## Chamberlain model

- The Chamberlin model assumes that an incumbent firm adjusts to an entrant to get
back to monopoly-level profit
- We start off with an incumbent firm in the industry gains monopoly profit
- A new firm enters, maximizing their profits off of the residual demand
- The new entrant drives down the price, leading to less total profit in the industry
- The incumbent firm notices that by cutting back its output, they can raise total profits to the monopoly level
- These total profits can then be split between the two firms

### Example

Continuing on the former example, we start off with monopoly model. That is we assume that when the entrant first enters, it expects the incumbent to produce the output that a monopolist would have produced (\(q_m=300\)). Then the entrant faces following inverse residual demand curve: \[ \begin{aligned} P &= 100-0.1(q_m) - 0.1(q_E) \\ &= 70-q_E \end{aligned} \]

The entrant’s will produce \[ \begin{aligned} MR_E &= 70-2q_E = 40 \\ q_E&=150 \end{aligned} \]

The market price will then be \[ P = 70-15 = 55 \]

Then the entrant’s profit is

\[
\Pi_E = (P - ATC)q_E = (55-40)150 = 2250
\]

The incumbent’s profit is \[ \Pi_I = (P - ATC)q_I = (55-40)300 = 4500 \]

Notice that due to incumbent’s production, the market price falls down which decreases the incumbent’s profit in half. The total industry profit also decreased from $9000 (monopoly) to $6750 (Chamberlain model).

Chamberlain claimed that the incumbent’s optimal response to the entrant’s output (150 units) is to cut down its production from 300 to 150 so that it could restore the industry equilibrium to the monopoly solution. Then the total industry output is 300 with price equal to $70 (an monopoly solution).

Then the entrant’s profit is

\[
\Pi_E = (P - ATC)q_E = (70-40)150 = 4500
\]

The incumbent’s profit is \[ \Pi_I = (P - ATC)q_I = (70-40)150 = 4500 \]

In Cournot duopoly, there is neither overt nor tacit collusion. Each firm reacts to the other’s output to maximize each firm’s profit. In Chamberlain’s case however, it could be argued that there is a tacit collusion. When incumbent shows a good faith by reducing its output, and the entrant accomodates such gesture and try to not take advantage of the incumbent’s output reduction, then both parties could not be worse off.

### Problems

- If we use game theory to analyze the Chamberlin model, we notice a few problems
- Cutting back production to reach the joint monopoly output is equivalent to accomodating the entrant
- The entrant firm can expand its production to increase its own profit, at the expense of the incumbent’s firm
- At this point, the entrant can make a higher profit by fighting back against the incumbent

Continuing on the previous example, there is a chance that the entrant cheats the incumbent about “tacit agreement” and produce more than 150. This is because with the incumbent’s output reduction, the entrant’s inverse residual demand now becomes \[ P =100-0.1q_I - 0.1q_E = 85-0.1q_E \]

Then the profit maximizing output of the entrant becomes \[ \begin{aligned} MR_E&= 85-0.2q_E = 40 \\ q_E&=225 \end{aligned} \]

This is greater than the tacitly agreed upon output. The total industry output will then be 375 and thus the market price will be 62.5.

Each firm’s profit will be \[ \begin{aligned} \Pi_I &= (62.5-40)150 = 3375\\ \Pi_E &=(62.5-40)225 = 5062.5 \end{aligned} \]

This gives us following payoff matrix

Incumbent | |||
---|---|---|---|

Fight | Accommodate | ||

Entrant | Fight | \(\Pi_E=4000\), \(\Pi_I=4000\) | \(\Pi_E=5062.5\), \(\Pi_I=3375\) |

Accommodate | \(\Pi_E=3375\), \(\Pi_I=5062.5\) | \(\Pi_E=4500\), \(\Pi_I=4500\) |

When both firms decide to fight, they will reach Cournot equilibrium. When one firm decides to accommodate while the other takes advantage of the other firm’s good faith and decides to fight, then the cheater will have higher profit than it might have had if it did not cheat. If both firms decide to accommodate, then they will reach Chamberlain’s analysis.

Looking at the payoff matrix closely, you will notice that the strategy profile of both firms accommodate never happens. The dominant solution for both party is to fight even though each would be better off with an accommodation. This is a classic “prisoner’s dilemma” case.