Topic 3.1 Notes – The Production Function

1. The Production Function

A production function shows how a firm transforms inputs (factors of production) into output.

We usually write it as:

$$Q = f(L, K)$$

• $Q$ = quantity of output
• $L$ = labor (workers)
• $K$ = capital (machines, tools, buildings)
• Sometimes land and entrepreneurship are included too

If you picture a pizza shop:

• Labor = workers
• Capital = ovens
• Land = building
• Output = pizzas

Short Run vs. Long Run

This distinction matters for everything that follows.

• Short run → at least one input is fixed (usually capital). Labor is variable.
• Long run → all inputs are variable.

In the short run, you can hire more workers but can’t instantly build a new factory. In the long run, you can adjust everything.

Production decisions determine costs, which determine profit. That connection drives the rest of Unit 3.

2. Total, Marginal, and Average Product

These three always move together. You should be able to calculate and interpret all three.

Total Product (TP)

Total Product is total output produced with a given amount of labor.

Example table:

TP usually:

• Increases at first
• Increases at a decreasing rate
• Eventually can fall

Here’s what that looks like in a graph. Focus on panel (a), the Total Product curve.

Total product, marginal product, and average product curves

Marginal Product (MP)

Marginal Product of Labor (MPL) is the additional output from hiring one more worker.

$$MP = \frac{\Delta TP}{\Delta L}$$

Using the table above:

• Worker 1 → MP = 8
• Worker 2 → MP = 10
• Worker 3 → MP = 9
• Worker 4 → MP = 5
• Worker 5 → MP = 2

Key relationships:

• If MP > 0 → TP is increasing
• If MP = 0 → TP is at maximum
• If MP < 0 → TP is decreasing

Average Product (AP)

Average Product of Labor (APL) is output per worker.

$$AP = \frac{TP}{L}$$

For 3 workers:

$$AP = \frac{27}{3} = 9$$

The big relationship students forget:

• If MP > AP, AP rises
• If MP < AP, AP falls
• MP intersects AP at AP’s maximum

On a graph like this, the MP curve crosses the AP curve at the highest point of the AP curve.

Marginal product and average product curves

If you see that intersection on a quiz, that point is where AP is maximized.

3. The Law of Diminishing Marginal Returns

Law of Diminishing Marginal Returns
In the short run, as more of a variable input is added to a fixed input, marginal product eventually decreases.

This only applies in the short run.

Typical stages:

1. Increasing marginal returns

   • Specialization and teamwork
   • MP rises

2. Diminishing marginal returns

   • Workers crowd fixed capital
   • MP falls but stays positive

3. Negative marginal returns

   • Too many workers
   • MP becomes negative
   • TP falls

The AP loves asking what happens to TP when MP is falling but positive. TP is still increasing, just more slowly. Students mix that up all the time.

This law explains why short-run marginal cost will eventually rise later in the unit.

4. Returns to Scale in the Long Run

Now we change all inputs.

If a firm doubles labor and capital, what happens to output?

Increasing Returns to Scale

• Output more than doubles
• Caused by specialization, efficiency, large-scale technology
• Associated with falling long-run average cost

Constant Returns to Scale

• Output exactly doubles
• Inputs and output change proportionally

Decreasing Returns to Scale

• Output less than doubles
• Caused by coordination and management problems
• Associated with rising long-run average cost

Do not confuse this with diminishing marginal returns.

• Diminishing marginal returns → short run, one input changes
• Returns to scale → long run, all inputs change

That distinction shows up constantly in multiple choice.

5. Production, Costs, and Profit

Production drives costs.

If MP is rising:

• Each worker adds more output
• Cost per unit falls

If MP is falling:

• Each worker adds less output
• Cost per unit rises

Revenue reminder:

$$TR = P \times Q$$

Profit:

• Accounting profit = TR − explicit costs
• Economic profit = TR − (explicit + implicit costs)

Opportunity cost matters for economic profit. If someone gives up a 200,000 dollar job to run a business that earns 150,000 dollars, accounting profit is positive but economic profit is negative.

Firms care about economic profit.