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# Production Function and Constraints

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Author: Kate Eskra
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This lesson will explain Production Function and Constraints

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Tutorial

## PRODUCTION FUNCTION AND CONSTRAINTS

Source: Image of Production Function created by Kate Eskra, Image of Utility Function created by Kate Eskra

## Video Transcription

Hi. Welcome to economics. This is Kate. This tutorial is called Production Function and Constraints. As always, my key terms are in red, and my examples are in green.

In this tutorial will be comparing fixed and variable inputs. You'll be able to understand how fixed and variable costs actually define the long run versus the short run. You'll look at a production function, and you'll see how it is similar to a utility function.

OK. So let's talk about firm costs. We know that factors of production are land, labor, and capital, and firms have to purchase them, obviously, to produce anything. They are categorized as either fixed or variable. So let's look at that. OK, so a fixed input is a factor of production or input that cannot be changed in the short run. That makes sense. Anything that's fixed doesn't change. It stays the same. So no matter how much the firm decides to produce, like I said, they don't change. And their cost is going to stay the same as well. So an example of a fixed input would be the factory or building office space, something like that. So typically, a business would lease this for a period of time. You have a contract, and you have to pay. The cost is exactly the same every month. So let's say the business decides to produce absolutely nothing. They can't say to their landlord, oh, you know what? I'm not going to pay this month. They still have to pay their rent. If they decide to produce a whole lot more, it's not like they have to pay more in rent. The rent stays the same. It's the same thing as with you, if you have a mortgage or if you pay rent. If you vacation for an entire month, it's not like you can call the bank and say, hey, you know what? I didn't live there at all this month. I'm just not going to pay my mortgage this month. It doesn't work that way. So that's what we mean by a fixed input.

Whereas a variable input is a factor of production or input that depends upon the level of production. Variable inputs do change depending on how much we choose to produce. So, businesses very often are making production decisions-- should we produce more? Should we produce less? And these inputs are going to change, as well as their cost, as they produce more. So an example of a variable input would be labor or machinery. If they decide to produce more, they can quickly hire more workers, they could buy some new machines. If they decide to produce less, they can lay off some workers. Again, with the example of your mortgage payment, which was fixed, let's say you do go on vacation that entire month. Well, now that you're not living in your house that month, your utility bills would be a little bit less, right? When you come home, your utility bills will be more. So something like utility bills would be a variable cost.

So we use these to actually define the long run and the short run. The short run is actually defined not in terms of, oh, it's exactly three months, or six months, or a year, but instead we define it as there is at least one fixed input or cost. So if something is fixed that you cannot change. So you cannot immediately expand your factory. You cannot immediately get out of your current lease.

Whereas the long run is the time in which all of these inputs or costs do become variable. So you can renegotiate your lease, or you can make some major changes to your firm, in the long run, when things become variable and you can change them.

All right. So now that we've talked about the difference between fixed and variable, we need to look for a minute here about how firms have to decide how they're going to go about producing their good or service. So they have to decide, how many workers should we hire? And that would be the labor input. How many machine should we purchase? That would be their capital input. The idea here is that they have to consider what combination of labor and capital are going to minimize their costs and make them the most profitable.

So they use something called a production function, which is a mathematical relationship between input and output. In the context of economics of the firm, cost and profit can be determined through cost and profit functions, which is what we're going to look at. So here's what a production function looks like. Let's say, I don't know, maybe this is for cars. And what we have on the axes are two different inputs. So this is machines, and this is the input of labor. What a production function does is, it lists all of the possible combinations here of workers and machines that can make different quantities, let's say, again, of cars. So A, B, and C all would represent different quantities of cars being produced. So if the firm, let's say, decides that they want to produce 500 cars, maybe that's here. That's along this curve. If they want to produce 1,000 cars, maybe that's here. If they want to produce 2,000 cars, that would be here. OK? So these represent different quantities of output that they're producing.

The green curves are what we call isoquants. And they represent combinations of labor and machines that will produce a certain amount of output, like I said. So anywhere along this curve would be the output of 2,000 cars. Anywhere along here would be, for example, the 1,000 cars. And anywhere around here would be producing the 500. But they're combinations of the labor and machines.

Whereas that blue line represents combinations of labor and machines for a given cost. So I only put one isocost curve on here, but that would be-- this would all cost the same amount. So up here would represent if you spent your entire budget on only machines and no workers. Down here would be spending your entire budget on all workers and hiring no machines or having no capital input at all. So anywhere along here are just combinations that all cost the same amount.

So we can actually compare this to a utility function. I think that's kind of helpful. This, as a utility function-- these were our indifference curves and the line here was our budget constraints. Anything in here was within our budget. So that looked at consumer choices and preferences between two goods. Here, for this person, it's trips to the movies and Chinese take-out. And the idea of looking at a utility function was to figure out how consumers can maximize their utility within a budget constraint. It's very similar with the production function. Instead of looking at consumer preferences, it's looking at firm choices between two inputs. And firms are going to maximize their returns by minimizing cost according to their production function.

So, again, how do consumers maximize their utility using utility functions? Well, they consume according to-- they would consume here, maximize their utility, at point C, because that's the highest indifference curve that is within their cost, or within their budget constraint. It's going to be the same thing with a production function. Producers are going to minimize cost by selecting the combination of machines and labor that are right there. So that's on the highest isoquant curve where it is within their budget, or on this isocost curve.

OK. So in this tutorial, what we talked about was, we looked at different types of inputs for firms-- fixed and variable. We talked about how those help us to define the short run and long run. The short run, there is at least one fixed input. In the long run, everything becomes variable. And we just saw how production functions are actually really similar to utility functions, but they help firms to maximize their returns.

Thank you so much for listening. Have a great day.

Terms to Know
Fixed Input

A factor of production (input) that cannot be changed in the short-run.

Production Function

A mathematical relationship between inputs and output. In the context of economics of the firm, cost/profits can be determined through cost/profit functions.

Variable Input

A factor of production (input) that depends upon the level of production. Variable inputs change depending upon how much we choose to produce.