What is Cost Function?

A cost function is a mathematical formula used to calculate the total cost of production for a given quantity of output. It represents the relationship between the cost of production and the level of output, incorporating various factors such as fixed costs, variable costs, and total costs.

What is Cost Function?

Cost Function refers to the relationship between input costs and output. In simple terms, the functional relationship between cost and output is referred to as the cost function. It is written as:

C = f(q)

Where, 

C = Cost of Production; 
q = Quantity of Production; and
f = Functional Relationship

Geeky Takeaways:

• Cost Function is a fundamental concept of microeconomics that is used to analyze the relationship between the production of goods and services and the cost incurred on their production.
• Three different types of costs are incurred during the production of a good or service. These are Total Cost, Average Cost, and Marginal Cost.
• Firms use cost functions to make decisions regarding production including output level, pricing, and resource allocation.
• The aim of using the cost function is to minimize costs and maximize the profits of the firm.

Types of Cost

There are different types of costs, including Average Costs, Marginal Costs, Fixed Costs, and Variable Costs.

A. Total Costs

In the short run, some of the factors are fixed, while other factors are variable. In the same way, the short-run costs are also categorized into two different kinds: Fixed and Variable Costs. The sum total of these costs is equal to the Total Cost.

1. Total Fixed Cost (TFC) or Fixed Cost (FC)

The costs on which the output level does not have a direct impact are known as Fixed Costs. For example, salary of staff, rent on office premises, interest on loans, etc. Other names of fixed costs are Supplementary Cost, Overhead Cost, Unavoidable Cost, Indirect Cost, or General Cost. 

2. Total Variable Cost (TVC) or Variable Cost (VC)

The costs on which the output level has a direct impact are known as Variable Costs. For example, fuel, power, payment for raw materials, etc. Other names of variable costs are Prime Cost, Avoidable Cost, or Direct Cost. 

3. Total Cost (TC)

The total expenditure incurred by an organisation on the factors of production which are required for the production of a commodity is known as Total Cost. In simple terms, total cost is the sum of total fixed cost and total variable cost at different output levels.

TC = TFC + TVC

B. Average Costs

Average Costs are the per unit costs which explain the relationship between the cost and output in a realistic manner. These per-unit costs are obtained from Total Fixed Cost, Total Variable Cost, and Total Cost. The three different types of per-unit costs are as follows:

1. Average Fixed Cost (AFC)

The per unit fixed cost of production is known as Average Fixed Cost. The formula for calculating Average Fixed Cost is:

Average~Fixed~Cost~(AFC)=\frac{Total~Fixed~Cost~(TFC)}{Quantity~of~Output~(Q)}     

2. Average Variable Cost (AVC)

The per unit variable cost of production is known as Average Variable Cost. The formula for calculating Average Variable Cost is:

Average~Variable~Cost~(AVC)=\frac{Total~Variable~Cost~(TVC)}{Quantity~of~Output~(Q)} 

3. Average Total Cost (ATC) or Average Cost (AC)

The per unit total cost of production is known as Average Total Cost or Average Cost. The formula for calculating Average Total Cost is:

Average~Cost~(AC)=\frac{Total~Cost~(TC)}{Quantity~of~Output~(Q)}     

Another way to define Average Total Cost is by the sum of Average Fixed Cost and Average Variable Cost; i.e., AC = AFC + AVC. 

C. Marginal Cost

The additional cost incurred to the total cost when one more unit of output is produced is known as Marginal Cost. For example, if the total cost of producing 2 units is ₹400 and the total cost of producing 3 units is ₹600, then the marginal cost will be 600 - 400 = ₹200.

MC_n = TC_n - TC_{n-1}

{Where, n = Number of units produced; MC_n = Marginal cost of the n^th unit; TC_n = Total cost of n units; TC_n-1 = Total cost of (n-1) units}

Another way to calculate Marginal Cost:

When the change in the units produced is more than one unit, then the previous formula of calculating MC will not work. In that case, the formula for calculating Marginal Cost will be:

MC=\frac{Change~in~Total~Cost}{Change~in~units~of~Output}=\frac{\Delta{TC}}{\Delta{Q}}     

For example, if the total cost of producing 4 units is ₹300 and the total cost of producing 2 units is ₹50, then the marginal cost will be:

MC=\frac{300-50}{4-2}=\frac{250}{2} 

Marginal Cost = ₹125

Importance of Cost Function in Business

The cost function is crucial for businesses as it helps in:

• Determining the cost-efficiency of production processes.
• Pricing products appropriately to ensure profitability.
• Making informed decisions on production levels.
• Identifying cost-saving opportunities.

Examples of Short-Run Costs

Example 1: 

From the table below, calculate the missing figures.

Solution:

At all output levels, TFC remains constant at ₹15.

Example 2: 

From the table below, calculate the missing number.

Solution:

Working Notes: 

1. In this example, fixed cost is assumed to be zero.

2. *TC = AC x Q

Example 3: 

From the table given below, calculate the weekly TC and AVC.

Solution:

TC = TVC + TFC

TVC = Raw materials used + Power + (Number of workers employed x Weekly wage)

TVC = 1,800 + 350 + (70 x 300)

TVC = ₹ 23,150

AVC = TVC/Units Produced per week

AVC = 23,150/200

AVC = ₹ 115.75

Applications of Cost Function

Break-Even Analysis

The cost function is essential in break-even analysis to determine the production level at which total revenue equals total cost.

Marginal Cost

Marginal cost, the cost of producing one additional unit, can be derived from the cost function: MC(Q) = \frac{dC(Q)}{dQ}

Conclusion

The cost function is a fundamental concept in both mathematics and commerce, providing essential insights into the relationship between production costs and output levels. By understanding and utilizing cost functions, businesses can optimize their production processes, set competitive prices, and enhance profitability.