Harold Hotelling
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Harold Hotelling (; September 29, 1895 – December 26, 1973) was an American
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
statistician A statistician is a person who works with Theory, theoretical or applied statistics. The profession exists in both the private sector, private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, a ...
and an influential
economic An economy is an area of the Production (economics), production, Distribution (economics), distribution and trade, as well as Consumption (economics), consumption of Goods (economics), goods and Service (economics), services. In general, it is ...
theorist, known for Hotelling's law, Hotelling's lemma, and Hotelling's rule in economics, as well as Hotelling's T-squared distribution in statistics. He also developed and named the principal component analysis method widely used in finance, statistics and computer science. He was associate professor of mathematics at
Stanford University Leland Stanford Junior University, commonly referred to as Stanford University, is a Private university, private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth ...
from 1927 until 1931, a member of the faculty of
Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...
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.


Statistics

Hotelling is known to statisticians because of Hotelling's T-squared distribution which is a generalization of the Student's t-distribution in multivariate setting, 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 " cumulants" for Thiele's Danish "semi-invariants". Fisher's emphasis on the sampling distribution of a statistic was extended by
Jerzy Neyman Jerzy Spława-Neyman (April 16, 1894 – August 5, 1981; ) was a Polish mathematician and statistician who first introduced the modern concept of a confidence interval into statistical hypothesis testing and, with Egon Pearson, revised Ronald Fis ...
and
Egon Pearson Egon Sharpe Pearson (11 August 1895 – 12 June 1980) was one of three children of Karl Pearson and Maria, née Sharpe, and, like his father, a British statistician. Career Pearson was educated at Winchester College and Trinity College ...
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 In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data is evaluated as it is collected, and further sampling is stopped in accordance with a pre-defi ...
and statistical decision theory, which Hotelling described as "pragmatism in action". In the United States, 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 in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...
and the University of North Carolina.


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 The University of Washington (UW and informally U-Dub or U Dub) is a public research university in Seattle, Washington, United States. Founded in 1861, the University of Washington is one of the oldest universities on the West Coast of the Uni ...
, he was encouraged to switch from pure mathematics toward mathematical economics by the famous mathematician Eric Temple Bell. Later, at
Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...
(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 Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
. Hotelling was influenced by the writing of Henry George and was an editorial adviser for the Georgist journal AJES.


Spatial economics

One of Hotelling's most important contributions to economics was his conception of " spatial economics" in his 1929 article. Space was not just a barrier to moving goods around, but rather a field upon which competitors jostled to be nearest to their customers. Hotelling considers a situation in which there are two sellers at point A and B in a
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
of size l. The buyers are distributed uniformly in this line segment and carry the merchandise to their home at cost c. Let p1 and p2 be the prices charged by A and B, and let the line segment be divided in 3 parts of size a, x+y and b, where x+y is the size of the segment between A and B, ''a'' the portion of segment to the left of A and ''b'' the portion of segment to the right of B. Therefore, a+x+y+b=l. Since the product being sold is a commodity, the point of indifference to buying is given by p1+cx=p2+cy. Solving for x and y yields: :x=\frac\left( l-a-b+\frac \right) :y=\frac\left( l-a-b+\frac \right) Let q1 and q2 indicate the quantities sold by A and B. The sellers profit are: :\pi_=p_q_=p_\left( a+x \right)=\frac\left( l+a-b \right)p_-\frac+\frac :\pi_=p_q_=p_\left( b+y \right)=\frac\left( l-a+b \right)p_-\frac+\frac By imposing profit maximization: :\frac=\frac\left( l+a-b \right)-\frac+\frac=0 :\frac=\frac\left( l+a-b \right)-\frac+\frac=0 Hotelling obtains the economic equilibrium. Hotelling argues this equilibrium is stable even though the sellers may try to establish a price cartel. Hotelling extrapolates from his findings about spatial economics and links it to not just physical distance, but also similarity in products. He describes how, for example, some factories might make shoes for the poor and others for the rich, but they end up alike. He also quips that, "Methodists and Presbyterian churches are too much alike; cider too homogenous."


Market socialism and Georgism

As an extension of his research in spatial economics, Hotelling realized that it would be possible and socially optimal to finance investment in public goods through a Georgist land value tax and then provide such goods and services to the public at marginal cost (in many cases for free). This is an early expression of the Henry George theorem that Joseph Stiglitz and others expanded upon. Hotelling pointed out that when local public goods like roads and trains become congested, users create an additional marginal cost of excluding others. Hotelling became an early advocate of Georgist congestion pricing and stated that the purpose of this unique type of toll fee was in no way to recoup investment costs, but was instead a way of changing behavior and compensating those who are excluded. Hotelling describes how human attention is also in limited supply at any given time and place, which produces a rental value; he concludes that billboards could be regulated or taxed on similar grounds as other scarcity rents. Hotelling reasoned that rent and taxation were analogous, the public and private versions of a similar thing. Therefore, the social optimum would be to put taxes directly on rent. Kenneth Arrow described this as market socialism, but Mason Gaffney points out that it is actually Georgism. Hotelling added the following comment about the ethics of Georgist value capture: "The proposition that there is no ethical objection to the confiscation of the site value of land by taxation, if and when the nonlandowning classes can get the power to do so, has been ably defended by he Georgist H. G. Brown."


Non-convexities

Hotelling made pioneering studies of non-convexity in economics. In
economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
, ''non-convexity'' refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences and convex budget sets and on producers with convex production sets; 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 failures, where supply and demand differ or where market equilibria can be inefficient.


Producers with increasing returns to scale: marginal cost pricing

In " oligopolies" (markets dominated by a few producers), especially in " monopolies" (markets dominated by one producer), non-convexities remain important.Page 1: () 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.


Consumers with non-convex preferences

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 Hotelling: 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 failures, 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,Pages 106, 110–137, 172, and 248: and with market failures, and public economics.Pages 63–65: Starrett discusses non-convexities in his textbook on public economics (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): Non-convexities occur also with information economics, and with stock markets (and other incomplete markets).Page 371: Such applications continued to motivate economists to study non-convex sets.


Works

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Harold Hotelling's review of Fishers' ''Statistical methods for research workers''.
* * * * * * * * * * * * * * * * * *


Papers

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See also

* Cournot competition


References

* * I. Olkina and A. R. Sampsonb (2001). "Hotelling, Harold (1895–1973)," '' International Encyclopedia of the Social & Behavioral Sciences'', pp. 6921–6925
Abstract.


External links

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American Statistical Association: Harold Hotelling
* Harold Hotelling The following have photographs:
Harold Hotelling
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page.
National Academy of Sciences Biographical Memoir
{{DEFAULTSORT:Hotelling, Harold Presidents of the Institute of Mathematical Statistics Fellows of the American Statistical Association 20th-century American social scientists Psychometricians Statistics educators 20th-century American economists American social democrats 1895 births 1973 deaths University of Washington alumni Fellows of the Econometric Society Presidents of the Econometric Society Mathematical economists American operations researchers American econometricians Environmental economists Regional economists Stanford University Department of Mathematics faculty Columbia University faculty University of North Carolina at Chapel Hill faculty American United Methodists People from Fulda, Minnesota Distinguished fellows of the American Economic Association Economists from Minnesota 20th-century Methodists Georgist economists American mathematical statisticians