Profit maximization for the price-taking firm (short run)
NOTES

Profit = total revenue – total cost
Profit will be maximized when the difference between TR and TC is the greatest.
In other words, at max-profit output, Qπ, marginal profit (i.e., ∆profit/∆output) is equal to zero.

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 price taker
When a firm can take the prevailing market price as constant, the firm is known as a price taker. For example, a firm in a perfectly competitive industry is a price taker.

When the seller can sell as much as he wants at a constant price, the total revenue curve (TR) is a straight line from the origin.

Since price is constant, MR is equal to price (P) because:
By definition, MR = P + (∆P/∆Q)*Q
i.e., MR is the additional revenue from selling one more unit at the new price minus the lost revenue from selling all units at the lower new price.
Since (∆P/∆Q) is zero for a price taker because price does not change with output,
MR = P

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

Measuring profit with P (Price) and ATC under horizontal demand
With TR and TC, maximum profit is simply the vertical distance between the two curves at the maximum-profit point.

With P and ATC, the distance between P and ATC represents profit per unit output.
Maximum total profit = Qπ * (P – ATC)

When P is equal to minimum ATC, total profit is zero.

When P is equal to minimum AVC, TR covers only TVC, with nothing left over to cover fixed cost. Since fixed cost is irrecoverable sunk cost in the short run, the firm is indifferent between staying in business and quitting altogether. Therefore, minimum TVC is known as the shut down point for a perfectly competitive firm.

 
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