The Production Function and Diminishing Returns - Online Tutor, Practice Problems & Exam Prep

The production function illustrates the relationship between inputs, like labor, and outputs, such as pizzas produced. The marginal product of labor (MPL) measures the additional output from hiring one more worker. Initially, adding workers increases output significantly, but due to the law of diminishing returns, the MPL eventually declines as fixed inputs, like ovens, limit productivity. Total costs comprise fixed costs (e.g., ovens) and variable costs (e.g., wages), affecting average total cost per pizza. Understanding these concepts is crucial for analyzing production efficiency and cost management in economics.

The Production Function and Diminishing Returns

Alright, now we're going to discuss the production function and see what happens as we add more and more workers into production. Let's check it out. The production function is going to relate the amount of inputs, in this case workers, to the amount of output, which is the quantity that we produce. So we're going to talk about the marginal product of labor. This term might sound technical, but we've got "marginal" here, meaning what happens when we add one more worker. I like to think of the marginal product of labor as the fruits of our labor.

Increase in output from adding one more worker, that's the marginal product of labor. We're going to use MPL. Actually, let me write that in blue so it stands out a little more, Marginal Product of Labor, MPL with a little 'l' there for labor. So, let's go ahead and calculate the marginal product of labor in this example. We've got a local pizza shop that leases two pizza ovens at a total daily cost of $100, and the pizza company is deciding how many employees to hire at a wage of $80 per day.

Let's look at what happens on this table. On the left-hand side, we have the number of pizza ovens. They told us we have two ovens. We're always going to have two ovens. We're going to add more and more workers. So, when we have zero workers, no pizzas are produced; there are just two ovens there, and nobody's making any pizzas. As we add workers, we start seeing pizzas being produced. Let's calculate our marginal product of labor. When there are zero workers, there's no production, so we can't calculate the marginal product yet.

When we add one worker, we start from zero pizzas and get up to 30 pizzas. That means by hiring one worker, we've increased our production by 30 pizzas. What happens when we hire a second worker? With one worker, we already made 30 pizzas, and by adding one more worker, we go up to 80 pizzas. The second worker thus made an extra 50 pizzas. You can imagine that maybe when there was one worker, he had to do everything. Now with two, they can split up tasks and become more productive. The third worker increases the number from 80 to 150 pizzas, making the marginal product of labor from the third worker 70 pizzas.

Let's add a fourth worker. We go from 150 to 180 pizzas. With the addition of the fourth worker, we only gain another 30 pizzas. And look at this last worker; we go from 180 to 190 pizzas. By adding one more worker, we only get 10 extra pizzas. This is because we still only have two ovens, so maybe those workers are now kind of stepping over each other. Their marginal product ends up getting smaller and smaller, leading us to the law of diminishing returns. We discuss this concept as we continue to add more and more workers to the same amount of fixed inputs. Eventually, the marginal product will decrease.

Now, when there are so many workers, they're all crammed in there and can't be as productive as if there were fewer workers. That's the law of diminishing returns. Before we finish this discussion, let's look at a graph of the production function. Here we have the quantity of workers on one axis and the total pizzas on the other. The slopes of the graph show the marginal product of labor at different points, illustrating the diminishing returns. As we keep adding workers, the graph becomes less and less steep.

So, as we add a few workers, we initially have positive returns, but eventually, it becomes worse and worse. That's going to be it for the production function. Let's pause right here, and in the next video, we'll go ahead and finish completing this table.

Calculating Fixed Cost, Variable Cost, and Average Total Cost

Alright, cool. So we're back here on the table, let's go ahead and finish filling it in. So the next column here was fixed costs and that's the cost of our ovens, right? In this problem, we just had the cost of the ovens and the cost of the workers, right? So the fixed cost is going to be the ovens. No matter how many pizzas we make, those ovens are going to cost us the same amount every day. Right? So it doesn't matter how many workers we have, how many pizzas we have, the ovens are always going to cost us $100. Right? That was what it told us in the problem with the daily cost of $100. So all of these all the way down is going to be $100. Right?

Now what about our variable costs? So our variable costs go up as we make more and more output, right? So as we, as we want to make more pizzas, we're going to need to hire more workers. So when we were hiring 0 workers, right, our variable costs were 0 because we didn't hire any workers, but then we hired our first worker and it tells us that the wage is $80 per day per worker. So the first worker is going to get $80 here. Two workers, right now we're paying 2 workers $80 and that's going to get us a $160. Right? 80 times 2 and the next one, we have 3 workers so we're just going to do 80 times 3 workers, right? So that's 240 and then 4 workers. It's going to be 80 times 4 workers, right? So you see what's happening here? We're just multiplying the number of workers. Right? It's going to be the number of workers times the wage. That's going to be our fixed cost. So this last one here, we've got 5 workers. So we got 5 workers times the wage of 80. That's going to get us up to oops. That gets us up to 400. Right? 80 times 5 is 400. So there we go.

To get our total cost, we just have to add our fixed cost plus our variable cost. Right? So that's going to be our total cost. So we just have to add across here. 100 plus 0 is 100. 100 +80 is 180. 100 +160 is 260. 100 +240 is 340. Right? So this is pretty easy here, 420 and last but not least, 100 +400 is 500. Cool.

Alright. So the last column here is the average cost per pizza. Right? How do we find average cost? Well, that's when we divide by quantity. Right? And this is average total cost. Right? Average total cost. I'm going to put this T here for average total cost. So we're just going to take our total cost column and divide it by quantity. Alright? And that's how we're going to get these numbers here. So the first one, we had a 100 total cost but we didn't make any pizzas. So if we didn't make any pizzas, we're not going to have an average cost; we're just going to leave that one empty.

How about the next one? We had $180, for 30 pizzas. So we would do 180 divided by 30, and we're going to get $6 as our average cost per pizza there. Right? So if we did the $6 times the 30 pizzas that we made, it would get us to that total cost of 180, right? So next one, we had $260 in total cost divided by 80. Right, we had 80 pizzas when we had 2 workers, so 260 divided by 80 gets us $3.25 per pizza. Right? Let's go on to the next one. We've got $340 in total cost divided by 150 pizzas in this case. Right? We've made 150 pizzas. So each pizza cost us $2.27. So let's keep this up. 420. Right? The next one, 4 workers, we have $420 in total cost divided by 180, the amount of pizzas, and we're going to get a total cost here of $2.33. The last but not least, we've got, $500 in total cost divided by 190 and that gets us a total cost, or average cost rather, of $2.63.

So look what's been happening here. As the marginal product was rising, right, we saw this marginal product going up, well here, the average cost is going down as that marginal product is rising and then that marginal product starts falling again, right? And as that marginal product falls, that price goes up, right? And the reason here is that we're getting less from each worker, right? Once that marginal product starts falling, we're getting less value from adding a worker. So that's causing the price to increase, right? So as we're getting more and more value from adding workers, the price goes down and then we start getting less value from the diminishing returns and the price starts going back up. Cool? Alright, let's go ahead and move on to the next video.

Here’s what students ask on this topic:

What is the production function in microeconomics?

The production function in microeconomics illustrates the relationship between the quantity of inputs used in production and the quantity of output produced. It is typically expressed as Q = f(L, K), where Q represents the quantity of output, L is the quantity of labor, and K is the quantity of capital. The production function helps in understanding how different combinations of inputs affect the level of output, and it is crucial for analyzing production efficiency and cost management. By examining the production function, firms can determine the optimal mix of inputs to maximize output and minimize costs.

What is the marginal product of labor (MPL) and how is it calculated?

The marginal product of labor (MPL) measures the additional output produced by adding one more unit of labor, holding other inputs constant. It is calculated by the change in total output divided by the change in the quantity of labor. Mathematically, it is expressed as:

$\mathrm{MPL}=\frac{\mathrm{\Delta Q}}{\mathrm{\Delta L}}$

where ΔQ is the change in total output and ΔL is the change in the quantity of labor. Initially, MPL may increase due to specialization, but it eventually decreases due to the law of diminishing returns, as fixed inputs like machinery limit productivity.

What is the law of diminishing returns?

The law of diminishing returns states that as more units of a variable input (e.g., labor) are added to a fixed input (e.g., capital or land), the additional output produced by each additional unit of the variable input will eventually decrease. This occurs because the fixed input becomes a limiting factor, causing workers to become less productive as they crowd around the same resources. For example, in a pizza shop with a fixed number of ovens, adding more workers initially increases pizza production, but eventually, the ovens become a bottleneck, and the marginal product of labor declines.

How do fixed and variable costs affect total cost in production?

Fixed costs are expenses that do not change with the level of output, such as the cost of machinery or rent. Variable costs change with the level of output, such as wages for workers or raw materials. Total cost (TC) is the sum of fixed costs (FC) and variable costs (VC), expressed as:

$\mathrm{TC}=\mathrm{FC}+\mathrm{VC}$

Understanding the distinction between fixed and variable costs is crucial for managing production expenses and setting prices. As production increases, variable costs rise, impacting the total cost and average cost per unit of output.

What is the average total cost (ATC) and how is it calculated?

Average total cost (ATC) is the total cost per unit of output produced. It is calculated by dividing the total cost (TC) by the quantity of output (Q). Mathematically, it is expressed as:

$\mathrm{ATC}=\frac{\mathrm{TC}}{Q}$

ATC helps firms determine the cost efficiency of their production process. By analyzing ATC, firms can identify the optimal level of production that minimizes costs and maximizes profitability. Changes in fixed and variable costs directly impact the ATC, influencing pricing and production decisions.