Hotelling (1929) - Stability In Competition

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• Hotelling, H. (1929), "Stability in competition", Economic Journal 39, 41-57. pdf
@article{hotelling1929stability,
title={Stability in competition},
author={Hotelling, H.},
journal={The economic journal},
volume={39},
number={153},
pages={41--57},
year={1929},
publisher={JSTOR}
}


Abstract

After the work of the late Professor F. Y. Edgeworth one may doubt that anything further can be said on the theory of competition among a small number of entrepreneurs. However, one important feature of actual business seems until recently to have escaped scrutiny. This is the fact that of all the purchasers of a commodity, some buy from one seller, some from another, in spite of moderate differences of price. If the purveyor of an article gradually increases his price while his rivals keep theirs fixed, the diminution in volume of his sales will in general take place continuously rather than in the abrupt way which has tacitly been assumed. A profound difference in the nature of the stability of a competitive situation results from this fact. We shall examine it with the help of some simple mathematics. The form of the solution will serve also to bring out a number of aspects of a competitive situation whose importance warrants more attention than they have received. Among these features, all illustrated by the same simple case, we find (1) the existence of incomes not properly belonging to any of the categories usually discussed, but resulting from the discontinuity in the increase in the number of sellers with the demand; (2) a socially uneconomical system of prices, leading to needless shipment of goods and kindred deviations from optimum activities; (3) an undue tendency for competitors to imitate each other in quality of goods, in location, and in other essential ways.

The Hotelling Model With Prices

As per Hotelling, there is a two stage game:

1. Firms choose locations
2. Firms choose prices

However, Hotelling on informally discusses location choice. In this version we will use game theory tools (unavailable to Hotelling) to solve for it.

Assumptions:

• There is a line of length 1 and firms chooses locations $a\,$ and $1-b\,$ such that $1-b \ge a \ge 0\,$.
• $mc=0\,$
• unit demands
• consumers are homogeneous with valuation $v\,$ (assumed large enough to cover the market)
• there are transportation costs

Specifically the transportation costs are:

$t(x-a)^2\,$ to buy from firm 1
$t(1-b-x)^2\,$ to buy from firm 2

Solving for prices and quantities

There is an indifferent consumer at location $x\,$ such that:

$p_1 + t(x-a)^2 = p_2 +t(1-b-x)^2 \,$

Rearranging for $x\,$ gives us the demand function for $q_1\,$, and likewise $1-x = q_2\,$.

Doing comparative statics on the demand we find:

• Demand is less sensitive to the price differential as $t\,$ (transport cost) increases
• Equal prices gives the firm captive demand plus half of intermediate segment $\frac{(1-b-a)}{2}\,$
• Taking $\frac{dq_1}{da}\,$ shows that (for equal prices) getting closer steals demand: the demand effect
• Taking $\frac{dq_1}{dp_2}\,$ shows prices are strategic complements - increases in rivals price increases your demand

To find prices:

$max_{p_i} \pi_i = p_i\cdot q_i\,$

where $q_i\,$ is given as above by the demand function.

This solves to:

$p_1^*(a,b) = t(1-b-a)\left( 1 + \frac{a-b}{3} \right )\,$
$p_2^*(a,b) = t(1-b-a)\left( 1 + \frac{b-a}{3} \right )\,$

Doing comparative statics on prices we find:

• Prices increase with transportation costs
• Taking $\frac{dp_1}{da}\,$ shows closeness stiffens competition (i.e. prices must fall): the competition effect
• And likewise if the rival moves closer.

Solving for locations

Given prices and quantities we can write profits:

$\pi_1 = p_1^*(a,b)q_1(a,b,p_1^*(a,b),p_2^*(a,b))\,$
$\pi_2 = p_2^*(a,b)q_2(a,b,p_2^*(a,b),p_1^*(a,b))\,$

Firms select locations taking the location of their rival as given, while implicitly considering the demand effect and competition effect.

Therefore:

$max_{p_1} \pi_1 \to \frac{dp_1^*}{da} \left(q_1 + p_1^* \cdot \frac{dq_1}{dp_1^*} \right) + p_1^* \left( \frac{dq_1}{da} + \frac{dq_1}{dp_2^*}\frac{dp_2^*}{da} \right) =0\,$

By the envelope theorem:

$\frac{d \pi}{dp_i} = \left(q_i + p_i^* \cdot \frac{dq_i}{dp_i^*} \right) = 0\,$

$\therefore p_1^* \left( \underbrace{\frac{dq_1}{da}}_{\mbox{Demand Effect}} + \underbrace{\frac{dq_1}{dp_2^*}\frac{dp_2^*}{da}}_{\mbox{Competition Effect}} \right) = 0\,$

Plugging in the values shows that increasing a beyond 0 can only decrease profits. Therefore the competition effect dominates, and firms locate at extremes.

Summary

This gives us a theory of product differentiation - firms differentiate to create monopoly and soften price competition. However, this is inefficient from a social welfare standpoint. The prices are transfers, but the transportation costs are lost. The welfare maximizing location would be $\left (\frac{1}{4}, \frac{3}{4} \right) \ne (0,1).\,$

If prices are fixed, because of regulation, or as in politics, the competition effect is gone and only the demand effect remains. This leads firms to locate in the center, which is also inefficient (as in Downsian competition for political parties). The `Political' Hotelling Model is the same as choosing firm location without prices.