Sequential entry and merger in spatial price discrimination

Abstract

This paper newly introduces sequential entry into previous models of spatial price discrimination that examine location choices made in anticipation of a potential merger. While merger with simultaneous entry routinely reduces social welfare, we show that mergers frequently improve welfare. The extent of welfare improving mergers varies inversely with the timing advantage of an excluded rival. When the rival locates alone in the first stage, the merger harms welfare. When the rival locates alone in the last stage, the merger improves welfare. These results hold true when allocating three firms across either two or three location stages. Thus, allowing staged entry dramatically alters and often reverses the conclusion previously drawn on the basis of simultaneous entry.

Appendices

Appendix 1: The benchmark cases without merger in two-stage entry

In this section, we identify the locations and welfare from two benchmark cases without the possibility of merger.

1.1 Benchmark case 1: One firm enters in stage one

A profit maximizing first entrant locates at the corner, and we assign it $L_1$. The two followers’ best responses in terms of leader’s location are

$$\begin{aligned} \begin{aligned} L_2&=\frac{2+3L_1 }{5}\\ L_3&=\frac{4+L_1 }{5} \end{aligned} \end{aligned}$$

(25)

Thus, the profit of the firm in the middle becomes $\pi _2 =\frac{2(1-L_1)^{2}t}{25}$. In order to maintain its profitable corner location, the leader must locate such that no firm earns greater profit locating on its left (the “no jump condition”). The profit of firm 2 if it jumps to the leader’s left corner is $\pi _2^j =\frac{(L_1)^{2}t}{3}$.⁶ The no jump condition requires that this profit just equal that when it is in the middle, $\pi _2^ =\pi _2^j$. This implies that $L_1^{2j} =0.329$. The unconstrained profit maximization, $L^u =\frac{4}{9}$, is such that $L_1^u \ge L_1^j$. Thus, the no-jump condition implies the equilibrium locations:

$$\begin{aligned} \begin{aligned} L_1^{B1}&=0.329 \\ L_2^{B1}&=0.597 \\ L_3^{B1}&=0.866 \end{aligned} \end{aligned}$$

(26)

The profit and welfare of the three firms follow immediately:

$$\begin{aligned} \begin{aligned} \pi _1^{B1}&=0.106t \\ \pi _2^{B1}&=0.036t \\ \pi _3^{B1}&=0.054t \\ W^{B1}&=r-0.099t \end{aligned} \end{aligned}$$

(27)

This result exactly mimics Heywood and Ye (2009) when $n=3$.

1.2 Benchmark case 2: Two firms enter in stage 1

When two firms enter in stage one, each locates at a corner to obtain the extra profit. They also move toward the center as much as possible such that firm 2 (in stage two) is indifferent between locating in the middle and jumping to a corner. The symmetry of two leaders implies that $L_1=1-L_3$ and firm 2 maximizes profit at $L_2 =\frac{L_1+L_3 }{2}$. The no-jump condition equates the profit of this middle location, $\pi _2 =\frac{(1-2L_1)^{2}t}{8}$, with that if it jumped, $\pi _2^j =\frac{(L_1)^{2}t}{3}$. This implies that

$$\begin{aligned} \begin{aligned} L_1^{B2}&=0.275\\ L_2^{B2}&=0.500\\ L_3^{B2}&=0.725 \end{aligned} \end{aligned}$$

(28)

Firms’ profit and welfare are

$$\begin{aligned} \begin{aligned} \pi _1^{B2}&=\pi _3^{B2} =0.074t\\ \pi _2^{B2}&=0.025t \\ W^{B2}&=r-0.101t \end{aligned} \end{aligned}$$

(29)

This result exactly mimics Gupta (1992).

Appendix 2: The benchmark case without merger in three stage entry

In this section, we identify the locations and welfare from the benchmark case without the possibility of merger. When first two firms have timing advantage, each locates at a corner to obtain the extra profit. They also move toward the center as much as possible such that the third firm is indifferent between locating in the middle and jumping to a corner. The symmetry of two early movers implies that $L_1 =1-L_2$, and the middle firm maximizes profit at $L_3 =\frac{L_1^ +L_2 }{2}$. The no-jump condition equates the profit of the third firm in the middle, $\pi _3 =\frac{(1-2L_1 )^{2}t}{8}$, with that if it jumped, $\pi _3^j =\frac{(L_1)^{2}t}{3}$. This implies that

$$\begin{aligned} \begin{aligned} L_1^{B3}&=0.275 \\ L_2^{B3}&=0.725 \\ L_3^{B3}&=0.500 \end{aligned} \end{aligned}$$

(30)

Please note that subscripts denote the timing of firms in the rest of this paper, and thus, firm 3 may locate on the left to firm 2. Firms’ profit and welfare are

$$\begin{aligned} \begin{aligned} \pi _1^{B3}&=\pi _2^{B3} =0.074t\\ \pi _3^{B3}&=0.025t \\ W^{B3}&=r-0.101t \end{aligned} \end{aligned}$$

(31)

This result exactly mimics Gupta (1992). We will now examine the three possible divisions of our firms comparing each possibility with merger to the relevant benchmark.