Total Cost Curve

Production Function

Let’s assume that things became great, and years after that, your passion and natural talent for selling lemonades led to the opening of your first lemonade factory. For the sake of example, we are going to keep things simple and we will analyze the short-run production mechanisms in the beginning. What do we need for production? Obviously, we need lemons, sugar, workers, and a factory in order to produce the lemonade. The physical capital in the factory can be considered the cost of the factory or the total fixed cost.

But what about the workers? How can we calculate their costs? We know that workers are paid since they are offering labor. Nonetheless, if you would hire more workers, the cost of production will be higher. For example, if a worker’s wage is $10 per hour, that means that hiring five workers will cost you $50 per hour. These costs are called variable costs. They change with respect to your production preferences. Now we can calculate the total costs under the different number of workers in the following table.

Table. 1 - Cost of producing lemonades with different combinations

So we can see that due to diminishing marginal returns, every additional worker adds less to the production of lemonades. We draw our production curve in Figure 1 below.

Fig. 1 - The production curve of the lemonade factory

As you can see, due to diminishing marginal returns, our production curve becomes flatter as we increase the number of workers. But what about the costs? We have calculated our total costs as the sum of our fixed costs and variable costs. Therefore we can graph it as follows.

Fig. 2 - The total cost curve of the lemonade factory

As you can see, due to diminishing marginal returns, as our costs increase, our production doesn’t increase by the same amount.

Derivation of the Total Cost Curve Formula

Derivation of the total cost curve formula can be done via multiple methods. Nonetheless, as we have seen, it is directly linked to production costs. First of all, we know that total costs are the sum of fixed costs and variable costs. Therefore we can most basically, from the definition:

\(\text {Total costs (TC)} = \text {Total fixed costs (TFC)} + \text {Total variable costs (TVC)} \)

As we have mentioned before, total fixed costs are fixed. Meaning that they are stable for any amount of production in the short run. Nonetheless, total variable costs change with respect to the production level. As we have shown before, you have to pay additional costs for every additional unit that you produce. TVC varies with respect to the unit of production.

Total Cost Curve and Marginal Cost Curve

The total cost curve and the marginal cost curve are closely linked. Marginal costs represent the change in the total costs with respect to the amount of production.

Since we represent changes with "\(\Delta\)", we can denote the marginal costs as follows:

\(\dfrac{\Delta \text{Total Costs}} {\Delta Q} = \dfrac{\Delta TC}{\Delta Q}\)

It is important to grasp the relationship between marginal costs and total costs. Therefore, it is better to explain it with a table as follows.

Table. 2 - The marginal costs of producing lemonades at different quantities

As you can see, due to diminishing marginal returns, the marginal costs increase as production increases. It is simple to calculate the marginal costs with the mentioned equation. We state that marginal costs can be calculated by:

\(\dfrac{\Delta TC}{\Delta Q}\)

Thus, if we want to show the marginal costs between two production levels, we can substitute values where it belongs. For example, If we want to find the marginal costs between 270 bottles of lemonade produced per hour and 340 bottles of lemonade produced per hour, we can do it as follows:

\(\dfrac{\Delta TC}{\Delta Q} = \dfrac{90-80}{340 - 270} = 0.143\)

Therefore, producing one additional bottle will cost $0.143 at this production level. Due to diminishing marginal returns, if we increase our output, marginal costs will also increase. We graph it for different levels of production in Figure 3.

Fig. 3 - The marginal cost curve for the lemonade factory

As you can see, the marginal costs increase with respect to increased total output.

How to Derive Marginal Costs from Total Cost Function

It is rather easy to derive marginal costs from the total cost function. Remember that marginal costs represent the change in total cost with respect to the change in total output. We have denoted marginal costs with the following equation.

\(\dfrac{\Delta TC}{\Delta Q} = \text {MC (Marginal Cost)}\)

Indeed, this is exactly the same thing as taking the partial derivative of the total costs function. Since the derivative measures the rate of change in an instant, taking the partial derivative of the total costs function with respect to the output will give us the marginal costs. We can denote this relationship as follows:

\(\dfrac{\partial TC}{\partial Q} = \text{MC}\)

We should keep in mind that the amount of production \(Q\) is a defining characteristic of the total costs function due to variable costs.

Average Cost Curves

Average cost curves are necessary for the next section, where we introduce the differences between long-run cost curves and short-run cost curves.

Remember that total costs can be denoted as follows:

\(TC = TFC + TVC\)

Intuitively, average total costs can be found by dividing the total cost curve by the amount of production. Thus, we can calculate the average total costs as follows:

\(ATC = \dfrac{TC}{Q}\)

Furthermore, we can calculate the average total costs and average fixed costs with a similar method. So in what manner do average costs change as production increases? Well, we can find out by calculating the average costs of your lemonade factory in a table.

Table. 3 - The average total costs of producing lemonades

As highlighted in the cells, after some point (between the 6th and 7th workers), your average costs stop decreasing and then start increasing after the 7th worker. This is an effect of diminishing marginal returns. If we graph this, we can clearly observe how these curves behave in Figure 4.

Fig. 4 - Average Costs of the Lemonade Factory

As you can see, due to diminishing marginal returns or increased marginal costs, after some point in time, average variable costs will be higher than average fixed costs, and the amount of change in the average variable costs will increase drastically after some point in time.

Short Run Total Cost Curve

Characteristics of the short-run total cost curve are highly important for grasping the nature of the total cost curve.

The most important aspect of the short run is its fixed decisions. For example, you can not alter your production structure in the short run. Furthermore, it is impossible to open new factories or close already existing ones in the short run. Thus, in the short run, you can hire workers to change the amount of production. Until now, all we have mentioned about total cost curves exists in the short run.

Let’s elaborate a little further and assume that you have two lemonade factories. One is larger than the other one. We can denote their average total costs with the following graph.

Fig. 5 - Average Total Costs of Two Factories in the Short Run

This is rather realistic since a bigger factory would be more efficient while producing the lemonades in higher quantities. In other words, the large factory will have lower average costs at higher quantities. Nonetheless, in the long run, things will change.

Long Run Total Cost Curve

The long-run total cost curve differs from the short-run total cost curve. The main difference arises due to the possibility to change things up in the long run. Unlike in the short run, fixed costs are no longer fixed in the long run. You can close factories, bring in new technologies, or change your business strategy. The long run is flexible compared to the short run. Therefore, average costs will become more optimal. In the long run, the firm reaches its equilibrium with the information gained in the short run.

Fig. 6 - Average Total Costs in the Long Run

You can imagine the long-run curve as a pocket that contains all possible short-run curves. The firm reaches equilibrium with respect to the information or tryouts made in the short run. Thus, it will produce at the optimum level.