Production function; short run and long run
Production = turning inputs (land, labour, capital) into output. A production function shows the maximum output from each combination of inputs, with given technology: q = f(L, K).
Fixed input: cannot be changed quickly (land, factory building, big machines). Variable input: can be changed quickly (labour, raw material).
- Short run: at least one input is fixed. Output changes by changing the variable input only. Studied by returns to a factor.
- Long run: all inputs can be changed. Studied by returns to scale (all inputs changed in the same proportion).
Short and long run are not fixed calendar periods; they depend on how quickly inputs can change.
TP, AP and MP
Total product (TP) = total output from a given amount of the variable input (with fixed inputs). Also called total physical product.
Average product (AP) = TP ÷ units of variable input.
Marginal product (MP) = change in TP from one more unit of variable input = TPn − TPn−1 = ΔTP/ΔL. TP = ΣMP.
| Workers (L) | TP | MP | AP |
|---|---|---|---|
| 1 | 10 | 10 | 10 |
| 2 | 24 | 14 | 12 |
| 3 | 42 | 18 | 14 |
| 4 | 56 | 14 | 14 |
| 5 | 65 | 9 | 13 |
| 6 | 70 | 5 | 11.7 |
| 7 | 70 | 0 | 10 |
| 8 | 66 | −4 | 8.25 |
Relationship between TP and MP
- MP rising → TP rises at an increasing rate (curve gets steeper).
- MP falling but positive → TP rises at a decreasing rate.
- MP = 0 → TP is maximum.
- MP negative → TP falls.
Relationship between AP and MP
- MP > AP → AP rises.
- MP = AP → AP is maximum (MP cuts AP at its highest point).
- MP < AP → AP falls.
- MP can be zero or negative; AP stays positive as long as TP is positive.
Both AP and MP curves are inverted-U shaped.
Returns to a factor (law of variable proportions)
When more and more units of a variable input are used with fixed inputs, MP first rises, then falls, and finally becomes negative.
| Phase | TP | MP | Why |
|---|---|---|---|
| 1. Increasing returns (L = 1 to 3) | rises at increasing rate | rises | fixed input was under-used; better division of work |
| 2. Diminishing returns (L = 4 to 7) | rises at decreasing rate, max at MP = 0 | falls, positive | fixed input becomes scarce per worker |
| 3. Negative returns (L = 8) | falls | negative | overcrowding, workers get in each other's way |
A rational producer works in phase 2: in phase 1 adding labour still raises MP, and in phase 3 extra labour reduces output.
Assumptions: technology is unchanged, one input is variable and others fixed, units of the variable input are identical.
Try it at home
Fold paper boats for 2 minutes alone, then with 2, 3, 4, 5 people sharing one small table and one pair of scissors. Record TP each time, find MP and AP, and see where MP starts to fall.
Key formulas and definitions
- q = f(L, K)
- AP = TP ÷ L
- MP = TPn − TPn−1 = ΔTP/ΔL
- TP = ΣMP; TP = AP × L
- MP > AP ⇒ AP↑; MP = AP ⇒ AP max; MP < AP ⇒ AP↓
- MP = 0 ⇒ TP max
Worked examples
1. TP with 3 workers = 42 and with 4 = 56. Find MP of the 4th worker and AP at 4.
MP = 56 − 42 = 14. AP = 56 ÷ 4 = 14.
2. AP of 5 workers is 13. Find TP.
TP = AP × L = 13 × 5 = 65.
3. MP of the first four workers: 10, 14, 18, 14. Find TP at 4.
TP = 10 + 14 + 18 + 14 = 56.
4. TP at 6 = 70 and at 7 = 70. What is MP of the 7th worker and what does it mean?
MP = 0. TP is at its maximum; adding the 7th worker adds nothing.
5. AP at 7 workers is 10, AP at 8 workers is 8.25. Find MP of the 8th worker.
TP7 = 70, TP8 = 66. MP = 66 − 70 = −4 (negative returns).
6. At some L, MP = 12 and AP = 15. Is AP rising or falling?
MP < AP, so AP is falling.
Common mistakes
- Thinking TP is maximum when MP is maximum. TP is maximum when MP = 0.
- Saying AP and MP meet at the top of MP. They meet at the top of AP.
- Calling the law of variable proportions "returns to scale". Returns to scale are about the long run when all inputs change.
- Believing AP can be negative. AP = TP/L stays positive while TP is positive.