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Harold Hotelling (; September 29, 1895 – December 26, 1973) was a mathematical statistician and an influential economic 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.^{[1]}
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.
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 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.
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 Bell. 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.
One of Hotelling’s most important contributions to economics was his conception of “spatial economics” in his 1929 article.^{[2]} Space was not just a barrier to moving goods around, but rather a field upon which competitors jostled to be nearest to their customers.^{[3]}
Hotteling considers a situation in which there are two sellers at point A and B in a line segment 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:
x=\frac{1}{2}\left( l-a-b+\frac{p_{2}-p_{1}}{c} \right)
y=\frac{1}{2}\left( l-a-b+\frac{p_{1}-p_{2}}{c} \right)
Let q_{1} and q_{2} indicate the quantities sold by A and B. The sellers profit are:
\pi_{1}=p_{1}q_{1}=p_{1}\left( a+x \right)=\frac{1}{2}\left( l+a-b \right)p_{1}-\frac{p_{1}^{2}}{2c}+\frac{p_{1}p_{2}}{2c}
\pi_{2}=p_{2}q_{2}=p_{2}\left( b+y \right)=\frac{1}{2}\left( l-a+b \right)p_{2}-\frac{p_{2}^{2}}{2c}+\frac{p_{1}p_{2}}{2c}
By imposing profit maximization:
\frac{\partial \pi}{\partial p_{1}}=\frac{1}{2}\left( l+a-b \right)-\frac{p_{1}}{c}+\frac{p_{2}}{2c}=0
\frac{\partial \pi}{\partial p_{2}}=\frac{1}{2}\left( l+a-b \right)-\frac{p_{1}}{2c}+\frac{p_{2}}{c}=0
Hotteling obtains the economic equilibrium. Hotteling argues this equilibrium is stable even though the sellers may try to establish a price cartel.
Hotelling made pioneering studies of non-convexity 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 sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood.^{[4]}^{[5]} 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,^{[6]}^{[7]} where supply and demand differ or where market equilibria can be inefficient.^{[4]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}
In "oligopolies" (markets dominated by a few producers), especially in "monopolies" (markets dominated by one producer), non-convexities remain important.^{[11]} 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,^{[12]} after which Hotelling wrote about marginal cost pricing in 1938.^{[13]} Both Sraffa and Hotelling illuminated the market power of producers without competitors, clearly stimulating a literature on the supply-side of the economy.^{[14]}
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:
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.^{[15]}
according to Diewert.^{[16]}
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.^{[4]}^{[7]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]} Non-convex sets arise also with environmental goods (and other externalities),^{[9]}^{[10]} and with market failures,^{[6]} and public economics.^{[8]}^{[17]} Non-convexities occur also with information economics,^{[18]} and with stock markets^{[11]} (and other incomplete markets).^{[19]}^{[20]} Such applications continued to motivate economists to study non-convex sets.^{[4]}
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