The Production Function

Understand how firms transform inputs into outputs, and discover the economic laws that govern diminishing returns and productivity.

AP Microeconomics

What Is a Production Function?

A production function describes the relationship between the quantity of inputs a firm uses and the quantity of output it produces. It answers a fundamental question: how much can a firm produce given its resources?

In the short run, at least one input (usually capital) is fixed. The production function then simplifies to $ Q = f(L) $, showing how output changes as the firm hires more workers.

Key Definitions

Total Product (TP)

The total quantity of output produced by a firm: $ TP = Q = f(L) $

Marginal Product (MP)

The additional output from one more unit of input: $ MP_L = \frac{\Delta TP}{\Delta L} $

Average Product (AP)

Output per unit of input: $ AP_L = \frac{TP}{L} $

Why Does It Matter?

1

Hiring decisions: Firms use MP to decide whether adding another worker is worth the cost.

2

Cost structure: The production function directly determines a firm's cost curves (MC, ATC, AVC).

3

Profit maximization: Understanding how output responds to inputs helps firms find the profit-maximizing level of production.

4

AP Exam essential: Production functions underpin nearly every cost and supply topic on the AP Micro exam.

The Law of Diminishing Marginal Returns

As a firm adds more of a variable input (like labor) to a fixed input (like capital), the marginal product of the variable input eventually decreases.

TP, MP, and AP Relationship

1

When $ MP > AP $, average product is rising.

2

When $ MP = AP $, average product is at its maximum.

3

When $ MP < AP $, average product is falling.

The Core Formulas

Total Product

$$ TP = f(L) $$

Marginal Product

$$ MP_L = \frac{\Delta TP}{\Delta L} $$

Average Product

$$ AP_L = \frac{TP}{L} $$

Short Run vs. Long Run

	Short Run	Long Run
Fixed inputs	At least one (e.g., $ K $)	None: all inputs are variable
Function form	$ Q = f(L) $	$ Q = f(L, K) $
Key law	Diminishing marginal returns	Returns to scale
Decision focus	How much labor to hire	What scale of production
Cost curves	MC, AVC, ATC	LRAC (long-run average cost)

Returns to Scale

In the long run, a firm can adjust all inputs. When it scales all inputs by the same proportion, we observe one of three outcomes:

Increasing

Doubling inputs more than doubles output. Specialization and efficiency gains.

Constant

Doubling inputs exactly doubles output. Output scales proportionally.

Decreasing

Doubling inputs less than doubles output. Coordination problems emerge.

How to Apply the Production Function

1

Build the TP schedule: List labor inputs $ (L) $ and corresponding total outputs $ (TP) $.

2

Calculate MP: For each unit of $ L $, compute $ MP = TP_n - TP_{n-1} $.

3

Calculate AP: Divide $ TP $ by $ L $ at each level: $ AP = \frac{TP}{L} $.

4

Identify diminishing returns: Find where MP begins to decline.

5

Connect to costs: When MP falls, MC rises. The production function drives the cost curves.

Example 1: Computing the Schedule

A bakery has a fixed oven (capital) and hires workers (labor). Given the following TP data, compute MP and AP:

$ L $	TP	MP	AP
0	0	—	—
1	10	10	10.0
2	25	15	12.5
3	45	20	15.0
4	60	15	15.0
5	70	10	14.0
6	75	5	12.5
7	75	0	10.7

Example 2: Graphing the Curves

Using the bakery data, let's graph TP, MP, and AP:

Example 3: Connecting to Costs

If each worker costs $ w = \$100 $ per day, we can derive the marginal cost (MC) of producing additional output:

$ L $	$ MP $	$ MC $
1	10	\$10.00
2	15	\$6.67
3	20	\$5.00
4	15	\$6.67
5	10	\$10.00
6	5	\$20.00

The Mirror Image: MP and MC

Common Mistakes

✕ Confusing DMR with DRS

Diminishing marginal returns is a short-run concept (one input varies). Decreasing returns to scale is long-run (all inputs vary). They are not the same.

✕ Negative MP = Negative Output

When $ MP < 0 $, total product falls: adding another worker actually reduces output. TP does not become negative; it just declines.

✕ MP and MC Move Together

They move in opposite directions. Rising MP causes falling MC. Falling MP causes rising MC. The relationship is inverse.

Summary Table

Condition	What Happens	Cost Implication
$ MP $ is rising	Each worker adds more	MC is falling
$ MP $ is at max	Peak productivity per worker	MC is at minimum
$ MP $ is falling	Diminishing returns begin	MC is rising
$ MP = AP $	AP is at its maximum	AVC is at its minimum
$ MP = 0 $	TP is at its maximum	MC approaches infinity
$ MP < 0 $	TP is falling	Firm should never operate here

Question 1 of 5

A firm's total product increases from 40 to 55 units when it hires a 6th worker. What is the marginal product of the 6th worker?

$ MP = \Delta TP / \Delta L = (55 - 40) / 1 = 15 $. Marginal product measures the change in total product, not the total level or the average.

Question 2 of 5

The law of diminishing marginal returns states that:

The law specifically states that marginal product eventually falls when a variable input increases with a fixed input. Choice (a) describes negative MP (a later stage). Choice (d) describes decreasing returns to scale, a long-run concept.

Question 3 of 5

If marginal product is greater than average product, then average product must be:

When $ MP > AP $, the marginal worker is pulling the average up. Think of it like exam scores: if your latest score (marginal) is above your current average, your average rises. AP is at its max only when $ MP = AP $.

Question 4 of 5

When a firm's marginal product of labor is falling, what happens to its marginal cost?

Since $ MC = w / MP_L $, a falling MP (smaller denominator) means a rising MC. The relationship between MP and MC is always inverse.

Question 5 of 5

A firm doubles both labor and capital, and output increases by 150%. This firm is experiencing:

Doubling inputs (100% increase) produced a 150% increase in output: more than proportional. This is increasing returns to scale. Choice (d) is wrong because diminishing marginal returns is a short-run concept with only one input varying.

	Short Run	Long Run
Fixed inputs	At least one (e.g., \( K \))	None: all inputs are variable
Function form	\( Q = f(L) \)	\( Q = f(L, K) \)
Key law	Diminishing marginal returns	Returns to scale
Decision focus	How much labor to hire	What scale of production
Cost curves	MC, AVC, ATC	LRAC (long-run average cost)

Condition	What Happens	Cost Implication
\( MP \) is rising	Each worker adds more	MC is falling
\( MP \) is at max	Peak productivity per worker	MC is at minimum
\( MP \) is falling	Diminishing returns begin	MC is rising
\( MP = AP \)	AP is at its maximum	AVC is at its minimum
\( MP = 0 \)	TP is at its maximum	MC approaches infinity
\( MP < 0 \)	TP is falling	Firm should never operate here