Profit Maximization in Microeconomics, Summaries of Calculus

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Intermediate Microeconomics

PROFIT MAXIMIZATION

BEN VAN KAMMEN, PHD

PURDUE UNIVERSITY

Profit

Profit (Π) : The amount by which a firm’s revenues exceed its costs.

Revenue: (TR) The amount that the firm receives for the sale of its output.

Π = 𝑇𝑇𝑇𝑇 − 𝑇𝑇𝑇𝑇

Both TR and TC depend on the quantity ( q ) of output the firm produces. ◦ The firm chooses an output level such that Π is maximized.

Profit maximization

Firms can potentially have increasing returns to scale over a range of output. ◦ Think of a very small firm that doubles it scale and takes advantage of additional specialization by its workers and tools by more than doubling output.

But eventually as firms get large, they experience decreasing returns to scale. ◦ And, hence, increasing marginal costs.

The firm will choose L and K in such a way as to minimize TC for any given level of output.

TC, TR, Π (graphically)

MR = MC

Considering what you know about calculus, this should come as no big surprise.

The profit maximizing output level can be identified by the level at which marginal profit equals zero , i.e., ◦ the slope of the profit function is flat.

Since the profit function is just comprised of the Revenue and Cost functions, its derivative is the MR minus the MC: 𝜕𝜕Π 𝜕𝜕𝜕𝜕

=

𝜕𝜕𝑇𝑇𝑇𝑇 𝜕𝜕𝜕𝜕

−

𝜕𝜕𝑇𝑇𝑇𝑇 𝜕𝜕𝜕𝜕

= 𝑀𝑀𝑇𝑇 − 𝑀𝑀𝑇𝑇.

When you set this to zero to get the optimal q , you get: 𝑀𝑀𝑇𝑇– 𝑀𝑀𝑇𝑇 = 0 → 𝑀𝑀𝑇𝑇 = 𝑀𝑀𝑇𝑇.

Example

Say that Revenue and Costs are given by: 𝑇𝑇𝑇𝑇 = 9 𝜕𝜕 and 𝑇𝑇𝑇𝑇 = 3𝜕𝜕^2.

The Profit function, then, is: Π = 𝑇𝑇𝑇𝑇– 𝑇𝑇𝑇𝑇 = 9 𝜕𝜕 − 3 𝜕𝜕^2.

Differentiating it with respect to q : ◦ Marginal Profit = 9 − 6𝜕𝜕.

Setting it equal to zero to solve for optimal q :

9– 6 𝜕𝜕 = 0 → 𝜕𝜕∗^ =

9 6 = 1.5 units.

Profit level

Finally it is possible to substitute the optimal quantity back into the profit function to solve for the firm’s profit level.

Π 𝜕𝜕∗^ = 9 1.5 – 3 1.5 2 = $6.75.

This is the amount of profit the firm makes if it chooses its output level optimally.

Summary

The firm’s objective is to maximize profits.

Profit is defined as Revenue minus Cost.

Profit is maximized when the firm chooses an output level at which marginal revenue equals marginal cost.