21.2 Hotelling Model

(KIM and SERFES 2006): A location model with preference variety

(Hotelling 1929)

• 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

$$\pi_2 (p_1, p_2) = \begin{cases} bp_2 + 0.5(l-a-b) p_2 + \frac{1}{2c}p_1 p_2 - \frac{1}{2c}p_2^2& \text{if } |p_1 - p_2| \le c(l-a-b) \\ lp_2 & \text{if } p_2 < p_1 - c(l-a-b) \\ 0 & \text{if } p_2 > p_1 + c(l-a-b) \end{cases}$$

References

d’Aspremont, C., J. Jaskold Gabszewicz, and J.-F. Thisse. 1979. “On Hotelling’s "Stability in Competition".” Econometrica 47 (5): 1145. https://doi.org/10.2307/1911955.

Hotelling, Harold. 1929. “Stability in Competition.” The Economic Journal 39 (153): 41. https://doi.org/10.2307/2224214.

KIM, HYUNHO, and KONSTANTINOS SERFES. 2006. “A LOCATION MODEL WITH PREFERENCE FOR VARIETY.” Journal of Industrial Economics 54 (4): 569–95. https://doi.org/10.1111/j.1467-6451.2006.00300.x.