1) Profit functions are convex in output price that is :
π (tp + (1- t)p′) ≤ tπ(p) + (1- t)π(p′)
Intuitively this implies that if price of output increases by one unite the profit will rise by
exactly or more than one unite.
Proof.
Let p′′ = tp + (1- t)p′ and y, y′ and y′′ maximise profits at p, p′ and p′′ respectively, then
π (tp + (1- t)p′y) = (tp + (1- t)p′) y′′= tpy′′ + (1- t)p′y′′
By definition of profit maximization we have
tpy′′ ≤ tpy = tπ(p) and (1- t)p′y′′ ≤ (1- t)p′y′ = (1- t)π(p′)
Hence, π (tp + (1- t)p′y) ≤ tπ(p) + (1- t)π(p′)