Total Cost, Marginal Cost, and Average Cost [Microeconomics I Studied #36]

Wait???? Cost curves??? Didn’t we already do this one????

You might be thinking that — but what we did before was the isocost line, and this is the cost curve!!!

The (total) isocost line is about the total cost determined by $(L, K)$. The cost curve is about the (minimum) Total Cost determined by $Q$!!!

So let’s start in the long run — the world where decisions are flexible (or, if you like, where there are no fixed costs). That perspective first.

OK so — how is the minimum long-run TC determined by $Q$?!

A K–L plane with one curved isoquant labelled Q_0 and a straight isocost line touching it at a single point

The isocost that’s tangent to the $Q_0$ isoquant — touching it at exactly one point like above — is the one that tells us. I listened closely to its answer, and the Total Cost this isocost line is telling us is TC_0 , apparently.

We’ll jot down “the minimum cost to produce quantity Q_0 is TC_0 ” as (Q_0, TC_0) .

Now let’s poke at yet another production quantity, Q_1 , too.

The same K–L plane with a second, higher isoquant Q_1 in red and its tangent isocost line, the first pair now dotted

And the minimum cost this time is TC_1 , apparently…. Let’s jot down (Q_1, TC_1) , that is…. heh

We repeat this process infinitely many times and end up with infinitely many little notes@@@ Now where do we take all those infinite notes —

Empty axes with TC on the vertical axis and Q on the horizontal axis

We bring ’em over here and plot all the infinite points@@@ Plot every single one, draw a ’line’ through them, and supposedly it comes out looking like this!!!

The total cost curve through the origin: red and concave at first, then blue and convex

Why on earth does it come out like that?!

Actually, this “total cost curve” is a mirror of the production function.

The production function, Q against L: red and convex at first, then blue and concave

I don’t mean a mirror in the strict inverse-function sense — I mean the shape mirrors it. The reason?!??!?!

Q against L with a dotted line at the inflection point. Blue note: after the inflection point a concave production function means the output can’t keep up with the input’s rate of increase. Red note: before it, a convex production function means the output shoots up even faster than the input

TC against Q with a dotted line at the inflection point. Blue note: after the inflection point a convex total cost curve means total cost goes up faster than Q. Red note: before it, a concave total cost curve means total cost only goes up a little however much Q is cranked up

Stare at the two for a sec — you can see they correspond to each other, right?!? Let’s just accept that and roll with it.

OK so now let’s mess around with the $TC(Q)$ function a bit.

What does the ‘derivative (slope of the tangent line)’ mean for $TC(Q)$?

The derivative is written like this:

and reading that out loud: “When $Q$ changes by a tiiiny bit, $dQ$, how much~~~ does $TC$ change?”

In other words — that’s the marginal cost.

We’re going to draw the marginal cost curve. We don’t know how $TC(Q)$ depends on $Q$ algebraically, so we can’t just differentiate the formula — instead we’ll eyeball the derivative geometrically, straight off the original curve.

The total cost curve with short red and blue dotted tangent lines along it and one green dashed tangent line where the red part meets the blue part

The slope keeps decreasing up to the spot where the red meets the blue. It bottoms out at the slope of the green dotted line — and after that, it just keeps climbing!!!

Let’s transfer that onto the derivative graph.

MC against Q: a U-shaped curve, red going down to a green point and blue going up after it

On the y-axis I’ve abbreviated marginal cost as MC. So the number on the y-axis is a number that represents a slope.

Now we get a graph that decreases~~~ all the way down to the y-value (remember: the y-value here is the slope) that the green point is talking about, and then after that, the y-value (again — is the slope) goes whooshing back up.

OK, this time let’s pull out AC instead of MC.

AC stands for average cost, and AC is asking “on average, how much did one unit cost to make?” In formula, it can be written like this:

and so, geometrically,

The total cost curve again, TC against Q

on this graph, the slope from the origin is what gives you AC.

Now let’s think about how the slope-from-the-origin behaves as $Q$ changes!!!

TC against Q with lines drawn from the origin to points on the curve: their slope falls down to the green line, which just touches the curve, and rises again after it

Oho ho ho! The slope from the origin goes down until it hits the slope of that green line — and after that, the slope goes back up!!! So let’s slap that onto the $Q$–AC axes!!!

AC against Q: a U-shaped curve, red going down to a green point and blue going up after it

It looks like this!!! One more thing!!! That point there, the one with the green slope — “MC and AC have the same slope.”

In other words, at the $Q$ where the green point sits, MC = AC!!!

The reason I’m hammering on this is because I’m about to plot MC and AC together on one set of axes!!!

Now now now now now now now now — we know that green point is the intersection of AC and MC.

But what about before and after………?

Let’s check the $Q$ where the MC curve hits its extreme point, and the $Q$ where the AC curve hits its extreme point.

The total cost curve with two dotted lines down to the Q axis: the magenta one, at the inflection point, marks the Q where MC is at its minimum, and the orange one further right marks the Q where AC is at its minimum

I’m not even gonna bother explaining — just look at the picture above and you’ve got it. So: at the point where AC is minimized, $MC = AC$, and on top of that, the $Q$ where MC is minimized is smaller.

Given these two facts, let’s plot them together on one set of axes.

MC and AC against Q: two U-shaped curves, the blue MC curve bottoming out first and then rising through the lowest point of the red AC curve, marked by a green dot

With these two facts pinned down, on one set of axes, only this shape is possible.

Summarizing what we’ve got:

MC passes through the extreme point of AC.

‘The MC value before it passes through AC’s extreme point is smaller than AC.’ “The MC value after it passes through AC’s extreme point is larger than AC.” Yeah, I think that’s the summary.

OK so things have been going maybe a little too smoothly — just breeeeezing along graphically and geometrically. As a last step, let me sketch a little scenario to make TC, MC, and AC feel like real things, and then we’ll move on.

TC : the actual total cost

Boss: Hey hey hey hey hey hey, we’ve produced $Q$ of these now — how much money has actually gone into making the thing????

Employee: Yes yes yes yes yes yes, exactly $TC$ went in!!!

Boss: What the heck?!?!?!?! Then how much did one unit cost to make?!?!?!

Employee: You can think of it as $AC$ per unit!!!!!

(kind of a total-average vibe??? AC)

Boss: Oh is that so??!?!?! Then how about we make just one more? If we made one more unit right now, how much extra cost would that take?????!!!

Employee: To make one more in the current situation, looks like $MC$ more should do it^^heh heh heh

Boss: Oho, is that so?~?~?~?! Sounds good~ So I’ll go ahead and make one more, or I’ll stop making them right here.

And — if cost per unit is what we’re watching — whether to make more or stop is decided by whether $AC \gt MC$ or $AC \lt MC$.

If the current AC is 3,000 won and MC is 2,000 won, isn’t it a no-brainer to throw in another 2,000 won and crank out one more of those AC-of-3,000-won units?!?!?!

Flip it: if AC is currently 2,000 won and MC is 3,000 won — if you were the boss, would you sink another 3,000 won into making one more of something that’s only running you 2,000 won per unit on average?? Nobody’s doing that.

Alright, see you in the next post heh

Comments