Isoquants and returns to scale

A teaching model — a Cobb-Douglas form is assumed; these are not real generation technology values.

Function Q = 2·L^a·K^b
0.6
0.3

a + b = 0.9

Point on the curve · scale
10
×2
MRTS = (a/b)·(K/L)4 L 10 · K 20

Decreasing returns to scale — output grows slower than inputs: inputs ×2 → output ×1.866

An isoquant joins the labour–capital mixes that give the same output. As labour rises, the capital you can drop while holding output keeps shrinking — diminishing MRTS, which is why the curve bulges toward the origin. Scale every input up together: more than in proportion is increasing returns, equal is constant, less is decreasing.

010203040020406080Labour LCapital KQ ×1.87outward = more output

Base isoquant 0.5× · 1.5× Tangent (slope = MRTS) All inputs ×2

Model assumption: Cobb-Douglas Q = A·L^a·K^b (elasticity of substitution 1). Convexity, diminishing MRTS and returns to scale are in the study guide; the functional form is a learning supplement. Curves never touching the axes is a property of this form only. Returns to scale is about all inputs together, not diminishing marginal product of one.

Not in this calculation

  • Perfect substitutes (straight) and complements (L-shaped), and forms in between — only Cobb-Douglas is drawn
  • Cost minimisation (tangency with an isocost) and the expansion path
  • Input prices
  • Real values for generation technology
  • Changing one input in the short run — see the product curves tool

Using the example values

Tool: Renuary (CC BY 4.0) · Numbers are examples. Method and limits on the original page.