Average and Marginal Product of Labor

OK so up to now we’ve been on the consumer side — preferences, utility, demand theory. The whole picture from the buyer’s angle.

From here on out? Supplier side.

But honestly — you’ll just get it as you go, same as how demand theory clicked once you’d put the work in. So I’d say: lock in the earlier stuff first. Don’t skip ahead.

And one thing I noticed studying that earlier stuff — economics is basically just math…. lol. Still, since this is undergrad level, the math you actually need is pretty tame. Don’t panic.

Alright, let’s go.

A supplier — whether that’s a firm or just some guy — needs stuff in order to sell anything. The usual suspects: machines, money, labor.

These are called factors of production.

You’ve seen the Input → Output thing before, right? Yeah, the stuff above is the input. Feed those in, output comes out, and we’re going to call the quantity of that output $Q$.

Labor we’ll call $L$ (for labor).

Physical capital and human capital — we’ll just lump them together for now and call the whole thing $K$ (for capital).

Obviously, how much $Q$ comes out depends on $L$ and $K$. So in the spirit of “$L$ and $K$ are the independent variables, $Q$ is the dependent variable hanging off them,” we write:

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

And… that’s it. Writing this down tells us nothing yet. We don’t know if $Q$ goes like $L^2$, or linearly, or like a square root, or what. All we know right now is: the $Q$ the supplier produces depends only on those two things, $L$ and $K$.

Now, yes — production really does depend on both $L$ and $K$. But let’s hold $K$ fixed for a sec and just watch what $Q$ does as we crank $L$ up and down.

Machines, money, office, factory… let’s say there’s some fixed amount of all of that, and let’s start from $L=0$.

If $L=0$, there are zero workers. Could anything possibly come out? Nope. So I think it’s pretty fair to say $Q = 0$ when $L = 0$.

But as $L$ climbs, $Q$ climbs too — and the size of each bump in $Q$ keeps getting bigger. Why? Because efficiency is going up alongside the labor.

Then, at some point, in the interval from $L = \alpha$ to $L = \beta$, $Q$ is still increasing — yes — but the rate it’s increasing at is starting to shrink. Efficiency is gradually losing steam.

Two pictures should be enough to explain why efficiency drops off:

(Situation 1.)

(Situation 2.)

There’s a saying that fits perfectly: too many cooks spoil the broth. Yeah. That.

So the graph ends up looking like this.

And from $L = \beta$ onwards, $Q$ actually starts to fall. The drop in efficiency more than cancels out the gain from adding more labor, so $\Delta Q$ goes negative.

Let me reorganize all that.

Adding $L$ on its own buys you a certain $\Delta Q$. But adding $L$ also juices up efficiency, which buys you more $\Delta Q$ on top. So in the red section, the increment is increasing — it’s an increasing function.

Past $L = \alpha$ (where efficiency peaks), adding more $L$ still bumps efficiency up, but not by as much as before. Efficiency’s contribution to $\Delta Q$ is shrinking — heading in the negative direction as we walk from $\alpha$ toward $\beta$.

Eventually we hit $L = \beta$. At that point, the negative pull from declining efficiency exactly cancels the positive push you get from more labor — $\Delta Q$ hits zero.

Past $\beta$? The efficiency drop wins outright, and total $\Delta Q$ goes negative.

So the graph being convex, then passing through an inflection point, then becoming concave — that’s the efficiency story. And apparently this graph isn’t just hand-wavy: it’s been backed up by actual data from industrial engineering. So it’s legit.

OK, let’s chew on this graph mathematically for a bit.

First, what does the slope on this graph even mean?

$$\frac{\Delta Q}{\Delta L}$$$$\frac{Q}{L}$$

When $L$ goes up by $\Delta L$, how much does $Q$ go up by? That number, right?

So $Q$ at a given $L$, divided by that $L$, is basically answering: “how much $Q$ did one unit of labor produce?” Seems like a fair thing to call it.

This slope — the average slope — is called the average product of labor. Apparently.

OK so when is

$$AP_L = \frac{Q}{L}$$

at its max?

One picture says it all.

The slope is max at point 4. (Look at the slopes of the dotted lines.)

That is — up until the line from the origin becomes tangent to the production function, the average slope keeps creeping up.

Meaning: the $Q$ that one unit of labor cranks out keeps growing and growing, all the way up to point 4. After that, the slope keeps falling.

So the $Q$ per unit of labor keeps shrinking. heh heh heh.

Let’s plot all that on a fresh set of axes.

This new set of axes here: the $y$-axis is the slope value.

Slope hits its max at the $L$-value of point 4, so the graph rises like this up to there. After that, as we just saw, it keeps falling — that’s the blue part.

This curve

$$AP_L$$

is what’s called the average product of labor. Apparently.

OK now — instead of the average slope, let’s look at the slope of the tangent line at each point.

What does that even mean?

$$\frac{dQ}{dL}$$

When $L$ goes up by thiiiis much $(dL)$, how much does $Q$ change (dQ)? That number.

Because of what it means, it’s called the marginal product of labor:

$$MP_L = \frac{dQ}{dL}$$

That’s how you write it. Apparently.

Same as we did for $AP_L$ — let’s figure out when $MP$ is at its max.

I’ll cram it all into one picture. Sorry it’s a little rough — as long as you can read it, that’s what matters.

Up until $L = L_3$, the slope of the tangent line keeps growing. (Mathematically, that point is called the inflection point.)

Meaning: up to $L = L_3$, every time you add labor, the increment in $Q$ keeps getting bigger.

Past $L = L_3$, the tangent slope shrinks, and eventually the increment from $L$ goes negative.

(I know, I know — you’re going to say, this is literally all stuff I already said earlier ^^)

Alright, same drill — let’s plot the value of the tangent slope at each $L$ on

$$\frac{dQ}{dL}$$

the graph.

Oh wait — this is also “the derivative of $Q = f(L)$,” isn’t it?? Right???

So based on high school calculus, let’s just extract the derivative.

We all learned how to sketch the derivative in high school, so let me just rough it in.

Looks alright, yeah?! That curve right there is $MP_L$.

OK now — let’s overlay $AP$ and $MP$ and draw them together?!

Since they’re both slopes, we can throw them on the same set of axes, no problem.

Now now now now now, first let’s think about where $MP$ and $AP$ intersect.

When is the slope of the tangent line equal to the slope of the line from the origin?!

When else would it be? At point 4, of course — that’s where the line from origin and the tangent line have the same slope.

Remember those problems from math class?

“You’ve got some function $y = f(x)$. Draw a line from $(0,0)$ to the function such that it’s tangent. Find the point of tangency.”

If a problem like that rings a bell, just think of it as, oh, that problem. Same vibe.

Anyway, the point where tangent slope = origin slope is point 4.

And on top of that — point 4 was also “the extremum of the $AP$ curve — the spot where its slope hits zero.”

And one more thing — the $L$ at point 4 has to be bigger than the $L$ at the inflection point.

Let me wrap up.

The extremum of $AP$ sits at the intersection of $AP$ and $MP$, and that intersection has to be after the inflection point — which is the extremum of the derivative ($MP$).

Plus the fact that $AP$ has the shape it has.

So when you draw $MP$ and $AP$ together, this is what you get.

This post has gotten way longer than I meant. I’ll pick it up in the next one.

Next post takes us through isoquants.

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Originally written in Korean on my Naver blog (2016-07). Translated to English for gdpark.blog.