If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Introduction to production functions

Production functions describe how output is determined by various inputs. The short run is defined as the period of time in which at least one input is fixed. Anything longer than that is considered the long run.

Want to join the conversation?

Video transcript

- [Instructor] You will hear the term production function thrown around in economic 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, and then you put them in, their 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 Two might recognize this. Or we could say the output, it's often use the letter Q in economic circle, 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 go, you could have as many inputs as is necessary to produce that good. And these inputs, if you wanted to categorize them, these are the classic factors of production that we would have talked about before. These would be, these would be your land, labor, capital, and entrepreneurship. And 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. And so 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. And 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 be, needs 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, they can process one slice per minute, one slice per minute. I know many of you 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 piece of bread, we could try to construct a production function here. So let's do that. So let's say then the output is going to be the number of slices of toasted bread. And it's going to be equal to, and I'm gonna write this as, well, I'm gonna make our production function as being the minimum of several values. And what you're going to see, it's going to be based on what's going to be our rate-limiting factor? And I 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, slices of bread. And why does that make sense? Well, you're only, 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 gonna 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 time six to get to an hour. That's gonna be 24 slices per hour. So we could do 24 times the number of toasters, times the 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'll do 60 times, times, let's call them workers, I was gonna call 'em toasters, but we are using that for the equipment, times the number of workers. And so it's worth, at this point, just 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. This is the number of slices of bread per hour, the number of toasters we have at our disposal, the number of workers. Toasters you could view as capital. Workers you could view as labor. And 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? And production functions are useful for thinking about the long run in the short run because the short run is defined, 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, 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 bread, 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 gotta order them and it takes a month. So let's say that there's a one-month lead time on this input, one month lead time. And let's say, for workers, there's just not a line of people ready to toast 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. 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, 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 and then you 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. In that case, the long run would be the time period greater than amount 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 it made it hard for us to get our toasters. So I will leave ya 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.