Cobb–Douglas
Output, marginal products, factor shares and returns to scale from Y = AKᵅLᵝ.
Y = 1.5 · K0.3 · L0.7
Total product
Doubling both inputs multiplies output by 2
with 0.667 of capital per worker
| Marginal product of capital | 0.59769 | αY ÷ K — output from one more unit of capital |
| Marginal product of labour | 0.92974 | βY ÷ L — output from one more worker |
| Average product of capital | 1.9923 | Y ÷ K |
| Average product of labour | 1.3282 | Y ÷ L |
| MRTS of labour for capital | 1.55556 | MPL ÷ MPK — units of capital one worker replaces |
| Capital’s share | 30% | equals α when markets are competitive |
| Labour’s share | 70% | equals β on the same assumption |
Factor payments exhaust output exactly: K × MPK + L × MPL = 398.4604, which is Y. With constant returns there is nothing left over, which is why the function is the workhorse of growth theory.
The Cobb–Douglas function Y = AKᵅLᵝ turns capital and labour into output, with A standing for productivity. Its exponents do double duty: they are the output elasticities, and under competition they are also the factor shares. α + β decides returns to scale.
Why the exponents carry so much meaning
Raise capital by 1% and output rises by α%. That is what it means to say α is the output elasticity of capital, and it falls straight out of the power form — the exponent of a power function is its elasticity. Since it does not depend on how much capital there is, the elasticity is constant everywhere, which is the function’s great convenience and its main simplification.
The second meaning follows from competition. A firm paying each factor its marginal product pays capital K × MPK, and MPK = αY ÷ K, so the wage bill for capital is exactly αY — a share α of output. Labour takes β. This is why estimates of labour’s share of national income, historically around two thirds, are quoted as evidence that β ≈ 0.7.
Returns to scale is just the sum
Multiply both inputs by t and output is multiplied by t^(α+β). So α + β = 1 gives constant returns — double everything and output doubles — while a sum above 1 means scale pays and below 1 means it does not. Nothing else in the function affects this, which is why the sum is the first thing to look at.
α + β > 1 increasing returns, = 1 constant, < 1 decreasing
- 1 Compute output. With A = 1.5, K = 200, L = 300, α = 0.3 and β = 0.7: Y = 1.5 × 200⁰·³ × 300⁰·⁷ = 398.46.
- 2 Add the exponents for returns to scale. 0.3 + 0.7 = 1, so doubling both inputs exactly doubles output.
- 3 Get the marginal products. MPK = 0.3 × 398.46 ÷ 200 = 0.5977, and MPL = 0.7 × 398.46 ÷ 300 = 0.9297.
- 4 Read off the factor shares. Capital takes 30% of output and labour 70% — exactly the exponents, under competition.
- 5 Check that the shares add up. K × MPK + L × MPL = 119.54 + 278.92 = 398.46, which is Y. With constant returns, factor payments exhaust output exactly.
What the sum of the exponents does
Effect of doubling both capital and labour, for Y = AKᵅLᵝ.
| α + β | Returns to scale | Doubling inputs multiplies output by | Factor payments |
|---|---|---|---|
| 0.7 | Decreasing | 1.62 | Leave a surplus over output |
| 0.9 | Decreasing | 1.87 | Leave a surplus over output |
| 1.0 | Constant | 2.00 | Exhaust output exactly |
| 1.2 | Increasing | 2.30 | Exceed output — competition cannot survive |
| 1.5 | Increasing | 2.83 | Exceed output — competition cannot survive |
Euler’s theorem, and why it matters
The last column of that table is the most interesting one. Euler’s theorem says that for a function homogeneous of degree α + β, the factor payments K × MPK + L × MPL come to exactly (α + β) × Y. With constant returns that is Y itself — the firm can pay both factors their marginal product and the books balance precisely, with nothing left over and no shortfall. This is the adding-up result that makes constant returns the standard assumption in growth theory.
Depart from it and something has to give. With increasing returns, paying marginal products costs more than the firm earns, so perfect competition is impossible and the industry tends toward a few large firms — which is the formal version of the intuition about natural monopoly. With decreasing returns there is a surplus left over, usually attributed to a factor the function does not name, such as land or entrepreneurship.
Two cautions about using it. The function assumes a constant elasticity of substitution of exactly one between capital and labour, which is a strong restriction rather than an empirical finding — the CES family generalises it precisely because that assumption often fails. And A absorbs everything not explained by measured capital and labour, which is why the Solow residual is sometimes called a measure of our ignorance: it is what is left after the two inputs have done what they can.