What happens if a firm doubles everything: twice the workers, twice the machines, twice the floor space? The answer is a property of the production function, and it is what shapes the long-run cost curve in Economies and Diseconomies of Scale.
Key idea
Multiply every input by $t > 1$. If output rises by more than $t$, the technology has increasing returns to scale; by exactly $t$, constant; by less, decreasing. For $Q = AK^\alpha L^\beta$: $$f(tK, tL) = t^{\alpha + \beta}f(K, L),$$ so returns are increasing, constant or decreasing as $\alpha + \beta$ is greater than, equal to or less than 1.
Why it works
$A(tK)^\alpha(tL)^\beta = t^\alpha t^\beta AK^\alpha L^\beta$. The exponents add because both inputs are scaled together. Returns to scale and diminishing marginal product answer different questions. Diminishing $MP_L$ asks what happens when labor alone grows, with the building and machines fixed: workers crowd each other. Returns to scale asks what happens when all inputs grow together: the firm builds a second plant as well as hiring. A technology can have both diminishing marginal products and constant returns.
Worked example: Doubling a Cobb-Douglas firm
$Q = K^{0.3}L^{0.5}$ now makes 100 units. What happens if both inputs double?
- Output is multiplied by $2^{0.3 + 0.5} = 2^{0.8} \approx 1.74$.
- About 174 units: less than double, decreasing returns to scale.
Answer: About 174 units.
Worked example: Diminishing products, constant returns
Show that $Q = K^{1/2}L^{1/2}$ has diminishing marginal products and constant returns to scale.
- $MP_L = \tfrac{1}{2}K^{1/2}L^{-1/2}$ falls as $L$ rises with $K$ fixed; likewise $MP_K$. Diminishing marginal products.
- $f(tK, tL) = \sqrt{t^2KL} = t\sqrt{KL}$: constant returns.
Answer: Both hold.
Worked example: A sum of square roots
What kind of returns does $Q = \sqrt{K} + \sqrt{L}$ have?
- $f(tK, tL) = \sqrt{t}\sqrt{K} + \sqrt{t}\sqrt{L} = \sqrt{t}\,f(K, L)$.
- $\sqrt{t} < t$ for $t > 1$: decreasing returns to scale.
Answer: Decreasing returns to scale.
Common mistake
Concluding that diminishing marginal products mean decreasing returns to scale. The second example has diminishing marginal products and constant returns. Check: to test returns to scale, scale every input by the same factor and compare output with that factor. Never test it by changing one input.
Returns to scale drive long-run costs. With constant returns and fixed input prices, doubling output doubles cost, so long-run average cost is flat; increasing returns give economies of scale (falling average cost); decreasing returns give diseconomies.