I need to solve the following equation, but I'm not sure how to proceed.
Profit cartel/ (1-δ) ≥ profit deviation + δ(profit Nash eq. / (1-δ))
I've got the following info:
Profit cartel = ((a-c)^2)/8b
Profit Nash eq. = ((a-c)^2)/9b
Profit deviation = (9(a-c)^2)/64b
How do I solve for δ?
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Answers & Comments
Verified answer
Note that you don't have an equation, but an inequality.
Not necessary, but let k = (a-c)²/b. Then
Profit cartel = k/8
Profit Nash eq. = k/9
Profit deviation = 9k/64
Then k / 8(1-δ) ≥ 9k/64 + δk / 9(1-δ)
Assuming δ < 1 and b > 0, we can multiply through by 576(1-δ)/k and not flip the inequality:
72 ≥ 81(1-δ) + 64δ = 81 - 81δ + 64δ = 81 - 17δ
17δ ≥ 9
δ ≥ 9/17 ≈ 0.53