The Marginal Product Of Labour Is Equal To The

The Marginal Product of Labour is Equal to the Average Product of Labour at Its Peak

When a firm hires workers, it wants to know how much extra output each new worker will generate. Worth adding: that extra output is called the marginal product of labour (MPL). In practice, in micro‑economics, a key relationship ties the MPL to the average product of labour (APL): the MPL is equal to the APL precisely at the point where the APL reaches its maximum. Understanding this relationship clarifies why firms stop hiring more workers once the MPL falls below the average and helps explain the shape of production curves.

Introduction

In any production process, labour is one of the primary inputs. Because of that, by adding more workers, a firm raises its total output, but the incremental benefit of each additional worker is not constant. Here's the thing — the marginal product of labour measures this incremental benefit. Plus, the average product of labour shows the output per worker on average. The intersection of these two concepts—where the marginal product equals the average product—occurs at the peak of the average product curve. This article explores why this equality holds, how it emerges mathematically, and what practical implications it has for managers and students of economics.

The Production Function and Key Definitions

A production function, Q = f(L, K), expresses total output (Q) as a function of labour (L) and capital (K). For simplicity, we often hold capital constant and examine how output changes with labour alone.

• Total Product (TP): The total quantity of output produced with a given amount of labour.
• Average Product of Labour (APL): TP divided by the number of workers, APL = TP / L. It represents the output per worker on average.
• Marginal Product of Labour (MPL): The additional output produced by adding one more worker, MPL = ΔTP / ΔL.

Graphically, TP rises at an increasing rate, then at a decreasing rate, and eventually declines if labour is the only input. Plus, the APL curve rises, reaches a maximum, and then falls. The MPL curve is the slope of the TP curve; it starts high, decreases, and may become negative.

Why MPL Equals APL at the Peak of APL

1. Calculus Insight

Mathematically, the average product is a function of labour:

[ APL(L) = \frac{TP(L)}{L} ]

The marginal product is the derivative of TP with respect to labour:

[ MPL(L) = \frac{d,TP(L)}{dL} ]

To find where APL is maximized, take the derivative of APL with respect to L and set it to zero:

[ \frac{d}{dL}\left(\frac{TP(L)}{L}\right) = 0 ]

Using the quotient rule:

[ \frac{L \cdot TP'(L) - TP(L)}{L^2} = 0 ]

Multiplying both sides by (L^2) (which is positive) yields:

[ L \cdot TP'(L) - TP(L) = 0 ]

Rearranging:

[ TP'(L) = \frac{TP(L)}{L} ]

But (TP'(L) = MPL(L)) and (\frac{TP(L)}{L} = APL(L)). Therefore:

[ MPL(L) = APL(L) ]

Thus, mathematically, the condition that the average product is at its maximum is exactly that the marginal product equals the average product.

2. Economic Intuition

Think of hiring workers one by one. Even so, when the first worker arrives, the average product is the same as the marginal product because no other workers exist to dilute the output. Even so, as more workers join, the average output per worker may rise if the MPL is still above the current average. When the MPL equals the current average, each new worker contributes exactly the same amount as the current average output, so the average does not change. Any further worker who adds less than the average will pull the average down, and any worker who adds more will lift it. The point where the MPL equals the APL is the tipping point: after this, adding workers reduces the average output per worker.

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Visualizing the Relationship

• APL peaks at L = 2 where APL = 11.
• MPL at L = 2 is also 12, slightly above APL, indicating that the average is still rising.
• At L = 3, MPL (8) falls below APL (10), signaling that adding a third worker would reduce the average output.
• The exact equality occurs between L = 2 and L = 3, where the incremental output equals the average output. In a continuous model, this equality would be exact.

Practical Implications for Firms

1. Hiring Decisions

A firm should continue hiring new workers as long as the marginal revenue product (MRP)—the additional revenue generated by an extra worker—exceeds the wage rate. Since the MRP is the MPL multiplied by the price of output, the firm’s hiring rule is:

[ MRP = MPL \times P \geq w ]

When MPL falls below the average product, it signals diminishing returns. If the wage rate is fixed, the firm may stop hiring once MPL < w, even if the average product remains high.

2. Productivity Management

Understanding the MPL–APL relationship helps managers identify the optimal labor intensity for a given capital stock. And if the firm is operating at a point where MPL > APL, it can increase output by hiring more workers. Conversely, if MPL < APL, it should consider reducing labour or investing in capital to shift the production function.

3. Cost Analysis

The average variable cost (AVC) curve is inversely related to the APL. Day to day, when APL is at its maximum, AVC is at its minimum. Thus, the point where MPL = APL also marks the lowest average cost per unit of output, a critical benchmark for pricing and competitive strategy.

Common Misconceptions

Frequently Asked Questions

Q1: How does capital affect the MPL–APL relationship?

Holding capital constant, the relationship holds. On the flip side, increasing capital can shift the production function upward, raising both MPL and APL. The equality point still occurs where MPL = APL, but the values will be higher.

Q2: What happens if labour is the only input and you keep hiring?

If labour is the sole input, the law of diminishing returns eventually forces MPL to become negative. Once MPL < 0, adding more workers reduces total output, and the average product will also decline sharply.

Q3: Can technology change the point where MPL equals APL?

Yes. Which means technological improvements can make each worker more productive, shifting the TP curve upward. Because of this, the peak of APL rises and occurs at a higher level of labour, potentially allowing the firm to hire more workers before diminishing returns set in.

Conclusion

The equality of the marginal product of labour and the average product of labour at the peak of the average product curve is a cornerstone of production theory. Which means it emerges from simple calculus and offers powerful intuition for managerial decisions. And by recognizing this point, firms can optimize hiring, manage costs, and strategically invest in capital or technology. For students, grasping this relationship deepens their understanding of how inputs translate into outputs and how economies of scale and diminishing returns shape real-world business practices.