What Is The Relationship Between Marginal Product And Average Product

Understanding the Interplay Between Marginal Product and Average Product

The concepts of marginal product (MP) and average product (AP) are fundamental to understanding production economics and firm behavior. They describe the relationship between the inputs a firm uses and the outputs it produces, providing crucial insights into efficiency and optimal resource allocation. While seemingly distinct, MP and AP are intrinsically linked, with the marginal product driving the movement and shape of the average product curve. This article will delve into the definitions of both, explore their relationship graphically and mathematically, and analyze their implications for businesses making production decisions.

Defining Marginal Product and Average Product

Before exploring their relationship, let's clearly define each term:

Marginal Product (MP)

The marginal product of an input (like labor or capital) is the additional output produced by employing one more unit of that input, holding all other inputs constant. It represents the change in total output resulting from a one-unit change in the input. Mathematically, it's expressed as:

MP = ΔTP / ΔInput

Where:

ΔTP = Change in Total Product (output)
ΔInput = Change in the input quantity (e.g., one more worker)

For example, if a factory produces 100 units with 5 workers and 105 units with 6 workers, the marginal product of the 6th worker is 5 units (105 - 100 = 5). It's important to note that MP can be positive, negative, or even zero.

Average Product (AP)

The average product, also known as average productivity, measures the average output produced per unit of input. It represents the total output divided by the total quantity of the input used. Mathematically:

AP = TP / Input

Where:

TP = Total Product (total output)
Input = Total quantity of the input

Using the same example, if the factory produces 100 units with 5 workers, the average product of labor is 20 units per worker (100 / 5 = 20). AP always represents a positive value, as it's a ratio of total output and input quantity.

The Relationship Between MP and AP: A Graphical Illustration

The relationship between MP and AP is best understood graphically. Typically, both MP and AP curves are plotted against the quantity of the variable input (e.g., labor). The curves exhibit a characteristic shape, often described as having three stages:

Stage 1: Increasing Marginal Returns

At the beginning of production, adding more units of the input (e.g., hiring more workers) leads to a more-than-proportional increase in output. This is due to factors like specialization, improved coordination, and better utilization of fixed inputs. In this stage:

MP rises: Each additional worker contributes more than the previous one.
AP also rises: The average output per worker increases as the marginal product exceeds the average product.
MP < AP: The marginal product is less than the average product initially. The average product is being pulled upward by the higher marginal product.

Stage 2: Diminishing Marginal Returns

As more and more units of the input are added, the rate of increase in output begins to slow down. This is the stage of diminishing marginal returns. This happens because the fixed inputs (e.g., capital equipment) become increasingly constrained. In this stage:

MP falls: The additional output from each additional worker decreases.
AP continues to rise initially, then falls: Initially, the average product continues to rise as long as MP remains above AP, but eventually it will fall as MP falls below AP. The average is pulled down by the lower marginal product.
MP = AP: The point where MP intersects AP marks the maximum point of the AP curve. At this point, the marginal product equals the average product.

Stage 3: Negative Marginal Returns

This stage represents a situation where adding more inputs actually leads to a decrease in total output. This can be due to overcrowding, decreased efficiency, and negative externalities arising from having too many workers. In this stage:

MP is negative: Adding more workers reduces total output.
AP continues to fall: The average output per worker continues its decline.
Rational firms avoid this stage: No profit-maximizing firm would ever operate in this range because it's inefficient.

Graphical Representation:

A typical graph showing the relationship between MP and AP would show the AP curve rising, reaching a peak, and then falling. The MP curve would intersect the AP curve at the peak of the AP curve. The MP curve would always be below the AP curve in stage 3 (Negative Marginal Returns).

The Mathematical Relationship Between MP and AP

The relationship between MP and AP can also be expressed mathematically. Consider a firm using labor (L) as its variable input and producing output (Q). The total product (TP) is a function of labor: TP = f(L).

Average Product (AP): AP = TP / L = f(L) / L
Marginal Product (MP): MP = dTP/dL = f'(L) (the derivative of the total product function)

The following relationships hold:

When MP > AP, AP is rising. Adding a worker whose productivity is higher than the average raises the average productivity of all workers.
When MP < AP, AP is falling. Adding a worker whose productivity is lower than the average reduces the average productivity of all workers.
When MP = AP, AP is at its maximum. This is the point where the marginal product curve intersects the average product curve.

Implications for Business Decisions

Understanding the relationship between MP and AP is crucial for businesses making production decisions.

Optimal Input Levels: Firms aim to operate in the range of diminishing marginal returns (Stage 2) where AP is still relatively high. This ensures efficient resource allocation and maximizes output per unit of input.
Hiring Decisions: Analyzing MP and AP helps determine the optimal number of workers (or units of any other input) to hire. Hiring should cease when the marginal product of an additional worker falls below the average product, which causes decrease in average product.
Cost Minimization: MP and AP provide insights into cost minimization. By understanding the productivity of inputs at different levels, firms can minimize the cost of producing a given output.
Production Planning: The curves help in efficient production planning, allowing businesses to optimize output levels given available resources.

Conclusion

The relationship between marginal product and average product is a cornerstone of production theory. The marginal product drives the movement of the average product, with the MP curve intersecting the AP curve at the maximum point of the average product curve. Understanding this interplay is crucial for firms to make informed decisions about input usage, production efficiency, and overall profitability. By carefully analyzing the MP and AP curves, businesses can optimize their operations and achieve greater success. The three stages of production – increasing marginal returns, diminishing marginal returns, and negative marginal returns – provide a comprehensive framework for understanding how productivity changes as more inputs are used. Mastering these concepts is critical for any business aiming to maximize its output and minimize its costs.