Harold Hotelling | |
|---|---|
 | |
| Born | (1895-09-29)September 29, 1895 Fulda, Minnesota, U.S. |
| Died | December 26, 1973(1973-12-26) (aged 78) |
| Alma mater | University of Washington (BA,MA) Princeton University (PhD) |
| Known for | Hotelling's T-square distribution Canonical correlation analysis Hotelling's law Hotelling's lemma Hotelling's rule Hotelling's location model Working–Hotelling procedure |
| Awards | North Carolina Award 1972 |
| Scientific career | |
| Fields | Statistics Economics |
| Institutions | Univ. of North Carolina 1946–1973 Columbia University 1931–1946 Stanford University 1927–31 |
| Doctoral advisor | Oswald Veblen |
| Doctoral students | Kenneth Arrow Seymour Geisser Ralph A. Bradley |
Harold Hotelling (/ˈhoʊtəlɪŋ/; September 29, 1895 – December 26, 1973) was an Americanmathematicalstatistician and an influentialeconomic theorist, known forHotelling's law,Hotelling's lemma, andHotelling's rule in economics, as well asHotelling's T-squared distribution in statistics.[1][2] He also developed and named theprincipal component analysis method widely used in finance, statistics and computer science.
He was associate professor of mathematics atStanford University from 1927 until 1931, a member of the faculty ofColumbia University from 1931 until 1946, and a professor of Mathematical Statistics at theUniversity of North Carolina at Chapel Hill from 1946 until his death. A street inChapel Hill bears his name. In 1972, he received theNorth Carolina Award for contributions to science.
Hotelling is known to statisticians because ofHotelling's T-squared distribution which is a generalization of theStudent's t-distribution in multivariate setting, and its use in statisticalhypothesis testing and confidence regions. He also introducedcanonical correlation analysis.
At the beginning of his statistical career Hotelling came under the influence ofR.A. Fisher, whoseStatistical 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" forThiele's Danish "semi-invariants". Fisher's emphasis on the sampling distribution of a statistic was extended byJerzy Neyman andEgon Pearson with greater precision and wider applications, which Hotelling recognized. Hotelling sponsored refugees from European anti-semitism and Nazism, welcomingHenry Mann andAbraham Wald to his research group at Columbia. While at Hotelling's group, Wald developedsequential analysis andstatistical 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 atColumbia University and theUniversity of North Carolina.
Hotelling has a crucial place in the growth of mathematical economics; several areas of active research were influenced by his economics papers. While at theUniversity of Washington, he was encouraged to switch from pure mathematics toward mathematical economics by the famous mathematicianEric Temple Bell. Later, atColumbia University (where during 1933-34 he taughtMilton Friedman statistics) in the '40s, Hotelling in turn encouraged youngKenneth Arrow to switch from mathematics and statistics applied to actuarial studies towards more general applications of mathematics in general economic theory. Hotelling is theeponym ofHotelling's law,Hotelling's lemma, andHotelling's rule ineconomics.
Hotelling was influenced by the writing ofHenry George and was an editorial adviser for theGeorgist journalAJES.[3]
One of Hotelling's most important contributions to economics was his conception of "spatial economics" in his 1929 article.[4] Space was not just a barrier to moving goods around, but rather a field upon which competitors jostled to be nearest to their customers.[5]
Hotelling considers a situation in which there are two sellers at point A and B in aline segment of size l. The buyers aredistributed 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 andb the portion of segment to the right of B. Therefore, a+x+y+b=l. Since the product being sold is acommodity, the point of indifference to buying is given by p1+cx=p2+cy. Solving for x and y yields:
Let q1 and q2 indicate the quantities sold by A and B. The sellers profit are:
By imposingprofit maximization:
Hotelling obtains theeconomic equilibrium. Hotelling argues this equilibrium isstable even though the sellers may try to establish a pricecartel.
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."[4]
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 aGeorgistland 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 theHenry George theorem thatJoseph 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 Georgistcongestion pricing and stated that the purpose of this unique type oftoll 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.[6]Kenneth Arrow described this asmarket socialism, butMason Gaffney points out that it is actually Georgism.[7] Hotelling added the following comment about the ethics of Georgistvalue 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 [the Georgist]H. G. Brown."[6]
Hotelling made pioneering studies ofnon-convexity in economics. Ineconomics,non-convexity refers to violations of theconvexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers withconvex preferences and convexbudget sets and on producers with convexproduction sets; for convex models, the predicted economic behavior is well understood.[8][9] When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated withmarket failures,[10][11] wheresupply and demand differ or wheremarket equilibria can beinefficient.[8][11][12][13][14][15]
In "oligopolies" (markets dominated by a few producers), especially in "monopolies" (markets dominated by one producer), non-convexities remain important.[15] Concerns with large producers exploiting market power initiated the literature on non-convex sets, whenPiero Sraffa wrote about firms with increasingreturns to scale in 1926,[16] after which Hotelling wrote aboutmarginal cost pricing in 1938.[17] Both Sraffa and Hotelling illuminated themarket power of producers without competitors, clearly stimulating a literature on the supply-side of the economy.[18]
When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is notconnected. 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.[19][20]
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 withmarket failures, where anyequilibrium need not beefficient or where no equilibrium exists becausesupply and demand differ.[8][11][12][13][14][15] Non-convex sets arise also withenvironmental goods and otherexternalities,[13][14] and withmarket failures,[10] andpublic economics.[12][21]Non-convexities occur also withinformation economics,[22] and withstock markets[15] (and otherincomplete markets).[23][24] Such applications continued to motivate economists to study non-convex sets.[8]
nonconvex OR nonconvexities.
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