The production function and short-run production – Production Theory (Senior 4 Economics) | Ganzaa

Production Theory

Lessons

Senior 4 · Economics

Production Theory

5 lessons

Lesson 3 of 5

The production function and short-run production

30 min

More hands, the same hillside

Rwanda's fifth population census counted 13,246,394 people in August 2022, after growth of 2.3% a year since 2012 (NISR). The country's land did not grow at all. In 1798 Thomas Malthus warned that when more and more labour works a fixed area of land, each extra worker adds less food than the one before, so food could not keep up with population.

Malthus's fear rested on a real law of production, the law of diminishing returns. His forecast was wrong for most of the world because farmers also changed what they worked with: better seed, fertiliser, terraces and irrigation. This lesson separates the two ideas: what happens when only one input grows (the short run), and why that is not the whole story.

Definition

Production function: the relationship between the quantities of inputs a firm uses and the maximum output it can produce with a given technology:

Q = f(L, K, N, E) \qquad\text{or, for two inputs,}\qquad Q = f(L, K)

where $Q$ is output, $L$ labour, $K$ capital, $N$ land and $E$ entrepreneurship.

Fixed factor: an input whose quantity cannot be changed in the period studied (the plot of land, the factory building). Variable factor: one that can (workers, seed, fuel).

Example: a cooperative with 2 hectares that can hire more pickers this season but cannot buy more land has land fixed and labour variable.

1. Planning periods

Period	What can change	Example
Market period (very short run)	Nothing: output is what has already been produced	Tomatoes already at the market this morning
Short run	Variable factors only; at least one factor is fixed	A bakery hires another worker but keeps the same oven
Long run	All factors, including plant size and land	The bakery moves to a bigger building with three ovens
Very long run	Technology itself	A new kind of oven halves baking time

These periods are not fixed lengths of time. A moto rider can double his "plant" in a week by buying a second motorcycle; a cement factory may need five years. The short run is simply the time in which at least one input cannot be changed.

Definition

Total product (TP): the whole output produced with a given quantity of the variable factor.

Average product (AP): output per unit of the variable factor, $AP_L = \frac{TP}{L}$ .

Marginal product (MP): the change in total product from one more unit of the variable factor, $MP_L = \frac{\Delta TP}{\Delta L}$ , or $TP_n - TP_{n-1}$ for one extra worker.

Example: if 4 workers harvest 176 bags and 5 workers harvest 250 bags, then $M P_{5} = 250 - 176 = 74$ bags and $AP_5 = \frac{250}{5} = 50$ bags.

2. Total, average and marginal product

A cooperative farms a fixed 5-hectare plot of maize and varies only the number of workers. Its total product follows $T P = 15 L^{2} - L^{3}$ (bags per season, illustrative).

Workers, L	Total product, TP (bags)	Marginal product, $MP = \Delta TP$	Average product, $AP = \frac{TP}{L}$
0	0	–	–
1	14	14	14
2	52	38	26
3	108	56	36
4	176	68	44
5	250	74	50
6	324	74	54
7	392	68	56
8	448	56	56
9	486	38	54
10	500	14	50
11	484	−16	44

Reading the graphs. The marginal products in the table belong to the step from one worker to the next, so they are plotted at the middle of each step; they lie almost exactly on the smooth MP curve. Notice:

While MP is rising (up to about 5 workers), TP rises at an increasing rate. Once MP starts to fall, TP still rises but more slowly: this is where diminishing returns set in, and TP has a point of inflexion.
When MP is above AP, AP rises; when MP is below AP, AP falls; so MP cuts AP at the maximum of AP (between the 7th and 8th worker, both AP = 56).
When MP = 0, TP is at its maximum (10 workers, 500 bags). When MP is negative, TP falls: the 11th worker gets in the way.

Worked example 1: reading a product schedule

A tea estate varies the number of pluckers on a fixed block. Daily output (kg of green leaf, illustrative):

Pluckers	1	2	3	4	5	6	7	8
TP	20	50	90	120	140	150	150	140
MP	20	30	40	30	20	10	0	−10
AP	20	25	30	30	28	25	21.4	17.5

(a) Diminishing marginal returns set in with the 4th plucker: MP falls from 40 to 30.

(b) Maximum AP is 30 kg, with 3 or 4 pluckers; there MP (40, then 30) passes through AP.

(c) Maximum TP is 150 kg, reached with 6 pluckers; the 7th adds nothing (MP = 0) and the 8th reduces output (MP = −10): negative returns.

(d) Advice: the estate should employ between 4 and 6 pluckers (stage II). Exactly where depends on wages and the price of tea, which unit 7 adds.

3. The law of diminishing returns

The law (the law of variable proportions): if more and more units of a variable factor are added to a fixed quantity of another factor, with technology unchanged, the marginal product of the variable factor will eventually fall.

Assumptions. At least one factor is fixed (it is a short-run law); the units of the variable factor are identical; technology and methods are unchanged; and the factors can be combined in varying proportions.

Why it happens. At first the fixed factor is under-used, so extra workers allow specialisation and MP rises. Beyond a point, each worker has less of the fixed factor to work with: less land, less machine time, more crowding. The fixed factor becomes the bottleneck.

Stage	Range	What happens	Should a firm produce here?
I: increasing returns to the variable factor	From 0 to maximum AP (7.5 workers)	AP rises; MP above AP; fixed factor under-used	No: each extra worker raises output per worker, so it pays to add more
II: diminishing returns	From maximum AP to maximum TP (7.5 to 10 workers)	AP and MP both fall; MP still positive	Yes: the rational stage; the exact point depends on costs and prices
III: negative returns	Beyond maximum TP (over 10 workers)	MP negative; TP falls	No: removing workers would raise output

Importance. It explains why short-run costs rise as output grows (unit 7), why a fixed area of farmland cannot feed an ever larger population with unchanged methods, and why producers look for better technology and more capital, which shift the whole TP curve upwards and postpone diminishing returns.

Worked example 2: using the equation

For $T P = 15 L^{2} - L^{3}$ : $AP = \frac{TP}{L} = 15L - L^2$ and $MP = \frac{d(TP)}{dL} = 30L - 3L^2$ .

(a) MP is greatest where $\frac{d(MP)}{dL} = 30 - 6L = 0 \Rightarrow L = 5$ : $M P = 150 - 75 = 75$ bags.

(b) AP is greatest where $\frac{d(AP)}{dL} = 15 - 2L = 0 \Rightarrow L = 7.5$ : $A P = 112.5 - 56.25 = 56.25$ . Check: $M P (7.5) = 225 - 168.75 = 56.25 = A P$ ✔.

(c) TP is greatest where $MP = 30L - 3L^2 = 3L(10 - L) = 0 \Rightarrow L = 10$ : $T P = 1,500 - 1,000 = 500$ bags.

So stage I runs from 0 to 7.5 workers, stage II from 7.5 to 10, and stage III beyond 10.

Warning

Diminishing returns is not negative returns. Diminishing returns means each extra worker adds less than the one before; total product is still rising. Only when MP is negative does TP fall. And the law is a short-run law: it needs a fixed factor. When all inputs grow together (the long run, lesson 4) the right idea is returns to scale.

Tip

"The newcomer pulls the average." If a new player scores more than the team's average, the average rises; if less, it falls. In the same way MP above AP pulls AP up, MP below AP pulls it down, and MP crosses AP at AP's peak.

Important

Discuss. Malthus expected diminishing returns on fixed land to keep most people near starvation. Using a TP diagram, show how fertiliser, irrigation or better seed shift the TP curve, and discuss whether Rwanda can rely on them to feed a growing population.

Khan Academy, "Total product, marginal product and average product" (about 7 minutes). Sal Khan uses an ice-cream factory to calculate the three measures; watch for the point where each extra worker starts to add less, and compare it with the maize schedule above.

✦

Practice

Define a production function, and distinguish between the short run and the long run.
Copy and complete for workers 1 to 6 with TP = 10, 24, 39, 52, 60, 60: find MP and AP for each.
From question 2, state where diminishing marginal returns begin and where TP is greatest.
State the law of diminishing returns and three of its assumptions.
Explain the relationship between average product and marginal product.
A firm's total product is $T P = 12 L^{2} - L^{3}$ . Find the number of workers at which (a) MP, (b) AP and (c) TP is greatest, and the value of each.
Explain why a rational producer will not operate in stage I or stage III.
"Diminishing returns mean that Rwanda's farms cannot feed a growing population." Evaluate this statement.

Show answersHide answers

A production function shows the maximum output from given inputs with given technology, $Q = f (L, K)$ . In the short run at least one factor is fixed; in the long run all factors can be varied.
MP: 10, 14, 15, 13, 8, 0. AP: 10, 12, 13, 13, 12, 10.
Diminishing marginal returns begin with the 4th worker (MP falls from 15 to 13). TP is greatest, 60, with 5 or 6 workers; the 6th adds nothing.
As more of a variable factor is added to a fixed factor, with technology unchanged, its marginal product eventually falls. Assumptions: at least one factor fixed; identical units of the variable factor; unchanged technology; factors can be combined in varying proportions.
When MP > AP, AP rises; when MP < AP, AP falls; MP = AP where AP is at its maximum.
(a) $M P = 24 L - 3 L^{2}$ ; $24 - 6L = 0 \Rightarrow L = 4$ , $M P = 96 - 48 = 48$ . (b) $A P = 12 L - L^{2}$ ; $12 - 2L = 0 \Rightarrow L = 6$ , $A P = 72 - 36 = 36$ . (c) $MP = 3L(8 - L) = 0 \Rightarrow L = 8$ , $T P = 768 - 512 = 256$ .
In stage I output per worker is still rising, so adding workers lowers cost per unit: it pays to expand. In stage III extra workers reduce total output, so cutting workers would raise output and lower costs.
The law applies when technology and other inputs are fixed. Better seed, fertiliser, irrigation, terracing and training shift the TP curve upwards, and more workers can move into industry and services. But on small plots with unchanged methods diminishing returns are real, so farm productivity must rise.

Quick check

Key points

The production function $Q = f (L, K)$ links inputs to maximum output; in the short run at least one factor is fixed.
$AP = \frac{TP}{L}$ and $MP = \frac{\Delta TP}{\Delta L}$ . MP cuts AP at AP's maximum, and TP is greatest where MP = 0.
The law of diminishing returns: adding a variable factor to a fixed one eventually lowers its marginal product.
Stage II, from maximum AP to maximum TP, is the only rational range of production.

Sources and credits

Population figures: Main indicators: 5th Rwanda Population and Housing Census (PHC5), National Institute of Statistics of Rwanda (NISR).
Photo: Rwanda maize lo, Neil Palmer (CIAT), Wikimedia Commons, CC BY-SA 2.0.
Video: Total product, marginal product and average product, Khan Academy, embedded from YouTube.

Written by Ganzaa: the text, the product diagrams (drawn to scale from $T P = 15 L^{2} - L^{3}$ ), the tables, the worked examples and the practice items. The maize and tea figures are illustrative.

This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1More hands, the same hillside

21. Planning periods

32. Total, average and marginal product

43. The law of diminishing returns

More hands, the same hillside

1. Planning periods

2. Total, average and marginal product

3. The law of diminishing returns