Hotelling's spatial duopoly model

Why are New Labour and the Tory Party so much alike? And are they really reflecting public opinion? Or, why are Coke and Pepsi alike?

This is a two part problem: a qualitative problem of product differentiation and a quantifiable problem of competitiveness. Harold Hotelling's spatial duopoly model, of 1929, attempts to solve this problem by substituting this qualitative differentiation with a type of quantifiable differentiation: hypothetically the spatial distance between rival firm's plants. Consequently we can consider a purely mathematical system relating product differentiation to competitiveness. So if we know something about one of these factors, we can make inferences about the other.

Suppose for instance two stalls selling flowers are set up alongside a highway, and alternately decide their prices and location along this highway. Potential customers are evenly distributed along the highway but the distance they are prepared to walk to buy flowers depends on two factors: price and distance*. Each flower seller is therefore in a position of seeking to set their price and location along the line to maximise their profits. But their strategy is ultimately dependant on what the other seller does

(*I've tried to make this picturesque: Hotelling talks about unit cost and transport cost)

Hotelling's mathematical solution of this problem gave the conclusion that both firms would acheive higher profits than under perfect competition and would tend to minimise the degree of product differentiation. In Hotelling's famous phrase they would compete "back-to-back" in the centre of the market.

Now, unfortunately subsequent studies, following d'Aspremont's revisions, have demolished the mathematical basis of Hotelling's reasoning. There isn't a pure strategy solution to the game Hoteling actually specifies; and variations to the Hotelling game are seen to result in deliberate product differentiation, and again profits above those determined by perfect competition.

It's possible Hotelling was thinking the transport cost accrued to the firms (but didn't state his presuppositions this way), in which case I believe his conclusions about product differentiation are justified, though it's hard to imagine real world circumstances to which this model could apply.