Tags: production function, marginal product, average product, law of diminishing marginal returns, returns to scale, constant returns to scale, increasing returns to scale, decreasing returns to scale, isoquant, isocost, MRTS, marginal rate of technical substitution, expansion path, cost minimisation, ECON 323, Texas A&M, microeconomics
Production theory examines how firms turn inputs (labour, capital) into output. The key concepts are marginal and average product, diminishing returns, returns to scale, and the geometry of isoquants and isocost lines. Cost minimisation requires the firm to find the input mix where the isoquant is tangent to the isocost line, and the expansion path traces these optimal bundles as output changes.
Production function
A mathematical relationship showing the maximum output a firm can produce from a given combination of inputs. Common form: F(K, L), where K is capital and L is labour.
Marginal product of labour (MPL)
The additional output from employing one more unit of labour, holding other inputs constant. MPL = ΔQ / ΔL (or the partial derivative ∂F/∂L).
Average product of labour (APL)
Total output divided by the quantity of labour used: APL = Q / L.
Law of diminishing marginal returns
As the use of one input increases while other inputs are held fixed, the marginal product of that input will eventually decrease. This is a short-run concept (at least one input is fixed).
Returns to scale
A long-run concept describing what happens to output when all inputs are scaled by the same factor.
Constant returns to scale (CRS): output scales proportionally. If inputs double, output doubles.
Increasing returns to scale (IRS): output more than doubles when inputs double.
Decreasing returns to scale (DRS): output less than doubles when inputs double.
Isoquant
A curve showing all combinations of inputs that produce the same level of output. Analogous to an indifference curve in consumer theory.
Marginal rate of technical substitution (MRTS)
The absolute value of the slope of an isoquant. It measures the rate at which the firm can substitute one input for another while keeping output constant. MRTS = MPL / MPK.
Isocost line
A line showing all combinations of inputs the firm can purchase for a given total cost. Its slope is the ratio of input prices: w/r (wage rate over rental rate of capital).
Expansion path
The curve on an isoquant map connecting all the cost-minimising input bundles as the level of output changes. It shows how the firm's optimal input mix evolves with scale.
When MPL > APL, average product is rising. The marginal "pulls up" the average.
When MPL < APL, average product is falling.
When MPL = APL, average product is at its maximum.
This is a purely mathematical relationship, identical in logic to the MC/ATC relationship for cost curves.
Applies in the short run only, because at least one input must be fixed
Does not say total output falls, only that the rate of increase in output slows down
Example: adding more workers to a factory with a fixed number of machines. Initially each new worker adds a lot of output; eventually the factory becomes crowded and each additional worker contributes less
To determine returns to scale for a production function F(K, L), multiply all inputs by a constant t > 1 and compare F(tK, tL) to t · F(K, L).
If F(tK, tL) = t · F(K, L), the function exhibits CRS
If F(tK, tL) > t · F(K, L), the function exhibits IRS
If F(tK, tL) < t · F(K, L), the function exhibits DRS
Cobb-Douglas shortcut: for F(K, L) = K^a · L^b, check the sum a + b.
a + b = 1 → CRS
a + b > 1 → IRS
a + b < 1 → DRS
Worked example (Cobb-Douglas):
F(K, L) = K^0.5 · L^0.5. Sum of exponents: 0.5 + 0.5 = 1 → Constant returns to scale.
Double inputs: F(2K, 2L) = (2K)^0.5 · (2L)^0.5 = 2^0.5 · K^0.5 · 2^0.5 · L^0.5 = 2 · K^0.5 · L^0.5 = 2 · F(K, L). Output exactly doubles, confirming CRS.
Worked example (linear):
F(K, L) = 10K + 5L. Double inputs: F(2K, 2L) = 10(2K) + 5(2L) = 20K + 10L = 2(10K + 5L) = 2 · F(K, L). Output exactly doubles → CRS.
The firm minimises cost for a given output where the isoquant is tangent to the isocost line
At the tangency point: MRTS = w/r (the ratio of input prices)
Equivalently: MPL/w = MPK/r (the marginal product per pound spent is equal across inputs)
Connects all tangency points (cost-minimising bundles) across different output levels
Shows the firm's long-run input combination strategy
If the expansion path is a straight line through the origin, the firm uses inputs in a constant ratio regardless of scale
Concept | Formula / Definition |
|---|---|
MPL | ∂F / ∂L |
APL | Q / L |
MRTS | MPL / MPK = absolute slope of isoquant |
Isocost slope | w / r |
Cost-minimisation condition | MRTS = w/r, or MPL/w = MPK/r |
CRS test (Cobb-Douglas) | a + b = 1 |
General RTS test | Compare F(tK, tL) to t · F(K, L) |
⚠️ Diminishing marginal returns is a short-run concept (one input fixed). Returns to scale is a long-run concept (all inputs vary). Do not confuse them. The exam may offer "diminishing marginal returns" as a distractor in a returns-to-scale question.
⚠️ When MPL > APL, APL is rising. This relationship is tested directly in multiple choice.
⚠️ MRTS is the slope of the isoquant, not the isocost line. The isocost line's slope is the input price ratio.
⚠️ For the returns-to-scale proof questions, show your working: substitute (tK, tL), simplify, and compare explicitly to t · F(K, L). Stating the answer without the algebra loses marks.
⚠️ The expansion path question is a definition recall, but make sure you can distinguish it from the isocost line and the marginal product curve.
Q: If MPL > APL, what is happening to the average product of labour?
A: APL is increasing. When the marginal exceeds the average, it pulls the average upward.
Q: A production function is F(K, L) = K^0.5 · L^0.5. What returns to scale does it exhibit?
A: Constant returns to scale. The exponents sum to 1, and doubling inputs exactly doubles output.
Q: Show that F(K, L) = 10K + 5L has constant returns to scale.
A: F(2K, 2L) = 10(2K) + 5(2L) = 20K + 10L = 2(10K + 5L) = 2F(K, L). Output doubles when inputs double, so CRS.
Q: What is the MRTS, and how is it represented graphically?
A: The MRTS is the rate at which the firm can substitute one input for another while holding output constant. Graphically, it is the absolute value of the slope of an isoquant.
Q: What is the cost-minimisation condition on an isoquant map?
A: The firm minimises cost where the isoquant is tangent to the isocost line, i.e., MRTS = w/r.
Q: What does the expansion path show?
A: It traces all cost-minimising input bundles as the output level changes on an isoquant map.
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