In This Article
Introduction
A production function describes the technical relationship between inputs (factors of production) and output. It shows the maximum output that can be produced with given quantities of inputs, assuming efficient use of resources.
Definition and Key Concepts
General Production Function:
Q = f(L, K, N, T)
Where: Q = Output, L = Labor, K = Capital, N = Land, T = Technology
Key Assumptions
- Technology is given (held constant)
- Inputs are used efficiently
- Inputs can be substituted to some degree
Short-Run Production
In the short run, at least one input is fixed (typically capital). Only variable inputs can be changed.
Key Measures
| Measure | Formula | Meaning |
|---|---|---|
| Total Product (TP) | Q | Total output |
| Average Product (AP) | TP / L | Output per unit of labor |
| Marginal Product (MP) | ΔTP / ΔL | Additional output from one more unit of labor |
Law of Diminishing Returns
As more units of a variable input are added to a fixed input, eventually the marginal product of the variable input will decline.
Stage I: MP > AP, AP rising (increasing returns)
Stage II: MP < AP, both positive (optimal range)
Stage III: MP < 0 (negative returns)
Long-Run Production
In the long run, all inputs are variable. The firm can change its scale of operations.
Returns to Scale
| Type | Effect | Example |
|---|---|---|
| Increasing Returns | Doubling inputs more than doubles output | Specialization, bulk buying |
| Constant Returns | Doubling inputs exactly doubles output | Simple replication |
| Decreasing Returns | Doubling inputs less than doubles output | Management complexity |
Isoquants
An isoquant shows all combinations of inputs that produce the same level of output. Properties:
- Downward sloping
- Convex to the origin
- Higher isoquants represent higher output
- Isoquants don't intersect
Cobb-Douglas Production Function
Q = A × Lα × Kβ
Where: A = technology parameter, α = output elasticity of labor, β = output elasticity of capital
Properties
- If α + β = 1: Constant returns to scale
- If α + β > 1: Increasing returns to scale
- If α + β < 1: Decreasing returns to scale
- α represents labor's share of output
- β represents capital's share of output
Conclusion
Key Takeaways
- Production function shows input-output relationship
- Short run: At least one input fixed; law of diminishing returns applies
- Long run: All inputs variable; returns to scale matter
- Marginal product is additional output from one more input unit
- Isoquants show input combinations for same output
- Cobb-Douglas is most widely used functional form
- Optimal production is in Stage II where MP > 0 and AP is positive