Free entry and social inefficiency under co-opetition

Abstract

We investigate the social desirability of free entry under co-opetition where firms compete in a homogeneous product market while sharing common property resources that affect industry-wide demand. Our findings indicate that free entry leads to socially excessive or insufficient market entry, depending on the commitment of investment in common property resources. In the non-commitment case, where quantities and investment are simultaneously chosen, there is a possibility of insufficient entry. However, in the pre-commitment case, where investment is chosen at a prior stage, free entry leads to excess entry and a reduction in common property resources. Interestingly, in this case, the excess entry results of Mankiw and Whinston (RAND J Econ 17:48–58, 1986) hold even without entry costs or economies of scale. These results have important policy implications for entry regulation.

Appendices

Appendix 1: Constant elasticity demand

In this appendix, we consider a constant elasticity demand (CED) instead of a linear one (2), and check the robustness of our results obtained in the main body. We specify a constant elasticity demand expressed by

$$\begin{aligned} P(Q,Z)=\left( \frac{a+Z}{Q} \right) ^{1/ \epsilon }, \end{aligned}$$

(16)

where a is a positive constant and $\epsilon$ is price elasticity of demand. We further assume $\epsilon \ge 1$ to satisfy $P_{ZZ}\le 0$. The CED has the property of $P_{QZ}<0$, indicating that the higher the quality of common property resources, the steeper is the slope of the inverse demand curve.

1.1 Simultaneous co-opetition

The simultaneous co-opetition game comprises two stages: in the first stage, firms make entry decisions; in the second stage, firms simultaneously decides their own investment and output.

Using (16), we obtain the symmetric equilibrium of the second stage as follows:

$$\begin{aligned} \hat{q}=\frac{\varLambda ^\epsilon (a \epsilon g +\varLambda ^{ \epsilon -1})}{\epsilon g n},\quad \hat{z}=\frac{\varLambda ^{ \epsilon -1}}{\epsilon g n}, \end{aligned}$$

where $\varLambda \equiv (\epsilon n -1)/(c \epsilon n)>0$ and $d\varLambda /dn>0$. Then, we have the following comparative statics:

$$\begin{aligned} \frac{d\hat{q}}{dn}= & {} -\frac{\varLambda ^{2 \epsilon -1} \big \{ (n-2)+\left[ \left( n-1 \right) \epsilon -1 \right] a\,g\,\varLambda ^{1- \epsilon }\big \}}{ (\epsilon n-1)g n^2}<0,\\ \frac{d\hat{z}}{dn}= & {} -\frac{(n-1)\varLambda ^{ \epsilon -1}}{ (\epsilon n-1)g n^2}<0,\\ \frac{d\hat{Q}}{dn}= & {} \frac{\varLambda ^ {2 \epsilon -1} \left[ (2 \epsilon -1)+ a \epsilon ^2 g \varLambda ^{1- \epsilon } \right] }{(\epsilon n -1)\epsilon g n}>0,\quad \frac{d\hat{Z}}{dn}\, =\, \frac{( \epsilon -1)\varLambda ^{ \epsilon -1}}{(\epsilon n-1)\epsilon g n}>0. \end{aligned}$$

As the number of firms increases, individual output and investment decrease, whereas the total output and investment increase. Furthermore, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\hat{q}=\lim _{n\rightarrow \infty }\hat{z}=0, \quad \lim _{n\rightarrow \infty }\hat{Q}=\frac{c^{-2 \epsilon }(c+a c^{ \epsilon }\epsilon g)}{\epsilon g}>0,\quad \lim _{n\rightarrow \infty }\hat{Z}=\frac{c^{1- \epsilon }}{\epsilon g}>0, \end{aligned}$$

which implies that as the number of firms approaches infinity, individual output and investment converge to zero, while the total output and investment converge to positive and finite values. These properties are the same as in the linear demand case.

In the first stage, firms enter the market until their expected profits fall to zero. Therefore, the free-entry equilibrium number of firms, $\hat{n}_f$, satisfies

$$\begin{aligned} \hat{\pi }(\hat{n}_f)=\frac{\left( 1+2a \epsilon g\varLambda ^{1- \epsilon }\right) c \varLambda ^{2 \epsilon -1}}{2( \epsilon n-1)\epsilon g \hat{n}_f}-K=0. \end{aligned}$$

Because $\hat{\pi }(n)$ is strictly decreasing in n and $\lim _{n\rightarrow \infty }\hat{\pi }=-K$, we confirm $\lim _{K\rightarrow 0}\hat{n}_f=\infty$. Thus, the free-entry number of firms goes to infinity when there are no entry costs, as in the linear demand case.

The socially optimal number of firms, $\hat{n}_{sb}$, satisfies

$$\begin{aligned}&\widehat{W}'(\hat{n}_{sb})=\hat{\pi }(\hat{n}_{sb})\\&\quad -\frac{\left\{ \left( \epsilon \hat{n}_{sb}-1\right) \left[ \left( \hat{n}_{sb}-1\right) \epsilon -1\right] ag - \left[ \left( \hat{n}_{sb}+1\right) \epsilon -\hat{n}_{sb} \right] c \hat{n}_{sb}\varLambda ^{\epsilon }\right\} c \varLambda ^{\epsilon }}{( \epsilon \hat{n}_{sb}-1)^3 g \hat{n}_{sb}}=0. \end{aligned}$$

We also find that $\lim _{K\rightarrow 0}\hat{n}_{sb}=\infty$, which implies that the second-best number of firms goes to infinity as K approaches zero.

Then we have

$$\begin{aligned} \widehat{W}'(\hat{n}_f)= & {} -\frac{\left\{ \left( \epsilon \hat{n}_f\!-\!1\right) \left[ \left( \hat{n}_f\!-\!1\right) \epsilon \!-\!1\right] ag \!-\! \left[ \left( \hat{n}_f+1\right) \epsilon -\hat{n}_f \right] c \hat{n}_f \varLambda ^\epsilon \right\} c \varLambda ^{ \epsilon }}{( \epsilon \hat{n}_f-1)^3 g \hat{n}_f} \lesseqgtr 0 \\\Leftrightarrow & {} g\gtreqless \frac{ [ \epsilon (\hat{n}_f+1)-\hat{n}_f]c\hat{n}_f\varLambda ^{ \epsilon }}{(\epsilon \hat{n}_f -1)[ (\hat{n}_f-1)\epsilon -1]a}\equiv \varOmega . \end{aligned}$$

Therefore, the excess (insufficient) entry theorem applies for a larger (smaller) market size (a) and greater (smaller) production and investment costs (c and g). We cannot analytically derive the impact of $\epsilon$ on the sign of $\widehat{W}'(\hat{n}_f)$, but the numerical examples demonstrate that the excess entry theorem is more likely to hold for less elastic demand (i.e., smaller value of $\epsilon$).

In his simple Cournot model with endogenous entry, Varian (1995) demonstrates that excessive entry results hold for constant elasticity demand cases. Our result extends this by allowing firms co-opetitive investment and shows that free entry leads to either excessive or insufficient entry depending on the value of price elasticity and investment cost.

We provide the following numerical examples. In the case of $a=2$, $c=0.1$, $\epsilon =2$, and $K=0.1$, we find $\hat{n}_f\approx 14$ and $\hat{n}_{sb}\approx 19$ for $g=1$, and $\hat{n}_f\approx 10$ and $\hat{n}_{sb}\approx 8$ for $g=8$. Therefore, free entry leads to excess (insufficient) entry when g is large (small). In the case of $a=2$, $c=0.1$, $g=3$, and $K=0.1$, we find that $\hat{n}_f\approx 49 < \hat{n}_{sb}\approx 74$ for $\epsilon =3$, and $\hat{n}_f \approx 5 > \hat{n}_{sb}\approx 4$ for $\epsilon =1.2$. Figure 3 depicts these situations. In each panel, profits and welfare in the second-stage equilibrium are depicted. The left (right) panel depicts the case in which price elasticity of demand is high (low), and indicates that insufficient (excess) entry occurs.

Fig. 3

Number of firms and profits/welfare under simultaneous co-opetition: the CED case for $\epsilon =3$ (left) and $\epsilon =1.2$ (right)

1.2 Sequential co-opetition

A sequential co-opetition game comprises three stages: in the first stage, firms make entry decisions; in the second stage, firms non-cooperatively decide on their investment; in the last stage, firms compete in quantity.

Specifying the inverse demand as (16), we obtain the third-stage equilibrium:

$$\begin{aligned} \tilde{q}_i=\tilde{q}_j=\tilde{q}(n,Z)=\frac{(a+Z)\varLambda ^{ \epsilon }}{n},\quad \tilde{Q}(n,Z)=(a+Z)\varLambda ^{ \epsilon }, \end{aligned}$$

where $\varLambda \equiv (\epsilon n -1)/(c\epsilon n)>0$, as in the simultaneous co-opetition game. Furthermore, we obtain $\partial \tilde{q}/\partial n<0$, $\partial \tilde{q}/\partial z_i>0$, $\partial \tilde{Q}/\partial n>0$, and $\partial \tilde{Q}/\partial z_i=\partial \tilde{Q}/\partial z_j>0$.

Solving for the second-stage problem, we have

$$\begin{aligned} \bar{z}=\frac{c \varLambda ^{ \epsilon }}{(\epsilon n -1)g n}, \quad \bar{q}=\frac{a\varLambda ^{ \epsilon }}{n}+\frac{c \varLambda ^{2 \epsilon }}{(\epsilon n -1)g n}. \end{aligned}$$

The comparative static yields the following:

$$\begin{aligned} \frac{d\bar{q}}{dn}= & {} -\frac{\left[ \left\{ 2 (n-1) \epsilon -1 \right\} + \left\{ (n-1)\epsilon -1 \right\} a \epsilon g n\varLambda ^{1- \epsilon } \right] \varLambda ^{2 \epsilon -1}}{(\epsilon n -1)\epsilon g n^2}<0,\\ \frac{d\bar{z}}{dn}= & {} -\frac{ [\epsilon (2n-1)-1]\varLambda ^{ \epsilon -1}}{ (\epsilon n -1)\epsilon g n^2 }<0, \\ \frac{d\bar{Q}}{dn}= & {} \frac{ \left[ a \epsilon g n \varLambda ^{1- \epsilon }-(n-2)c\right] \epsilon \varLambda ^{ 2 \epsilon -1}}{ (\epsilon n -1)\epsilon g n^2 }\gtreqless 0, \quad \frac{d\bar{Z}}{dn}\,=\,-\frac{(n-1)\varLambda ^{ \epsilon -1}}{(\epsilon n-1)g n^2 }<0. \end{aligned}$$

We find that $d\bar{Z}/dn<0$ also holds under constant elasticity demand, indicating that an increase in the number of firms decreases total investment. The sign of $d\bar{Q}/dn$ is ambiguous: however, it is more likely to be negative when $\epsilon$ is smaller, c is larger, and/or d is smaller.¹⁴ Furthermore, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\bar{q}=\lim _{n\rightarrow \infty }\bar{z}=\lim _{n\rightarrow \infty }\bar{Z}=0, \quad \lim _{n\rightarrow \infty }\bar{Q}={a c^{- \epsilon }}>0, \end{aligned}$$

Therefore, we find that as the number of firms increases, the total output converges to the perfect competition outcome without investment activities and the total investment converges to zero, as in the linear demand case.

In the first stage, the number of firms is endogenously decided. The free-entry equilibrium number of firms, $\bar{n}_f$, satisfies

$$\begin{aligned} \bar{\pi }(\bar{n}_f)=\frac{\left[ (2\bar{n}_f-1)c+2 (\epsilon \bar{n}_f -1)a g \bar{n}_f \varLambda ^{-\epsilon }\right] c \varLambda ^{ 2 \epsilon } }{2 (\epsilon \bar{n}_f -1)^2 g \bar{n}_f^2 }-K=0. \end{aligned}$$

On the basis of the fact that $\bar{\pi }(n)$ is strictly decreasing in n and $lim_{n\rightarrow \infty }\bar{\pi }=-K$, we confirm $\lim _{K\rightarrow 0} \bar{n}_f=\infty$, which indicates that the free-entry number of firms goes to infinity when there are no entry costs. After some tedious manipulation, we find that the second-best number of firms, $\bar{n}_{sb}$, satisfies

$$\begin{aligned} \overline{W}'(\bar{n}_{sb})= & {} \bar{\pi }(\bar{n}_{sb})\!-\! \frac{c\varLambda ^{2 \epsilon }}{(\epsilon \bar{n}_{sb}-1)^3 g \bar{n}_{sb}^2}\\&\times \left[ \begin{array}{l} \left( \epsilon \bar{n}_{sb} \!-\!1 \right) \left\{ \epsilon (\bar{n}_{sb}-1)-1 \right\} a g \bar{n}_{sb} \varLambda ^{- \epsilon }\\ \ +\ \{ \epsilon \bar{n}_{sb} (\bar{n}_{sb}^2-4)+\bar{n}_{sb}(\epsilon \bar{n}_{sb} -1)+1+ \epsilon \}c\end{array}\right] =0. \end{aligned}$$

Then, we find that

$$\begin{aligned} \overline{W}'(\bar{n}_f)=-\frac{c\varLambda ^{2 \epsilon }}{(\epsilon \bar{n}_f -1)^3 g \bar{n}_{f}^2}\left[ \begin{array}{l} \left( \epsilon \bar{n}_{f} -1 \right) \left\{ \epsilon (\bar{n}_{f}-1)-1 \right\} a g \bar{n}_{f} \varLambda ^{- \epsilon }\\ \ +\ \{ \epsilon \bar{n}_{f} (\bar{n}_{f}^2-4)+\bar{n}_{f}(\epsilon \bar{n}_{f} -1)+1+ \epsilon \}c\end{array}\right] <0, \end{aligned}$$

which indicates that free entry necessarily results in excessive entry. Therefore, we find that, under constant elasticity demand, the free-entry equilibrium of the sequential co-opetition game yields excessive entry because $\overline{W}'(\bar{n}_f)$ is always negative. Furthermore, this result holds even when there are no entry costs.

In the constant elasticity demand case, the second-best number of firms when $K=0$ cannot be analytically derived; therefore, we provide numerical examples to show that the excess entry property holds even when $K=0$. In the case of $a=20$, $c=0.1$, $g=4$, and $\epsilon =2$, $\bar{n}_f\approx 31>\bar{n}_{sb}\approx 7$ holds for $K=0.1$ and $\bar{n}_f \approx \infty >\bar{n}_{sb}\approx 9$ holds for $K=0$.

Appendix 2: Investment cooperation and obligation

Here, we consider two additional cases of investment: (1) investment cooperation among firms in which firms invest cooperatively in common property resources to maximize industry profits but remain rivals in Cournot competition, and (2) investment obligation in which the government requires firms to provide a specified level of investment that maximizes social welfare. In particular, we examine the social desirability of free entry in an extended sequential co-opetition game in which firms make entry decisions in the first stage, determine the level of investment required from all firms in the second stage, and compete in a Cournot fashion in the third stage. The inverse demand is linear as given by (2).

We first characterize the equilibrium of the investment cooperation case. The third stage equilibrium is the same as that derived in Sect. 4, so the Nash equilibrium output and investment per firm are given by (8). In the second stage, the level of investment ($z=Z/n$) is determined to maximize joint profits:

$$\begin{aligned} \max _{z} \sum _{i=1}^n \pi _i=P(\tilde{Q}, Z)\tilde{Q}-n C(\tilde{q})-n G \left( z\right) -n K. \end{aligned}$$

Arranging the first-order condition and using (3), we have

$$\begin{aligned} \left[ P_Q\frac{\partial \tilde{q}}{\partial z}(n-1) + nP_Z \right] \tilde{q}-G'=0, \end{aligned}$$

(17)

which implies that firms can internalize positive external effects of investment for market expansion (i.e., $n P_Z$ in the first bracket). Therefore, the level of cooperative investment is greater than that of non-cooperative investment. Solving (17), we characterize the second-stage equilibrium as

$$\begin{aligned} \breve{q}(n)=\frac{(a-c)(n+1)g}{\varPsi }, \quad \breve{z}(n)=\frac{2(a-c)n}{\varPsi }, \end{aligned}$$

where $\varPsi \equiv (n+1)^2 bg-2n^2>0$ (the inequality follows from the second-order condition). Because $\varPsi <\varTheta$, from (10), we have $\breve{q}(n)>\bar{q}(n)$ and $\breve{z}(n)>\bar{z}(n)$: given that the number of firms is fixed, the individual output and investment under the cooperative investment case are greater than those under the non-cooperative investment case. Then, we have the following comparative statics:

$$\begin{aligned} \frac{d\breve{Q}}{dn}=\frac{(a-c)\left[ (n+1)^2 bg + 2n^2 \right] g}{\varPsi ^2}>0,\quad \frac{d\breve{Z}}{dn}=\frac{(a-c)(n+1)bgn}{\varPsi ^2}>0. \end{aligned}$$

In contrast to the non-cooperative investment case analyzed in Sect. 4, an additional entry in this case increases the total investment and total output because entry does not increase the incentive to free ride on investments in common property resources. Furthermore, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\breve{q}=\lim _{n\rightarrow \infty }\breve{z}=0, \ \lim _{n\rightarrow \infty }\breve{Q}=\frac{(a-c)g}{bg-2}>0,\ \lim _{n\rightarrow \infty }\breve{Z}=\frac{a-c}{b g-2}>0. \end{aligned}$$

Note that these limiting values are similar to those in the case of simultaneous co-opetition. The total investment converges to positive and finite values as the number of firms approaches infinity.

In the first stage, firms enter the market until their profits fall to zero. The free-entry number of firms is defined as $\breve{n}_f$ such that $\breve{\pi }(\breve{n}_f)=0$. The associated level of welfare is defined by $\breve{W}(\breve{n}_f)$. Then, we have

$$\begin{aligned} \breve{W}'(\breve{n}_f)=-\frac{(a-c)^2 \left[ \left( \breve{n}_f+1 \right) ^3 b^2 g^2 -2 \left( \breve{n}_f +1 \right) \left( 5 \breve{n}_f+2\right) bg\breve{n}_f +8 \breve{n}_f^3 \right] g\,\breve{n}_f }{ \left[ (\breve{n}_f+1)^2 b g -2\breve{n}_f^2 \right] ^3}, \end{aligned}$$

where

$$\begin{aligned} \breve{W}'(\breve{n}_f)\lesseqgtr 0 \quad \mathrm {if}\quad bg \gtreqless \frac{\breve{n}_f\sqrt{17\breve{n}_f^2+12\breve{n}_f+4}+5\breve{n}_f^2+2\breve{n}_f}{ \left( \breve{n}_f+1 \right) ^2}. \end{aligned}$$

This indicates that the greater (smaller) the value of b and/or d, the more likely free entry would be to lead to excess (insufficient) entry. In other words, even when investment has commitment value, investment cooperation may lead to insufficient entry.

We provide the following numerical examples: In the case of $a=10,\ b=1,\ c=1,$ and $K=2$, we find that $\breve{n}_f\approx 7<\breve{n}_{sb}\approx 10$ for $g=5$, which corresponds to the insufficient entry theorem. On the other hand, we find that $\breve{n}_f\approx 6 > \breve{n}_{sb}\approx 3$ for $g=20$, which corresponds to the excess entry theorem.

Then, we characterize the equilibrium of the investment obligation case in which the government requires firms to provide a specified level of investment that maximizes social welfare.

The third stage equilibrium is the same as the case of investment cooperation. The level of investment in the second stage is determined to maximize the social surplus:

$$\begin{aligned} \max _z W=\int _0^{\tilde{Q}}P(s,Z)ds-nC(\tilde{q})-nG(z)-nK \end{aligned}$$

The second-stage equilibrium is characterized as

$$\begin{aligned}&\check{q}(n)=\frac{(a-c)(n+1)g}{\varXi },\quad \check{z}(n)=\frac{(a-c)(n+2)n}{\varXi },\\&\check{\pi }(n)=\frac{(a-c)^2[2(n+1)^2 bg-(n+2)^2 n^2]}{2\varXi ^2}-K,\\&\check{W}(n)=\frac{(a-d)^2(n+2)gn}{2\varXi }-nK, \end{aligned}$$

where $\varXi \equiv bg(n+1)^2-(n+2)n^2>0$ (the inequality follows the second-order condition).

After some mathematical manipulations, we have the following condition:

$$\begin{aligned} \check{W}'(\check{n}_f)\gtreqless 0 \quad \mathrm {if} \quad bg \lesseqgtr \frac{n(n+2)^2}{n+1}, \end{aligned}$$

where $\check{n}_f$ is a free-entry number of firms that satisfies $\check{\pi }(\check{n}_f)=0$. Therefore, we find that the greater (smaller) the value of b and/or g, the more likely is free entry to be lead to excess (insufficient) entry. The level of investment in the investment obligation case is greater than that in the investment cooperation case because the government decides the level of investment by considering the positive effect on consumer surplus. This implies that free entry under investment obligation is more likely to result in insufficient entry compared to the case of investment cooperation.