Unit 5: Production Theory

Production theory studies how a firm converts inputs (factors of production) into output, and how it chooses input quantities to produce efficiently. It rests on the production function, which maps the maximum output obtainable from given inputs with a given technology.

• Production function: A technical relation Q = f(L, K, ...), where Q is output per period, L is labour, K is capital. It states the maximum Q for each input bundle, assuming technical efficiency.
• Fixed vs variable inputs: A fixed input cannot be altered in the period considered (plant, machinery); a variable input can (labour, raw material).
• Short run vs long run: Short run = at least one input fixed; long run = all inputs variable, so scale itself can change.
• Assumptions: Given, unchanging technology; inputs are divisible; the firm seeks technical efficiency; homogeneous input units.

II. Production Function with One Variable Input

Law of Variable Proportions (short-run analysis)

With capital fixed and only labour varied, output responds through three measurable concepts.

• Total product (TP): Total output from a given quantity of the variable input: TP = f(L) with K fixed.
• Average product (AP): Output per unit of variable input: AP = TP / L.
• Marginal product (MP): Change in total output from one more unit of the variable input:

MP = ΔTP / ΔL   (or dTP/dL for continuous input)

• Symbols: TP total product, AP average product, MP marginal product, L units of labour.

The behaviour of these curves is governed by the Law of Variable Proportions.

• Statement: As more units of a variable input are added to a fixed input, the marginal product first rises, reaches a maximum, then falls, eventually turning negative.
• Stage I – increasing returns: MP rises and AP rises; better utilisation of the fixed factor as more variable units are added. TP increases at an increasing rate.
• Stage II – diminishing returns: MP falls but stays positive; TP rises at a decreasing rate to its maximum. Rational firms produce here.
• Stage III – negative returns: MP becomes negative and TP falls; the fixed factor is overcrowded by the variable input.
• Key relations between AP and MP:
  • When MP > AP: AP is rising.
  • When MP = AP: AP is at its maximum.
  • When MP < AP: AP is falling.
• Numeric illustration: With K fixed, if labour rises 1→2→3 giving TP 10→24→36, then MP = 10, 14, 12 and AP = 10, 12, 12. MP peaks (14) before AP, confirming the sequence.

III. Production Function with Two Variable Inputs

Isoquant Analysis (long-run choice of factor mix)

When both labour and capital vary, output combinations are shown by isoquants, the two-input parallel of the consumer's indifference curve.

• Isoquant: The locus of all (L, K) combinations yielding the same output level, e.g. every mix producing Q = 100.
• Isoquant map: A set of isoquants; a higher curve represents greater output.
• Properties: Downward sloping, convex to the origin, non-intersecting, and higher isoquants mean more output.

Marginal Rate of Technical Substitution

The slope of an isoquant measures how one input substitutes for another while holding output constant.

• Definition (MRTS): The units of capital a firm can give up for one more unit of labour with output unchanged:

MRTS(L,K) = − ΔK / ΔL = MP_L / MP_K

• Symbols: MP_L marginal product of labour, MP_K marginal product of capital.
• Diminishing MRTS: As more L replaces K, each extra labour unit substitutes less capital, giving the isoquant its convex shape.
• Anchor: If moving along Q = 100, L rises by 1 and K falls by 2, then MRTS = 2; at the next step K may fall by only 1, so MRTS = 1.

Isocost Line

Costs bound the input choice, represented by the isocost line.

• Isocost line: All (L, K) combinations that cost the same total outlay:

C = w·L + r·K

• Symbols: C total cost, w wage rate of labour, r rental price of capital.
• Slope: = w / r, the market rate at which labour trades for capital.
• Shifts: A larger budget C shifts the line outward parallel; a change in w or r rotates it.

IV. Optimal Combination of Inputs

Producer Equilibrium (least-cost input mix)

The optimal combination is where the firm produces a target output at least cost, or gets maximum output from a given cost, found where isoquant and isocost meet.

• Tangency condition: Equilibrium occurs where the isocost line is tangent to the highest attainable isoquant:

MRTS(L,K) = MP_L / MP_K = w / r

• Rearranged (equimarginal principle): Least-cost output requires equal marginal product per rupee across inputs:

MP_L / w = MP_K / r

• Interpretation: The last rupee spent on labour and on capital adds the same output; otherwise the firm reallocates spending toward the higher-yield input.
• Two decision framings:
  1. Cost minimisation: Fix the target isoquant (output), choose the lowest isocost touching it.
  2. Output maximisation: Fix the isocost (budget), reach the highest isoquant touching it.
• Expansion path: The line joining successive equilibrium points as the budget expands, showing how the optimal (L, K) mix scales with output at constant input prices.
• Worked example: If MP_L = 20, MP_K = 10, w = 4, r = 4, then MP_L/w = 5 but MP_K/r = 2.5. Labour yields more per rupee, so the firm hires more labour until falling MP_L restores equality.

V. Returns to Scale

Long-Run Response to Proportional Input Changes

Returns to scale describe how output changes when all inputs rise in the same proportion — a strictly long-run idea since every factor is variable.

• Set-up: Multiply every input by a factor t > 1. Compare the resulting output multiple with t.

If f(tL, tK) = t^n · f(L, K):
   n > 1 → increasing returns to scale
   n = 1 → constant returns to scale
   n < 1 → decreasing returns to scale

• Symbols: t common scaling factor, n degree of homogeneity of the production function.

Increasing Returns to Scale

• Meaning: Output rises more than proportionately; doubling all inputs more than doubles output.
• Causes: Economies of scale — specialisation of labour, indivisibility of large efficient machines, dimensional and technical economies.
• Anchor: Inputs ×2 yield output ×2.5; on the isoquant map, successive equal-output isoquants lie closer together.

Constant Returns to Scale

• Meaning: Output rises in exact proportion; doubling inputs doubles output (n = 1, linear homogeneous function).
• Cause: Internal economies are offset by diseconomies; the process is simply replicated at a larger size.
• Anchor: The Cobb–Douglas form Q = A·L^a·K^b shows constant returns when a + b = 1.

Decreasing Returns to Scale

• Meaning: Output rises less than proportionately; doubling inputs less than doubles output.
• Causes: Diseconomies of scale — managerial control problems, coordination difficulties, communication overload as the firm grows too large.
• Anchor: Inputs ×2 yield output ×1.7; equal-output isoquants spread farther apart.

Returns to Scale versus the Law of Variable Proportions

The two laws answer different questions and must not be confused.

1. Law of variable proportions: Short run; only one input varies while others stay fixed, so input proportions change. It explains diminishing marginal product.
2. Returns to scale: Long run; all inputs vary together in fixed proportion, so scale changes but the input ratio is constant. It explains economies and diseconomies of scale.

Significance and Limitations

• Managerial use: Guides plant-size decisions, staffing, and the cost curve's shape — increasing returns underlie a falling long-run average cost, decreasing returns a rising one.
• Least-cost sourcing: The equimarginal condition tells managers when to substitute machinery for labour as relative prices w/r change.
• Limitations: Assumes constant technology and divisible, homogeneous inputs; real firms face lumpy investment, technological change, and measurement difficulty in isolating one input's marginal product.