Average and Marginal Cost: Meaning, Solved Examples - Embibe
• Written By Ritu_Kumari
• Written By Ritu_Kumari

# Average and Marginal Cost: Definition, Formulas, Relationship, Difference

Average and marginal cost: Differentiation has significant applications in estimating various quantities of interest. We use differentiation in a different subject other than mathematics. In Economics, differentiation has tools line marginal cost and marginal revenue, which are necessities for calculating the change in demand for a product to change in the price to estimate the rate of change of revenue with an increase or decrease in selling price.

The marginal cost plays a significant role in all businesses’ decision-making processes. Whenever a company performs the financial analysis of companies revenue, the management evaluates the pricing of each product and makes necessary changes using marginal costing analysis. Let us learn about the average and marginal costs in detail.

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## Average and Marginal Cost – Types of Cost

Let us discuss types of cost under average and marginal cost:

Fixed cost $$(F(x))$$

Fixed cost is the investment required to produce the firm, and it does not depend on the volume of the production. This means that the fixed cost is independent of the number of production.

For example, the property rent, maintenance charges and wages paid to the workers can be independent of the number of items produced in a given month.

Variable cost $$(v(x))$$

Variable costs include the costs related to the volume of production. It depends on the number of products being produced.

For example, the cost of transportation of the produced goods, packaging and storage costs depend on the number of goods being produced in a given month.

## Cost Function

The total cost of producing $$x$$ number of products is represented by $$C(x)$$. It can be written as –

$$C\left( x \right) = F\left( x \right) + V\left( x \right)$$

Here,

$$F(x) \to$$ fixed cost

$$V(x) \to$$ variable cost

## Marginal Cost

Marginal cost is the additional cost resulting from an increase in the production of one more unit of the product. The marginal cost plays a significant role in all the business’ decision-making processes. It helps in deciding the number of production cycles. Whenever a company performs the financial analysis of companies revenue, the management used to evaluate the pricing of each product offered to the consumers and makes necessary pricing changes. To perform this task, the management uses marginal costing analysis.

If $$C(x)$$ is the total cost for producing $$x$$ units, then a change in the total cost of producing one additional unit, at an output level of $$x$$ units, is given by

$${\text{Marginal}\;{\text{cost}}} = \frac{{{\text{Change}\;{\text{in}}\;{\text{total}}\;{\text{cost}}}}}{{{\text{Change}\;{\text{in}}\;{\text{number}}\;{\text{of}}\;{\text{production}}}}}$$

$$MC = \frac{{dC\left( x \right)}}{{dx}}$$

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## Shape of the Marginal Cost Curve

Most businesses’ production cost eventually results from diminishing marginal product of labor and the diminishing marginal product of capital. This means that most businesses reach a point of production where each additional unit of labour or capital is not as valuable as before.
Once the marginal products start increasing, the marginal cost of producing each additional unit will become more significant than the marginal cost of the previous unit. In simpler words, we can say that the marginal cost curve for more production processes will eventually be an upward slope curve, as shown below.

## Average Cost

The total cost of making a single product is known as the average cost. It is calculated by dividing the total cost by the number of products produced. The two most essential components in average cost are fixed and variable costs. It is also called Unit cost.

The average cost can be calculated as

$${\text{Average}\;{\text{Cost}}} = \frac{{{\text{Total}\;{\text{cost}}}}}{{{\text{Number}\;{\text{of}}\;{\text{units}}\;{\text{of}}\;{\text{production}}}}}$$

$$AC = \frac{{\;\left( {F\left( x \right) + V\left( x \right)} \right)}}{x}$$

Note that the average cost is directly proportional to the total cost of production and inversely proportional to the number of productions. Hence, we can say that the average cost decreases when the number of productions increases.

Average cost per product, $$AC = \frac{{C\left( x \right)}}{x}$$

Average fixed cost per product, $$AFC = \frac{{F\left( x \right)}}{x}$$

Average variable cost per product, $$AVC = \frac{{V\left( x \right)}}{x}$$

## Shape of the Average Cost Curve

The average cost curve starts falling due to decreasing fixed costs but rises when the average variable cost increases.

## Relationship Between Average and Marginal Costs

The average cost (AC), also called the average total cost, is the ratio of total cost and the number of quantities produced. In contrast, the marginal cost (MC) is the increased cost of the last unit produced.

Average cost and marginal cost are related to each other as:

• When average cost decreases, $$MC < AC$$
• When average cost increases, $$MC > AC$$

We can understand this relationship via a simple analogy.

Assume that the average grade of a student in a course is $$75$$.

1. If they get a score of $$70$$ on the next exam, then their average score will drop less than $$75$$.
2. Assuming they score $$80$$ in that exam, their new average will be greater than $$75$$.
3. Alternatively, if they score $$75$$ in that exam, their average will remain the same.

As per the context of production costs, let the average cost for the particular production quantity be the current average grade of the student. The marginal cost at that quantity is the grade obtained in the next exam.

From the grade analogy, we can understand that

• The average cost of quantity produced will be decreased when $$MC < AC$$.
• As quantity increases, when $$MC > AC$$.
• If $$MC = AC$$ then the average cost will be neither reducing nor increasing.

This relation is shown in the graph below:

## Marginal Cost vs Average Cost

The difference between marginal cost and average cost is given below:

## Solved Examples on Average and Marginal Cost

Q.1. The total cost $$C(x)$$ associated with the production of $$x$$ units of an item is given by $$C\left( x \right) = 0.007{x^3} + 0.03{x^2} + 15x + 4000$$. Find the marginal cost when $$7$$ units are produced.
Ans:
Given: $$C\left( x \right) = 0.007{x^3} + 0.03{x^2} + 15x + 4000$$
Marginal cost, $$MC\left( x \right) = \frac{{dC\left( x \right)}}{{dx}}$$
$$= \frac{{d\left( {0.007{x^3} + 0.03{x^2} + 15x + 4000} \right)}}{{dx}}$$
$$= \left( {0.007} \right)3{x^2} + \left( {0.03} \right)2x + 15$$
$$= 0.021{x^2} + 0.06x + 15$$
When $$x = 7$$, $$MC\left( x \right) = \left( {0.021} \right) \times {7^2} + 0.06 \times 7 + 15$$
$$\Rightarrow MC\left( 7 \right) = 0.021 \times 49 + 0.42 + 15$$
$$\therefore \,MC\left( 7 \right) = 16.449$$

Q.2. Find the average cost of the bags whose prices are $${\mathbf{Rs}}.500,\;{\mathbf{Rs}}.550,\;{\mathbf{Rs}}.\;450,\;{\mathbf{Rs}}.\;510,\;{\mathbf{Rs}}.\;530,\;{\mathbf{Rs}}.\;540,\;{\mathbf{Rs}}.\;520,\;{\mathbf{Rs}}.\;470,\;{\mathbf{Rs}}.\;460,\;{\mathbf{Rs}}.\;490$$ and $${\mathbf{Rs}}.480$$
Ans:

Average cost, $$AC = \frac{{Total\;cost\;}}{{Number\;of\;units\;of\;production}}$$
$$AC = \frac{{C\left( x \right)}}{x}$$
$$\Rightarrow AC = \frac{{500 + 550 + 450 + 510 + 530 + 540 + 520 + 470 + 460 + 490 + 480}}{{11}}$$
$$AC = \frac{{5500}}{{11}}$$
$$\therefore \,AC = 500$$
Hence, $$Rs. 500$$ is the average cost of the bags.

Q.3. The total cost $$C(x)$$ associated with the production of $$x$$ units of an item is given by $$C(x) = 13 x^2 + 26 x + 15$$. Find the marginal cost when $$10$$ units are produced.
Ans:
Given: $$C(x) = 13 x^2 + 26 x + 15$$
Marginal cost $$MC\left( x \right) = \frac{{dC\left( x \right)}}{{dx}}$$
$$= \frac{{d\left( {13{x^2} + 26x + 15} \right)}}{{dx}}$$
$$= 26 x + 26$$
When $$x = 10$$, $$MC\left( x \right) = 26 \times 10 + 26$$
$$MC\left( 10 \right) = 260 + 26$$
$$\therefore MC = 286$$.

Q.4. If the total cost $$C(x)$$ of $$x$$ units of an item is given by $$C\left( x \right) = {x^3} + 3{x^2} + 17x + 40$$ and the variable cost for the same units is $$V\left( x \right) = {2x^2} + 5{x} + 20$$ then find the fixed cost for $$17$$ units.
Ans:
We know that, $$C\left( x \right) = F\left( x \right) + V\left( x \right)$$
Here,
$$C\left( x \right) = {x^3} + 3{x^2} + 17x + 40$$
$$V\left( x \right) = {2x^2} + 5{x} + 20$$
$$\therefore \,{x^3} + 3{x^2} + 17x + 40 = F\left( x \right) + 2{x^2} + 5x + 20$$
$$\Rightarrow F\left( x \right) = {x^3} + 3{x^2} + 17x + 40 – \left( {2{x^2} + 5x + 20} \right)$$
$$\Rightarrow F\left( x \right) = {x^3} + {x^2} + 12x + 20$$
The fixed cost of $$17$$ units is $$F\left( {17} \right) = {17^3} + {17^2} + 12 \times 17 + 20$$
$$\therefore \,F\left( {17} \right) = 5426$$

Q.5. An employee gets wages in a factory, depending on how he works for the day. For 6 days, he gets wages of $$100,\,150,\,220,\,300,\,250$$ and $$210$$ rupees. Find the average cost of his wages.
Ans:

Average cost formula, $$AC = \frac{{Total\;cost\;}}{{Number\;of\;units\;of\;production}}$$
$$AC = \frac{{C\left( x \right)}}{x}$$
$$AC = \frac{{100 + 150 + 220 + 300 + 250 + 210}}{6}$$
$$AC = \frac{{1230}}{6}$$
$$\therefore\, AC = 205$$
Hence, $$Rs. 205$$ is the average wage.

## Summary

The total cost of producing $$x$$ products is $$C(x)$$. It can be calculated as$$C(x) = F(x) + V(x)$$. Marginal cost occurs when we need to ascertain the change in total cost, i.e., dependent variable with a unit change of product, the independent variable. The additional cost or increase in the dependent variable results from an increase in the production or the independent variable by one more unit. The average cost is the total cost of making a single product, and it is the ratio of the total cost and the number of products produced. The two most important components in average cost are fixed cost and variable cost. The shape of the curve for marginal cost is concave and convex, while that of average cost is the shape of the letter $$U$$.

## FAQs on Average and Marginal Cost

The frequently asked questions regarding average and marginal cost are explained below:

Q.1. What is the relation between average cost and marginal cost?
Ans: The average cost (AC) is also called average total cost. It is the ratio of the total cost and the number of quantities produced, whereas the marginal cost (MC) is the increased cost of the last quantity (unit) produced. The average cost of quantity produced will be decreased when $$MC < AC$$, and quantity increases when $$MC > AC$$. If $$MC = AC$$ then the average cost will be neither decreasing nor increasing.

Q.2. Explain the relation between average cost and marginal cost with a diagram?
Ans:

From the diagram given above, we can see that:

• The average cost of quantity produced will be decreased when $$MC < AC$$.
• As quantity increases, when $$MC > AC$$.
• If $$MC = AC$$ then the average cost will be neither reducing nor increasing.

Q.3. What is the marginal cost example?
Ans: Marginal cost is the additional cost resulting from an increase in the production of one more unit of the product.
For example, if a company wants to build a new factory to produce more goods to increase its profit, making a building is the marginal cost for the company.

Q.4. What is an example of average cost?
Ans: The total cost of making a single product is known as the average cost. It is calculated by dividing the total cost by the total number of output.
Example: Total cost of producing $$100$$ shirt is $$Rs. 4000$$, then the average cost of producing each shirt is $$\frac {4000}{100} = 40$$ i.e., $$Rs. 40$$.

Q.5. What are total cost, average cost, and marginal cost?
Ans: Total Cost: The total cost of producing $$x$$ number of products is represented by $$C(x)$$. It can be written as-
$$C\left( x \right)\; = \;F\left( x \right)\; + \;V\left( x \right)$$
Where $$F(x)$$ is a fixed cost and $$V(x)$$ is a variable cost.
Average Cost: The average cost is the total cost of making a single product. It is quotient of the total cost and the number of products produced. The two most important components in average cost are fixed cost and variable cost. It is also called Unit cost.
The average cost can be calculated by $$AC = \frac{{Total\;cost\;\left( {F\left( x \right) + V\left( x \right)} \right)}}{{Number\;of\;units\;of\;production}}$$
Marginal Cost: Marginal cost occurs or comes into play when we need to ascertain the change in total cost, i.e., dependent variable with a unit change of product, i.e., independent variable. The additional cost/increase in the dependent variable is the result of an increase in the production/independent variable of one more unit of product.
If $$C(x)$$ is the total cost for producing $$x\,units$$, then change in the total cost of producing one additional unit, at an output level of $$x\,units$$, is given by the formula $$\frac {dC(x)}{dx}$$
Marginal cost $$= \frac {dC(x)}{dx}$$
Marginal cost $$= \frac{{Change\;in\;total\;cost\;\left( {c\left( x \right)} \right)}}{{Change\;in\;number\;of\;production\;\left( x \right)}}$$.

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