1 Economic Concepts · 1.1 Microeconomics: demand, production, markets and factor pricing
1.1.3 Production: factors, law of variable proportions and returns to scale
Production turns inputs into output: in the short run adding a variable factor to a fixed one gives first rising, then diminishing, then negative returns (law of variable proportions), while in the long run changing all inputs together gives increasing, constant or decreasing returns to scale.
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Core facts
- Factors of production and their rewards:
| Factor | Reward | Key feature |
|---|---|---|
| Land | Rent | Free gift of nature; fixed in supply |
| Labour | Wages | Human effort; perishable, cannot be stored |
| Capital | Interest | Produced means of production |
| Organisation (entrepreneur) | Profit | Bears risk and uncertainty |
- Production function: Q = f(L, K) – the technical relation between inputs and the maximum output.
- Short run: at least one factor fixed. Long run: all factors variable.
- TP = total product; AP = TP ÷ L; MP = ΔTP ÷ ΔL (TPₙ − TPₙ₋₁).
- Law of variable proportions (short run; also "law of diminishing returns"): as more units of a variable factor are added to a fixed factor, MP first rises, then falls, and finally becomes negative.
- Three stages:
- Stage I (increasing returns to the factor): from origin to where AP is maximum (AP = MP).
- Stage II (diminishing returns): from AP maximum to MP = 0 (TP maximum). A rational producer operates here.
- Stage III (negative returns): MP negative, TP falls.
- MP cuts AP at the maximum point of AP. When MP > AP, AP rises; when MP < AP, AP falls.
- Returns to scale (long run, all inputs changed in the same proportion):
- output rises more than proportionately → increasing returns;
- in the same proportion → constant returns;
- less than proportionately → decreasing returns.
- Cobb–Douglas: Q = A·Lᵅ·Kᵝ. α + β > 1 increasing, = 1 constant, < 1 decreasing returns to scale. α and β are output elasticities of L and K.
- Isoquant (iso-product / equal-product curve): input combinations giving the same output; slope = MRTS_LK = MPL/MPK.
- Producer equilibrium (least cost): isoquant tangent to the isocost line: MRTS = w/r, i.e. MPL/w = MPK/r.
Explained
1. The three stages in a table.
Example 1. Land is fixed; labour is added one worker at a time.
| Labour | TP | MP | AP |
|---|---|---|---|
| 1 | 10 | 10 | 10 |
| 2 | 24 | 14 | 12 |
| 3 | 39 | 15 | 13 |
| 4 | 52 | 13 | 13 |
| 5 | 60 | 8 | 12 |
| 6 | 66 | 6 | 11 |
| 7 | 66 | 0 | 9.43 |
| 8 | 64 | −2 | 8 |
Reading. MP is highest at the 3rd worker (15). AP peaks at 13 (3rd and 4th workers), where MP of the 4th worker = AP = 13: Stage I ends. TP is maximum (66) where MP = 0 at the 7th worker: Stage II ends. The 8th worker has MP = −2: Stage III. Answer: rational range = 4th to 7th worker (Stage II)
2. Why returns diminish. The fixed factor is spread over more units of the variable factor, so each extra worker has less land or machinery to work with. Early increasing returns come from better use of the fixed factor and division of labour.
3. Why Stage I and Stage III are avoided. In Stage III, MP of labour is negative – fewer workers would give more output. In Stage I, the fixed factor is under-used (its MP is negative), and AP of the variable factor is still rising, so it pays to keep adding labour.
4. Returns to scale with Cobb–Douglas.
Example 2. Q = 10 L^0.5 K^0.5 with L = 16 and K = 25. Find Q, and Q when both inputs double. Solution. Q = 10 × 4 × 5 = 200. Doubling: L = 32, K = 50 → Q = 10 × √1,600 = 400. Output doubles. Answer: 200; 400 – constant returns (α + β = 1)
Example 3. Q = L^0.6 K^0.6. By what factor does output rise when both inputs double? Solution. 2^(0.6 + 0.6) = 2^1.2. Answer: about 2.30 times – increasing returns
5. Causes of returns to scale. Increasing returns come from internal economies – technical (bigger, specialised machines), managerial, marketing (bulk buying), financial (cheaper credit), risk-bearing. Decreasing returns come from diseconomies – management becomes hard to coordinate, and supervision costs rise. External economies (a cluster of firms sharing roads, skilled labour, suppliers) benefit every firm in an industry.
Deeper
1. Law of variable proportions vs returns to scale.
| Point | Variable proportions | Returns to scale |
|---|---|---|
| Period | Short run | Long run |
| Inputs changed | One, others fixed | All, same proportion |
| Factor ratio | Changes | Constant |
| Diagram | TP, AP, MP curves | Spacing of isoquants |
On an isoquant map, increasing returns show as isoquants (for 100, 200, 300 units) getting closer along a ray from the origin; constant as equal spacing; decreasing as wider spacing.
2. Cobb–Douglas results.
- MPL = α·Q/L and MPK = β·Q/K. In Example 2, MPL = 0.5 × 200/16 = 6.25, MPK = 0.5 × 200/25 = 4.
- Elasticity of substitution is 1. The original 1928 study of US manufacturing found output shares close to 0.75 (labour) and 0.25 (capital).
- Euler's theorem: with constant returns, paying each factor its MP exactly exhausts the output (product exhaustion).
3. Least-cost combination.
Example 4. Wage w = ₹10, rental r = ₹20. At the present mix MPL = 30 and MPK = 50. Is the firm minimising cost? Solution. MPL/w = 3 and MPK/r = 2.5. A rupee on labour gives more output, so shift spending from capital to labour until the ratios are equal. Answer: No – use more labour, less capital
Example 5. Q = L^0.5 K^0.5, w = ₹4, r = ₹16. Find the cheapest way to make 100 units. Solution. MRTS = K/L = w/r = 1/4, so K = L/4. Then √(L × L/4) = L/2 = 100 → L = 200, K = 50. Cost = 4 × 200 + 16 × 50 = 800 + 800. Answer: L = 200, K = 50, cost ₹1,600
4. Isoquant properties. Slope downward, convex to the origin (diminishing MRTS), never intersect, higher isoquant = higher output. Ridge lines join the points where isoquants turn upward (MP of one factor = 0); the economic region of production lies between them – the long-run counterpart of Stage II. Expansion path: the line joining least-cost points as outlay rises.
Common traps
- ✗ "The law of variable proportions is a long-run law." ✓ It is short run (one factor fixed); returns to scale is long run.
- ✗ "Stage I ends where MP is maximum." ✓ It ends where AP is maximum (MP = AP).
- ✗ "TP is maximum where MP is maximum." ✓ TP is maximum where MP = 0.
- ✗ "A rational producer works in Stage I, where returns rise." ✓ He works in Stage II.
- ✗ "α + β = 1 in Cobb–Douglas means increasing returns." ✓ It means constant returns; > 1 increasing.
- ✗ "Rent is the reward of capital." ✓ Rent → land, wages → labour, interest → capital, profit → entrepreneur.
- ✗ "Least cost: MPL = MPK." ✓ MPL/w = MPK/r (MRTS = factor price ratio).
Quick revision
- Land–rent, labour–wages, capital–interest, organisation–profit.
- AP = TP/L; MP = ΔTP/ΔL.
- MP cuts AP at AP's maximum.
- Stage I to AP max; Stage II to MP = 0; Stage III MP < 0.
- Rational producer: Stage II.
- Variable proportions: short run; returns to scale: long run.
- Cobb–Douglas α + β: > 1 IRS, = 1 CRS, < 1 DRS.
- Isoquant slope = MRTS = MPL/MPK.
- Least cost: MPL/w = MPK/r.
- Internal economies → increasing returns.
Practise this topic. Questions on Production, costs & market structures, with answers explained.
All Microeconomics: demand, production, markets and factor pricing notes · APPSC Paper I syllabus · APPSC Paper I Mock 1 with answers · Start free
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