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Agricultural economics and marketing

The law of diminishing returns and the production function

Why each extra unit of fertiliser or labour adds less output, the three stages of production, and how to find the most profitable input level.

Exam essential6 min readUpdated
On this page
  1. The production function
  2. The three stages of production
  3. The economic optimum
  4. The other production relationships
  5. Not to be confused with returns to scale
  6. Key points
  7. What the exams ask
  8. Questions people ask
  9. Sources
  10. Related on Laisha
  11. Discussion

Add fertiliser to a field and yield rises. Add more and it rises again, but by less. Keep adding and, eventually, the extra fertiliser adds nothing, and then yield falls. This is the law of diminishing returns, also called the law of variable proportions: when more and more of one input is added to fixed amounts of others, the extra output from each additional unit eventually declines.

It is probably the single most important idea in farm production economics. It explains why the highest yield is rarely the most profitable one, and it underlies fertiliser recommendations, feeding rations and labour planning. Exam questions ask for the three stages of production and for the economic optimum.

The production function

A production function describes how output depends on inputs:

A production function with one variable input
Y = f(X1 | X2, X3, …, Xn)

Here Y is output (say, grain), X1 is the variable input (say, nitrogen), and the inputs after the bar (land, labour, machinery, other nutrients) are held fixed. The relationship is described by three measures:

  • Total physical product (TPP): the total output at each level of input.
  • Average physical product (APP) = TPP ⁄ X: output per unit of input.
  • Marginal physical product (MPP) = ΔTPP ⁄ ΔX: the extra output from one more unit of input.

The law of diminishing returns is a statement about MPP: beyond some point it declines.

The three stages of production: total product rises at an increasing then a decreasing rate and finally falls; marginal product peaks first and falls, crossing average product at its maximum, which ends stage one; stage two ends where marginal product reaches zero and total product is at its maximum; rational production lies in stage two Stage I Stage II rational zone Stage III TPP APP MPP MPP = APP (APP at maximum) MPP = 0, TPP at maximum Variable input (for example fertiliser, labour) Total output Output per unit
The three stages of production for a variable input. Marginal product peaks first, then falls through average product (at its maximum) and through zero (where total product peaks).Diagram: Laisha, original.

The three stages of production

Stage I: increasing average returns. From zero input to the point where APP is at its maximum (where MPP = APP). Each unit added raises the average productivity of all units. The fixed inputs are under-used. A rational producer who uses any of the input at all should not stop in Stage I, because adding more raises output per unit of input.

Stage II: diminishing but positive returns. From maximum APP to the point where MPP = 0 (maximum TPP). Output still rises, but by less with each unit. This is the rational stage of production: the profit-maximising input level always lies here, exactly where depending on prices.

Stage III: negative returns. Beyond maximum TPP, MPP is negative: more input actually reduces output, as when excess nitrogen causes lodging, or overstocking a pasture reduces total liveweight gain. No rational producer operates here, even if the input were free.

The elasticity of production, Ep = MPP ⁄ APP, summarises the stages: Ep > 1 in Stage I, between 0 and 1 in Stage II, and below 0 in Stage III.

Why returns diminish

The cause is the fixed factors. A field of one hectare has only so much light, water and root space. The first kilograms of nitrogen correct the most limiting deficiency and produce large responses; later kilograms meet other limits (water, phosphorus, the crop's genetic potential) and produce less. Agronomists recognised the same pattern independently: E. A. Mitscherlich's "law of diminishing increments" (1909) described yield response to fertiliser as approaching a maximum asymptotically.

The economic optimum

The yield-maximising level of input (where MPP = 0) is not the profit-maximising level, because the last units cost money and add almost nothing. Profit is maximised where the value of the extra output from one more unit of input equals that unit's cost:

Profit-maximising input level
MPP × Py = Px, or equivalently MPP = Px ⁄ Py

Py is the price of output and Px the price of input. In words: keep adding input as long as the value of the marginal product (VMP = MPP × Py) exceeds the input's price; stop where they are equal. Because MPP falls through Stage II, a higher input price or a lower output price moves the optimum to a lower input level.

Worked example

Finding the economic optimum rate of nitrogen

Trials give a yield response that fits a quadratic curve:

Y = 3,000 + 30N − 0.1N2 (Y in kg grain per hectare, N in kg nitrogen per hectare)

Grain sells at 0.2 currency units per kg and nitrogen costs 1.0 currency unit per kg.

  1. Marginal product: MPP = dY ⁄ dN = 30 − 0.2N.
  2. Yield maximum: MPP = 0 → 30 − 0.2N = 0 → N = 150 kg/ha, yield = 3,000 + 4,500 − 2,250 = 5,250 kg/ha.
  3. Economic optimum: MPP = Px ⁄ Py = 1.0 ⁄ 0.2 = 5 → 30 − 0.2N = 5 → N = 125 kg/ha, yield = 3,000 + 3,750 − 1,562.5 = 5,187.5 kg/ha.

Going from 125 to 150 kg N adds 62.5 kg of grain, worth 62.5 × 0.2 = 12.5 currency units, but the extra 25 kg of nitrogen costs 25 currency units. The farmer is better off stopping at 125.

If the nitrogen price doubles to 2.0, the price ratio becomes 10, and 30 − 0.2N = 10 gives N = 100 kg/ha. Higher input prices push the optimum down the curve.

When capital is limited, a farmer may not even reach this optimum on every field. The equi-marginal principle says that limited money should be spread so that the last unit spent in each use (each crop, field or enterprise) returns the same amount; otherwise moving money from the lower-return use to the higher one raises total profit.

The other production relationships

Diminishing returns is the first of three relationships in farm production economics:

RelationshipQuestionDecision rule
Factor–producthow much of one input to use?MPP = Px ⁄ Py (the case above)
Factor–factorwhich combination of inputs produces a given output at least cost? (for example labour versus machinery, grain versus fodder in a ration)marginal rate of technical substitution = ratio of input prices, where the isoquant touches the isocost line
Product–productwhich combination of outputs to produce from given resources? (for example wheat versus barley on the same land)marginal rate of product transformation = ratio of output prices, where the production possibility curve touches the iso-revenue line

Products can be competitive (more of one means less of the other), complementary (a legume in rotation raises the next cereal's yield) or supplementary (backyard poultry using spare family labour), and the type of relationship shapes enterprise choice.

Not to be confused with returns to scale

Diminishing returns is about changing one input while others stay fixed: a short-run idea. Returns to scale is about changing all inputs in the same proportion: a long-run idea. Doubling every input on a farm might double output (constant returns), more than double it (increasing returns) or less (decreasing returns). A farm can face diminishing returns to fertiliser and constant returns to scale at the same time.

Key points

  • Law of diminishing returns: adding more of one variable input to fixed inputs eventually gives smaller increments of output.
  • TPP is total output; APP = TPP ⁄ X; MPP = ΔTPP ⁄ ΔX.
  • Stage I ends where APP is at its maximum (MPP = APP); Stage II ends where MPP = 0 (TPP at its maximum); Stage III has negative MPP.
  • Rational production is in Stage II; Ep = MPP ⁄ APP lies between 0 and 1 there.
  • Profit-maximising input level: MPP = Px ⁄ Py, below the yield-maximising level whenever the input costs money.
  • Diminishing returns (one input varied) differs from returns to scale (all inputs varied together).

What the exams ask

  • School-leaving and entrance papers: statement of the law, definitions of TPP, APP and MPP, and the shape of the curves.
  • Degree and research-fellowship entrances: the three stages and their boundaries, elasticity of production, the optimum input rule, factor–factor and product–product relationships, and the difference from returns to scale.
  • Professional and civil-service examinations: calculating economic optimum fertiliser or feed rates from a response function, and the equi-marginal principle in farm planning.

Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.

Questions people ask

Why not aim for the highest yield?

Because the last kilograms of input that push yield to its maximum cost more than the extra grain is worth. The most profitable yield is a little below the maximum, and it moves with prices.

Does the law apply to labour and machinery as well as fertiliser?

Yes, to any variable input used with fixed ones: workers on a fixed area, feed given to a fixed herd, irrigation water on a fixed field. Each extra unit eventually adds less.

Can new technology escape diminishing returns?

It shifts the whole production function upward, as new varieties did in the Green Revolution, so that each level of input produces more. But for the new function, diminishing returns still apply.

Sources

  1. Heady, E. O. Economics of Agricultural Production and Resource Use. Prentice-Hall, 1952.
  2. Debertin, D. L. Agricultural Production Economics, 2nd edition. University of Kentucky, 2012 (open edition). Chapters 2–4.
  3. Doll, J. P. and Orazem, F. Production Economics: Theory with Applications, 2nd edition. Wiley, 1984.
  4. Mitscherlich, E. A. "Das Gesetz des Minimums und das Gesetz des abnehmenden Bodenertrages." Landwirtschaftliche Jahrbücher 38, 537–552 (1909).
  5. Kay, R. D., Edwards, W. M. and Duffy, P. A. Farm Management, 9th edition. McGraw-Hill, 2019.

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