Theory of Production

1Why this matters

Behind every price you have studied so far sits a firm deciding how much to make. A tomato trader at Nyabugogo, Bralirwa, a tailor in Kimironko — each one must answer the same question: given what I have to work with, how much should I produce? Before a firm can weigh costs against revenue, it has to understand the physical relationship between the resources it puts in and the output it gets out. That relationship is the theory of production, and every cost curve in the rest of this unit is built directly on it.

This lesson stays on the physical side of the story: workers and land go in, kilograms of beans come out. We attach money to it in the next lesson.

2The factors of production

Economists group everything a firm uses to produce into four factors of production:

• Land — natural resources: the plot itself, water, minerals.
• Labour — human effort, both physical and mental.
• Capital — man-made aids to production: tools, machines, buildings.
• Enterprise — the entrepreneur who combines the other three and carries the risk.

The production function is simply the relationship between the quantity of factors used and the maximum output they can produce. We write it as Q = f(land, labour, capital), which reads: output Q depends on (‘is a function of’) the amounts of land, labour and capital employed. For now we will change one factor at a time and watch what happens to Q.

3Short run and long run

These two terms are not about the calendar. They are about which factors a firm is able to change.

In the short run, at least one factor is fixed — its quantity cannot be changed. A bean farmer with one hectare can hire more workers this week, but she cannot acquire more land this week; the land is her fixed factor and labour is her variable factor.

In the long run, every factor is variable. Given enough time she can rent a second hectare, build a store, buy an ox-plough — nothing is fixed. How long the long run actually takes differs by industry: for a street-food seller it may be a matter of weeks; for a cement factory building a new kiln, several years.

4A worked example: one field, more and more workers

Mukamana farms a fixed plot of one hectare of beans. The land is fixed; she varies only the number of workers. Here is what she harvests, in kilograms, as she adds workers one at a time.

Read the three measures:

• Total product (TP) is the total output from all the workers together.
• Marginal product (MP) is the extra output from adding one more worker: MP = the change in TP when labour rises by one. Going from 2 to 3 workers, TP rises from 50 to 90, so MP = 90 − 50 = 40 kg.
• Average product (AP) is output per worker: AP = TP ÷ number of workers. With 4 workers, AP = 120 ÷ 4 = 30 kg.

Now follow the marginal product column, because it tells the whole story:

• Workers 1 to 3 — increasing returns. MP rises: 20, then 30, then 40. Each extra worker adds more than the one before, because they can specialise — one digs, one weeds, one waters — instead of one person doing everything alone.
• From worker 4 — diminishing returns. MP is still positive but now falling: 30, 20, 10. The fixed hectare is getting crowded; each new worker has less land to work with, so adds less than the last.
• Worker 7 — MP = 0. Total product is at its maximum, 150 kg. The seventh worker adds nothing.
• Worker 8 — MP is negative. Total product actually falls, to 140 kg. The workers are now getting in each other’s way; there are simply too many bodies on one hectare.

5The law of diminishing returns

What you have just watched has a name. The law of diminishing returns (also called the law of variable proportions) states: as successive units of a variable factor are added to a fixed factor, beyond a certain point the marginal product of the variable factor begins to fall.

Notice the three conditions built into that sentence, because examiners test them:

• It is a short-run law — it only works because one factor (the hectare) is fixed.
• The added units must be identical (equally capable workers).
• Technology is held constant — no new tool or method appears part-way through.

One more relationship is worth locking in now, because it returns in the next lesson with costs. Compare the MP and AP columns. While MP is above AP, AP rises; once MP falls below AP, AP falls. The marginal always pulls the average towards itself. Think of your own test marks: sit a test above your current average and your average rises; sit one below it and your average drops. Marginal cost and average cost will behave in exactly this way.

6The long run: returns to scale

Diminishing returns was a short-run idea: we changed one factor and held the rest fixed. In the long run nothing is fixed, so we ask a different question: if we change all factors together, in the same proportion, what happens to output? The answer is called returns to scale.

Suppose a bakery in Musanze doubles everything at once — twice the ovens, twice the floor space, twice the staff, twice the flour (a ×2 increase in every input). Three things could happen to output:

• Increasing returns to scale — output more than doubles, say from 200 to 500 loaves. The larger scale let the bakery specialise and run bigger, more efficient ovens. Inputs rose by a factor of 2 while output rose by a factor of 2.5.
• Constant returns to scale — output exactly doubles, 200 to 400. Twice the inputs give twice the output.
• Decreasing returns to scale — output less than doubles, say 200 to 350. The firm has grown so large that management and coordination become difficult, and each new batch of inputs adds less.

The single most common mistake in this topic is to confuse the two ideas. Keep them apart like this: diminishing returns is short run, one factor changed with the rest fixed. Returns to scale is long run, all factors changed together. If a question changes only labour, it is about diminishing returns; if it doubles the whole firm, it is about returns to scale.

Practice

Work each one out before reading the solution beneath it.

Solution. MP = change in TP = 63 − 48 = 15 units. AP with 7 workers = 63 ÷ 7 = 9 units per worker. Notice MP (15) is above AP (9), so average product is still rising at this point.

Solution. The marginal product is falling (20, 12, 4) while total output is still rising, so this is the law of diminishing returns. It must be the short run because one factor — the plot — is being held fixed while only labour changes. If the plot could grow as well, the law would not apply.

Solution. Inputs rose ×3. Proportional (constant) returns would give 400 × 3 = 1,200 litres. Actual output is only 900, which is less than 1,200, so output rose less than proportionately: decreasing returns to scale. It is not diminishing returns of a single factor, because every input was changed, not just one.

Solution. True. The long run is defined as the period long enough for every factor, land included, to be varied — she can rent more land or give some up. It is only in the short run that the land is fixed.

Solution. The 4th worker’s marginal product (22) is below the current average (30), so it drags the average down: average product falls. Check it: new TP = (3 × 30) + 22 = 112, so new AP = 112 ÷ 4 = 28 units, down from 30.