# What was that about tacit collusion?

Yesterday when discussing why supermarkets may discount Nurofen during cold season I put to the side the possibility that it was the result of a collapse in tacit collusion between Nurofen providers. My reason for ignoring this explanation was purely selfish – I was tired and that explanation required more thought than I had capacity for 🙂 . However, today I will attempt to shed some light on the tacit collusion explanation, even though my capacity is still extremely limited 😉 .

The tacit collusion argument that supports the Nurofen case was originally provided in this paper by Rotemberg and Saloner. In the paper they illustrate that, when demand fluctuates and the level of market demand is observable, tacit collusion will break down when we are in “high demand” states.

High state defection model

If you have done this model before – or have done any maths just look at the paper for a clear indication. Also if you have access to “The Theory of Industrial Organization” by Tirole go to page 248 instead of reading my attempt to explain it with virtually no maths 🙂

Think of it this way. Assume that there are two states of the world Dh, where demand for the product is high, and Dl, where demand for the product is low. Furthermore, assume that the game is played over an infinite horizon, and that both firms are playing “grim trigger” strategies – which is where any deviation from the collusive outcome will be punished until the end of time (for those that have done some game theory, this assumption is subgame perfect as it involves playing the static nash equilibrium repeatedly, but it is still a big assumption. However, like perfect competition it provides us with one extreme which we can then move away from to incorporate more realistic assumptions).

Given this, we have to think about how the firms will behave in the two different states of the world.

Say that our firm is in the low demand state. If we “deviate” from the collusive agreement we could get some payoff from doing so – deviate(L). This static payoff is greater than the payoff from colluding – deviate(L)>collude(L), implying that if this is a one period game, this is what they would do if they expect the other person to collude.

However, this is a infinitely repeated game, which implies that they also think of the payoff they receive when playing this game repeatedly in the future, as a result they wish to maximise the whole value of the game. Now, they have some discounted value of the future when they do not deviate, and a discounted value of the future when they do deviate – deviate(f) and collude(f). These do not need state subscripts as they are the same in either state.

There would be no point colluding if the deviation game gave a higher payoff, so we have that collude(f)>deviate(f). Now the high state game will also have pretty looking bits – deviate(H)>collude(H).

Now, in the low state game the firm will only collude if:

collude(L)+collude(f)>deviate(L)+deviate(f)

Similarly for the high state, collusion holds if:

collude(H)+collude(f)>deviate(H)+deviate(f)

From here we can tell that the incentive to deviate (the potential for a collapse in tacit collusion) is higher in the high state of the world. How?

Take a simple bertrand game. Here when you deviate you take all the surplus, but if you collude you only take half the surplus. As the surplus increases with the level of demand the gain from deviating in the high demand state (deviate(H)-collude(H)) is greater than in the low demand state (deviate(L)-collude(L)). This result generalizes for standard assumptions.

As the reward for deviating is higher in the high demand state, firms are more likely to deviate in high demand states.

The low demand deviation

Now the previous model made sense – however, the common train of thought at the time was that price wars occurred during recession, not during booms. One reason given for this was liquidity constraints – if a recession smashes the firms cashflow, they may be willing to trigger a price war in order to get some funds in the door. However, this explaination didn’t seem terribly satisfying.

Before Rotemberg and Saloner, in 1983, another couple of guys called Green and Porter, wrote about price wars.

In their model, demand fluctuated, but the source of the demand fluctuation was unobservable. This raised the question, did my opponent defect on me or are we in a recession? The introduction of this uncertainty is sufficient to get the result of low demand causing price wars.

Why? Well, when playing your strategy you want to be able to punish your opponent for deviating. Now if you can’t tell if they deviated or if the economy slowed punishing them might still be optimal, if it can help to sustain collusion in the high demand states (by illustrating that you as a firm have a credible threat). In this case, price wars will begin even though no-one is deviating, and they will be used to ensure that collusion is kept during the good times.

So supermarkets …

Which model is more descriptive of the supermarket industry? Well I don’t think that supermarkets are particularly liquidity constrained, they pay alot of money for reports on the state of the economy, and they tend to spy on each other – implying that they will know if there is some deviation going on.

As a result, I think the Rotemberg-Saloner model better describes the supermarket industry, and as a result it is entirely possible that the cheap Nurofen in the face of lots of sick people could be the result of a price war over Nurofen. However, looking at the different explanations, I still believe that the complementarity argument is the strongest argument for supermarkets price cutting behaviour in this case.

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