ECON5001 · Microeconomic Theory
Production & Costs
This is consumer theory turned inside out — the supply side of the model. Where a household turned a budget into the best bundle, a firm now turns inputs into output and asks what the cheapest way to hit any output target costs. Almost every tool transfers: isoquants play the role of indifference curves, the technical rate of substitution (TRS) replaces the MRS, and the isocost line replaces the budget line. You build the chapter in two halves — first the technology (marginal and average product, the three production forms, the TRS, and returns to scale), then the costs (cost minimisation, which gives the cost function; the short-run cost family with MC cutting AVC and ATC at their minima; and the long-run cost envelope). These cost curves become the supply side of the perfect-competition and monopoly chapters, so getting them clean here pays off twice. The exam rewards setting up and grinding the canonical model — differentiate for marginal products, add exponents for returns to scale, set MC equal to the average curve to find its minimum.
What this chapter covers
- 011. Production function y = f(x) & isoquants — the indifference curves of production
- 022. Marginal product MPᵢ = ∂f/∂xᵢ, average product & diminishing MP (short run)
- 033. The three production forms — Cobb-Douglas, fixed proportions, perfect substitutes
- 044. Technical rate of substitution TRS = −MP₁/MP₂ — the isoquant slope
- 055. Returns to scale (constant/increasing/decreasing) vs diminishing MP
- 066. Cost minimisation — isocost tangency TRS = −w₁/w₂ → conditional input demands → c(w,y)
- 077. Short-run cost family — F, VC, AFC, AVC, ATC, MC; MC cuts AVC & ATC at their minima
- 088. The long run — LRAC is the lower envelope of SRAC (but LMC is NOT the envelope of SRMC)
Cost minimisation: conditional input demands & the cost function
- +2Set up the tangency. Cost minimisation needs the isoquant tangent to an isocost line: MRTS = MP_L/MP_K = w/r. For this technology MP_L = ½L^(−1/2)K^(1/2) and MP_K = ½L^(1/2)K^(−1/2), so MP_L/MP_K = K/L.
- +2Apply the price ratio. K/L = w/r = 4/9, so K = (4/9)L. (Capital is dearer, so the firm leans toward labour.)
- +3Impose the output constraint. √(LK) = 60 ⇒ LK = 3600. Substitute K = (4/9)L: (4/9)L² = 3600 ⇒ L² = 8100 ⇒ L* = 90, and K* = (4/9)(90) = 40.
- +1Total cost. C = wL* + rK* = 4(90) + 9(40) = 360 + 360 = 720. (Check: LK = 90·40 = 3600, so √3600 = 60 ✓.)
Key terms
- Production function & isoquant
- The production function y = f(x₁, x₂) gives the maximum output from an input bundle. An isoquant is one of its level curves — all input bundles that yield the same output ȳ. It is the 'indifference curve of production': further from the origin means more output, isoquants do not cross, and for smooth technologies they are convex.
- Marginal & average product
- Marginal product MPᵢ = ∂f/∂xᵢ is the extra output from one more unit of input i, holding the others fixed; average product APᵢ = y/xᵢ is output per unit of that input. MP cuts AP at AP's maximum — the same average–marginal geometry that governs cost curves.
- Diminishing marginal product
- ∂²f/∂xᵢ² < 0 — each extra unit of one input adds less output than the last, because the fixed inputs become crowded. It is a SHORT-RUN idea (one input varies). It is distinct from, and can coexist with, returns to scale.
- Technical rate of substitution (TRS)
- The slope of an isoquant, TRS = −MP₁/MP₂: the rate at which input 2 can be swapped for input 1 while keeping output constant. It is the production analogue of the consumer's MRS. For Cobb-Douglas A·x₁ᵃx₂ᵇ, TRS = −(a/b)(x₂/x₁).
- Returns to scale
- What happens to output when ALL inputs are scaled by a factor t > 1: constant (f(tx) = t·f(x)), increasing (> t·f(x)) or decreasing (< t·f(x)). For Cobb-Douglas it is decided entirely by the exponent sum a + b (=1 constant, >1 increasing, <1 decreasing). It is a long-run, whole-firm concept — not the same as diminishing MP.
- Cost minimisation, conditional input demands & the cost function
- Choosing inputs to minimise w₁x₁ + w₂x₂ subject to f(x₁,x₂) = y. The isocost line (slope −w₁/w₂) is tangent to the isoquant at the optimum: TRS = −w₁/w₂, equivalently 'equal bang per dollar' MP₁/w₁ = MP₂/w₂. Solving this with the output constraint gives the conditional input demands xᵢ*(w, y), and substituting them back gives the cost function c(w, y) — the object every later supply curve is built from. (For fixed-proportions or perfect-substitute technologies the tangency fails — use corner / fixed-ratio logic.)
- Short-run cost family (F, VC, AFC, AVC, ATC, MC)
- Short-run total cost splits as c(y) = F + c_v(y): fixed cost F (paid whatever you produce) plus variable cost. Then AFC = F/y, AVC = c_v/y, ATC = AFC + AVC, and MC = dc/dy = dc_v/dy (F drops out). MC cuts both AVC and ATC from below at their minimum points.
- Long-run cost as an envelope
- In the long run every input adjusts, so there is no fixed cost. LRAC is the LOWER ENVELOPE of all the short-run average-cost curves — the cheapest plant for each output. Crucially, LMC is NOT the envelope of the SRMCs: LMC equals the SRMC of the plant the firm actually chooses at that output.
Production & Costs FAQ
What's the difference between diminishing marginal product and decreasing returns to scale?
They answer different questions. Diminishing marginal product is a SHORT-RUN idea — you increase ONE input while holding the others fixed, and each extra unit adds less output (∂²f/∂xᵢ² < 0). Decreasing returns to scale is a LONG-RUN idea — you scale ALL inputs up together and output rises less than proportionally. They can coexist: a constant-returns Cobb-Douglas (exponents summing to 1) still has diminishing marginal product in each input. Answering 'decreasing returns' just because the marginal products diminish is one of the most-tested traps in this chapter.
How do I find returns to scale for a Cobb-Douglas function quickly?
Add the exponents. For f = A·x₁ᵃx₂ᵇ, returns to scale are constant if a + b = 1, increasing if a + b > 1, and decreasing if a + b < 1. To prove it from scratch, substitute (tx₁, tx₂) and factor out the power of t: f(tx) = t^(a+b) f(x). You almost never need the full definition in an exam — the exponent sum settles it.
Why does fixed cost disappear from marginal cost?
Because MC is the derivative of total cost, MC = dc/dy, and the fixed cost F is a constant — its derivative is zero. So MC = dc_v/dy depends only on variable cost. This is exactly why fixed (sunk) cost is irrelevant to the marginal output decision: it does not change as you produce one more unit. It is also why the short-run supply decision in Chapter 7 depends on AVC, not ATC.
Why does marginal cost cut AVC and ATC at their minimum points?
It is pure averaging. While MC is below an average curve, the next unit costs less than the running average, so the average is pulled DOWN; while MC is above it, the average is pulled UP. The average can therefore only stop falling and start rising — i.e. reach its minimum — exactly where MC crosses it. The same logic applies to BOTH AVC and ATC, so state it for both; quoting it only for ATC drops a mark.
Is long-run average cost just the envelope of short-run average costs — and does the same hold for marginal cost?
LRAC is the lower envelope of the SRAC curves — for each output the firm picks the cheapest plant, so LRAC touches each SRAC at the output where that plant is best and lies below them elsewhere. But this does NOT carry over to marginal cost: LMC is NOT the envelope of the SRMCs. Instead LMC equals the short-run marginal cost of the plant actually chosen at that output (the plant whose SRAC is tangent to LRAC there). Writing 'LMC is the envelope of the SRMCs' is a guaranteed lost mark.
When does the tangency rule fail in cost minimisation?
The tangency TRS = −w₁/w₂ assumes smooth, convex isoquants — true for Cobb-Douglas. It fails for the two corner technologies. With fixed proportions (L-shaped isoquants) inputs are used in a rigid ratio, so you read inputs straight off the corner. With perfect substitutes (straight-line isoquants) the firm uses only the cheaper input per unit of output, a corner solution. This mirrors perfect complements and perfect substitutes in consumer theory.
Exam move
Treat this chapter as two reflexes you can run on autopilot, because the exam tests setup speed under time pressure. Reflex one — technology: given f(x₁, x₂), differentiate for the marginal products, form TRS = −MP₁/MP₂, and settle returns to scale by adding the Cobb-Douglas exponents (never confuse diminishing MP with decreasing returns to scale). Reflex two — costs: any cost-minimisation problem is 'tangency + constraint' (two equations, two unknowns) yielding conditional input demands and then the cost function c(w, y); any cost-curve problem is 'decompose c(q) into F, VC, AFC, AVC, ATC, MC, then set MC equal to the average you want to minimise and solve.' Memorise the average–marginal rule (MC cuts AVC and ATC at their minima) and its averaging reason, and memorise the long-run envelope result with its caveat (LRAC is an envelope, LMC is not). Drill both reflexes on numerical problems until the setup is automatic — and remember these cost curves are the firm's supply side, so they reappear immediately in perfect competition (P = MC, above min AVC short-run / min AC long-run) and monopoly.
