By definition, marginal profit = marginal revenue – marginal cost.
When marginal profit = zero at the max-profit output Qπ
marginal revenue (MR) = marginal cost (MC)

Maximum-profit output for single-price searcher
When a firm must search for a single price for each output level, it is known as a single-price searcher. For example, a firm under monopoly or monopolistic competition that does practice price discrimination is a single-price searcher.

The familiar straight-lined downward-sloping demand curve generates a dome-shaped TR.

If the single price (P) must be lowered to sell more, MR is below P for every output because:
By definition, MR = P + (∆P/∆Q)*Q

Since (∆P/∆Q) < 0 because a single-price searcher must lower price to sell more,
MR > P

For a straight line demand curve, MR is half the value of P.

For a single-price searcher, therefore, at its max-profit output Qπ, P > MR = MC.

Measuring profit with P (Price) and ATC under downward-sloping demand curve
Single-price searchers must charge a price higher than MC because MR (additional revenue for selling an extra unit) is below P.

Even with P above MC, positive profit is not guaranteed:
• When P is equal to ATC, total profit is zero.
• When P is higher than ATC, total profit is positive.
• When P is lower than ATC, total profit is negative.

Single-pricing under downward-sloping demand curve
Because P > MR under downward-sloping demand
and MR = MC is the condition for profit maximization,
P > MC when profit is maximized.

If a single-price searcher is forced to price its output equal to MC where MR=MC, there will be excess demand and loss if MC is below ATC (or lower profit if MC is above ATC).

If a single-price searcher is forced to price its output equal to MC where MC intersects the demand curve, there will be lower profit if MC is above ATC or greater loss if MC is below ATC.