their products as similar as possible. This is also referred to as the principle of minimum differentiation as well as Hotelling's "linear city model"...
, Hotelling's lemma
Hotelling's lemma
Hotelling's lemma is a result in microeconomics that relates the supply of a good to the profit of the good's producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm. The lemma is very simple, and can be stated:...
, and Hotelling's rule
Hotelling's rule
Hotelling's rule states that the most socially and economically profitable extraction path of a non-renewable resource is one along which the price of the resource, determined by the marginal net revenue from the sale of the resource, increases at the rate of interest...
in economics
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-convexities
Hotelling made pioneering studies of non-convexity in economics.In economics
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
Convexity in economics
Convexity is an important topic in economics. In the Arrow-Debreu model of general economic equilibrium, agents have convex budget sets and convex preferences: At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. The profit function is the convex conjugate...
. 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"...
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...
.
Producers with increasing returns to scale: Marginal cost pricing
In "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...
" (markets dominated by a few producers), especially in "monopolies
Monopoly
A monopoly exists when a specific person or enterprise is the only supplier of a particular commodity...
" (markets dominated by one producer), non-convexities remain important. 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 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 Hotelling wrote about marginal cost 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.
Consumers with non-convex preferences
When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connectedConnected 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 Hotelling:
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.
according to Diewert.
Following Hotelling's pioneering research on non-convexities in economics, research in economics has recognized non-convexity in new areas of economics. In these areas, 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 any equilibrium
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 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, 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.
Works
- "A General Mathematical Theory of Depreciation", 1925, Journal of ASA.
- "Differential Equations Subject to Error", 1927, Journal of ASA
- Review of R. A. Fisher's Statistical Methods for Rearch Workers,1927. Journal of ASA Harold Hotelling’s review of Fishers’ Statistical Methods.
- "Applications of the Theory of Error to the Interpretation of Trends", with H. Working, 1929, Journal of ASA.
- "Stability in Competition", 1929, EJ.
- "The Economics of Exhaustible Resources", 1931, JPE.
- "The Generalization of Student's Ratio", 1931, Annals of Mathematical Statistics.
- "Edgeworth's Taxation Paradox and the Nature of Supply and Demand Functions", 1932, JPE.
- "Analysis of a Complex of Statistical Variables with Principal Components",1933, Journal of Educational Psychology
- "Demand Functions with Limited Budgets", 1935, EconometricaEconometricaEconometrica is a peer-reviewed academic journal of economics, publishing articles not only in econometrics but in many areas of economics. It is published by the Econometric Society and distributed by Wiley-Blackwell. Econometrica is one of the most highly ranked economics journals in the world...
. - "The most predictable criterion", 1935, Journal of Educational Psychology
- "Relation Between Two Sets of Variates", 1936, Biometrika.
- "Rank Correlation and Tests of Significance Involving no Assumption of Normality", in "American Mathematical Statistics", 1936 (coauthor M. R. Pabst)
- "The General Welfare in Relation to Problems of Taxation and of Railway and Utility Rates", 1938, EconometricaEconometricaEconometrica is a peer-reviewed academic journal of economics, publishing articles not only in econometrics but in many areas of economics. It is published by the Econometric Society and distributed by Wiley-Blackwell. Econometrica is one of the most highly ranked economics journals in the world...
. - "A generalized T-Test and measure of multivariate dispersion", Proc. Second Berkeley Symposium of Mathematical Statistics and Probability, 1951
External links
These entries have photographs. There is another at
- Harold Hotelling on the Portraits of Statisticians page.
