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 Cobb Douglas production function. Let’s look again at the marginal product of X1, which we find by taking the derivative of Q with respect to X1 to get a * b * X1^(b-1) * X2 ^ c * X3 ^ d … which now depends on how much of the other factors we are using. If b We can also use the Cobb Douglas production function to illustrate the concept of returns to scale or economies of scale. What happens to output if we use twice the amount of every input? There are three possibilities decreasing, constant and increasing returns to scale if we get less than twice, exactly twice or more than twice the amount of output. In the Cobb Douglas case it is easy to see that if the exponents b, c, d … add up to less than 1 we have the decreasing case, exactly 1 gives us constant and greater than 1 increasing. Another important question to ask is to what degree we can substitute one factor of production for another. For instance, it is pretty clear that in many cases we can either use more machines (capital) and fewer humans (labor) or the other way round and achieve the same output. The quantity which captures this tradeoff is the so-called Marginal Rate of Technical Substitution (sorry, I didn’t pick the terms, just explaining them). We find it by taking the ratio of the marginal products for the two factors. With just a tad of math you can see that for the Cobb Douglas case, the rate between x1 and x2 is (b/c) * (x2/x1), meaning if we use 1 unit of x1 less then we need to use (b/c) * (x2/x1) units more of x1 to maintain the same output. You can also go back to the linear case and check that the rate of substitution there is simply b/c, i.e. doesn’t depend on how much of the inputs we are currently using. What about factors of production that cannot be substituted for each other? For instance, if you make a bicycle you generally can’t substitute the rubber of the tires for the metal in the frame. Instead if you make more bicycles you will always need more rubber and more metal. These two factors would be called technological complements. The Leontief production function captures this notion in the extreme with Q = a * min(x1/b, x2/c, x3/d …). The marginal product of any factor by itself is 0 as increasing it does nothing when the other factors don’t increase also. The rate of substitution is undefined (it goes off to infinity), instead factors x1 and x2 will always be used in the ratio b/c. So far we have not said anything at all about the cost for each of these factors of production. That will be the subject of a separate post, which will introduce the cost minimization problem at the level of the firm. Our eventual goal with all of this is to link changes in production technology to the determination of the factors prices in the economy overall. That will allow us to examine the impact of technology on wages and the return on capital.

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 Cobb Douglas production function. Let’s look again at the marginal product of X1, which we find by taking the derivative of Q with respect to X1 to get a * b * X1^(b-1) * X2 ^ c * X3 ^ d … which now depends on how much of the other factors we are using. If b We can also use the Cobb Douglas production function to illustrate the concept of returns to scale or economies of scale. What happens to output if we use twice the amount of every input? There are three possibilities decreasing, constant and increasing returns to scale if we get less than twice, exactly twice or more than twice the amount of output. In the Cobb Douglas case it is easy to see that if the exponents b, c, d … add up to less than 1 we have the decreasing case, exactly 1 gives us constant and greater than 1 increasing. Another important question to ask is to what degree we can substitute one factor of production for another. For instance, it is pretty clear that in many cases we can either use more machines (capital) and fewer humans (labor) or the other way round and achieve the same output. The quantity which captures this tradeoff is the so-called Marginal Rate of Technical Substitution (sorry, I didn’t pick the terms, just explaining them). We find it by taking the ratio of the marginal products for the two factors. With just a tad of math you can see that for the Cobb Douglas case, the rate between x1 and x2 is (b/c) * (x2/x1), meaning if we use 1 unit of x1 less then we need to use (b/c) * (x2/x1) units more of x1 to maintain the same output. You can also go back to the linear case and check that the rate of substitution there is simply b/c, i.e. doesn’t depend on how much of the inputs we are currently using. What about factors of production that cannot be substituted for each other? For instance, if you make a bicycle you generally can’t substitute the rubber of the tires for the metal in the frame. Instead if you make more bicycles you will always need more rubber and more metal. These two factors would be called technological complements. The Leontief production function captures this notion in the extreme with Q = a * min(x1/b, x2/c, x3/d …). The marginal product of any factor by itself is 0 as increasing it does nothing when the other factors don’t increase also. The rate of substitution is undefined (it goes off to infinity), instead factors x1 and x2 will always be used in the ratio b/c. So far we have not said anything at all about the cost for each of these factors of production. That will be the subject of a separate post, which will introduce the cost minimization problem at the level of the firm. Our eventual goal with all of this is to link changes in production technology to the determination of the factors prices in the economy overall. That will allow us to examine the impact of technology on wages and the return on capital.