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 p
1 and p
2 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 p
1+cx=p
2+cy. Solving for x and y yields:
:
:
Let q
1 and q
2 indicate the quantities sold by A and B. The sellers profit are:
:
:
By imposing
profit maximization:
:
:
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.
[
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Works
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Harold Hotelling's review of Fishers' ''Statistical methods for research workers''.*
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Papers
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See also
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Cournot competition
References
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* 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
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Harold Hotelling
The following have photographs:
Harold Hotellingon th
page.
National Academy of Sciences Biographical Memoir
{{DEFAULTSORT:Hotelling, Harold
Presidents of the Institute of Mathematical Statistics
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1973 deaths
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