Credible spatial preemption in a mixed oligopoly

Abstract

In this paper, we consider a public incumbent firm who produces multiple products and a private potential foreign entrant who contemplates on entering the market. Contrary to Judd (Rand J Econ 16:153–166, 1985) that preemption of the private incumbent by overcrowding the product space is not a credible threat of intense postentry competition if the exit cost of the incumbent is low, we show that if the incumbent is a public firm who cares about consumer surplus as well, the public incumbent’s preemption can credibly threaten an entrant with intense postentry competition without exiting after entry into the same location as the incumbent’s. Paradoxically, entering the identical location as the incumbent does not occur in the case of a public incumbent who relatively favors entry (inducing low consumer prices), whereas entry can occur in the case of a private incumbent who does not want entry. However, we also show that the incumbent can encourage entry by choosing its locations to leave enough room for the entrant so that the entrant can profitably enter locations far from the incumbent.

Appendix

In this appendix, we consider the case that the public incumbent locates $n (\ge 2)$ products equidistantly and the entrant enters with a single product. The interaction between the public incumbent and the private entrant is the same as in Sect. 5. Because of the analytic complexity as put in Footnote 3, we will assume that price discrimination among its substitutable products is not allowed.

From Lemma 1, we know that the entrant will not enter one of the locations of the incumbent, because the public incumbent will not leave the location even after the entry. Rather, it will enter the midpoint between two adjacent locations which are farthest apart. We assume that the entrant enters between the first two products of the incumbent if it enters. If the incumbent has n products, it will locate their products at ${\mathbf {x}}_{1}=(x_{11}, x_{12}, \ldots , x_{1n})=(0, 2x, 2x +\frac{1-2x}{n-1}, \ldots , 2x+(n-2)\frac{1-2x}{n-1})$ where the distance between two neighboring locations is $\frac{1-2x}{n-1}$ except the distance between $x_{11}$ and $x_{12}$, knowing that it is socially optimal for $(n+1)$ products to be located equidistantly after entry. Assume that $2x >\frac{1-2x}{n-1}$. By abusing notation, we denote the marginal consumer between 0 and x by $z^{**}$. Then, $z^{**}$ is determined by

$$\begin{aligned} tz +p_1 = t(x-z)+ p_{2}. \end{aligned}$$

(45)

So, we have $z^{**}=\frac{x}{2}+\frac{p_{2}-p_{1}}{2t}$. Then, the demand for the entrant is

$$\begin{aligned} q=2(x-z^{**})=x-\frac{p_{2}-p_{1}}{t}, \end{aligned}$$

(46)

and the profits are $\pi _2 =p_2 q$ and $\pi _1 =p_1 (1-q)$. The average transportation cost and consumer surplus are appropriately modified as

$$\begin{aligned}&\tilde{T}=\frac{1}{4}\left[ \frac{(1-2x)^{2}}{n-1}+4x^2 -4xq+2q^2 \right] , \end{aligned}$$

(47)

$$\begin{aligned}&CS = r-\tilde{p}-t\tilde{T}, \end{aligned}$$

(48)

where $\tilde{p}=p_1 (1-q)+p_2 q$. Thus we have

$$\begin{aligned} SW^E (n) = r-\left[ \frac{(1-2x)^{2}}{4(n-1)}+x^2 \right] t -(p_2 -xt)q -\frac{1}{2}tq^2 -nf. \end{aligned}$$

(49)

Given n, 2x is determined by

$$\begin{aligned} \frac{\partial SW^{E}(n)}{\partial x}=-\frac{t}{8}\left[ 14x - \frac{8(1-2x)}{n-1}\right] = 0. \end{aligned}$$

(50)

Therefore, $x=\frac{4}{7n+1} (<\frac{1}{n+1})$, implying that $1-2x=\frac{7(n-1)}{7n+1}$. Thus, we have

$$\begin{aligned} SW^{E}(n) &= {} r-\frac{t}{8}\left[ \frac{112}{(7n+1)^{2}}+\frac{98(n-1)}{(7n+1)^{2}}\right] -nf \nonumber \\ &= {} r-\frac{t}{4}\left[ \frac{7}{7n+1}\right] -nf \nonumber \\ & >{} r-\left[ \frac{t}{4n}+nf \right] =SW (n). \end{aligned}$$

(51)

Let $n^{**}$ be the socially optimal number of products provided by the incumbent. Then, $n^{**}$ is determined by

$$\begin{aligned} \frac{\partial SW^{E}(n)}{\partial n}=\frac{49t}{4(7n+1)^{2}}-f =0. \end{aligned}$$

(52)

Therefore, $n^{**}=\frac{1}{2}\left( \frac{t}{f }\right) ^{1/2}-\frac{1}{7}<n^{*}$.¹³ As a result, we have $SW^{E}(n^{**})=r-(tf )^{1/2}+\frac{f }{7}>SW (n^{*})$.

Proposition 3

(i) $n^{**}\le n^{*}$ and (ii) $SW^E (n^{**})>SW (n^{*})$.

This proposition confirms the main insight of this paper that it is possible for the incumbent to encourage entry by under-crowding the product space.