21.3 Positioning Models
Tabuchi and Thisse (1995)
Relax Hotelling’s model’s assumption of uniform distribution of consumers to non-uniform distribution.
Assumptions:
Equal cost
Consumers distributed over [0,1]
\(F(x)\) = cumulative distribution of consumers where \(F(1) = 1\) = total population
2 distributions:
Traditional uniform density: \(f(x) =1\)
New: triangular density: \(f(x) = 2 - 2|2x-1|\) which represents consumer concentration
Transportation cost = quadratic function of distance.
Hence, marginal consumer is
\[ \bar{x} = (p_2 - p_1 + x^2_2-x_1^2)/2(x_2-x_1) \]
then when \(x_1 < x_2\) the profit function is
\[ \Pi_1 = p_1 F(\bar{x}) \]
and
\[ \Pi_2 = p_2[1-F(\bar{x})] \]
and vice versa for \(x_1 >x_2\), and Bertrand game when \(x_1 = x_2\)
If firms pick simultaneously their locations, and then simultaneously their prices, and consumer density function is log-concave, then there is a unique Nash price equilibrium
Under uniform distribution, firms choose to locate as far apart as possible (could be true when observing shopping centers are far away from cities), but then consumers have to buy products that are far away from their ideal.
Under triangular density, no symmetric location can be found, but two asymmetric Nash location equilibrium can still be possible (decrease in equilibrium profits of both firms)
If firms pick sequentially their locations, and pick their prices simultaneously,
- Under both uniform and triangular, first entrant will locate at the market center
Sajeesh and Raju (2010)
Model satiation (variety-seeking) as a relative reduction in the willingness to pay of the previously purchased brand. also known as negative state dependence
Previous studies argue that in the presence of variety seeking consumers, firms should enjoy higher prices and profits, but this paper argues that average prices and profits are lower.
- Firms should charge lower prices in the second period to prevent consumers from switching.
Assumptions:
Period 0, choose location simultaneously
Period 1, choose prices simultaneously
Period 2, firms choose prices simultaneously
K. S. Moorthy (1988)
- 2 (identical) firms pick product (quality) first, then price.
Tyagi (2000)
Extending Hotelling (1929) Tyagi (1999b) Tabuchi and Thisse (1995)
Two firms enter sequentially, and have different cost structures.
Paper shows second mover advantage
KIM and SERFES (2006)
Consumers can make multiple purchases.
Some consumers are loyal to one brand, and others consume more than one product.
Shreay, Chouinard, and McCluskey (2015)
- Quantity surcharges from different sizes of the same product (i.e., imperfect substitute or differentiated products) can be led by consumer preferences.