Cobb-Douglas Production Function Calculator
From total factor productivity, capital, labour and elasticities, compute output and the returns-to-scale property.
輸入資料
計算結果
重點速覽：Cobb-Douglas: Y = A · K^α · L^β, with A total factor productivity and α, β the output elasticities of capital and labour. The sum α+β gives returns to scale: =1 constant, >1 increasing, <1 decreasing. Example: A=1.2, K=100, L=50, α=0.3, β=0.7 → Y ≈ 77.8, α+β=1 → constant returns. It is a standard micro tool for input-mix and output analysis. WARNING: A smooth, simplified form; real production may not fit; ignores tech change over time and fixed factors. Education, not advice.
計算公式
Y = A · K^α · L^β。
α + β = 1：規模報酬不變；> 1 遞增；< 1 遞減。
$$Y = A \, K^{\alpha} \, L^{\beta}$$使用說明
- Enter A, K and L.
- Enter the elasticities α and β.
- View output and the returns-to-scale property.
柯布—道格拉斯生產函數計算範例 (α=0.3、β=0.7)
| 情境 | 技術 A | 資本 K | 勞動 L | 總產出 Y |
|---|---|---|---|---|
| 基準 | 1 | 100 | 100 | 100.00 |
| 資本加倍 | 1 | 200 | 100 | 123.11 |
| 技術加倍 | 2 | 100 | 100 | 200.00 |
| 規模加倍 | 1 | 200 | 200 | 200.00 |
α+β=1 時規模報酬不變：K、L 同時加倍 (規模加倍) 則產出恰好加倍；技術 A 加倍則產出直接加倍。
理財情境案例
案例一：計算總產出並驗證規模報酬
假設技術水平 A = 1、資本 K = 100、勞動 L = 100，資本彈性 α = 0.3、勞動彈性 β = 0.7 (α+β = 1)。
套用公式：Y = A × K^α × L^β = 1 × 100^0.3 × 100^0.7 = 1 × 100^(0.3+0.7) = 100^1 = 100。
若把 K 與 L 同時加倍到 200：Y = 1 × 200^0.3 × 200^0.7 = 200，產出恰好加倍。這驗證了『α+β=1 代表規模報酬不變』— 所有要素同比例擴張，產出等比例增加。
案例二：單一要素的邊際報酬遞減
從基準 (A=1、K=100、L=100、Y=100) 出發，若只把資本 K 加倍到 200、勞動 L 不變：Y = 1 × 200^0.3 × 100^0.7 ≈ 123.11。
資本增加了 100% (從 100 到 200)，但產出只增加了約 23.11% (從 100 到 123.11) — 遠低於資本的增幅。
這體現了『邊際報酬遞減』:在其他要素 (勞動) 不變下，單獨增加一種要素，其對產出的貢獻會越來越小。這也是為何長期經濟成長不能只靠堆資本或堆人力，而必須仰賴技術進步 (提升 A) — 技術 A 加倍會讓產出直接加倍 (Y = 2 × 100 = 200)，不受遞減規律限制。
常見問題
What does the Cobb-Douglas function mean?
It is a classic way to model production: output Y comes from capital K and labour L, scaled by efficiency A. The exponents α and β show how sensitive output is to each input — α is the share of output from an extra unit of capital (roughly), β from labour.
What is returns to scale?
If you scale both K and L by the same factor, output scales by (factor)^(α+β). α+β=1 → output scales equally (constant returns); >1 → more than proportionally (increasing); <1 → less (decreasing). It tells whether growing the firm's inputs yields proportionally more output.
Does α+β=1 mean something special?
Yes — it implies constant returns to scale and, under competition, that α and β approximate the cost shares of capital and labour (the function is 'homothetic' and consistent with factor incomes). Many textbook versions set α+β=1 for simplicity.
How do I use this for a business?
Estimate α and β from your data, then test scenarios: adding capital vs labour, or scaling both, to see the output response. It helps decide where the marginal gain is larger. But treat it as a planning abstraction, not a precise forecast — real processes have fixed factors and limits.
What are the limits of the model?
It imposes a specific smooth, substitution-friendly form that may not match reality; it ignores technological progress over time, fixed inputs (land, management), and the fact that elasticities may vary by scale. Use it to understand relationships and ballpark effects, not as a literal production blueprint.
相關工具
參考資料
內容審核：香港計算器財經團隊。計算邏輯與公式參考香港金融管理局（HKMA）及投資者及理財教育委員會（IFEC）之個人理財計算指引，結果僅供參考，實際以相關機構公佈為準。