Entry, market structures and welfare

Abstract

Conventionally, entry is thought to enhance welfare by enhancing competition and hence lowering prices and increasing the output. Contrary to the conventional wisdom, working with an $n$-firm Cournot oligopoly set up and using the trigger strategies, we show that entry may or may not impact welfare. However, entry has the potential to alter the market structure from collusion to Cournot competition, and when it does so, there is a discontinuous rise in welfare.

Appendix A: Generalizations

1.1 Bertrand Oligopoly

With the same demand and cost as in Sect. 2 (an inverse demand function of the form $p\left( q \right) = 1 - q$; $n$ firms with symmetric cost function of the form $c_{i} \left( {q_{i} } \right) = cq_{i}$; $0 \le c < 1$, $i = 1,2, \ldots ,n$, and $q = \mathop \sum \limits_{i = 1}^{n} q_{i}$ where $q_{i}$ is the output produced by firm $i$ and $q$ is the industry output), consider collusion in a Bertrand oligopoly set up. If all the $n$ firms decide to collude, they will sell at the agreed collusive price, say $P^{c}$, in all the time periods, and $P^{c} \in \left( c \right.,\left. {P^{m} } \right]$ where $P^{m}$ is the monopoly price (as in Eq. (3)). When each firms chooses the same price, i.e., $P^{c}$, it earns the collusive profit every time-period (as in Eq. (4)). Thus, as above, the present discounted value of the lifetime profit from collusion is $\Pi_{i}^{c} = \frac{{\pi_{i}^{c} }}{1 - \delta } = \frac{{\left( {1 - c} \right)^{2} }}{{4n\left( {1 - \delta } \right)}}$.

However, if the $i^{th}$ firm deviates it charges a price $P^{d} = P^{c} - \varepsilon ; \varepsilon > 0$ and captures the entire market and earns a profit of $\pi \left( {P^{c} } \right)$ and a profit of zero thereafter as all the firms revert to the Nash equilibrium of the one-shot Bertrand game. Hence, collusion would be an SPE iff,

$$\frac{{\pi_{i}^{c} }}{1 - \delta } = \frac{{\left( {1 - c} \right)^{2} }}{{4n\left( {1 - \delta } \right)}} \ge \pi \left( {P^{c} } \right) = n\pi_{i}^{c} = \frac{{\left( {1 - c} \right)^{2} }}{4}$$

(A-1)

Finally, $\delta \ge \frac{{\pi_{i}^{d} - \pi_{i}^{c} }}{{\pi_{i}^{d} - \pi_{i}^{o} }} = \frac{{\left( {n - 1} \right)}}{n} \equiv \underline {\delta }^{b}$. As in Sect. 2, $\frac{{\partial \underline {\delta }^{b} }}{\partial n} > 0$ and $\frac{{\partial^{2} \underline {\delta }^{b} }}{{\partial n^{2} }} < 0$. We denote the critical discount factor under Bertrand oligopoly by superscripting it with “b” to distinguish it from its Cournot counterpart.

Proposition (A.1)

$\forall \delta \in \left( {\frac{1}{2},1} \right),$ $\exists$ a unique $\hat{n}$ satisfying $\underline {\delta }^{b} \left( n \right) = \delta$, such that $\forall n \le \hat{n}$ collusion is the SPE outcome and Cournot competition otherwise.

Proof

$\mathop {\lim }\limits_{n \to 2} \underline {\delta }^{b} = \frac{1}{2}$ while $\mathop {\lim }\limits_{n \to \infty } \underline {\delta }^{b} = 1$. Hence, $\forall \delta \in \left( {\frac{1}{2},1} \right)$, there exists a unique $n$, say $\hat{n}$, such that $\underline {\delta }^{b} \left( n \right) = \delta$. If $n \le \hat{n}$, then $\delta \ge \underline {\delta }^{b}$ and collusion is the SPE, else, Cournot competition is the SPE. Q.E.D.

Hence all the propositions of Sect. 3 are valid here as well.

1.2 General demand function

Consider a general inverse demand function of the form $p = \left( {1 - q} \right)^{\beta }$. This form of the demand function allows for the fuller analysis of (the different curvatures) the demand curve. $0 < \beta < 1$ corresponds to the concave demand curve, $\beta = 1$ to the linear demand curve while $\beta > 1$ to the convex demand curve. We retain the constant marginal cost assumption, however, we assume it to be zero.

Cournot competition

The individual firm’s output in equilibrium is $q_{1}^{o} = q_{2}^{o} = \ldots = q_{i}^{o} = \ldots = q_{n}^{o} = \frac{1}{n + \beta }$ while the total output and price in equilibrium are

$$q^{o} = \frac{n}{n + \beta }\,{\text{and}}\,p^{o} = \left( {\frac{\beta }{n + \beta }} \right)^{\beta }$$

(A–2)

The profit of the $i^{th}$ firm is,

$$\pi_{i}^{o} = \beta^{\beta } \frac{1}{{\left( {n + \beta } \right)^{1 + \beta } }}$$

(A–3)

Corchón and Torregrosa (2020) consider an inverse demand function of the form $p = a - bq^{\beta }$ and show that $n + \beta > 0$ is a necessary and sufficient condition for existence of a Cournot equilibrium. We also get the same result as summarized in proposition A.2 as follows.

Proposition (A.2):

$n + \beta > 0$ is a necessary and sufficient condition for existence of a Cournot equilibrium.

Proof: $q_{1}^{o} = q_{2}^{o} = \ldots = q_{i}^{o} = \ldots = q_{n}^{o} = \frac{1}{n + \beta } > 0$ if $n + \beta > 0$ and $q^{o} > 0$, $p^{o} > 0$ and $\pi_{i}^{o} > 0$ if $n + \beta > 0$ (Eqs. A–2 and A–3). Q.E.D.

Collusion

In equilibrium each firm produces $q_{i}^{c} = \frac{1}{{n\left( {1 + \beta } \right)}}$, hence the total output and price are,

$$q^{c} = \frac{1}{1 + \beta }\,{\text{and}}\,p^{c} = \left( {\frac{\beta }{1 + \beta }} \right)^{\beta }$$

(A–4)

The profit of the $i^{th}$ firm is,

$$\pi_{i}^{c} = \beta^{\beta } \frac{1}{{n\left( {1 + \beta } \right)^{1 + \beta } }}$$

(A–5)

Deviation/cheating

While each firm produces the collusive output, $q_{i}^{c} = \frac{1}{{n\left( {1 + \beta } \right)}}$, the deviating firm’s output (which it chooses as per its reaction function) and the price in deviation are,

$$q_{i}^{d} = \frac{{\left( {1 + n\beta } \right)}}{{n\left( {1 + \beta } \right)^{2} }}\,{\text{and}}\,p_{i}^{d} = \left( {\frac{{\beta \left( {1 + n\beta } \right)}}{{n\left( {1 + \beta } \right)^{2} }}} \right)^{\beta }$$

(A–6)

The profit of the deviating firm is,

$$\pi_{i}^{d} = \beta^{\beta } \left( {\frac{{\left( {1 + n\beta } \right)}}{{n\left( {1 + \beta } \right)^{2} }}} \right)^{1 + \beta }$$

(A–7)

Thus, $\delta \ge \frac{{\pi_{i}^{d} - \pi_{i}^{c} }}{{\pi_{i}^{d} - \pi_{i}^{o} }} = \frac{{\beta^{\beta } \left( {\frac{{\left( {1 + n\beta } \right)}}{{n\left( {1 + \beta } \right)^{2} }}} \right)^{1 + \beta } - \beta^{\beta } \frac{1}{{n\left( {1 + \beta } \right)^{1 + \beta } }}}}{{\beta^{\beta } \left( {\frac{{\left( {1 + n\beta } \right)}}{{n\left( {1 + \beta } \right)^{2} }}} \right)^{1 + \beta } - \beta^{\beta } \frac{1}{{\left( {n + \beta } \right)^{1 + \beta } }}}} \equiv \underline {\delta }$.

Finally, $\delta \ge \frac{{\left( {n + \beta } \right)^{1 + \beta } \left( {\left( {1 + n\beta } \right)^{1 + \beta } - n^{\beta } \left( {n + \beta } \right)^{1 + \beta } } \right)}}{{\left( {\left( {1 + n\beta } \right)\left( {n + \beta } \right)} \right)^{1 + \beta } - \left( {n\left( {1 + \beta } \right)^{2} } \right)^{1 + \beta } }} \equiv \underline {\delta }^{g}$. Similarly, as above, $\frac{{\partial \underline {\delta }^{g} }}{\partial n} > 0$ and $\frac{{\partial^{2} \underline {\delta }^{g} }}{{\partial n^{2} }} < 0$.

We denote the critical discount factor under the general demand function by superscripting it with “g” to distinguish it from its Cournot and Bertrand counterparts. Following proposition has the results for various values of $\beta$ (highlighting the demand curvatures) and $\underline {\delta }^{g}$.

Proposition (A.3)

For $\beta = \frac{1}{2}$ and $\forall \delta \in \left( {.5571, 1} \right)$, for $\beta = 1$ and $\forall \delta \in \left( {\frac{9}{17},1} \right)$ and for $\beta = 2$ and $\forall \delta \in \left( {.5018, 1} \right)$, $\exists$ a unique $\hat{n}$ respectively satisfying $\underline {\delta }^{g} \left( {n,\beta } \right) = \delta$, such that $\forall n \le \hat{n}$ collusion is the SPE outcome and Cournot competition otherwise.

Proof

$\mathop {\lim }\limits_{{\left( {n,\beta } \right) \to \left( {2, \frac{1}{2}} \right)}} \underline {\delta }^{g} = .5571$ and $\mathop {\lim }\limits_{{\left( {n,\beta } \right) \to \left( {\infty , \frac{1}{2}} \right)}} \underline {\delta }^{g} = 1$, $\mathop {\lim }\limits_{{\left( {n,\beta } \right) \to \left( {2,1} \right)}} \underline {\delta }^{g} = \frac{9}{17}$ and $\mathop {\lim }\limits_{{\left( {n,\beta } \right) \to \left( {\infty ,1} \right)}} \underline {\delta }^{g} = 1$ while $\mathop {\lim }\limits_{{\left( {n,\beta } \right) \to \left( {2, 2} \right)}} \underline {\delta }^{g} = \frac{1088}{{2168}} = .5018$ and $\mathop {\lim }\limits_{{\left( {n,\beta } \right) \to \left( {\infty 2} \right)}} \underline {\delta }^{g} = 1$. Hence, for the above respective values of $\beta$ and the indicated respective ranges of $\delta$, there exists a unique respective $n$, say $\hat{n}_{\beta }$, such that $\underline {\delta }^{g} \left( {n,\beta } \right) = \delta$. If $n \le \hat{n}_{\beta }$, then $\delta \ge \underline {\delta }^{b}$ and collusion is the SPE, else, Cournot competition is the SPE. Q.E.D.

Hence, in both Bertrand oligopoly as well as with the general demand function, the results hold.