Entry and social efficiency under Bertrand competition and asymmetric information

Abstract

This paper explores the welfare implications of free entry when firms face known entry costs, but production costs are privately known. Upon entering, firms compete in prices to supply a homogeneous good. Our framework yields results that are more nuanced than those of the literature on social efficiency and entry, where there is either insufficient or excessive entry for all parameter values. With asymmetric information, depending on the distribution of costs, and the magnitude of entry costs, it is possible to have both excessive and insufficient entry, as well as the optimal level of entry. We also show that the existence of entry costs fundamentally changes one of the key results of Spulber (J Ind Econ 43(1):1–11) on the convergence of the equilibrium price to the competitive equilibrium.

Appendix

Proof of Proposition 1

First we calculate the probability that firm i’s price is the lowest among others. In a symmetric, increasing equilibrium this probability is equal to the lowest order statistics conditional on the number of firms that entered. That is:

$$\begin{aligned} Pr(p^{-1}(p_i)<c_{m}\vert m) = \frac{\big (F({\tilde{c}}) -F (p^{-1}(p_i))\big )^{m-1}}{F({\tilde{c}})^{m-1}} \end{aligned}$$

(24)

Note that only firms with types lower than ${\tilde{c}}$ would enter. Therefore, ${\tilde{c}}$ is the cost of the firm that is indifferent between entering or not the market when the m firms entered. Also, as we later show in Lemma 1, ${\tilde{c}}$ is explicitly defined as the solution to $\Pi ({\tilde{c}},{\tilde{c}},k)=0$.

We can then rewrite (4) as follows,

$$\begin{aligned} \begin{aligned}&\left( D(p_i)+D^{\prime }(p_i)p_i- D^{\prime }(p_i) \frac{\partial C(D(p_i),c_i) }{\partial p_i}\right) \frac{\big (F({\tilde{c}}) -F (p^{-1}(p_i)\big )^{m-1}}{F({\tilde{c}})^{m-1}} \\&\quad -(m-1)(p_iD(p_i)-C(D(p_i),c_i)) \frac{f(c_i)\big (F({\tilde{c}}) - F(c_i)\big )^{m-2}}{p'(c_i)F({\tilde{c}})^{m-1}}=0 , \end{aligned} \end{aligned}$$

(25)

Rearranging we obtain the following differential equation.

$$\begin{aligned} p'(c_i) =\frac{(m-1)(p_iD(p_i)-C(D(p_i),c_i))}{ D(p_i)+D^{\prime }(p_i)p_i- D^{\prime }(p_i) \frac{\partial C(D(p_i),c_i) ) }{\partial p_i}} \frac{f(c_i)}{F({\tilde{c}}) - F(c_i)} \end{aligned}$$

(26)

Note that $\pi ^2(p^*(c),c)=p_iD(p_i)-C(D(p_i),c_i))$ and $\frac{ \partial \pi ^2(p^*(c),c)}{\partial p^*}=D(p_i)+D^{\prime }(p_i)p_i- D^{\prime }(p_i) \frac{\partial C(D(p_i),c_i) )}{\partial p_i}$. It follows from Spulber (1995, Proposition 2) that there is a unique increasing solution to the above differential equation with a boundary condition $\pi ^2(p^*({\bar{c}}),{\bar{c}})=0$.

We now show that $p^{*}(c_{i})<p_{o}(c_{i})$. Suppose $p^{*}(c_{i})\ge p_{o}(c_{i})$. Note that the monopoly price $p_o$ is profit maximizing. Therefore, we must have,

$$\begin{aligned} D(p_o)+D^{\prime }(p_o)p_o- D^{\prime }(p_o) C^{\prime }(D(p_o),c_i) =0 \end{aligned}$$

(27)

However, $p^*$ cannot be equal to $p_o$ because the first order condition in (4) is not satisfied. Also for $p^*>p_o$ the first term in (4) becomes negative while the second term is also strictly negative. That is, we must have $p^*(c_{i})<p_{o}(c_{i})$. $\square$

Proof of Lemma 1

First note that the ex-ante profit function $\Pi (c,{\tilde{c}},k)$ is strictly decreasing in c. Also since $k<{\bar{k}}$, then $\Pi ({\underline{c}},{\tilde{c}},k) >0$ and $\Pi ({\bar{c}}, {\tilde{c}},k) = -k$. Thus, due to continuity, there exists a unique ${\hat{c}}$, such that $\Pi ({\hat{c}},{\tilde{c}},k)=0$. We next show that in fact, ${\hat{c}}={\tilde{c}}$. To see this suppose ${\hat{c}}\ne {\tilde{c}}$. In the case where ${\hat{c}}<{\tilde{c}}$ then for any ${\hat{c}}<c<{\tilde{c}}$ we must have $\Pi (c,{\tilde{c}},k)<0$ and the firm enters the market which is a contradiction. Also when ${\hat{c}}>{\tilde{c}}$ for all types ${\hat{c}}>c>{\tilde{c}}$ we must have $\Pi (c,{\tilde{c}},k)>0$ and the firm does not enter the market which is again a contradiction. Therefore we have ${\hat{c}}={\tilde{c}}$ and for all types below ${\tilde{c}}$ the ex-ante expected profit is positive while for all types above ${\tilde{c}}$ the ex-ante expected profit is negative.

$\square$

Proof of Proposition 2

The proof simply follows the proof of Proposition 1 and Lemma 1. $\square$

Proof of Proposition 3

First suppose the number of entrants is continuous. Differentiate (10) with respect to $\ell$ yields:

$$\begin{aligned} \frac{\partial {S(q_{\ell })}}{\partial {\ell }} = D'(p_\ell )p'_\ell \big (p(q_\ell )-c_1(q_\ell ,c_{\ell })\big ) - \int _{0}^{q_{\ell }} c'_1(t,c_{\ell })\frac{dc_{\ell }}{d\ell }dt - k \end{aligned}$$

(28)

Note that the first two terms of the RHS of (28) are positive, decrease with the number of firms that enter the market ($\ell$), and approach zero when $\ell$ goes to infinity. This means that with no entry cost, the surplus is maximized at the highest possible number of entrants. Also when the entry cost is ${\bar{k}}$, the surplus is zero. Therefore, with positive entry costs below ${\bar{k}}$, there exists a ${\hat{\ell }}$ that maximises the surplus. Now note that the number of entrants is an integer and discontinuous. Therefore we denote $\ell ^*$ as the integer value that is closest to ${\hat{\ell }}$. We can conclude that for any entry above $\ell ^*$ the total surplus decreases and for entry below $\ell ^*$ it is increasing in $\ell$ and the integer value of the solution to (28) when it is equal to zero, gives $\ell ^*$.

$\square$

Proof of Proposition 4

Given that $p_o$ is strictly greater than $p=mc$ it is enough to show that $p^*>mc$ for any given number of firms. We use a proof by contradiction. Suppose the price is equal to the marginal cost of the winner, that is, $p^*=mc$. This guarantees zero profit at the interim stage. Given that there is a positive entry cost k, any firm who takes the pricing strategy of $p^*=mc$ guarantees a negative payoff at the entry stage. Thus the equilibrium pricing strategy of each firm i, must be strictly larger than their marginal costs. $\square$

Proof of Lemma 2

When the entry cost is equal to ${\bar{k}}$, zero entry is optimal. In a close neighborhood, there is only a small subset of types that could make a positive profit from entry. Given that the payoffs are decreasing in the entry fee, it is straightforward to conclude that the lower the entry fee is the higher the number of potential entrants. On the other hand, when $k=0$, then we have a situation similar to Spulber (1995), and then there is always a positive marginal effect on the overall surplus with an extra entry. So the optimal level of entry is infinity. On a close neighborhood of $k=0$, the optimal number of entry is very large and again becomes smaller when k increases. $\square$

Proof of Proposition 5

According to Proposition 3 we know that there exist a unique $\ell ^*$ that maximizes the total surplus. Also note that from Lemma 1 we know that there exist a threshold type ${\tilde{c}}$ such that firms with realized costs below ${\tilde{c}}$ enter the market. The probability that exactly $\ell ^*$ firms have a realized value below ${\tilde{c}}$ is positive and equal to,

$$\begin{aligned} \text {Prob optimal entry} = \left( {\begin{array}{c}n\\ \ell ^*\end{array}}\right) F({\tilde{c}})^{\ell ^*} (1-F({\tilde{c}}))^{n-\ell ^*}.\end{aligned}$$

Therefore given the space of realized values and the existence of ${\tilde{c}}$ within this interval, with positive probability the entry will be optimal. Furthermore, it is possible to show the probability that more than $\ell ^*$ firms have realized values below ${\tilde{c}}$ is also positive. This probability is the sum of events where $\{\ell ^*+1, \ell ^*+2,\ldots , n\}$ firms have realized values below ${\tilde{c}}$ which is given by,

$$\begin{aligned} \text {Prob excess entry} = \sum \limits _{j=\ell ^*+1}^{n} \left( {\begin{array}{c}n\\ j\end{array}}\right) F({\tilde{c}})^j (1-F({\tilde{c}}))^{n-j}.\end{aligned}$$

Finally, if less than $\ell ^*$ firms have realized values below ${\tilde{c}}$ we will have insufficient entry. The probability is given by,

$$\begin{aligned} \text {Prob insufficient entry} = \sum \limits _{i=0}^{\ell ^*-1} \left( {\begin{array}{c}n\\ i\end{array}}\right) F({\tilde{c}})^i (1-F({\tilde{c}}))^{n-i}.\end{aligned}$$

Therefore, given the space of possible realized values in the interval, all three cases of optimal, excess and insufficient entries are possible. $\square$

Proof of Corollary 1

First note that according to the definition of ${\bar{k}}$ the change in the surplus as shown in (28) will be negative at ${\bar{k}}$. In particular, we can denote ${\bar{k}}$ as,

$$\begin{aligned} {\bar{k}} = \int _0^{q_o} p(q)dq. \end{aligned}$$

(29)

It is straightforward to check that for any extra entry the increase in the surplus by a reduction in price is less than ${\bar{k}}$ and therefore, the overall effect is negative. Also note that, when $k=0$ then moving from one to two entry would increase the overall surplus. Given that (28) is continuous and decreasing in k, there exists a ${\hat{k}}<{\bar{k}}$ such that the (28) is zero when the entry changes from one to two. Therefore, for all entry costs above ${\hat{k}}$ the optimal level of entry is one. $\square$