## Total Cost Curve Definition

It is better to define the total costs before introducing the definition of the total cost curve.

Let’s say that you are planning to buy a new phone. Nonetheless, you know that these days they are expensive! The amount of savings you have is $200. The phone you want is $600 dollars. So with basic algebra, you realize that you need to earn $400 more to buy the phone. So you decided to use the oldest trick in the book for earning money and opened a lemonade stand!

Intuitively we know that profit is the difference between your revenue and your costs. So if you gained revenue of $500 and your costs were $100, this means that your profit would be $400. We generally denote profit with \(\pi\). Therefore we can denote the relationship as follows:

\(\hbox{Total Profit} (\pi) = \hbox{Total Revenue} - \hbox{Total Costs} \)

\(\$400 = \$500 - \$100 \)

Nonetheless, your costs may not be as obvious as your profits. When we think of the costs, we generally think about **explicit costs,** such as the lemons you buy and the stand itself. On the other hand, we should consider **implicit costs** too.

What could you have done with the opportunity cost of opening a lemonade stand and working in there? For example, if you are not spending your time selling lemonade, can you earn more money? As we know, this is the **opportunity cost**, and economists take this into consideration when calculating the costs. This is the fundamental** difference between accounting profit and economic profit.**

We can state **accounting profit** as follows:

\(\pi_{\text{Accounting}} = \text{Total Revenue} - \text{Explicit Costs}\)

On the other hand, economic profit adds implicit costs to the equation too. We state the **economic profit** as follows:

\(\pi_{\text{Economic}} = \text{Total Revenue} - \text{Total Costs}\)

\(\text{Total Costs} = \text{Explicit Costs} + \text{Implicit Costs}\)

We have covered Opportunity Costs in detail! Don’t hesitate to check it out!

**Explicit costs** are payments we make directly with money. These generally include things such as wage payment for labor or the money you spend on physical capital.

**Implicit costs** are generally the opportunity costs that do not require explicit monetary payments. They are the costs due to the missed opportunities that arise from your choice.

This is why **we generally find economic profit to be lower than accounting profit**. Now we have an understanding of the total costs. We can elaborate our understanding with another simple example. In this scenario, it is time to open your first lemonade factory!

### 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.

Bottles of Lemonade Produced per Hour | Number of Workers | Variable Costs (Wages) | Fixed Cost(Infrastructure Cost of The Factory) | Total Cost per Hour |

0 | 0 | $0/hour | $50 | $50 |

100 | 1 | $10/hour | $50 | $60 |

190 | 2 | $20/hour | $50 | $70 |

270 | 3 | $30/hour | $50 | $80 |

340 | 4 | $40/hour | $50 | $90 |

400 | 5 | $50/hour | $50 | $100 |

450 | 6 | $60/hour | $50 | $110 |

490 | 7 | $70/hour | $50 | $120 |

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.

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.

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

The **total cost curve** represents total costs with respect to different output levels of production.

## 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.

For example, our previous total cost curve can be given as follows.

\(\text{TC}(w) = w \times $10 + $50 | w \in N\)

\(w\) is the number of workers, and the total costs function is a function of the number of workers. We should notice that $50 is the fixed costs for this production function. It doesn’t matter if you decide to hire 100 workers or 1 worker. The fixed costs will be the same for any number of produced units.

## 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.

**Marginal costs** can be defined as the change in the total costs when producing an additional quantity.

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.

Bottles of Lemonade Produced per Hour | Number of Workers | Variable Costs(Wages) | Fixed Cost(Infrastructure Cost of The Factory) | Marginal Costs | Total Cost per Hour |

0 | 0 | $0/hour | $50 | $0 | $50 |

100 | 1 | $10/hour | $50 | $0.100 per Bottle | $60 |

190 | 2 | $20/hour | $50 | $0.110 per Bottle | $70 |

270 | 3 | $30/hour | $50 | $0.125 per Bottle | $80 |

340 | 4 | $40/hour | $50 | $0.143 per Bottle | $90 |

400 | 5 | $50/hour | $50 | $0.167 per Bottle | $100 |

450 | 6 | $60/hour | $50 | $0.200 per Bottle | $110 |

490 | 7 | $70/hour | $50 | $0.250 per Bottle | $120 |

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.

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.

For example, let’s assume that we have a total costs function with one argument, quantity (\(Q\)), as follows:

\(\text{TC} = \$40 \text{(TFC)} + \$4 \times Q \text{(TVC)} \)

What is the marginal cost of producing one unit of an additional product? As we have mentioned before, we can calculate the change in costs with respect to the change in the amount of production:

\(\dfrac{\Delta TC}{\Delta Q} = \dfrac{$40 + $4(Q + 1) - $40 + $4Q}{(Q+1) - Q} = $4\)

In addition to this, we can directly take the partial derivative of the total cost function with respect to the amount of production since it is exactly the same process:

\(\dfrac{\partial TC}{\partial Q} = $4\)

Indeed, this is why the slope of the total cost curve (the rate of change in total costs with respect to production) is equal to the marginal cost.

## 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.

Bottles of Lemonade Produced per Hour | Number of Workers | Total Variable Costs (TVC) | Average Variable Costs (AVC) (TVC / Q) | Total Fixed Costs (TFC) | Average Fixed Costs (AFC) (TFC / Q) | Total Costs (TC) | Average Costs(AC)(TC/Q) |

0 | 0 | $0/hour | - | $50 | - | $50 | - |

100 | 1 | $10/hour | $0.100 Per Bottle | $50 | $0.50 Per Bottle | $60 | $0.6 Per Bottle |

190 | 2 | $20/hour | $0.105 Per Bottle | $50 | $0.26 Per Bottle | $70 | $0.37 Per Bottle |

270 | 3 | $30/hour | $0.111 Per Bottle | $50 | $0.18 Per Bottle | $80 | $0.30 Per Bottle |

340 | 4 | $40/hour | $0.117 Per Bottle | $50 | $0.14 Per Bottle | $90 | $0.26 Per Bottle |

400 | 5 | $50/hour | $0.125 Per Bottle | $50 | $0.13 Per Bottle | $100 | $0.25 Per Bottle |

450 | 6 | $60/hour | $0.133 Per Bottle | $50 | $0.11 Per Bottle | $110 | $0.24 Per Bottle |

490 | 7 | $70/hour | $0.142 Per Bottle | $50 | $0.10 Per Bottle | $120 | $0.24 Per Bottle |

520 | 8 | $80/hour | $0.153 Per Bottle | $50 | $0.09 Per Bottle | $130 | $0.25 Per Bottle |

540 | 9 | $90/hour | $0.166 Per Bottle | $50 | $0.09 Per Bottle | $140 | $0.26 Per Bottle |

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.

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.

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.

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.

## Total Cost Curve - Key takeaways

**Explicit costs**are payments we make directly with money. These generally include things such as wage payment for labor or the money you spend on capital.**Implicit costs**are generally opportunity costs that do not require monetary payments. They are the costs due to the missed opportunities arising from your choice.- If we sum up explicit and implicit costs, we can measure the
**total cost**(TC). The total economic costs are different from accounting costs since accounting costs only include explicit costs. Thus, accounting profit is generally higher than economic profit. - Total costs can be divided into two components, one is the total fixed costs (TFC) and the other component is total variable costs (TVC): \(TVC + TFC = TC\).
- Marginal costs can be defined as the change in the total costs when producing an additional quantity. Since we measure the rate of change with partial derivative marginal costs are equal to the partial derivative of total costs with respect to output:\(\dfrac{\partial TC}{\partial Q} = MC\).
- Average costs can be found by dividing total costs by the amount of production: \(\dfrac{TC}{Q} = ATC\). With a similar approach, we can find average fixed costs and average variable costs.
- In the long run, fixed costs can be changed. Therefore, the long-run total cost curve is different from the short-run one.

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##### Frequently Asked Questions about Total Cost Curve

How do you calculate the total cost curve?

The total cost curve can be calculated via the sum of the total fixed costs and total variable costs. Total fixed costs are fixed in the short run and they don’t change with respect to production amount. Total variable costs change with respect to the amount of production.

What is the total cost function formula?

Total Costs = Total Variable Costs + Total Fixed Costs

Total Costs = Average Total Costs x Quantity

Why is marginal cost a derivative of total cost?

Because marginal costs measure the rate of change in total costs with respect to the change in output. We can easily calculate this with a partial derivative. Since the derivative also measures the rate of change.

How do you derive variable cost from the total cost function?

We can derive the variable costs at a specific level of production by subtracting total fixed costs from the total costs at that level of production.

What happens to total cost in the short run?

Total costs in the short run are directly correlated with variable costs, such as the number of workers. Since technology or the method of production is fixed in the short run, our fixed costs remain the same.

What is the shape of a total cost curve?

We can’t say that every total cost curve will be the same. There are s-shaped curves, linear curves, etc. Nonetheless, the most common form is the “S” shaped total cost curve.

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