Cobb-Douglas Production Surface

Cobb-Douglas Inputs

Capital K 10.0

Labor L 10.0

Capital share α 0.35

Productivity A 1.4

Output & Marginal Products

Y = A·K^α·L^(1-α)—

MPK = ∂Y/∂K—

MPL = ∂Y/∂L—

Returns to Scale

Scale both inputs by the same factor and compare how much output actually moves.

How this works

The surface plots output Y for every combination of capital K and labor L, using the Cobb-Douglas production function — the workhorse of growth models like Solow and of firm-level output studies.

Y = A · K^α · L^(1-α)
MPK = α·Y/K   MPL = (1-α)·Y/L

K, L — move the marker across the surface; height at that point is output Y, read off the ridge line.
α (capital share) — tilts how much extra output an added unit of capital versus labor buys; higher α makes the surface steeper along the K axis and flatter along L.
A (productivity) — a pure scale-up: raising A lifts the whole surface without changing its curvature or the isoquant shapes.
Isoquant — the glowing ring is every (K,L) combination that yields today's exact output; move along it and Y stays fixed while the K/L mix trades off.
Double / Halve K & L — scales both inputs by the same factor at once. Because the exponents α and (1-α) sum to 1, Y scales by exactly the same factor — constant returns to scale — which is why this shows up as a straight ratio, not an approximation.

Diminishing marginal returns are visible directly: holding L fixed and sliding K up, the surface keeps rising but each extra unit of K buys a smaller output gain — the slope MPK falls as K grows.

Output surface Current (K,L,Y) Isoquant (constant Y)

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