The Production Function: Study Guide (Microeconomics 3.1)

From inputs to output

Unit 2 looked at markets from the outside: buyers, sellers, prices. Unit 3 looks inside the firm. Before a firm can decide how much to supply, it has to know how much it can produce with the resources it hires. That link is the production function.

A production function shows the maximum output a firm can produce from each combination of inputs (labour, capital, land and raw materials), with the technology it has. It is a physical relationship: it is measured in units of output, not in money. Costs come in the next topic.

Throughout this guide we use Harbourline Bakery, a fictional firm that bakes bread. It has 2 ovens (its capital) and can hire as many bakers as it likes. Output is measured in loaves per hour.

Short run and long run

Economists do not define the short run and the long run by the calendar. They define them by which inputs can change.

Period	What can change	Example at Harbourline
Short run	At least one input is fixed; others are variable	The bakery can hire more bakers today, but buying and installing a third oven takes months
Long run	All inputs are variable; nothing is fixed	The bakery can choose any number of ovens, a bigger building and any number of bakers

In this course the usual short-run story has labour as the variable input and capital as the fixed input. The production function exists in both periods. In the short run it tells you what happens as you add workers to fixed capital (this topic). In the long run it tells you what happens when the firm changes the scale of everything at once (Topic 3.3).

Total, marginal and average product

Three measures describe a short-run production function.

Measure	Meaning	Formula
Total product (TP)	Total output from a given number of workers	read from the table
Marginal product (MP)	Extra output from one more worker	MP = ΔTP ÷ ΔL
Average product (AP)	Output per worker	AP = TP ÷ L

Here L is the quantity of labour (number of workers) and Δ means “change in”. When workers are added one at a time, ΔL = 1, so MP is just the change in TP. Because they are measured per worker, MP and AP are often called the marginal product of labour (MPL) and the average product of labour (APL).

Harbourline Bakery, with its 2 ovens fixed:

Workers (L)	Total product (loaves per hour)	Marginal product	Average product
0	0	–	–
1	12	12	12
2	30	18	15
3	51	21	17
4	64	13	16
5	75	11	15
6	81	6	13.5
7	84	3	12
8	82	−2	10.25

Marginal product sits between two rows: the 13 loaves in the L = 4 row are what the fourth worker adds when L goes from 3 to 4. That is why MP is plotted halfway between worker numbers on a graph.

How marginal product changes: three stages

Read down the marginal product column and you see three stages.

1. Increasing marginal returns (workers 1 to 3). MP rises: 12, 18, 21. With only one baker, that person must mix, shape, bake, clean and serve. With two or three, they specialise: one mixes, one shapes, one runs the ovens. Each works on fewer tasks, wastes less time switching, and gets faster. The ovens, mostly idle with one baker, are now used more fully.
2. Diminishing marginal returns (workers 4 to 7). MP falls: 13, 11, 6, 3. MP is still positive, so total product still rises, but each extra baker adds less than the one before. The reason is the fixed input. There are still only 2 ovens and one kitchen. Extra bakers wait for oven space and get in each other’s way. Each worker has less capital to work with.
3. Negative marginal returns (worker 8). MP is −2. The kitchen is so crowded that adding a baker actually lowers output. No sensible firm would hire this worker, even if they worked for free.

Diminishing marginal returns happen when a firm adds more and more of one input while holding at least one other input constant. It is a short-run idea. The workers are not worse; the eighth baker could be the most skilled person in the building. MP falls because each additional worker has a smaller share of the fixed ovens and space.

Diminishing marginal returns begin where MP starts to fall (the fourth worker here), not where MP becomes negative and not where TP starts to fall.

The three curves and how they are linked

Figure 1 shows TP. Its slope between two worker numbers is the marginal product, so:

• while MP rises, TP gets steeper (workers 1 to 3);
• while MP falls but is positive, TP still rises but gets flatter (workers 4 to 7);
• where MP falls to zero, TP is at its maximum (84 loaves at 7 workers: the seventh worker still adds 3, and an eighth would subtract 2);
• when MP is negative, TP falls (worker 8).

Figure 2 shows MP and AP together. Two rules link them:

• When MP is above AP, AP rises. The second worker adds 18 loaves, more than the average of 12, so the average goes up to 15. The third adds 21, more than 15, so the average rises to 17.
• When MP is below AP, AP falls. The fourth worker adds 13, less than the average of 17, so the average falls to 16.

So MP crosses AP at the maximum of AP. In Figure 2, MP cuts through AP from above around the third worker, where AP peaks at 17.

A school example helps. If your average test score is 70 and your next test scores 80 (the “marginal” score), your average rises. If the next test scores 60, your average falls. The marginal score pulls the average towards it. It does not matter whether the marginal score itself is rising or falling: the fourth worker’s MP has already started to fall (21 to 13), but AP falls only because 13 is below the average.

Note also that AP is always positive while TP is positive, even when MP is negative: 82 loaves ÷ 8 workers = 10.25.

What shifts the production function

The table and curves assume fixed technology and fixed amounts of the other inputs. If either changes, the whole production function changes.

• More of the fixed input (a third oven): every number of bakers can now make more bread, so TP shifts up, and MP is higher at most levels of labour. Adding capital is a long-run change, after which the firm has a new short run.
• Better technology or better-trained workers (a faster mixer, a training course): TP and usually MP and AP shift up. Productivity has risen.
• Losing some of the fixed input (an oven breaks down): TP shifts down, and diminishing marginal returns usually set in after fewer workers.

Worked example 1: calculating marginal and average product

Question. Use the Harbourline Bakery table. (a) Calculate the marginal product of the third worker and of the fifth worker. (b) Calculate the average product of 4 workers and of 6 workers. (c) With which worker do diminishing marginal returns begin? Explain how you know. (d) Should the bakery ever hire an eighth worker? Explain.

(a) MP of a worker = TP with that worker − TP without them.

1. Third worker: 51 − 30 = 21 loaves per hour.
2. Fifth worker: 75 − 64 = 11 loaves per hour.

(b) AP = TP ÷ L.

1. Four workers: 64 ÷ 4 = 16 loaves per worker per hour.
2. Six workers: 81 ÷ 6 = 13.5 loaves per worker per hour.

(c) MP rises from 12 to 18 to 21, then falls to 13. The first worker whose MP is lower than the previous worker’s is the fourth. Diminishing marginal returns begin with the fourth worker. Total product is still rising there (51 to 64): diminishing returns are about the extra output, not the total.

(d) No. The eighth worker’s MP is 82 − 84 = −2 loaves per hour. Hiring them lowers total output, so the bakery would make less bread and pay an extra wage. Even at a wage of zero, it is better not to hire.

Check. MP is a change (“per extra worker”); AP is a ratio (“per worker”). Never divide TP by L to get MP, and never subtract to get AP.

Worked example 2: from marginal product back to total product

Question. Tallow Fields is a fictional orchard with a fixed 10 hectares of land. The marginal product of each extra apple picker, in kilograms of apples per hour, is:

Picker	1st	2nd	3rd	4th	5th	6th	7th
Marginal product (kg per hour)	40	50	45	35	20	10	0

(a) Find total product and average product for 1 to 7 pickers. (b) Where is average product highest? How does this relate to marginal product? (c) At what number of pickers is total product at its maximum? (d) The orchard buys a second tractor to carry crates. With 5 pickers, output rises from 190 kg to 250 kg per hour. Calculate the change in average product, and describe what has happened to the production function.

(a) Total product is the running sum of the marginal products. Average product is TP ÷ L.

Pickers	TP (kg per hour)	AP (kg per picker per hour)
1	40	40
2	40 + 50 = 90	45
3	90 + 45 = 135	45
4	135 + 35 = 170	42.5
5	170 + 20 = 190	38
6	190 + 10 = 200	33.33
7	200 + 0 = 200	28.57

(b) AP is highest at 45 kg, with 2 and with 3 pickers. The third picker’s MP is exactly 45, equal to the average, so the average does not change. This is the point where MP crosses AP. Before it, MP is above AP (50 > 40) and AP rises. After it, MP is below AP (35 < 45) and AP falls.

(c) The seventh picker’s MP is 0, so TP is the same with 6 or 7 pickers: 200 kg per hour, the maximum. TP is at its maximum where MP reaches zero.

(d) AP with 5 pickers rises from 190 ÷ 5 = 38 to 250 ÷ 5 = 50 kg per picker per hour, an increase of 12 kg. More capital means each picker has more to work with, so the whole TP curve shifts up. Buying a tractor is a change in the fixed input, so it is a long-run decision; afterwards the orchard faces a new short-run production function, which will still show diminishing marginal returns because land stays fixed.

Interpretation. Diminishing marginal returns begin with the third picker (MP falls from 50 to 45), yet AP does not start falling until the fourth. The two turning points are different: MP turns down first, and AP turns down only once MP has dropped below it.

Common misconceptions

• “Diminishing marginal returns means total product is falling.” TP still rises while MP is positive. TP falls only when MP is negative.
• “Diminishing returns begin where MP becomes negative.” They begin where MP starts to fall. Negative MP is a later, separate stage.
• “Later workers are less skilled.” The model assumes identical workers. MP falls because each extra worker shares the same fixed capital.
• “Diminishing marginal returns happen in the long run too.” They need a fixed input, so they are a short-run idea. In the long run, when all inputs change together, economists talk about returns to scale and economies of scale instead (Topic 3.3).
• “AP falls as soon as MP falls.” AP falls only when MP is below AP. In Worked example 2, MP fell with the third picker but AP did not fall until the fourth.
• “MP and AP are the same thing.” MP is the extra output from one more worker; AP is total output shared over all workers.
• “The short run is less than a year.” It is any period in which at least one input is fixed.

Where this leads

Every unit of output a firm makes has to be paid for. In Topic 3.2 you will turn these production numbers into costs: falling marginal product makes short-run production costs such as marginal cost rise. For the market context this unit builds on, look back at the previous topic, international trade and public policy. Test yourself with the practice questions, then use the revision notes and the checklist to consolidate.