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 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, 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 was influenced by the writing of Henry George and was an editorial adviser for the Georgist journal AJES Mason Gaffney claims that Hotelling kept his lifelong beliefs about land and tax reform secret because he feared ridicule.
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 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:
Let q1 and q2 indicate the quantities sold by A and B. The sellers profit are:
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 Georgistland 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 [the Georgist] H. G. Brown."
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. 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:
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.
^ abSalanié, 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. ISBN978-0-262-19443-3.
^ abcPages 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. ISBN978-0-521-31112-0.CS1 maint: multiple names: authors list (link)
^Hotelling, Harold (July 1938). "The General welfare in relation to problems of taxation and of railway and utility rates". Econometrica. 6 (3): 242–269. doi:10.2307/1907054. JSTOR1907054.
^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. ISBN978-0-19-506553-4.
^Page 270: Drèze, Jacques H. (1987). "14 Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. (ed.). Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press. pp. 261–297. doi:10.1017/CBO9780511559464. ISBN978-0-521-26484-6. MR0926685. (Originally published as Drèze, Jacques H. (1974). "Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. (ed.). Allocation under Uncertainty: Equilibrium and Optimality. New York: Wiley. pp. 129–165.)
^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.