Theory of Production
- Production is the process of transforming inputs into outputs. This unit covers total, average, and marginal product
- production functions including the Cobb-Douglas function
- the law of variable proportions in the short run
- isoquants and the marginal rate of technical substitution
- the optimal combination of inputs
- the laws of returns to scale in the long run.
In this chapter
Meaning of Production
Production is the process of transforming inputs into outputs — goods and services. Traditionally, production is also defined as the process of creating utility. Production is regarded as the mother of all economic activities because, without it, there would be no consumption, exchange, or distribution. The aim of every firm is to maximise profit or sales revenue, which is possible only through efficient production — and efficient production requires the optimum combination of inputs.
Total, Average, and Marginal Product
Three Product Concepts
- Total product (TP) — the total quantity of output produced using all units of inputs in a given period. TP = ΣMP = AP × L.
- Average product (AP) — output per unit of variable factor: AP = TP / L.
- Marginal product (MP) — the additional output from employing one more unit of the variable input: MP = ΔTP / ΔL.
Total, average, and marginal product
Production Function
A production function expresses the technological relationship between the quantity of inputs and the quantity of output. It shows the maximum output that can be produced from a given set of inputs, assuming technical efficiency. Production functions are classified into short-run (at least one input is fixed) and long-run (all inputs are variable). The general form is Q = f(L, K), where Q is output, L is labour, and K is capital.
General production function
Cobb-Douglas Production Function
The Cobb-Douglas production function is the most widely used specific production function in economic analysis. It was first proposed by Swedish economist Kunt Wicksell (1851–1926) and later tested empirically by American mathematician C.W. Cobb and economist P.H. Douglas in 1928. The function takes the form Q = A·L^α·K^β, where A is total factor productivity, α is the output elasticity of labour, and β is the output elasticity of capital. The sum (α + β) determines the nature of returns to scale.
Cobb-Douglas production function and returns to scale
Law of Variable Proportions (Short Run)
The law of variable proportions (also called the law of diminishing returns) operates in the short run when one factor is fixed (usually capital) and another is variable (usually labour). It states that as more units of the variable factor are added to a fixed factor, the marginal product first increases, then diminishes, and eventually becomes negative. This law gives rise to three stages of production.
Three Stages of Production
| Stage | MP Behaviour | TP Behaviour | Rational? |
|---|---|---|---|
| Stage I: Increasing returns | MP rises, then starts falling but stays above AP | TP increases at increasing, then decreasing rate | Not rational — AP still rising |
| Stage II: Diminishing returns | MP falls but stays positive; MP = AP at start | TP increases at decreasing rate, reaches max | ✓ Rational stage — produce here |
| Stage III: Negative returns | MP becomes negative | TP starts falling | Not rational — reduce labour |
Where to Produce
A rational producer always operates in Stage II — where TP is still rising but MP is falling. Stage I wastes the fixed factor (AP still rising means variable factor is underused); Stage III is wasteful (MP negative means too much variable factor). The exact point within Stage II depends on input and output prices.
Isoquant and MRTS
An isoquant (or equal-product curve) shows all combinations of two inputs (labour and capital) that produce the same level of output. The marginal rate of technical substitution (MRTS) is the rate at which one input can be substituted for another while keeping output constant — it is the slope of the isoquant. Due to diminishing MRTS, isoquants are convex to the origin.
Marginal rate of technical substitution
Optimal Combination of Inputs
The producer reaches equilibrium — the least-cost combination of inputs — at the point where the isoquant is tangent to the iso-cost line. The iso-cost line shows all combinations of inputs that cost the same total amount. The equilibrium condition is that the MRTS equals the ratio of input prices: MRTS_LK = P_L / P_K, or equivalently MP_L / P_L = MP_K / P_K. This is the producer's version of the equi-marginal principle.
Producer equilibrium — least-cost input combination
Laws of Returns to Scale (Long Run)
- Increasing returns to scale — output increases by a larger proportion than inputs (doubling inputs more than doubles output)
- Constant returns to scale — output increases by the same proportion
- Decreasing returns to scale — output increases by a smaller proportion
Returns to scale — output response to proportional input change
Practice Problem
A firm's production function is Q = 10L^0.5 · K^0.5. (a) What type of returns to scale does this function exhibit? (b) If L = 4 and K = 9, calculate output Q. (c) If both inputs are doubled (L = 8, K = 18), what is the new output? Verify your answer to (a).
Short-Run vs Long-Run Production
The short run is the period during which at least one factor of production is fixed (usually capital) — the firm can only vary variable factors (like labour) to change output. The long run is the period during which all factors are variable — the firm can build new plants, buy new machinery, or change technology. The distinction is not about calendar time but about input flexibility. In the short run, the law of variable proportions applies (diminishing returns); in the long run, returns to scale applies.
Properties of Isoquant (Detailed)
Five Key Properties of Isoquants
- Slope downward to the right — to maintain the same output, reducing one input requires increasing the other.
- Convex to the origin — due to diminishing MRTS; the curve flattens as you move along it.
- Higher isoquants represent higher output — more inputs (in efficient combinations) yield more output.
- Two isoquants never intersect — intersection would mean the same input combination produces two different output levels, which is impossible.
- They do not touch either axis — both inputs are essential; producing output with zero of either input is impossible (in most production functions).
Returns to Scale — Types and Examples
| Type | Condition | Output Change | Example |
|---|---|---|---|
| Increasing returns | Inputs doubled, output more than doubles | 2× inputs → 2.5× output | Small firm expanding |
| Constant returns | Inputs doubled, output exactly doubles | 2× inputs → 2× output | Mature industry |
| Decreasing returns | Inputs doubled, output less than doubles | 2× inputs → 1.7× output | Very large firm |
Real-Life Example: Nepali Agriculture
In Nepali agriculture, the law of variable proportions is clearly visible. A small farmer in the Terai with 1 hectare of land (fixed) adds more labour (variable). Initially, output rises sharply as workers clear weeds and harvest efficiently. But beyond a point, adding more workers to the same land causes overcrowding — MP falls and eventually turns negative. This is why Nepali farms have disguised unemployment — more family members work the land than needed, with near-zero marginal product.
Exam Tip
Remember: Stage II is the only rational stage of production. In Stage I, AP is still rising (variable factor is underused). In Stage III, MP is negative (too much variable factor). The exact point within Stage II depends on input and output prices — if labour is cheap relative to output, produce closer to the Stage II/III boundary.
Key Terms and Definitions
- Production: The process of transforming inputs into outputs — उत्पादन: आगतलाई निर्गतमा रूपान्तरण गर्ने प्रक्रिया।
- Production function: Technological relationship between inputs and output, Q = f(L, K) — उत्पादन प्रकार्य: आगत र निर्गतबीचको प्राविधिक सम्बन्ध।
- Total Product (TP): Total output produced with given inputs — कुल उत्पादन: दिइएका आगतबाट उत्पादन गरिएको कुल निर्गत।
- Marginal Product (MP): Additional output from one more unit of variable input — सीमान्त उत्पादन: एक थप परिवर्तनशील आगत एकाइबाट थप निर्गत।
- Law of Variable Proportions: MP first rises, then falls as variable input increases with fixed input — परिवर्तनशील अनुपातको नियम: स्थिर आगतसहित परिवर्तनशील आगत बढ्दा MP पहिले बढ्छ, त्यसपछि घट्छ।
- Isoquant: Curve showing all input combinations producing the same output — आइसोक्वान्ट: उही उत्पादन दिने सबै आगत संयोजन देखाउने वक्र।
- MRTS: Rate at which one input substitutes for another keeping output constant — सीमान्त प्राविधिक प्रतिस्थापन दर: उत्पादन स्थिर राख्दै एक आगतले अर्कोलाई प्रतिस्थापन गर्ने दर।
- Returns to scale: Output response when all inputs change proportionally — स्केल प्रतिफल: सबै आगत अनुपातमा परिवर्तन हुँदा निर्गत प्रतिक्रिया।
- Cobb-Douglas function: Q = A·L^α·K^β; α+β determines returns to scale — कब-डगलस प्रकार्य: α+β ले स्केल प्रतिफल निर्धारण गर्छ।
- Producer equilibrium: Least-cost input combination where MRTS = PL/PK — उत्पादक सन्तुलन: न्यूनतम-लागत आगत संयोजन जहाँ MRTS = PL/PK।
Practice Problem
A firm produces output using labour (L) and capital (K). The production function is Q = 20L + 10K. The prices are PL = Rs 4 and PK = Rs 2. The firm has a budget of Rs 200. (a) Write the iso-cost line equation. (b) Find the optimal combination of L and K that maximises output. (c) Calculate the maximum output.