Introduction to production functions | APⓇ Microeconomics | Khan Academy

You will hear the term production function thrown around in economics circles, and it might seem a little intimidating and a little mathy at first. But as you're about to see, it's a fairly basic idea.

It's this idea that you could have these various inputs. Let's call this input number one, and then you have input number two, and you can keep going. Then you put these inputs into some type of process, and then that function—let's just call that F—that's going to describe how much output you can get given that input.

We can also describe it a little bit more mathematically. Those of you who remember your algebra 2 might recognize this, where we could say the output, it's often used the letter Q in economic circles, it's going to be a function. It's going to be a function of the various inputs. So I'll put input number one, input number two, and you could have as many inputs as is necessary to produce that good.

If you wanted to categorize them, these are the classic factors of production that we would have talked about before. These would be your land, labor, capital, and entrepreneurship. It doesn't have to be all of them, but each of these inputs would likely be factored as one of these.

Now, this might still seem very abstract and very mathy. So to make things very tangible, let's give a well, let's give a tangible example. Let's say that we're trying to make a bread-toasting operation.

So what we need to do is we take bread, we stick it in a toaster, and then once it's toast, we're done. What are our inputs there? Well, you're definitely going to need some bread. So let me draw some bread right over here—my best attempt at drawing bread. So that right over there, that is bread. You could call that input number one.

Now, you're also going to need a toaster—at least one toaster, or toasters, I should say. Let's say the toasters that we use for this operation, they can toast four pieces of bread at a time, and it takes 10 minutes to do that—four slices in 10 minutes.

Now, you might say, well, aren't those going to be all of our inputs? But then the obvious question is that bread isn't just going to jump into the toaster on its own and then jump back out. Someone—there's going to need to be some labor to operate this operation. So we're going to need some toaster operators, and let's say that they can process one slice per minute.

I know many of y'all are thinking that you could do better than that, but try to do it all day—one slice per minute. Now based on this, if these are really all of the three inputs into producing the output, toasted pieces of bread, we could try to construct a production function here.

So let's do that. Let's say the output is going to be the number of slices of toasted bread, and it's going to be equal to—and I'm going to write this as well; I'm going to make our production function as being the minimum of several values. What you're going to see is going to be based on what's going to be our rate-limiting factor.

I don't want to get too much in the weeds with you on this, but just to help us understand. So it's going to be the minimum of, well, the amount of bread you have—so slices of bread. And why does that make sense? Well, the amount of toasted bread you can produce is always going to be limited by the amount of untoasted bread that you put into your process.

If you only have 60 that's going in per hour here, well then you can only produce a maximum of 60 right over here. And this is going to be per hour—per hour. So this is going to be the slices of bread per hour.

Now our other input—how much toast can one toaster toast in one hour? Well, if they do four slices in 10 minutes, we'll multiply that times six to get to an hour; that's going to be 24 slices per hour. So we could do 24 times the number of toasters.

And then last but not least, how much bread, or how many slices, can one person process per hour? Well, it's going to be 60 slices per hour. So we do 60 times—let's call them workers. I was going to call them toasters, but we are already using that for the equipment—times the number of workers.

So, it's worth at this point just to pause this video and really process what's going on. What are the inputs here and what are the outputs? Well, the inputs are right over here: the number of slices of bread per hour, the number of toasters we have at our disposal, and the number of workers. Toasters you could view as capital, workers you could view as labor.

Now, another interesting thing to think about—and we will talk a lot about this in economics—is what's going on in the long run and the short run. Production functions are useful for thinking about the long run and the short run because the short run is defined as the situation in which at least one of your inputs is fixed. Let me write this down: at least one input is fixed.

Now, what does that mean in our bread-toasting example right over here? Well, let's just say that we can, it's very easy to get slices of bread. If we have the capacity and we want to produce more, the slices of bread are, let's say, it's just never our rate-limiting factor, so that part isn't fixed.

But to get a new toaster, let's say these are special toasters and you've got to order them, and it takes a month. So let's say that there's a one-month lead time on this input—a one-month lead time. Let's say for workers, there's just not a line of people ready to toast. You have to put a job posting out there, and you're going to have to interview people, and so let's say that it takes two weeks to hire someone—so two weeks to hire, or I guess you could also say two weeks to hire or to fire someone if you want to reduce capacity.

And let's say it takes one month to either get a toaster or to remove a toaster. Well, in that case, the short run in this situation is a time period where at least one of the inputs is fixed. So pause this video and think about what would be the short run in our situation.

Well, the short run in our situation, the number of toasters we're going to have is going to be fixed for at least a month. So our short run in this situation is up to a month—so up to a month. And then the other side of it, what would the long run be?

Well, in the long run, by definition, none of your inputs are fixed. You can change the number you have of any of these things, so our long run is going to be greater than one month in this example.

Now, it's really worth noting that was just for this example. If we were talking about some type of automobile factory and the output is the number of automobiles produced per day or per month, you would have all these inputs: you would have your metal, you would have your labor, and then you would have the equipment for the factory itself.

Well, there, the long run—it might take another year, or even two years, or five years to build a factory, and in that case, the long run would be the time period greater than it takes to build another factory. Usually, capital is the thing that is most fixed for the longest period of time, and that's why I made it hard for us to get our toasters.

So I will leave you there. This is just an introduction to the idea of a production function, but hopefully with our bread-toasting example, it is not so intimidating.