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Marginal Product and Differentiation

Marginal product (MP) is the extra output you get from one more unit of input. When output is a smooth formula Q(L), MP is its derivative: MP = dQ/dL, the slope of the output curve. If MP falls as you add more input, we have diminishing returns. The same idea gives marginal cost MC = dC/dq.

🎬 Step-by-step story

  1. A small farm grows grain. The helpers are called workers (L). With 0 workers, the farm makes 0 sacks. Nothing to see yet.
  2. Add workers one by one. Each green bar is the total sacks made. 1 worker makes 11 sacks, 2 make 20, 3 make 27, and so on.
  3. Look at the gold block on top of each bar. It is the extra sacks from one more worker: +11, +9, +7, +5, +3, +1. This extra is the marginal product.
  4. The gold blocks keep getting smaller. The bars still grow, but more slowly. This is called diminishing returns. The purple line joins the tops of the bars into a smooth curve.
  5. Now a red tangent line touches the curve at 3 workers. Its slope is the marginal product. We find it with the derivative: Q = 12L − L², so dQ/dL = 12 − 2L = 6.
  6. Free play: move the slider. Watch the red line get flatter as workers increase. At 6 workers the line is flat, slope 0, and the farm cannot make any more sacks.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Is marginal product the same as average product?

No. Average product is Q divided by L. Marginal product is the extra from one more worker. In step 2 the gold block is the marginal product, the whole bar is the total.

Why are the gold blocks getting smaller?

Land and tools stay the same, so each new worker has less to work with. This is diminishing returns.

How does a slope turn into 'extra output'?

Slope means rise over run: change in Q for one step in L. That is exactly the extra output per extra worker.

What does a flat tangent mean?

MP = 0. One more worker adds nothing. This is the highest output.

Why does the constant vanish when we differentiate?

A constant does not change when L changes, so its rate of change is 0. Example: fixed cost in the cost formula.

What is marginal product?

Marginal means 'one more'. Marginal product is how much extra output you get when you add one more unit of an input, like one more worker.

Example: 3 workers make 27 sacks. 4 workers make 32 sacks. The fourth worker added 32 − 27 = 5 sacks. So the marginal product of the fourth worker is 5.

In symbols: MP = ΔQ ÷ ΔL. The triangle Δ (delta) means 'change in'.

From counting to the derivative

Counting works when we add whole workers. But many inputs are smooth: water in litres, fertiliser in grams, machine hours. For a smooth formula Q = f(L) we make the step ΔL very, very small. The fraction ΔQ ÷ ΔL then becomes the derivative:

MP = dQ/dL

On the graph this is the slope of the tangent line, the line that just touches the curve at one point.

Useful rule (power rule): the derivative of L² is 2L, the derivative of a number times L is just that number, and the derivative of a plain number is 0.

For Q = 12L − L²: dQ/dL = 12 − 2L.

Diminishing returns: why MP falls

When MP gets smaller as we add more input, we say there are diminishing marginal returns. The farm has only so much land and so many tools, so extra workers share them.

In maths: MP = 12 − 2L gets smaller as L grows. Its own derivative is −2, which is negative. So the curve bends downward (it is concave).

Where MP = 0, the curve is flat at its top. Here that is at L = 6 and Q = 36 sacks. Adding workers after this would make output fall (MP becomes negative).

Same idea: marginal cost and marginal revenue

All 'marginal' words in economics are derivatives.

Example: if the cost is C = 100 + 5q + q², then MC = 5 + 2q. At q = 10, MC = 25 rupees. The fixed 100 disappears in the derivative, because it does not change when you make one more item.

Try it: find the marginal product yourself

  1. Open the 3D and jump to the last step. Set the slider to 2 workers. Write down the slope. Now guess the slope at 5 workers before you move the slider. Check your guess.
  2. At home, take a pile of 20 sheets of paper and fold paper boats for 2 minutes alone, then with one friend, then with two friends. Write the number of boats each time. Subtract to find the marginal product of each friend.
  3. Does the third person help as much as the first friend? Why or why not?

Key formulas and definitions

Worked examples

1. 3 workers make 27 sacks and 4 workers make 32 sacks. Find the marginal product of the 4th worker.

MP = ΔQ / ΔL = (32 − 27) / (4 − 3) = 5 sacks.

2. Q = 12L − L². Find the output with 4 workers.

Q = 12 × 4 − 4² = 48 − 16 = 32 sacks.

3. For Q = 12L − L², find MP using the derivative.

Differentiate: dQ/dL = 12 − 2L. This is the formula for MP at any L.

4. For Q = 12L − L², find MP when L = 2.

MP = 12 − 2L = 12 − 2 × 2 = 12 − 4 = 8 sacks per extra worker.

5. For Q = 30L − 2L², find the MP when L = 5.

dQ/dL = 30 − 4L. At L = 5: 30 − 4 × 5 = 30 − 20 = 10.

6. For Q = 12L − L², at what L does output become highest, and what is the highest output?

Set MP = 0: 12 − 2L = 0, so L = 6. Then Q = 12 × 6 − 6² = 72 − 36 = 36 sacks.

7. The cost of making q toys is C = 100 + 5q + q². Find the marginal cost at q = 10.

MC = dC/dq = 5 + 2q. At q = 10: 5 + 20 = 25 rupees.

8. Q = 20L − L². Show that MP is falling, and find where it becomes zero.

MP = 20 − 2L. Each extra worker lowers MP by 2, so it falls. MP = 0 when 20 − 2L = 0, so L = 10.

Common mistakes

Practice quiz

1. Marginal product is:
2. Marginal product equals:
3. If Q = 10L − L², then MP is:
4. Diminishing returns means:
5. On the output graph, MP is the:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is marginal product in simple words?

It is the extra thing you make when you add one more worker or one more unit of an input.

How is the derivative linked to marginal product?

The derivative dQ/dL is the slope of the output curve. That slope tells how much output changes for one more unit of input, which is exactly the marginal product.

Why does marginal product fall?

Other things, like land and machines, stay fixed. More workers must share them, so each new worker adds less.

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