Types of Isoquants: Linear, L-Shaped, and Cobb-Douglas [Microeconomics I Studied #27]

OK so this time, let’s run through the various isoquants.

Not hard at all, because we already dealt with a bunch of utility functions back in demand theory. Compared to that, literally nothing’s different — except the variables are now just $L$ and $K$.

Back when we did utility functions, the shapes basically broke down into 4 types.

Four shapes side by side, each captioned: Linear utility function (a straight line sloping down), L-Shape utility function (an L), Cobb-Douglas utility function (a smooth bending curve) and Quasi-linear utility function (a flatter bending curve)

The same idea for isoquants, three shapes captioned: Linear isoquant (a straight line sloping down), L-Shape isoquant (an L) and Cobb-Douglas isoquant (a smooth bending curve)

That’s how isoquants get classified.

(There’s no quasi-linear isoquant in this lineup T_T)

Ehhh~~~~ so let’s just keep it short and simple.

Linear isoquant….. If the production function is linear, the isoquant is gonna be drawn as a straight line, so we’re good as long as the production function looks like this.

OK so, in the linear utility function, um… where was the solution that maximized utility again?????

A straight line sloping down on x and y axes, with a star where it meets the y axis and a triangle where it meets the x axis, labeled utility function

Either the star point or the triangle point.

(The thing that maxes out utility within whatever the budget line lets you do.)

That’s what we call a corner solution,

The same picture for an isoquant: a straight line sloping down on K and L axes, with the star on the K axis and the triangle on the L axis, labeled isoquant

so if the isoquant looks like this (we’ll get to it later, but: within whatever the cost-line allows, the combo that maxes out output), the solution is also a corner solution. The firm has no choice but to go to one of the extremes — star or triangle (saving the details for later).

The fact that the firm’s choice is one of two extremes means that when you tweak the $(K, L)$ combination, it doesn’t shift little by little — it flips all at once…. in this case $\sigma$ (elasticity of substitution) is $\infty$, you feel me????

Why does that make sense?

Well, the definition of $\sigma$ is

$The definition of sigma as a fraction: the % change of (K/L) over the % change of MRTS$

and for a linear function, the rate of change of MRTS is zero.

So $\sigma = \infty$. Not wrong^^

Now let’s look at the L-shape isoquant. This one’s also gonna run on the “same logic as before~~~” rails.

An L shape on x and y axes, a vertical line standing on a horizontal one, with a star at the corner, labeled utility function

When we had a utility function like this, where was the solution always???!?

Yep — always the star.

(Because nobody wants mismatched shoes.)

Same deal here.

The same L shape on K and L axes with a star at the corner, labeled isoquant

If the isoquant looks like that, the solution is always the star. The firm’s choice is only the $(L, K)$ combo at that point. Meaning the firm has no reason to go messing with things like the $K/L$ ratio.

Therefore@@@

The definition of sigma again: the % change of (K/L) over the % change of MRTS

The numerator is 0, so $\sigma = 0$.

So far we’ve covered $\sigma = \infty$ and $\sigma = 0$. Which means we’ve seen the two extremes that sigma can hit.

So,

The straight-line isoquant and the L-shaped isoquant on the same K and L axes, with the labels sigma = infinity beside the L and sigma = 0 beside the line

the two extremes look like this, and so, an isoquant with a value of $0 < \sigma < \infty$ — something landing in the middle of the two —

The same drawing with a thick red curve between the line and the L, labeled 0 ≤ sigma ≤ infinity

couldn’t you kinda guess it’d look like this??????

Alright, let’s check out the case where $\sigma = 1$.

(Now here’s where it diverges from utility functions. Well — no, since the definition itself is different, of course it diverges: the $\sigma$ of these isoquants is the same at every point, at all times.

Meaning, we’re about to look at the case where $\sigma = 1$, and that means $\sigma = 1$ at every single point, at all times….. heh heh heh that’s just how econ definitions roll, apparently. Same deal for linear and L-shape — $\sigma = \infty$ and $\sigma = 0$ at every point, respectively.)

The case where $\sigma = 1$ is exactly@@@@@

The production function that has $\sigma = 1$ everywhere, all the time, is called the “Cobb-Douglas production function”.

The Cobb-Douglas production function

is defined like this, !!!!!

So first, let me prove that $\sigma = 1$ at every $(L, K)$@@@@@

$The proof that sigma = 1: MRTS works out to (α/β)(K/L), then the definition of sigma, MRTS named g, the substitution X = K/L, and a last fraction that comes out to 1, Q.E.D$

So so so so so so so,

All three isoquants on one set of K and L axes: the straight line, the L, and a bending curve between them labeled sigma = 1

we can wrap it up like this — the shape is set by sigma@@@@

(I don’t even need to spell out that if you draw an isoquant from the Cobb-Douglas production function, it curves~~~??)

And of course, in between those, there exist isoquants with $0 < \sigma < \infty$@@@@@@@

Heads-up. The figures here are the original notes, slips and all. A few things to watch for:

In the isoquant drawings labelled with $\sigma$ (the two extremes, the same drawing with the red curve, and the summary at the end), $\sigma = \infty$ and $\sigma = 0$ sit beside the wrong shapes: $\sigma = \infty$ belongs to the straight line and $\sigma = 0$ to the L shape, as the text says. The red curve’s label should read $0 < \sigma < \infty$.

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