The relationship between the quantity of inputs (land, labour, capital) and the quantity of outputs (burgers) is called the production function. The production function is useful for helping us think on the margin, to realize when we should take a course of action. In the case below it helps us understand how many workers Bob's Burger Shack should hire. The production function is best utilized in a theoretical practice to help firms estimate when things would best be viable. The reason that it is largely theoretical is because of the amount of variables that exist in the real world. Consider this, not every worker is the same, Sally might be able to produce 20 burgers in an hour and Sam might be able to do 19. Leadership is another one, perhaps a workforce is 5% more efficient when they have a good leader. These inputs are nearly impossible things to perfectly quantify and therefore firms largely use the theory for rough estimates in their planning and decision making, likewise you can use it as such in your lives as well.
Let's look at chart below, what do we see? In the "Marginal Product of Labour" or in other words how many more burgers are made with each new employee hired we see that the number varies! Why is that? It climbs steadily early and then tapers off and drops at the bottom. Think about it like a production line in your home kitchen, if you're making burgers by yourself you have total freedom and easy mobility in the kitchen. If you add two more people, it is still beneficial as people can specialize in one area and produce more burgers. What happens though when you try to fit 8 people in a small kitchen space? Time is possibly wasted, people are tripping over each other and therefore your production of burgers drops. This is known as the "point of diminishing returns" or as the "diminishing marginal product". In this scenario we see that point occurs after the 5th worker when "Marginal Product of Labour" drops from 75 to 65.
The next thing we see is that there are "fixed costs" and "variables costs''. Fixed costs are costs that don't change relative to the increase in production in the short run. For example the cost of the factory would remain the same in the short run, but long run they might need to go to a bigger burger factory to meet demand. Variable costs are those that change in relation to the number of burgers produced and in this case the cost of each additional worker.
From here we can see using this production function that to achieve the lowest cost per burger Bob's Burger Shack should hire 6 people.
Bob's Burger Shack has the following Production Function:
Now let's look at this scenario through the lens of profit. A company's goal is to maximize profit. What do you notice below? Profit is maximized given a $2.00 price on burgers not where the cost of producing the burgers is the lowest, but rather where the "Total Revenue" minus the "Total Costs" is greatest at $435.00. This occurs at the point when there are 7 people and not 6. As you can see the production function is just one level of analysis useful for aiming the company in the right direction to achieve it's goals of profit maximization.