Note: Chapter XIX includes an Appendix, which concerns Professor Pigou’s "Theory of Unemployment." Since I'm unfamiliar with Pigou's work, and since the material adds little to what Keynes has stated in the chapter, we shall avoid discussing it.
Section I: Keynes seeks to define an employment function. This chapter, and this section in particular, thus becomes a confused collection of variables and algebraic simplifications. Keynes tells us that: "Those who (rightly) dislike algebra will lose little by omitting the first section of this chapter."
Although we would do well to take him at his word, at least in this instance, there is one point I wish to make regarding his "function." The term has a precise definition: "an ordered triple of sets, which may be written as (X, Y, F). X is the domain of the function, Y is the codomain, and F is a set of ordered pairs. In each of these ordered pairs (a, b), the first element a is from the domain, the second element b is from the codomain, and every element in the domain is the first element in exactly one ordered pair."
Thus, for the employment function, the domain would be wage-units, or whatever other variable Keynes believes he can track to give us: employment, the codomain, measured in men--or man hours, perhaps. If the employment function exists, a change in wage-units would necessitate a change in employment; moreover, we could determine beforehand what this change will be. That this is plainly not so should be obvious; no matter how sophisticated the functional models of the econometricians, they come no nearer to being able to predict the future. Empirical reality diverges wildly from what their algebra would suggest.
Now, this is not to say that there is no relationship between, say, wage rates and employment. It is only to say that the relationship is not a precise mathematical one. By using mathematical terminology, Keynes leads us to believe that a relationship possesses a precision which it can not have. I believe I have made this point earlier, but it bears repeating as the pseudo-precision is hitting us rather hard at this point in the book.
The remaining sections held little that caught my eye, so I'll pass them over to summarize the next chapter.