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Chapter 2 / Technology and Production Functions

2.1 Production Functions


As economists, we want to analyze the decisions made by billions of people all over the world, but at its core an economy is just like Chuck: it has resources, and technologies to produce goods from those resources. In this chapter we’ll look at how economists model the production process that transforms resources into goods: the “production function.”

Because producing a good requires multiple inputs — for example, catching fish requires Chuck’s time and a spear — this production function will be a multivariable function $f()$ of the sort we examined in the last chapter. For simplicity, we will analyze a production function that takes only two inputs: human effort, or “labor” (denoted $L$, and generally measured in hours), and tools and machines, or “capital” ($K$, after the German kapital). The output of this production function will be the quantity of a good produced ($q$, for “quantity”). Putting this all together, we will write \(q = f(L,K)\) which we may read as

"The quantity of output is a function of the hours of labor and level of capital devoted to its production."

or in the case of Chuck’s production function for fish,

"The number of fish Chuck catches is a function of the amount of time he spends fishing, and the quality of his fishing spear."

The first part of this chapter will be to look at each of the aspects of a multivariable function we introduce last time: level sets, partial derivatives, and the slope along a level curve — and see how they relate to important economic concepts relating to production.

We’ll then think about how we might model different kinds of production technologies using different functional forms: for example, what kind of mathematical function describes a production technology that uses labor and capital in conjunction with one another, and what kind of function describes a technology in which a good can be produced using either labor or capital?

Finally, we’ll analyze the characteristics of production functions as production is scaled up, either by increasing one input while the others are held constant, or by increasing all inputs proportionally.

For simplicity, we’ll assume that Chuck has just two inputs, capital and labor. Obviously, even for Chuck, we could analyze his resources in a more complex manner. But we’re going to try as much as we can in this course to analyze the simplest possible model in order to understand how it works, keeping in the back of our mind that the real world is a lot more complicated.

Next: Marginal Products
Copyright (c) Christopher Makler / econgraphs.org