# Harold Hotelling

Harold Hotelling
Born September 29, 1895
Fulda, Minnesota, U.S.
Died December 26, 1973 (aged 78)
Chapel Hill, North Carolina, U.S.
Nationality United States
Fields Statistics
Economics
Institutions Univ. of North Carolina 1946–73
Columbia University 1931–46
Stanford University 1927–31
Alma mater Princeton University PhD 1924
University of Washington BA 1919, MA 1921
Doctoral students Kenneth Arrow
Seymour Geisser
Known for Hotelling's T-square distribution
Canonical correlation analysis
Hotelling's law
Hotelling's lemma
Hotelling's rule
Influenced Kenneth Arrow, Seymour Geisser, Milton Friedman
Notable awards North Carolina Award 1972

Harold Hotelling (/ˈhtəlɪŋ/; 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.

## 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 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.

## 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 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.

### Spatial 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 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:

${\displaystyle x={\frac {1}{2}}\left(l-a-b+{\frac {p_{2}-p_{1}}{c}}\right)}$

${\displaystyle y={\frac {1}{2}}\left(l-a-b+{\frac {p_{1}-p_{2}}{c}}\right)}$

Let q1 and q2 indicate the quantities sold by A and B. The sellers profit are:

${\displaystyle \pi _{1}=p_{1}q_{1}=p_{1}\left(a+x\right)={\frac {1}{2}}\left(l+a-b\right)p_{1}-{\frac {p_{1}^{2}}{2c}}+{\frac {p_{1}p_{2}}{2c}}}$

${\displaystyle \pi _{2}=p_{2}q_{2}=p_{2}\left(b+y\right)={\frac {1}{2}}\left(l-a+b\right)p_{2}-{\frac {p_{2}^{2}}{2c}}+{\frac {p_{1}p_{2}}{2c}}}$

By imposing profit maximization:

${\displaystyle {\frac {\partial \pi }{\partial p_{1}}}={\frac {1}{2}}\left(l+a-b\right)-{\frac {p_{1}}{c}}+{\frac {p_{2}}{2c}}=0}$

${\displaystyle {\frac {\partial \pi }{\partial p_{2}}}={\frac {1}{2}}\left(l+a-b\right)-{\frac {p_{1}}{2c}}+{\frac {p_{2}}{c}}=0}$

Hotteling obtains the economic equilibrium. Hotteling argues this equilibrium is stable even though the sellers may try to establish a price cartel.

### Non-convexities

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]

#### 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.[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]

#### 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:

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]

## References

1. ^ Dodge, Y. (2008). The concise encyclopedia of statistics, Springer
2. ^ Hotelling, Harold (1929). Stability in Competition. Economic Journal 39(153):41-57
3. ^ Palda, Filip (2013). The Apprentice Economist: Seven Steps to Mastery. Toronto. Cooper-Wolfling Press.
4. ^ a b c d Mas-Colell, A. (1987). "Non-convexity". In Eatwell, John; Milgate, Murray; Newman, Peter. The New Palgrave: A Dictionary of Economics (PDF) (new ed.). Palgrave Macmillan. pp. 653–661. doi:10.1057/9780230226203.3173.
5. ^ Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in economics. 1. Amsterdam: North-Holland Publishing Co. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 634800.
6. ^ a b Salanié, Bernard (2000). "7 Nonconvexities". Microeconomics of market failures (English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris) ed.). Cambridge, MA: MIT Press. pp. 107–125. ISBN 978-0-262-19443-3.
7. ^ a b c d Salanié (2000, p. 36)
8. ^ a b c Pages 63–65: Laffont, Jean-Jacques (1988). "3 Nonconvexities". Fundamentals of public economics. MIT. ISBN 978-0-262-12127-9. External link in |publisher= (help)
9. ^ a b c Starrett, David A. (1972). "Fundamental nonconvexities in the theory of externalities". Journal of Economic Theory. 4 (2): 180–199. doi:10.1016/0022-0531(72)90148-2. MR 449575.
10. ^ a b c Pages 106, 110–137, 172, and 248: Baumol, William J.; Oates, Wallace E.; with contributions by V. S. Bawa and David F. Bradford (1988). "8 Detrimental externalities and nonconvexities in the production set". The Theory of environmental policy (Second ed.). Cambridge: Cambridge University Press. pp. x+299. doi:10.2277/0521311128. ISBN 978-0-521-31112-0.
11. ^ a b c d Page 1: Guesnerie, Roger (1975). "Pareto optimality in non-convex economies". Econometrica. 43: 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 443877. ("Errata". Econometrica. 43 (5–6): 1010. 1975. doi:10.2307/1911353. JSTOR 1911353. MR 443878.)
12. ^ Sraffa, Piero (1926). "The Laws of returns under competitive conditions". Economic Journal. 36 (144): 535–550. JSTOR 2959866.
13. ^ Hotelling, Harold (July 1938). "The General welfare in relation to problems of taxation and of railway and utility rates". Econometrica. 6 (3): 242–269. JSTOR 1907054.
14. ^ Pages 5–7: Quinzii, Martine (1992). Increasing returns and efficiency (Revised translation of (1988) Rendements croissants et efficacité economique. Paris: Editions du Centre National de la Recherche Scientifique ed.). New York: Oxford University Press. pp. viii+165. ISBN 0-19-506553-0.
15. ^ Hotelling (1935, p. 74): Hotelling, Harold (January 1935). "Demand functions with limited budgets". Econometrica. 3 (1): 66–78. JSTOR 1907346.
16. ^ Diewert (1982, pp. 552–553): Diewert, W. E. (1982). "12 Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in economics. 1. Amsterdam: North-Holland Publishing Co. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 648778.
17. ^ Starrett discusses non-convexities in his textbook on public economics (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): Starrett, David A. (1988). Foundations of public economics. Cambridge economic handbooks. Cambridge: Cambridge University Press. ISBN 978-0-521-34801-0.
18. ^ Radner, Roy (1968). "Competitive equilibrium under uncertainty". Econometrica. 36: 31–53. doi:10.2307/1909602.
19. ^ Page 270: Drèze, Jacques H. (1987). "14 Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press. pp. 261–297. ISBN 0-521-26484-7. MR 926685. (Originally published as Drèze, Jacques H. (1974). "Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. Allocation under Uncertainty: Equilibrium and Optimality. New York: Wiley. pp. 129–165.)
20. ^ Page 371: Magill, Michael; Quinzii, Martine (1996). "6 Production in a finance economy". The Theory of incomplete markets (31 Partnerships ed.). Cambridge, Massachusetts: MIT Press. pp. 329–425.