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Main content
Current time:0:00Total duration:9:20
PRD‑3 (EU)
PRD‑3.C.6 (EK)
PRD‑3.C.7 (EK)

Video transcript

in the last video we saw how there could be an industry that has two firms the duopoly and if those two farms coordinate they could behave as a monopolist and they could optimize their collective economic profit and in the last video we saw that would happen when they produced 50 units per period and they could split it assuming these were two identical firms by each producing half of it in the case of the last video it was 250 units per firm but then we saw that there was an incentive to cheat that by producing extra units the margin from a markets point of view the marginal economic or the economic profit on those incremental units would be negative so the whole economic profit would get would shrink a little bit as you produce units beyond that but the cheater would get a bigger chunk of those units or the bigger chunk of that economic profit and so the cheater could actually gain go from $250 per time period to 280 and it would be all at the expense of the non-cheater who and and then some who would lose even more than what the cheater gained and obviously the who is initially the non-cheater has an incentive now to cheat and they'll both keep increasing they'll both keep increasing production so that if they've wanted to keep doing this one-upsmanship and so they'll both they both have the incentive to keep going assuming that they don't hold to their cartel agreement until you get to a quantity until you get to a quantity where the where there is no economic profit left so right over here the way I've drawn it the demand curve intersects the average total cost curve right over here and there is no economic profit left we're producing a good quantity it looks like it's about 75 units combined 75 units for the whole market but at this point the market price the market price is equal to the average total cost and so there's no economic profit per unit on average now what I want to do is think about this in kind of a game theoretic way so let's look at a bunch of states and so this is the optimal state that we are starting off in and you can actually call it the Pareto optimal state Pareto optimal Pareto optimal State named after vilfredo pareto and all it means is that's the state where there's no other state where you can make someone better off without making the other person worse off so any of the states here there are states for example where blue is better off so for example in this state right over here blue is better off but green is worse off so that's why it's called Pareto Pareto optimality now what I want to think about is how these these characters will change their state due to their incentives and then we'll talk a little bit about Nash equilibrium as well so on this axis on the up here let's say that this is one of the competitors this is where they produce 25 and let's say on the ultimate cheating quantity of 75 where and this is somewhat close to kind of the market or that is the equilibrium quantity if this was a completely and this was perfect competition they produce half of that so this is then producing 37 and a half units and as we go from 25 to 30 seven and a half units they are cheating more so this is more more cheating and over here this was no cheating and we can do the same thing for the blue player and I'll write them as B this is them producing 25 this is them producing 37 and a half and as we go up and up and up they are cheating more so this is a lot of cheating or more more cheating so to think of it in the game theoretical way this is the Pareto optimal state right over here and it's it's optimal in many ways this is they've maximized the total economic profit here there's no other state that one person would benefit without making the other worse now let's think about whether this is a Nash equilibrium so let's remind ourselves what Nash what Nash equilibrium was this was a state where holding other all the other players constant so in this case there's only one other player holding others others constant others constant a player can't gain by changing strategy player player can't gain by changing strategy in this case changing strategy is changing your output by changing changing strategy so let's see if that is true of this state right over here well let's hold a constant if a is constant where in this column right over here is there something B can do is there change or strategy B can do that would allow B to game well sure B can increase production that's what we saw in the last video so we would go from this bottom right state to 1 right above it now B's economic profit is 280 a is 200 the pie has shrunk but B has got a larger chunk of it so that was not a Nash equilibrium there is there is holding all others constant of play there is a player that can gain the by changing their strategy and this is by holding the Nash equilibrium definition just to make sure they say it's a state we're holding others constant no player no player can gain by genic strategy we just show that at least one player can gain by changing strategy holding others constant and the same would be true if we went the other way around if we held B constant at 25 a could gain by changing his strategy could go right over there so this is not a Nash equilibrium so this is not a Nash equilibrium and then regardless of what state we go to if we go to this state it's still not a Nash equilibrium if we hold a constant B can improve by increasing his production or if we hold B constant then a can still improve by cheating even more so none of these are Nash equilibriums from any one of these state if you hold a constant B could produce more or if you hold B constant a could produce more and get and get some gain over here a is going from 130 to 160 and getting some gain and you could imagine this keeps happening incrementally they keep producing more and more and more we kind of go there then we go there then maybe we go there then we go there then then maybe a cheat some more then be cheat some more than a cheats a little bit more B cheats a little bit more maybe a little bit more past that then a cheats a little bit more the whole time the whole economic profit PI which is the sum of a and B gets getting smaller and smaller until finally a finally cheats and they are at zero economic profit now let's think about whether this is a Nash equilibrium clearly they won't want to move backwards if you hold a constant B would not want to go from would not want to move down then he would lose economic profit so that's that doesn't work he doesn't gain by doing that if you old B constant a wouldn't want to move to the right a would also lose economic profit now you might say well what if they produced beyond thirty seven and a half why can't they keep producing and go beyond there well if they if be holding a constant if B were to produce more than thirty seven and a half from this state right over here then the total pie will get negative and it doesn't matter if he's getting a larger or smaller chunk of that pie B's chunk is going to be negative he's going to drive down the price even more and you can see it over here if they increase quantity beyond this market quantity of seventy five thirty seven and a half each if we go beyond that the the price the the price that they would be selling at at that quantity over there is lower than the average total cost and so you're going to be the total economic the average economic profit per unit is going to be negative there will be a total of negative economic profit so neither of them will want to produce more from this state either so all of a sudden in this top left state holding others constant B if you hold a constant B can't gain by changing his strategy and if you hold B constant a can't gain by changing his strategy and so we are up here we are up here in a Nash equilibrium this is a Nash equilibrium and like the prisoner's dilemma equilibrium it was not the optimal state the optimal state was here but because they both wanted to cheat they both want to do this one-upsmanship they both broke their contracts they could end up in this state over here but this state is stable there's nothing holding the other party equal there's nothing that they could do to change to to optimize now what they could do and this is not what national applies to they could say oK we've been really ruining each other's business let's go coordinated n and I'm going to decrease production if you decrease production now that is not and they could maybe try to go back to this date and that does not mean that this is not a Nash equilibrium because by coordinating again we're not holding the others constant we're like we're saying I'm changing my strategy while you're changing your strategy and so maybe only through another agreement they could go over here but that still doesn't mean that this is not a Nash equilibrium this is a Nash equilibrium if you if there's no coordination if you hold one player constant the other player cannot change their strategy or change their production for a game