Questions: Returns to Scale and Production Function Homogeneity

Question 1

A firm uses production function f(K, L) = K^0.3 · L^0.3. The firm currently produces 1,000 units. If it doubles both capital and labor, how much will it produce?

• A. Exactly 2,000 units — constant returns to scale
• B. More than 2,000 units — increasing returns to scale
• C. Less than 2,000 units — decreasing returns to scale
• D. 4,000 units — because each exponent doubles the input's contribution

Answer: (C)

Check by computing f(2K, 2L) = (2K)^0.3 · (2L)^0.3 = 2^0.3 · K^0.3 · 2^0.3 · L^0.3 = 2^0.6 · f(K,L). Since 2^0.6 ≈ 1.516, doubling inputs only multiplies output by about 1.516 — less than doubling. The degree of homogeneity is α + β = 0.3 + 0.3 = 0.6 < 1, which directly signals decreasing returns to scale. Option D is a common error: the exponents don't 'each double' independently; the homogeneity degree is the sum of exponents, and you raise the scalar t to that power.

Question 2

A factory manager observes that each additional worker hired produces less additional output than the previous worker — the 100th worker adds less than the 99th. What does this tell us about the factory's returns to scale?

• A. The factory exhibits decreasing returns to scale
• B. The factory exhibits diminishing marginal returns to labor, but this tells us nothing definitive about returns to scale
• C. The factory exhibits constant returns to scale because only one input is changing
• D. The factory exhibits increasing returns to scale in capital, since labor is the bottleneck

Answer: (B)

Diminishing marginal returns (diminishing MPL) describes what happens when only one input (labor) increases while others (capital) are held fixed. Returns to scale asks what happens when ALL inputs scale proportionally. A production function can simultaneously have diminishing marginal returns to each individual input AND exhibit constant or even increasing returns to scale. The Cobb-Douglas function K^0.5 · L^0.5 has diminishing MPL and MPK but exhibits constant returns to scale (exponents sum to 1). The factory manager is observing a one-input phenomenon, not a scale phenomenon.

Question 3

A production function f(K, L) = K^0.5 · L^0.5 exhibits constant returns to scale.

• T. True
• F. False

Answer: True

For a Cobb-Douglas function f(K,L) = K^α · L^β, returns to scale equals the degree of homogeneity, which is α + β. Here α + β = 0.5 + 0.5 = 1, which means f(tK, tL) = t¹ · f(K,L) — exactly constant returns. If you double both inputs, output exactly doubles. This is the replication argument: a factory can be perfectly duplicated, producing a proportional increase in output.

Question 4

A firm that exhibits diminishing marginal returns to labor is expected to also exhibit decreasing returns to scale.

• T. True
• F. False

Answer: False

These are distinct concepts. Diminishing marginal returns to labor means ∂²f/∂L² < 0 — holding K fixed, each additional unit of labor adds less output than the last. Returns to scale measures what happens when both K and L increase proportionally. The Cobb-Douglas function K^0.5 · L^0.5 shows diminishing MPL (and diminishing MPK) but constant returns to scale. A firm can exhibit IRS, CRS, or DRS while simultaneously having diminishing marginal returns to each individual input. Conflating the two is one of the most common errors in producer theory.

Question 5

What is the difference between diminishing marginal returns to an input and decreasing returns to scale? Why does this distinction matter for understanding long-run costs?

Model answer: Diminishing marginal returns applies when only one input increases while others are held constant — it describes movement along a single isoquant's slope getting flatter. Decreasing returns to scale applies when all inputs increase proportionally — it describes how far apart isoquants are. A firm can have both diminishing marginal returns (a short-run concept) and constant returns to scale (a long-run concept) simultaneously. The distinction matters for costs because decreasing returns to scale implies rising long-run average cost (each unit becomes more expensive as the scale of all inputs grows), while diminishing marginal returns explains rising short-run marginal cost (as you add labor to a fixed factory). Misidentifying one as the other leads to incorrect predictions about whether large firms have cost advantages over small ones.

The long-run vs. short-run framing is key: short-run analysis holds some inputs fixed and observes diminishing returns to the variable input. Long-run analysis varies all inputs proportionally. Returns to scale is inherently a long-run concept about the technology, while diminishing marginal returns describes short-run constraints.