ECOS2001 · Intermediate Microeconomics
Production: Technology & Profit Maximisation
The producer side starts with technology: a production function q = f(L, K) describes the most output achievable from inputs. The marginal product of an input is its partial derivative, usually diminishing; the isoquant (constant-output curve) has slope equal to the marginal rate of technical substitution, MRTS = MP_L/MP_K. Returns to scale compare f(t·x) to t·f(x) — for Cobb-Douglas q = A·LᵃKᵇ the exponent sum a + b decides increasing, constant or decreasing returns. The firm minimises the cost of any target output by setting MRTS = w/r (the isoquant tangent to the isocost line), and a price-taking profit maximiser hires each input until its value of marginal product equals the input price.
What this chapter covers
- 01Production function q = f(L, K) and marginal products MP_L, MP_K
- 02Diminishing marginal product; isoquants and the MRTS = MP_L/MP_K
- 03Returns to scale; Cobb-Douglas a + b vs 1
- 04Cost minimisation: MRTS = w/r, i.e. MP_L/w = MP_K/r
- 05Isocost lines with slope −w/r
- 06Profit max for a price-taker: value of marginal product = factor price
Cost minimisation and the cost function
- 1 markMarginal products: MP_L = K and MP_K = L, so MRTS = MP_L/MP_K = K/L.
- 2 marksCost-minimisation condition MRTS = w/r: K/L = 2/8 = 1/4, so K = L/4.
- 2 marksImpose the output target L·K = q: L·(L/4) = q → L²/4 = q → L = 2√q. Then K = L/4 = (1/2)√q.
- 1 markFor q = 64: L = 2√64 = 2·8 = 16 and K = (1/2)·8 = 4. Check output L·K = 16·4 = 64. ✓
- 1 markCost at this output: C = wL + rK = 2·16 + 8·4 = 32 + 32 = 64.
- 1 markGeneral cost function: C(q) = wL + rK = 2·(2√q) + 8·((1/2)√q) = 4√q + 4√q = 8√q.
Key terms
- Marginal product (MP)
- The extra output from one more unit of an input, the partial derivative of f; typically diminishing as more of the input is used.
- Marginal rate of technical substitution (MRTS)
- The slope of an isoquant, MP_L/MP_K; how much capital can be cut per extra unit of labour while holding output fixed.
- Returns to scale
- How output responds to scaling all inputs by t: increasing if f(tx) > tf(x), constant if equal, decreasing if less. Cobb-Douglas keys off a + b vs 1.
- Cost minimisation
- Choosing inputs to produce a target output at least cost; the optimum sets MRTS = w/r so the isoquant is tangent to the isocost line.
Production: Technology & Profit Maximisation FAQ
What is the difference between the MRS and the MRTS?
The MRS is a consumer concept — the slope of an indifference curve, set by preferences. The MRTS is the producer analogue — the slope of an isoquant, set by technology (MP_L/MP_K). Both are matched to a price ratio at an optimum: p₁/p₂ for the consumer, w/r for the firm.
How do I get the cost function from a production function?
Solve the cost-minimisation conditions (MRTS = w/r plus the output target) for the conditional input demands L(q) and K(q), then substitute into C = wL + rK. The result is total cost as a function of q, which feeds the cost curves in the next chapter.
Exam move
Practise the two-step cost-minimisation drill: set MRTS = w/r, then impose the output target to solve for both inputs, and substitute to build C(q). Recognise Cobb-Douglas returns to scale instantly from the exponent sum.
