1.2.7.3 Mathematical approach to profit maximisation

The mathematical approach to profit maximization involves using mathematical techniques, particularly calculus, to analyze the profit function of a firm and determine the level of output that results in the highest possible profit. The key concept in this approach is to find the level of output where marginal revenue (MR) equals marginal cost (MC), as this is the condition for profit maximization.

Here’s a step-by-step mathematical approach to profit maximization:

1. Identify the Profit Function: The profit function represents the relationship between the firm’s output level (Q) and its profit (Π). It is typically expressed as follows:

Π(Q) = Total Revenue (TR) – Total Cost (TC)

Total Revenue (TR) is the product of the firm’s output (Q) and the price of each unit sold (P): TR = P * Q

Total Cost (TC) is the sum of the firm’s fixed costs (FC) and variable costs (VC), which are functions of output: TC = FC + VC(Q)

2. Find Marginal Revenue (MR): Marginal revenue (MR) is the change in total revenue resulting from selling one additional unit of output. In perfectly competitive markets, MR is equal to the price (P) because the firm is a price taker and can sell additional units at the market price. Therefore, MR = P.
3. Find Marginal Cost (MC): Marginal cost (MC) is the change in total cost resulting from producing one additional unit of output. It is calculated as the derivative of the total cost function (TC) with respect to output (Q): MC = dTC/dQ
4. Set MR = MC to Maximize Profit: To maximize profit, the firm should produce the level of output where MR equals MC:

MR = MC

In perfectly competitive markets, where MR = P, this condition simplifies to:

P = MC

5. Solve for the Optimal Output Level (Q*): Once you have the equation P = MC, you can find the optimal output level (Q*) that maximizes profit. Set the marginal cost (MC) equal to the market price (P) and solve for Q:

MC(Q*) = P

6. Calculate Profit: Finally, calculate the profit at the optimal output level (Q*) by plugging the value into the profit function:

Π(Q*) = TR(Q*) – TC(Q*)

Where TR(Q*) is the total revenue at output level Q* (TR = P * Q), and TC(Q*) is the total cost at output level Q* (TC = FC + VC).