Marginal (Calculus)

In calculus and economics, marginal means:

The change in an output caused by adding a very small amount of input.

Mathematically, the marginal value of $f(x)$ is its derivative:

$$\text{Marginal value}=f^{\prime }(x)$$

The derivative measures the instantaneous rate of change:

$$f^{\prime }(x)=\underset{\Delta x\to 0}{\lim }\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

For a small increase $\Delta x$,

$$f(x+\Delta x)-f(x)\approx f^{\prime }(x)\Delta x$$

Therefore, if $x$ increases by approximately one unit,

$$f(x+1)-f(x)\approx f^{\prime }(x)$$

Example: Marginal Cost

Suppose the cost of producing $x$ items is

$$C(x)=100+2x+0.01x^2$$

Then the marginal cost is

$$C^{\prime }(x)=2+0.02x$$

At $x=100$,

$$C^{\prime }(100)=4$$

This means that when the company is producing around $100$ items, producing one additional item costs approximately \4$.

Marginal vs. Average

• Average cost: total cost per item

$$\frac{C(x)}{x}$$

• Marginal cost: additional cost of producing slightly more

$$C^{\prime }(x)$$

Intuition

Marginal does not mean small or unimportant. It means at the margin: what happens when we slightly change the current amount.

Examples:

• Marginal cost: cost of producing a little more
• Marginal benefit: benefit from receiving a little more
• Marginal value of time: value of one additional unit of time
• Marginal revenue: revenue gained from selling a little more

A useful translation is:

Marginal = the effect of one more small increment, given where you currently are.