Unit 3 · Topic 3.1

3.1 The Production Function

A production function links a firm's inputs to its output. In the short run, when some inputs are fixed, adding more of a variable input like labor eventually brings diminishing marginal returns. That one idea explains the shape of the cost curves in the rest of the unit.

Key terms

production function
total product
marginal product
average product
diminishing marginal returns
fixed input

Short run and long run

The short run is a period in which at least one input is fixed. Usually that's capital: a restaurant can hire more cooks this week, but it can't build a bigger kitchen. The long run is a period long enough that the firm can change every input, including the size of its building. These aren't set lengths of time; they depend on the industry.

A production function shows the most output a firm can make from each combination of inputs. In this topic, capital is fixed and labor is the variable input.

Total, marginal and average product

Marginal product often rises at first. With only one or two workers in a big kitchen, adding another lets them specialize: one chops, one cooks, one plates. Each new worker adds more output than the last. That stage is called increasing marginal returns.

Measure	What it is	How to calculate it
Total product (TP)	Total output	Read it from the table
Marginal product (MP)	Extra output from one more worker	change in TP ÷ change in workers
Average product (AP)	Output per worker	TP ÷ number of workers

Diminishing marginal returns

Sooner or later, the fixed input gets crowded. The kitchen has only so many stoves. Each extra cook still adds output, but less than the cook before. This is the law of diminishing marginal returns: as you add more of a variable input to a fixed input, the marginal product of the variable input eventually falls.

Diminishing marginal returns don't mean total output falls. TP keeps rising as long as MP is positive; it just rises more slowly. If you add so many workers that they get in each other's way, MP can become zero (TP stops rising and is at its maximum) or negative (TP falls). No firm would knowingly hire a worker with negative marginal product.

Marginal and average: the grade rule

Marginal and average always relate the same way. If your next test score is above your average, your average rises; if it's below, your average falls. In the same way, when MP is above AP, AP rises, and when MP is below AP, AP falls. So the MP curve crosses the AP curve at AP's highest point.

On a graph with output on the vertical axis and workers on the horizontal axis, TP rises steeply at first, then more and more gently, levels off at its peak where MP = 0, and may turn down. On a separate graph below it, MP rises, peaks and falls, crossing zero where TP peaks, while AP is a hill that MP cuts through at its top.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Marginal and average product from a table
A bakery with one oven has the following total product (loaves per hour) for 0 to 7 workers: 0, 10, 25, 36, 44, 48, 48, 46. Find the marginal and average product of each worker. When do diminishing marginal returns begin?
Show the solutionHide the solution
1. Step 1: MP is the change in TP: worker 1 adds 10, worker 2 adds 15, worker 3 adds 11, worker 4 adds 8, worker 5 adds 4, worker 6 adds 0, worker 7 adds −2.
2. Step 2: AP is TP ÷ workers: 10, 12.5, 12, 11, 9.6, 8, about 6.57.
3. Step 3: MP rises from 10 to 15 with the second worker (increasing marginal returns), then falls starting with the third worker (11 < 15). Diminishing marginal returns begin with the third worker.
4. Step 4: Check the grade rule: worker 2's MP (15) is above the old AP (10), so AP rises to 12.5; worker 3's MP (11) is below 12.5, so AP falls to 12.
Answer: MP: 10, 15, 11, 8, 4, 0, −2. AP: 10, 12.5, 12, 11, 9.6, 8, 6.57. Diminishing marginal returns begin with the third worker.
Example 2
Diminishing vs. negative returns (classic trap)
Using the same bakery, a student says: 'Diminishing marginal returns start with the third worker, so after that, hiring more workers lowers output.' Is the student right? At what point does total product actually stop rising?
Show the solutionHide the solution
1. Step 1: Diminishing returns mean each extra worker adds less, not that output falls. Workers 3, 4 and 5 still add 11, 8 and 4 loaves, so TP keeps rising from 25 to 48.
2. Step 2: Worker 6 adds 0, so TP stays at 48: this is TP's maximum.
3. Step 3: Only worker 7, with MP = −2, makes total output fall (from 48 to 46).
Answer: No. TP rises through the fifth worker, peaks at 48 loaves with 5 or 6 workers (MP of the sixth is zero), and falls only with the seventh worker.

Common mistakes

Thinking diminishing marginal returns mean total output falls. TP falls only when MP is negative.
Confusing diminishing marginal returns (short run, one input varies) with diseconomies of scale (long run, all inputs vary).
Calculating MP as TP ÷ workers. That's average product; MP is the change in TP from one more worker.
Saying diminishing returns start where MP becomes negative. They start at the first worker whose MP is lower than the previous worker's.

On the exam

Expect a table of workers and output; you'll be asked for marginal or average product, or where diminishing marginal returns begin.
Questions about why marginal cost rises usually want the answer 'diminishing marginal returns', so connect this topic to topic 3.2.

Connected topics

Videos

Diminishing Returns and the Production Function- Micro Topic 3.1
Jacob CliffordWatch on YouTube (opens in a new tab)
Micro 3.1 The Production Function
ReviewEconWatch on YouTube (opens in a new tab)
Introduction to production functions | APⓇ Microeconomics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
Micro: Unit 3.1 -- Marginal Product and Diminishing Returns
You Will Love EconomicsWatch on YouTube (opens in a new tab)
Total product, marginal product and average product | APⓇ Microeconomics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)