Source: Microeconomic Theory, Texas A&M University
Tags: production function, factors of production, marginal product, average product, diminishing marginal returns, isoquant, MRTS, returns to scale, Cobb-Douglas, short run, long run
Production theory explains how firms turn inputs (labour and capital) into output, and how output responds as those inputs change. The chapter moves from one-variable production in the short run to two-variable production in the long run, introducing isoquants and returns to scale along the way.
Factors of production
Inputs into the production process, e.g. labour (L), capital (K), and materials. For simplicity, most problems use a two-input case: L and K.
Production function
A function showing the highest output a firm can produce for every specified combination of inputs: q = F(K, L). Production at this level is both feasible (technically possible) and efficient (no waste of resources).
Fixed input
A production factor that cannot be varied in the short run, e.g. a building or piece of land.
Variable input
A production factor that can be adjusted, e.g. the number of workers hired.
Short run
A period in which at least one production factor is fixed and cannot be changed.
Long run
The amount of time needed to make all production inputs variable. There is no specific calendar boundary between short run and long run; it is determined case by case.
Average product of labour (APL)
Output per unit of labour input.
APL = q / L = F(L) / L
Marginal product of labour (MPL)
The additional output produced when labour is increased by one unit.
MPL = Δq / ΔL = F'(L)
Note: MPL may depend on the level of the fixed input K.
Law of diminishing marginal returns
The principle that as the use of one input increases while other inputs are held fixed, the resulting additions to output will eventually decrease.
Isoquant
A curve showing all possible combinations of inputs that yield the same output. Think of it as the firm's indifference curve in production. For output level q, an isoquant plots all (L, K) such that F(K, L) = q.
Isoquant map
A graph combining several isoquants, used to describe a production function.
Marginal rate of technical substitution (MRTS)
The amount by which the quantity of one input (K) can be reduced when one extra unit of another input (L) is used, while output remains constant.
Graphical definition: the absolute value of the slope of an isoquant
Mathematical definition: MRTS = MPL / MPK
Increasing returns to scale
Output more than doubles when all inputs are doubled: F(2K, 2L) > 2F(K, L). Often associated with specialisation benefits.
Constant returns to scale
Output exactly doubles when all inputs are doubled: F(2K, 2L) = 2F(K, L).
Decreasing returns to scale
Output less than doubles when all inputs are doubled: F(2K, 2L) < 2F(K, L). Often associated with difficulties managing large-scale operations.
When K is fixed at some level K̄, the production function simplifies to F(L).
Visualising APL on a total product curve
Connect the point (L, F(L)) to the origin. The slope of that line equals APL.
Visualising MPL on a total product curve
Find the tangent line to the total product curve at (L, F(L)). The slope of that tangent equals MPL.
Key relationships between MPL and APL
When MPL > APL, the average product is rising
When MPL < APL, the average product is falling
The MPL curve crosses the APL curve at the maximum of APL
When MPL crosses zero
At the point where the marginal product curve crosses the horizontal axis from above (and does not rise again), total product is maximised.
Why diminishing marginal returns make sense in the short run
When K is fixed and L is increasing:
At low levels of L, adding workers can increase MPL through specialisation
At high levels of L, the fixed capital becomes a constraint that limits each additional worker's marginal output
Example 1: F(K, L) = K^0.5 · L^0.5, with K fixed at 1.
F(L) = L^0.5
APL = L^(−0.5)
MPL = 0.5L^(−0.5)
Example 2: F(K, L) = KL, with K fixed at 2.
F(L) = 2L
APL = 2
MPL = 2
Example 3 – Identifying diminishing marginal returns from a graph:
Look for the total product curve whose slope is eventually decreasing (the curve flattens out). Among the four typical shapes shown in the review, only the concave curve (Graph C) satisfies the law of diminishing marginal returns.
Example 4 – Finding largest APL from a graph:
Connect each labelled point on the total product curve to the origin. The steepest of those lines corresponds to the highest APL.
Example 5 – Finding largest MPL from a graph:
Draw the tangent line at each labelled point. The steepest tangent corresponds to the highest MPL.
Reading an isoquant map for diminishing marginal returns
Draw a horizontal line at a fixed level of capital. As you move rightward (adding more labour), the jumps in output between successive isoquants get smaller, confirming diminishing marginal returns.
Example: At K = 3, the MPL of the 2nd unit of labour is 75 − 55 = 20, while the MPL of the 3rd unit is 90 − 75 = 15.
MRTS worked example
If MRTS = K/L, and the firm currently uses L = 10 and K = 20:
MRTS = 20/10 = 2
This means the firm can reduce capital by 2 machine hours and increase labour by 1 hour, with no change in output.
Three types of isoquant maps
Case 1 (standard): smooth curves bending towards the origin, exhibiting diminishing MRTS
Case 2 (perfect substitutes): straight-line isoquants with constant MRTS
Case 3 (perfect complements / fixed-proportions): L-shaped (right-angle) isoquants
When both factors are variable, returns to scale describe how output responds to a proportional increase in all inputs.
Testing returns to scale
Compare F(2K, 2L) with 2F(K, L):
If F(2K, 2L) > 2F(K, L), increasing returns to scale
If F(2K, 2L) = 2F(K, L), constant returns to scale
If F(2K, 2L) < 2F(K, L), decreasing returns to scale
Example 1: F(K, L) = 2L + 3K
F(2K, 2L) = 4L + 6K
2F(K, L) = 4L + 6K
Equal, so constant returns to scale
Example 2 – Cobb-Douglas: F(K, L) = A · L^α · K^β
F(2K, 2L) = 2^(α+β) · A · L^α · K^β
Compare 2^(α+β) with 2:
α + β > 1 → increasing returns to scale
α + β = 1 → constant returns to scale
α + β < 1 → decreasing returns to scale
Production function: q = F(K, L)
Average product of labour: APL = F(L) / L
Marginal product of labour: MPL = ΔF(L) / ΔL = F'(L)
MRTS = MPL / MPK = |slope of isoquant|
Returns to scale test: compare F(2K, 2L) with 2F(K, L)
Cobb-Douglas returns to scale: determined by α + β relative to 1
⚠️ MPL crosses APL at APL's maximum. This relationship appears frequently on exams.
⚠️ Diminishing marginal returns is a short-run concept (at least one input fixed). Returns to scale is a long-run concept (all inputs variable). Do not confuse the two.
⚠️ On graphical questions, APL is the slope of the line from the origin to the curve; MPL is the slope of the tangent. Know how to use both techniques.
⚠️ For Cobb-Douglas, returns to scale hinge entirely on whether α + β is greater than, equal to, or less than 1.
⚠️ Perfect substitutes produce straight-line isoquants; perfect complements produce L-shaped isoquants. Be ready to identify the type from a graph.
Q: What is the difference between diminishing marginal returns and decreasing returns to scale?
A: Diminishing marginal returns applies in the short run when one input increases while another is held fixed, and refers to MPL eventually falling. Decreasing returns to scale is a long-run concept where all inputs increase proportionally and output rises by less than that proportion.
Q: If F(K, L) = K^0.5 · L^0.5 and K is fixed at 4, what are APL and MPL?
A: F(L) = 2L^0.5. APL = 2L^(−0.5). MPL = L^(−0.5).
Q: A production function has MRTS = K/L. If L = 5 and K = 15, how many units of capital can be given up when one more unit of labour is added, holding output constant?
A: MRTS = 15/5 = 3. Three units of capital can be given up per additional unit of labour.
Q: Does F(K, L) = 5K^0.3 · L^0.8 exhibit increasing, constant, or decreasing returns to scale?
A: α + β = 0.3 + 0.8 = 1.1 > 1, so increasing returns to scale.
Q: On a total product curve, at what point is total output maximised?
A: Where MPL = 0, i.e. where the tangent to the total product curve is flat.
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