The marginal product of labour (MPL) represents the extra output generated by adding more labor. Formally, it is the rate of change in output for an infinitesimal increase in labor, calculated as the derivative of the production function, $g'(h)$, and represented by the function's slope. In practice, especially in introductory contexts, MPL is often approximated as the output gain from a single additional unit of labor (e.g., one hour or one worker), calculated as $g(h+1) - g(h)$. It is crucial to understand that this one-unit approximation is generally not equal to the precise, calculus-based value.

Marginal Product of Labour

A core principle of production theory is that the average product of an input is diminishing if and only if the marginal product is less than the average product. This mathematical equivalence means the two conditions imply each other: a falling average product necessitates a marginal product below the average, and conversely, a marginal product below the average guarantees that the average product will fall. This relationship is formally established by differentiating the average product function.

The Condition for a Diminishing Average Product

A visual inspection of a concave production function provides evidence for the relationship between marginal and average product. For instance, at a given point P, the tangent line, representing the marginal product, is visibly less steep than the ray from the origin to P, which represents the average product. This graphical comparison demonstrates that the marginal product is less than the average product when the average product is diminishing.

Graphical Evidence of MP < AP on a Concave Function

The conceptual distinction between marginal and average product is visualized on the production function graph through two different slope measures. The marginal product at a given point is represented by the slope of the tangent to the production function at that point, mathematically denoted by the first derivative, $f'(x)$. The average product of labor (APL) at that same point is represented by the slope of a ray drawn from the origin to the point on the function, calculated as $APL = f(x)/x$.

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The average product of labor at any given point on a production function graph can be visually and mathematically determined by the slope of a ray drawn from the graph's origin to that specific point. This slope is calculated by dividing the total output (the vertical distance) by the total labor input (the horizontal distance). For example, within Angela's model, the average product at a specific point T is represented by the slope of the ray from the origin to T. This principle applies generally to production functions.

Average Product of Labor as the Slope of a Ray from the Origin

Marginal product and average product are two distinct, fundamental measures of productivity. The average product is the total output divided by the total quantity of an input, such as labor, representing the overall productivity per unit. In contrast, the marginal product measures the additional output generated from using one more unit of that input. They are not the same value.

Conceptual Distinction Between Marginal and Average Product

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The principle of diminishing average product can be visually confirmed from the production function graph without requiring calculations. For instance, the ray from the origin to point A is steeper than the ray to point B, showing a higher average product at A. This observation is generalizable: as one moves along the production function curve by increasing the number of farmers, the rays from the origin to these points become progressively flatter. This visual flattening of the rays directly illustrates the continuous decline in the average product of labor.

Graphical Evidence of Diminishing Average Product: Flatter Rays Along the Production Function

Graphical Comparison of Marginal and Average Product

A specific calculation on the production function graph from Figure E5.2a illustrates the concept of Average Product of Labor (APL). At point P, where the labor input is 5 hours and the corresponding output is 19.04 units, the APL is calculated by dividing the total output by the total input. This value, $19.04 / 5 = 3.8$, represents the slope of the ray drawn from the origin to point P.

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