Source: Lecture Notes of Prof. Guoqiang Tian, Texas A&M University
Tags: production function, isoquant, marginal product, average product, diminishing marginal returns, MRTS, returns to scale, CRS, IRS, DRS, total cost, fixed cost, variable cost, marginal cost, average cost, isocost, cost minimisation, long-run average cost, expansion path
Production theory examines how firms transform inputs (labour, capital) into output, using tools that closely parallel consumer theory. The short run features fixed inputs and the law of diminishing marginal returns; the long run allows all inputs to vary and introduces returns to scale. Cost theory translates the production function into cost curves, which become the foundation for analysing profit maximisation in later chapters. The key result: in the long run, cost minimisation requires MRTS = w/r (the ratio of input prices).
Inputs (factors of production)
The resources a firm uses to produce goods or services: labour (L), capital (K), land.
Production function
A relationship identifying the maximum output the firm can produce with each combination of inputs. Q = F(K, L).
Technological efficiency
A production process is technologically efficient if it produces the maximum output possible from a given combination of inputs.
Isoquant
A curve showing all combinations of inputs that yield the same level of output. Analogous to an indifference curve.
Short run
A time period in which at least one input is fixed (typically capital).
Long run
A time period in which the firm can vary all inputs, including plant capacity.
Total product (TP)
The total output produced by the firm.
Average product of labour (APL)
Total product per unit of variable input. APL = TP / L. Geometrically, it is the slope of a line from the origin to a point on the TP curve.
Marginal product of labour (MPL)
The change in total output from a one-unit change in labour, holding other inputs constant. MPL = delta TP / delta L. Geometrically, it is the slope of the TP curve itself.
Law of diminishing marginal returns (DMR)
As more and more of a variable input is used together with fixed amounts of other inputs and fixed technology, a point is reached beyond which the marginal product of the variable input begins to fall.
Marginal rate of technical substitution (MRTS)
The rate at which one input can be substituted for another without changing output. MRTS_LK = MPL / MPK. Equal to the absolute value of the slope of the isoquant.
Law of diminishing MRTS
As you move down an isoquant (more L, less K), the MRTS diminishes. This is why isoquants are convex to the origin.
Returns to scale
The effect on output of proportionately increasing all inputs.
Constant returns to scale (CRS)
Doubling all inputs exactly doubles output. F(lambda K, lambda L) = lambda F(K, L).
Increasing returns to scale (IRS)
Doubling all inputs more than doubles output. F(lambda K, lambda L) > lambda F(K, L).
Decreasing returns to scale (DRS)
Doubling all inputs less than doubles output. F(lambda K, lambda L) < lambda F(K, L).
Total fixed cost (TFC)
Costs of fixed factors that do not change with output level. Incurred even at zero output.
Total variable cost (TVC)
Costs that depend on how much output is produced.
Total cost (TC)
TC = TFC + TVC.
Average fixed cost (AFC)
AFC = TFC / Q. Always declining as output increases.
Average variable cost (AVC)
AVC = TVC / Q = w / APL.
Average total cost (ATC)
ATC = TC / Q = AFC + AVC.
Marginal cost (MC)
The change in total cost from a one-unit change in output. MC = delta TC / delta Q = delta TVC / delta Q = w / MPL.
Isocost line
Shows all combinations of inputs a firm can purchase for a given total cost. Equation: C = wL + rK. Slope = -w/r. Analogous to the budget line.
Expansion path
Connects all cost-minimising input combinations as output varies. Analogous to the ICC in consumer theory.
Opportunity cost
The cost of a resource measured in terms of the best alternative use forgone. Includes both explicit and implicit costs.
With capital fixed, varying labour produces the total product curve. Key relationships:
When MPL > APL, the APL is rising.
When MPL < APL, the APL is falling.
When MPL = APL, the APL is at its maximum.
When MPL = 0, the TP reaches its maximum.
The law of diminishing marginal returns means MPL eventually declines. This shapes the TP curve: it first rises at an increasing rate (MPL rising), then at a decreasing rate (MPL falling), and eventually turns downward (MPL negative).
Isoquant properties mirror indifference curve properties:
Downward sloping
Never intersect
Higher isoquants represent more output
Convex to the origin (diminishing MRTS)
MRTS = MPL / MPK. Along an isoquant, dQ = 0, so MPL dL + MPK dK = 0, which gives dK/dL = -MPL/MPK.
Special cases: perfect substitutes give straight-line isoquants (constant MRTS); perfect complements (fixed proportions) give L-shaped isoquants, Q = min(K, L).
To test: multiply all inputs by lambda and see how output changes.
F(lambda K, lambda L) = lambda^t * F(K, L)
t = 1: CRS
t > 1: IRS
t < 1: DRS
Examples:
F(K, L) = 5K^(1/3) * L^(2/3) gives t = 1 (CRS).
F(K, L) = 7K + 6L gives t = 1 (CRS).
F(K, L) = KL gives t = 2 (IRS).
In practice, IRS is common at small scale, CRS at intermediate scale, and DRS at large scale.
MC = w / MPL. Because of diminishing marginal returns, MPL eventually falls, so MC eventually rises. This gives MC its U-shape.
AVC = w / APL. Since APL rises then falls, AVC falls then rises (also U-shaped).
MC intersects both AVC and ATC at their respective minimum points. The logic is the same as the marginal-average relationship: when the marginal is below the average, it pulls the average down; when above, it pulls it up.
The firm minimises cost for a given output level by finding the point on the isoquant that is tangent to the lowest isocost line.
Tangency condition: MRTS = w/r, which can be rewritten as MPL/w = MPK/r.
This means the firm should employ inputs so that the marginal product per pound (or dollar) spent is equal across all inputs.
The long-run ATC curve is the envelope of all short-run ATC curves. For each output level, it shows the lowest achievable average cost when the firm is free to choose any plant size.
The U-shape of the long-run ATC reflects returns to scale: IRS causes AC to fall, DRS causes AC to rise.
Long-run MC intersects long-run AC at the minimum of AC.
A change in an input price shifts the entire family of cost curves. For example, a fall in the wage rate lowers MC and AC, and also changes the optimal input mix (more labour, less capital) along new tangency points.
When two firms have different marginal costs of pollution abatement, the cheapest way to achieve a total pollution reduction target is to equalise marginal abatement costs across firms. This is more efficient than requiring equal reductions from each firm.
Production function: Q = F(K, L)
APL = Q / L
MPL = delta Q / delta L
MRTS = MPL / MPK = absolute value of slope of isoquant
Returns to scale test: F(lambda K, lambda L) = lambda^t * F(K, L)
Isocost: C = wL + rK, slope = -w/r
Cost minimisation: MRTS = w/r, equivalently MPL/w = MPK/r
MC = delta TC / delta Q = w / MPL
AVC = TVC / Q = w / APL
ATC = AFC + AVC
Long-run AC = envelope of short-run ATCs
⚠️ The relationship MC = w/MPL is critical. It directly links the production function to cost curves. When MPL rises, MC falls, and vice versa.
⚠️ MC passes through the minimum of both AVC and ATC. Know why (marginal-average relationship).
⚠️ Returns to scale is a long-run concept (all inputs vary). Diminishing marginal returns is a short-run concept (one input varies). Do not confuse them.
⚠️ The cost-minimisation tangency condition (MRTS = w/r, or MPL/w = MPK/r) is the producer-theory analogue of MRS = px/py in consumer theory. Exam questions frequently test both.
⚠️ The long-run ATC is the envelope of short-run ATCs, not their intersection.
⚠️ A production function can exhibit IRS at low output, CRS at intermediate output, and DRS at high output.
Q: If F(K, L) = K^(0.5) * L^(0.5), what kind of returns to scale does this production function exhibit?
A: CRS. F(lambda K, lambda L) = (lambda K)^(0.5) (lambda L)^(0.5) = lambda K^(0.5) L^(0.5) = lambda F(K, L). Since the exponent on lambda is 1, it is constant returns to scale.
Q: Why is the marginal cost curve U-shaped in the short run?
A: Because MC = w/MPL. The law of diminishing marginal returns means MPL initially rises (MC falls) and then falls (MC rises), producing the U-shape.
Q: At the point where MC = AVC, what is happening to AVC?
A: AVC is at its minimum. When MC < AVC, AVC is falling; when MC > AVC, AVC is rising. They cross at AVC's lowest point.
Q: What condition must hold at the cost-minimising input combination in the long run?
A: MRTS = w/r, which is equivalent to MPL/w = MPK/r. The marginal product per pound spent must be equal across all inputs.
Q: Why does the long-run ATC curve have a U-shape even though diminishing marginal returns do not apply in the long run?
A: The long-run U-shape reflects returns to scale. At low output, increasing returns to scale drive AC down. At high output, decreasing returns to scale push AC up.
Q: A firm has MC of pollution abatement of $4,000 and another firm has MC of $2,000. Both currently abate 100 units each. How can total cost be reduced?
A: Shift abatement from the high-MC firm to the low-MC firm. Let the firm with MC = $2,000 abate one more unit (cost: $2,000) and the firm with MC = $4,000 abate one fewer unit (saving: $4,000). Net saving: $2,000. Continue until marginal abatement costs are equalised.
production function, inputs, factors of production, labour, capital, isoquant, isoquant map, short run, long run, total product, average product, marginal product, law of diminishing marginal returns, diminishing returns, marginal rate of technical substitution, MRTS, perfect substitutes in production, fixed proportions, Leontief production function, returns to scale, constant returns, increasing returns, decreasing returns, homogeneous function, total cost, fixed cost, variable cost, marginal cost, average cost, ATC, AVC, AFC, MC, isocost line, cost minimisation, expansion path, long-run average cost, envelope curve, economies of scale, diseconomies of scale, pollution abatement, ECON 323