What is Production Function?
A production function is a mathematical model that describes the relationship between physical inputs (labour, capital, materials) and the quantity of output a firm can produce. It is central to understanding efficiency, costs, and scale in production.
A production function Q = f(L, K) expresses the maximum output a firm can produce with given quantities of labour (L) and capital (K), assuming efficient production.
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Step-by-step worked examples
A firm has Q = 2 × L^0.6 × K^0.4. With L=8 and K=5, find output.
Q = 2 × 8^0.6 × 5^0.4 = 2 × 3.52 × 1.77 = 12.5 units (approx.)
Linear production function: Q = 3L + 2K. If L=10 and K=5, what is Q?
Q = 3(10) + 2(5) = 30 + 10 = 40 units
Cobb-Douglas: Q = 5 × L^0.7 × K^0.3. Labour elasticity is 0.7. If labour increases by 1%, how much does output increase?
Output elasticity w.r.t. labour = 0.7 A 1% increase in L → 0.7% increase in Q (ceteris paribus)
Flashcards
Quick quiz
Q1.A production function Q = f(L, K) shows…
Q2.In Cobb-Douglas Q = A × L^0.6 × K^0.4, if both L and K double, output…
Q3.What does 'A' (scalar) in production functions represent?
Q4.If a firm's production function shows diminishing marginal product of labour, then…
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Common mistakes
Production function is the same as cost function. — Correct: Production function is technological (input → output); cost function is economic (input cost → minimum cost for Q).
If all inputs double, output always doubles. — Correct: Depends on returns to scale: could double, less than double, or more than double.
Labour is the only variable input. — Correct: Both labour and capital (and materials) are variable in the long run.
Constant returns to scale means zero profit. — Correct: It relates to input scaling, not profitability; profits depend on prices and costs.
FAQ
What is a production function?
A mathematical relationship showing maximum output a firm can produce with given quantities of inputs, assuming efficient production.
What is Cobb-Douglas production function?
Q = A × L^α × K^(1−α), a widely used form with constant elasticity and constant returns to scale.
How do you find marginal product of labour?
MP_L = ∂Q/∂L — the partial derivative of output with respect to labour.
What is the difference between short-run and long-run production?
Short-run: some inputs fixed (capital fixed, labour variable). Long-run: all inputs variable.