Tough game theory question

[veganmav]

Let us consider a homogeneous good market. The yearly (inverse) demand function is P = a - Q where a is a positive constant and p and Q are re*spectively the market price and quantity for that year. Two identical firms, each with zero fixed costs, zero marginal costs and identical discount fac*tor X epsilon (0,1) are trying to set up a cartel in this market where the fims would try to share the monopoly profits equally each year. We restrict our attention to the grim trigger strategy. Any deviation from the coopera*tive arrangement would mean that firms would play the stage game Nash equilibrium every year in the future.
Consider two different version of this infinitely repeated game. In the first version, price is the strategic variable. In the second version, quantity is the strategic variable. In price competition, we assume that usual conditions hold, namely (a) customers buy only from the firm with lower price, and (b) if they charge an identical price, they split the market equally.
Which version of these competitions (Le., price or quantity) is better to sustain the cartel? Why?

[econophile]

it seems that price competition is going to best incentivize cooperation, because the difference in profits between the cooperative outcome and the Nash under price competition is greater than that under quantity competition. that is because price competition results in the perfectly competitive price and quantity for even two firms, whereas quantity competition results in some price above the perfectly competitive one and allows firms to earn profits. there could be something i'm missing.

[Alan3]

I'm am sure I don't have enough background to answer this correctly but I want to learn so I am going to make a lot of obvious statements and ask some stupid questions.

If the two firms form a cartel then they want the total profit from the market to be maximized. Does this mean that the cartel should offer a P=Q=.5a to maximize revenue? There are a lot of terms in the question that I don't know how to fit into the analysis.

Let a=2 for the rest of the argument.

A firm could cheat by lowering the price slightly. Instead of bringing in $0.5 (.25*a*a/2) in revenue, they could set the price at .4 and bring in $0.64 (.16*a*a/1). Then the other guy could lower it to .3 and make .36 instead of zero. This would lead to an equilibrium strategy where both companies sell at zero and both make zero.

Let's say that they both agree to set a fixed cost. Now both companies are responsible for managing their own quantity so that they do not lose money. If they both under produce then they both sell all of their product below market value. If one under produces and the other over produces, then the over-producer gets to use some of the other guy's market shortage makes more money. If they both over produce then they both don't sell some of their product and presumably lose money.

But there may be another side to over producing. Say that consumers pick a random supplier based on what is available in the market. If producer A makes N units and producer B makes 2N units and a consumer selects product randomly from the market, then producer B has twice the chance of selling a single unit than producer A. That means if you produce more you sell more. There seems to be an advantage to flooding the market with your product so sell more (although you have extra left for the next season). Although this argument violates one of your assumptions.

What if they collude using quantity? They both agree to produce a/2 units and the market sets the price at a/2. If the price drops below the expected cartel price then the non-cheating producer will know that the other is producing more and can immediately drop his price to equilibrium.

I think colluding on quantity is more stable if your (b) assumption is violated.

Based on strict interpretation of your assumptions I am not sure there is a difference.

[bobman0330]

[ QUOTE ]
it seems that price competition is going to best incentivize cooperation, because the difference in profits between the cooperative outcome and the Nash under price competition is greater than that under quantity competition. that is because price competition results in the perfectly competitive price and quantity for even two firms, whereas quantity competition results in some price above the perfectly competitive one and allows firms to earn profits. there could be something i'm missing.

[/ QUOTE ]

I think this is right. For quantity competition, the NE is the Cournot solution, which is still a premium above 0 profit. Price competition has the NE at 0 profit. So, the future downside to defecting in an earlier round is higher, so greater deviations from the NE should be possible in the price version. I refuse to do all the math that would be necessary to show that though.