Generally, when you do your microeconomics class you get to see isoquants. I mean, I hope you get to see them (some principles class dont introduce them and are left to intermediate classes). But when you do, they look like this:
Its a pretty conventional approach. However, there is a neat article in History of Political Economy by Peter Lloyd (2012) titled “the discovery of the isoquant“. The paper assigns the original idea, rather than to the usual suspect of Abba Lerner in 1933, to W.E. Johnson in 1913 as A.L. Bowley was referring to his “isoquant” in a work dated from 1924 (from which the image is drawn). But what is more interesting that the originator of the idea is how the idea has morphed from another of its early formulations. In the 1920s and 1930s, Ragnar Frisch was teaching his price theory classes in Norway and depicted isoquants in the following manner in his lectures notes.
Do you notice something different about Frisch’s 1929 (or 1930) lectures relative to the usual isoquants we know and love today? Watch the end of each isoquant. They seem to “arc” do they not? How could an isoquant have such a bent? Most economists are probably so used to using isoquants that do not bend (except for perfect complements) that it will take a minute to answer. Well, here is the answer: its because Frisch was assuming that the production function from which the isoquant is derived had a maximum which means that the marginal product of an input could become negative. This is in stark contrast with our usual way to assume production functions as smoothly declining (but never negative) marginal products. This is why Frisch includes an arc to this shape (a backward bend).
Why did we move away from Frisch’s depiction? Well think about the economic meaning of a negative meaning marginal product. It means that a firm would be better off scaling down production regardless of anything else. Its a straightforward proposition to understand why in all settings, a firm would automatically from such an “uneconomic” zone. In other words, we should never expect firms to continually operate in a zone of negative marginal product. Ergo, the “bend”/”arc” is economically trivial or irrelevant. Removing it simplifies the discussion and formulation but also does something subtle — it sneaks in a claim rationality of behavior from firmowners and operators.
This is a good setup for a question to ask your students in an advanced microeconomics class that isnt just about the mathematics but about what the mathematical formulations mean economically!