Question 2
In the game from question 1, suppose that both firms adopt a
tit-for-tat strategy. They initially collude. In future periods,
a firm colludes if its competitor did in the previous period,
and elects the lower price if its competitor cheated in the previous period.
What has to be true about the interest rate (r) for collusion to be sustainable?
SOLUTION:
If the firms employ a tit-for-tat strategy, then there are two types of
"cheating" we need to consider. The first involves a firm cheating in every period.
In order to prefer collusion to cheating in every period, we require r < 1/2.
This is equivalent to the solution for question 1. The second type of cheating
involves a firm cheating once, then selecting the high price in the second period
in order to regain the collusive outcome.
| PV(collude) | = 10 + 10/(1+r) + 10/(1+r)2 + 10/(1+r)3 + ... |
| PV(cheat) | = 20 + 0/(1+r) + 10/(1+r)2 + 10/(1+r)3 + ... |
In order for collusion to be sustainable, we must have PV(collude) > PV(cheat)
- 10 + 10/(1+r) > 20 + 0/(1+r)
- 10/(1+r) > 10
- 1+r < 1
- r < 0
But r<0 appears nonsensical. Recall that r represents the interest earned on our investment.
If r is less than zero, that implies that we expect our money to depreciate
quite quickly. Another way of interpreting this is to think that for collusion to be
sustainable, we have to value $1 a year from now more than a $1 today.
Why do we get this strange result. In this game, collusion results in $10
every year. On the other hand, I can adopt a strategy in which I cheat today,
agree to suffer some losses next year, and then resume colluding.
The tradeoff that I face is earning $10 this year and next, or $20
this year and $0 the next. Obviously, the latter is more preferable regardless
of the interest rate.
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