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. 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 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: :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 q1 and q2 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_{1}}{\partial p_{1}}=\frac{1}{2}\left( l+a-b \right)-\frac{p_{1}}{c}+\frac{p_{2}}{2c}=0 :\frac{\partial \pi_{2}}{\partial p_{2}}=\frac{1}{2}\left( l-a+b \right)-\frac{p_{1}}{2c}+\frac{p_{2}}{c}=0 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."
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." 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. 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: {{blockquote|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. and with
market failures, and
public economics. Non-convexities occur also with
information economics, and with
stock markets Such applications continued to motivate economists to study non-convex sets. == Works ==