21.2 Hotelling Model
(KIM and SERFES 2006): A location model with preference variety
Stability in competition
Duopoly is inherently unstable
Bertrand disagrees with Cournot, and Edgeworth elaborates on it.
- because Cournot’s assumption of absolutely identical products between firms.
seller try to \(p_2 < p_1 c(l-a-b)\)
the point of indifference
\[ p_1 + cx = p_2 + cy \]
c = cost per unit of time in each unit of line length
p = price
q = quantity
x, y = length from A and B respectively
\[ a + x + y + b = l \]
is the length of the street
Hence, we have
\[ x = 0.5(l - a - b + \frac{p_2- p_1}{c}) \\ y = 0.5(l - a - b + \frac{p_1- p_2}{c}) \]
Profits will be
\[ \pi_1 = p_1 q_1 = p_1 (a+ x) = 0.5 (l + a - b) p_1 - \frac{p_1^2}{2c} + \frac{p_1 p_2}{2c} \\ \pi_2 = p_2 q_2 = p_2 (b+ y) = 0.5 (l + a - b) p_2 - \frac{p_2^2}{2c} + \frac{p_1 p_2}{2c} \]
To set the price to maximize profit, we have
\[ \frac{\partial \pi_1}{\partial p_1} = 0.5 (l + a - b) - \frac{p_1}{c} + \frac{p_2}{2c} = 0 \\ \frac{\partial \pi_2}{\partial p_2} = 0.5 (l - a + b) - \frac{p_2}{c} + \frac{p_1}{2c} = 0 \]
which equals
\[ p_1 = c(l + \frac{a-b}{3}) \\ p_2 = c(l - \frac{a-b}{3}) \]
and
\[ q_1 = a + x = 0.5 (l + \frac{a -b}{3}) \\ q_2 = b + y = 0.5 (l - \frac{a-b}{3}) \]
with the SOC satisfied
In case of deciding locations, socialism works better than capitalism
(d’Aspremont, Gabszewicz, and Thisse 1979)
- Principle of Minimum Differentiation is invalid
\[ \pi_1 (p_1, p_2) = \begin{cases} ap_1 + 0.5(l-a-b) p_1 + \frac{1}{2c}p_1 p_2 - \frac{1}{2c}p_1^2 & \text{if } |p_1 - p_2| \le c(l-a-b) \\ lp_1 & \text{if } p_1 < p_2 - c(l-a-b) \\ 0 & \text{if } p_1 > p_2 + c(l-a-b) \end{cases} \]
and
\[ \pi_2 (p_1, p_2) = \begin{cases} bp_2 + 0.5(l-a-b) p_2 + \frac{1}{2c}p_1 p_2 - \frac{1}{2c}p_2^2& \text{if } |p_1 - p_2| \le c(l-a-b) \\ lp_2 & \text{if } p_2 < p_1 - c(l-a-b) \\ 0 & \text{if } p_2 > p_1 + c(l-a-b) \end{cases} \]