Output from two factors — capital and labour — whose exponents ARE the income shares. Y = A·Kα·Lβ. When α+β=1 the returns to scale are exactly constant: double both inputs, double the output — and Euler's theorem then makes the two factor payments exhaust the product, with capital earning share α and labour share β. Down the center, data flows: the inputs go in, the engine computes, the proven identities come out. The blue team builds and defends it; the red team tries to break it. A model of production, not investment advice.
source Cobb, C. W. & Douglas, P. H., A Theory of Production, American Economic Review 18(1), Suppl. (Mar. 1928), pp. 139–165 — jstor.org/stable/1811556 AMBER · paywalled. Rendered, not quoted.
Nothing here is looked up; it falls out of the algebra of exponents:
RTS scaling both inputs by t scales output by tα+β — constant iff α+β=1, increasing if >1, decreasing if <1. MP the marginal products are MPK=α·Y/K and MPL=β·Y/L. EULER under constant returns K·MPK+L·MPL=Y exactly — the factors exhaust the product, capital taking α, labour β. ISO the isoquants are convex (diminishing MRTS).
Live, for the current α, β, K, L:
| quantity | value |
|---|
Cobb-Douglas (1928) supplies the form of output from two factors: Y = A·Kα·Lβ with α+β=1, giving constant returns and Euler exhaustion — the exponents read straight off as income shares.
Drop the two factors into an accumulation equation and you get the growth engine next door: this is the production function inside the Solow growth model, where f(k)=A·kα drives capital to its steady state. Each sphere is the next one's premise.
The blue team's live check: re-derive the returns-to-scale and Euler-exhaustion identities for the canonical constant-returns case and confirm them against the known truth. If red tampers, this badge is where it shows.
A production function takes total factor productivity A, capital K, labour L, and two exponents:
| symbol | meaning | role |
|---|---|---|
| A | total factor productivity | scales all output |
| K | capital input | raised to α |
| L | labour input | raised to β |
| α | capital exponent | = capital's income share |
| β | labour exponent | = labour's income share |
The single number that decides the character of the whole function is α+β: below 1 it thins out, above 1 it snowballs, exactly 1 and it is constant returns — that is what you feed the panel below.
Every value below is computed live from Y = A·Kα·Lβ — never looked up. Move α or β and watch the returns to scale flip.
Isoquants — each curve is the set of (L,K) giving the same output. They bow toward the origin: convex, the diminishing marginal rate of technical substitution.
What the machine produces, proven exactly: returns to scale = tα+β (constant iff α+β=1), marginal products matching numeric derivatives to 1e-6, and — under constant returns — the Euler identity K·MPK+L·MPL=Y holding to 1e-9, so the factor shares are precisely α and β. The current form's readout is above; the identities are the output.
The blue team's witness (left) confirms these identities live; the red team (right) tries to make them false.
The reading "exponents = income shares" holds only under perfect competition and constant returns AMBER — idealizations, not facts. Economics is not physics: these are ceteris-paribus laws about rational agents, and any forecast built on them is AMBER, not a guarantee. Not investment advice.
"Cobb-Douglas always has constant returns to scale." Cut. Only when α+β=1. With α+β>1 it is increasing, <1 decreasing — the engine computes tα+β live.
"The exponents just measure how important each input is." Cut. They are elasticities — and under competition, the income shares. Euler's theorem is what ties them to payments, and only under constant returns.
"If it fits the data, the economy really is Cobb-Douglas." Kept, corrected. A good fit can be a national-accounts identity in disguise (Shaikh). Fit is not proof of the technology.
The red team's move: claim constant returns while quietly setting the canonical case to α=0.7, β=0.7 (α+β=1.4). The blue team's witness (window 7) is watching the Euler identity.
Break constant returns but keep claiming exhaustion, and K·MPK+L·MPL becomes 1.4·Y ≠ Y — the witness recomputes, disagrees with the known identity, and turns red. Nothing is faked; the attack is real and it is caught.