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For constant-elasticity demand the contribution-maximising price is the Lerner rule: price equals variable cost multiplied by E/(E-1), where E is the absolute elasticity. Volume is then projected with the constant-elasticity curve so you can see the profit trade-off rather than just the price.
Optimal price (Lerner)
P* = Variable cost x E / (E - 1), for E > 1
Projected volume
Q* = Q0 x (P*/P0)^-E
There is no interior optimum — the model says raise price indefinitely. In reality a competitive ceiling or a value cap binds first, so treat low elasticity estimates with caution.
No. Fixed costs do not change the optimal price; they only determine whether the resulting contribution is enough to be viable.