So I have been meaning to write more about what is happening to labor and capital as we transition from industrial society to information society. To do that well I need lay some groundwork first though by talking about the idea of a production function. A function as you may recall from math is something like f(x) = 3x + 1 which takes a value (here x) and transforms it into a different value (here into 3x - 5, so that f(1) = 4, f(2) = 7 …)

A production function then is the mathematical abstraction of the production processes of a single firm, or an industry, or even an entire economy into a function that captures how the inputs get transformed into the outputs. We will write Q = f(x1, x2, x3, …) where Q is the output, eg bicycles and x1, x2, x3 … are the inputs such as labor, raw materials, machines, buildings that go into making the bicycles. We will use the words inputs and factors (of production) interchangeably. At the level of the overall economy the output measure is generally taken to be GDP (which is problematic but that’s not the point today).

This mathematical abstraction turns out to be useful because it lets us give more precise definitions for a bunch of terms and capture some basic intuitions. Let’s start by considering a linear production function of the form Q = a + b*X1 + c*X2 + d*X3 … We will use it to define our first term, the marginal product of an input factor. This is simply the change in output that results from using more of a particular input. In the linear example it easy to see that when you use 1 more unit of X1 you always get b more units of output. That’s completely independent of how much of X1 you are already using and also independent of how much of the other factors you have. That seems wrong for most types of production because it is saying that you can make more bicycles simply by using more labor or more rubber or more steel or more buildings.

Now consider Q = a * X1^b * X2^c * X3^d …, the so called