Average Product and Marginal Product: Formulas and Worked Table

Marginal product is the extra output from adding one more unit of labor, MP = change in total product / change in labor. Average product is output per worker, AP = total product / labor. As workers are added to a fixed amount of capital, MP rises and then falls because of diminishing marginal returns, and MP crosses AP at the point where AP is at its maximum.

Total, marginal and average product together describe the short-run production function in AP Micro topic 3.1, and they are the mirror image of the cost curves that come next. This guide gives the formulas, works through a full table, explains why MP crosses AP at its peak, shows how to read the curves on a graph, and connects them to MC and AVC.

Total, marginal and average product

All three are short-run ideas. At least one input, usually capital such as a building or a machine, is fixed, and the firm changes output by changing labor (L).

Total product (TP) is the total quantity of output produced with a given number of workers.

Marginal product of labor (MP) is the change in total product when one more worker is added:

MP = ΔTP / ΔL

Average product of labor (AP) is total product divided by the number of workers:

AP = TP / L

MP answers "what did the last worker add?" AP answers "how much does the typical worker produce?" They are different questions, and the exam checks that you keep them apart. The marginal product vs average product comparison sets the two definitions side by side.

A worked table

A T-shirt print shop has one screen-printing press (the fixed input) and adds workers one at a time. Output is shirts per hour.

Computing MP. Going from 2 to 3 workers, total product rises from 20 to 36, so MP of the third worker is (36 - 20) / (3 - 2) = 16 shirts.

Computing AP. With 5 workers, total product is 55, so AP is 55 / 5 = 11 shirts per worker. With 7 workers it is 60 / 7, about 8.6.

Working backward. Total product is the running sum of the marginal products: 8 + 12 + 16 + 12 + 7 = 55 for five workers. Average product is the average of the marginal products so far: for three workers, (8 + 12 + 16) / 3 = 12. If a question gives you only one column, you can rebuild the other two this way. The average product calculator and the marginal product of labor calculator check each step.

Reading the MP column: three stages of returns

The marginal product column moves through three phases.

Increasing marginal returns (workers 1 to 3). MP rises from 8 to 12 to 16. One worker has to do every job alone. Adding a second and third lets them specialize, one loading screens, one printing, one folding, so each new worker adds more than the last.

Diminishing marginal returns (from worker 4). MP falls from 16 to 12, then 7, then 5, then 0. The press is fixed, so each new worker has less of it to use. This is where diminishing marginal returns begins: with the fourth worker, the first one whose MP is lower than the worker before.

Negative marginal returns (worker 8). MP is -4. The shop is so crowded that the eighth worker gets in the way and total product falls from 60 to 56.

Total product keeps rising as long as MP is positive, all the way to the sixth worker, and stays flat at the seventh, when MP is zero. Diminishing returns means output grows by smaller amounts, not that output falls. The diminishing returns guide covers the law itself with more examples.

Why MP crosses AP at the maximum of AP

This is the relationship the exam returns to most. It follows from a simple rule about any average.

If the next value is above the current average, the average rises. If it is below, the average falls. Think of the average height of a class. A new student taller than the average raises it; a shorter one lowers it; one exactly at the average leaves it unchanged.

For production, the "next value" is the marginal product of the next worker. The algebra makes it exact:

New AP = old AP + (MP - old AP) / L

Test it on the table:

• From 2 to 3 workers: 10 + (16 - 10) / 3 = 10 + 2 = 12. MP is above AP, so AP rises.
• From 3 to 4 workers: 12 + (12 - 12) / 4 = 12. MP equals AP, so AP stays the same.
• From 4 to 5 workers: 12 + (7 - 12) / 5 = 12 - 1 = 11. MP is below AP, so AP falls.

So AP rises while MP is above it and falls once MP drops below it. The only place AP can stop rising and start falling is where MP equals AP. That makes the crossing point the maximum of AP. In the table, AP peaks at 12 with 3 and 4 workers, and the fourth worker's MP is exactly 12. On smooth curves, MP cuts through AP at the very top of AP. This is the marginal-average rule, and the same rule explains why MC crosses ATC and AVC at their lowest points.

Notice also that MP peaks (16, with the third worker) no later than AP does: in the table AP first reaches 12 at the third worker and holds there at the fourth. On smooth curves MP always turns down first, and AP keeps rising for a while because MP is still above it.

The curves on a graph

Plot labor on the horizontal axis.

• TP curve. Rises steeply at first, then more slowly, flattens at its peak and turns down.
• MP curve. MP is the slope of TP. It peaks where TP is steepest (the inflection point), crosses zero where TP is at its maximum, and goes negative after that.
• AP curve. AP is the slope of a straight line from the origin to a point on TP. It peaks where that line is just tangent to TP. At that point the slope of the line (AP) equals the slope of TP (MP), which is the crossing again.

The diminishing returns graph explainer shows how to label these curves.

From product curves to cost curves

If labor is the only variable input and each worker earns a wage w, then:

• AVC = w / AP
• MC = w / MP

Suppose each worker at the print shop earns $96 per hour.

Check one row directly. Five workers cost 5 x 96 = $480 and make 55 shirts, so AVC is 480 / 55, about $8.73, the same as 96 / 11.

The table shows the mirror:

• When MP is at its maximum, MC is at its minimum. The third worker has MP = 16, and MC is $6.00.
• When AP is at its maximum, AVC is at its minimum. AP = 12 gives AVC = $8.00.
• Because MP crosses AP at the top of AP, MC crosses AVC at the bottom of AVC.

Diminishing marginal returns is the reason MC slopes upward. The average product vs average variable cost comparison and the production costs guide carry this into the full set of cost curves, and you can watch MC and AVC move in the production costs sandbox.

See it move

Make it move

The short-run cost curves, with marginal cost through the minimum of AVC and ATC. Tap it to drag the curves, open the full version, or put it on your own site free, or turn it into a five-minute class activity.

How it shows up on the AP exam

This material is AP Micro topic 3.1. Students are usually asked to:

• Calculate MP or AP from a table of total product, or rebuild total product from MP.
• Identify the worker at which diminishing marginal returns begins.
• Explain the relationship between MP and AP, and where they cross.
• Connect falling MP to rising MC.

Common mistakes:

• Dividing TP by L when the question asks for MP. That gives AP. MP is the change in TP.
• Saying diminishing returns begins when total product falls. It begins when MP starts to fall, while TP is still rising.
• Saying MP crosses AP at the maximum of MP. It crosses at the maximum of AP.
• Blaming diminishing returns on worse workers. The workers are identical; the cause is the fixed input.
• Applying diminishing returns to the long run. It needs a fixed input, so it is a short-run idea. Long-run cost shapes come from economies and diseconomies of scale.

For more practice, see the AP Micro 3.1 topic page.