| A | α | β | K | L | Action |
|---|---|---|---|---|---|
| 1 | 0.3 | 0.7 | 100 | 200 | |
| 0.9 | 0.4 | 0.5 | 120 | 180 | |
| 1.2 | 0.25 | 0.65 | 150 | 130 | |
| 0.8 | 0.45 | 0.55 | 90 | 240 | |
| 1.1 | 0.5 | 0.4 | 200 | 140 |
Click Load to populate the inputs above.
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| # | Capital K | Labor L | Output Q | MPK | MPL |
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The Cobb–Douglas production function is Q = A · Kα · Lβ, where Q is output, A represents technology (TFP), K is capital input, and L is labor input. Parameters α and β are output elasticities.
- Returns to scale: If α + β > 1 → increasing; = 1 → constant; < 1 → decreasing.
- Marginal products: MPK = ∂Q/∂K = α·A·Kα-1·Lβ = α·Q/K; MPL = β·A·Kα·Lβ-1 = β·Q/L.
- Elasticities: ∂lnQ/∂lnK = α and ∂lnQ/∂lnL = β.
- Enter values for technology A, elasticities α and β, and inputs K and L.
- Choose which input to vary for the chart and set a range with step count.
- Click Calculate to compute Q, marginal products, and returns to scale.
- Review the chart and the generated series table for scenario analysis.
- Use Download CSV to export the series table; use Download PDF to save results with the chart.
- Experiment with the example data to compare different technologies and elasticities.
- Document your assumptions on units (e.g., worker-hours, machine-hours) for clarity.
1) What do α and β represent?
They are output elasticities with respect to capital and labor, indicating percentage change in output from a one percent change in the respective input.
2) Must α + β equal 1?
No. When α + β = 1 the technology exhibits constant returns to scale. Values above or below one indicate increasing or decreasing returns to scale.
3) How should I choose A?
A scales the entire production function and captures technology or efficiency. You can calibrate it to match an observed baseline output for given K and L.
4) Are units important?
Yes. Keep units consistent across inputs and output. For example, if K is machine-hours and L is worker-hours, Q should be in units consistent with those inputs.
5) What are MPK and MPL?
They are marginal products of capital and labor, the additional output from a small increase in each input while holding the other constant.
6) Can I model diminishing returns?
Yes. For Cobb–Douglas, if 0 < α < 1 and 0 < β < 1, holding one input fixed delivers diminishing marginal product for the other.
7) Why does the chart sometimes look flat?
If A is small or elasticities are low, output may change slowly over the chosen range. Expand the axis range or increase steps for more visible variation.