Economic Production Functions and Cost Relationships
Key Relationships in Production and Costs
APL and AVC Relationship
If APL (Average Product of Labor) is growing, AVC (Average Variable Cost) and ATC (Average Total Cost) are decreasing. There is an inverse relationship between APL and AVC (and ATC). Specifically, you can check that:
AVC = w / APL
MC = w / MPL
Example Calculation: AVC = 2 / (0.25L) = 2 / (0.25 · 2Q0.5) = 4 / Q0.5
MPL and MC Relationship
If MPL (Marginal Product of Labor) is growing, MC (Marginal Cost) is decreasing. There is an inverse relationship between MPL and MC. Specifically, you can check that:
Q = 0.25L2
L = 2Q0.5
The relationship between MPL and MC is inverse: when MPL reaches its maximum (at a certain number of workers, L), MC is at its minimum (for a given output, Y).
Similarly, the relationship between APL and AVC is inverse: when APL is at its maximum, AVC is at its minimum.
Innovation Cost Analysis
Total Cost (TC) Before vs. After Innovation
For example, if producing 400 units (Q=400) costs 90€ before innovation and 50€ after innovation, this means producing Q(400) is 40€ less expensive after innovation. This 40€ represents the cost savings for the company with the new technology.
Should the Company Innovate if the Cost is 50€ or 30€?
The company will not incorporate the innovation if its cost is 50€, because the new machinery only saves 40€. However, if the incorporation cost is 30€, the company will still make an additional profit of 10€. The cost of implementing an innovation can be considered a fixed cost (FC) for the company, so this investment is only beneficial if the total cost of producing Q units is lower than before.
Maximum Investment for Innovation Acquisition
The maximum investment the firm is willing to make to acquire new machinery is 40€, which represents the total savings the company can achieve after the acquisition.
The Law of Diminishing Marginal Productivity (David Ricardo)
The production function for most outputs has three stages:
Stage 1: Increasing Marginal Returns
- The production function exhibits increasing marginal returns.
- Each additional unit of input yields more output than the previous unit.
Stage 2: Diminishing (but Positive) Marginal Returns
- The production function exhibits diminishing—but positive—marginal returns.
- Total production increases with each additional unit of input, but the rate of increase slows down.
Stage 3: Negative Marginal Returns
- The production function exhibits negative marginal returns.
- As the variable input used exceeds the capacity of the fixed inputs, total output may actually begin to decline.
APL (Average Product of Labor) is the output per worker.
MPL (Marginal Product of Labor) is the additional output produced by one more unit of labor.
Stages of Production Function in Relation to MPL and APL
- Stage 1: Initially, the production function exhibits increasing MPL and increasing APL.
- Stage 2: Then, the production function exhibits diminishing MPL and, initially, an increasing APL, followed by a decreasing APL.
- Stage 3: Finally, the production function exhibits negative MPL and decreasing APL.
The “law of diminishing returns” states that adding additional amounts of labor (L) to a fixed amount of capital (K) will eventually reduce the Marginal Product of Labor (MPL).
Our analysis typically focuses on the level of production where MPL decreases as the number of workers increases, and where APL reaches its maximum.
This concept is also sometimes referred to as the “Flowerpot Law.”