Maximisation of Total Revenue and Profit: We use differentiation in different subjects other than Mathematics. In Economics, differentiation is a tool used for marginal cost and marginal revenue, i.e. for calculating change in demand for a particular product and its price. We also use differentiation to estimate the rate of change of revenue with an increase or decrease in selling price. 

Every business has a vision of maximising the profit of shareholders. The primary role is handled by the revenue as profit is the difference between total revenue and total cost to achieve this objective. Increasing the revenue leads to profit maximisation of the company and ultimately increases the profit of shareholders. Nowadays, it is tough for an organisation to increase its revenue, as it is a customer-driven market. If a customer is cent percent satisfied, the product/service will get attention and boost the company’s revenue.

Maximisation of Total Revenue and Profit

Revenue is the total income obtained by the sale of goods and services related to the primary operations of the business. Commercial revenue may also be referred to as sales or turnover. Some companies receive revenue from interest, royalties, and other fees too.

Definition of Total Revenue (TR)

The total revenue is the total sales proceeds in the market. A firm sells different product quantities to its customers at the prevailing market price. So, the total revenue can be calculated by multiplying price by quantity.

\(T R=P \times Q\)

where

\(TR=\) Total revenue

\(P=\) Price

\(Q=\) Quantity

Average Revenue (AR)

Average revenue is defined as the ratio of total revenue to the quantity of the product.

\(A R=\frac{T R}{Q}\)

where

\(AR=\) Average revenue

\(TR=\) Total revenue

\(Q=\) Quantity

On substituting \(TR=P \times Q\) we get

\(A R=\frac{P \times Q}{Q}\)

\(\therefore A R=P\)

We can say that the average revenue is the same as the price of a product.

In other words, the average revenue is the revenue obtained by the firm on the sale of per unit product.

Marginal Revenue (MR)

The marginal revenue is the increase in the total revenue due to sales of an extra unit of the commodity by the firm in the market.

In simpler words, marginal revenue is the addition to the total revenue form selling an additional unit of goods.

\(M R=\frac{\Delta T R}{\Delta Q}\)

where

\(\Delta T R \rightarrow\) change in total revenue

\(\Delta Q \rightarrow\) change in quantity

Relationship Between Average, Marginal and Total Revenue

Let us see the table first in which the price of products and their quantities are given, and the corresponding \(TR, MR\) and \(AR\) are calculated.

Price(P)	Quantity(Q)	\(\mathbf{T R}[P \times Q]\)	\(\mathbf{MR}\left[\frac{\Delta T R}{\Delta Q}\right]\)	\(\mathbf{AR}\left[\frac{T R}{Q}\right]\)
\(10\)	\(0\)	\(0\)	\(0\)	\(0\)
\(10\)	\(1\)	\(10\)	\(10\)	\(10\)
\(10\)	\(2\)	\(20\)	\(10\)	\(10\)
\(10\)	\(3\)	\(30\)	\(10\)	\(10\)
\(10\)	\(4\)	\(40\)	\(10\)	\(10\)

Note that:

1. Since the price of goods or \(AR\) is given under the perfect competition and is constant. Therefore, \(AR\) and \(MR\) are always equal.
   \(AR=MR\)
2. Among \(MR\) and \(TR, MR\) is known as the rate of change in the total revenue. The value of \(M R\) for any quantity given by the value of \(TR\) at which rate it has increased from its previous unit.

What is Profit?

Profit of any company is the difference of the total revenue \((TR)\) and the total cost \((TC)\).

Profit\(=\)Total Revenue\(-\)Total Cost

Types of Profit

There are two types of profit often defined by the economists:

1. Supernormal Profit
   Supernormal profit is defined as the surplus of revenue over the total cost. This means total revenue is more significant than total cost. If the difference between total revenue and the total cost is positive or greater than zero, the firm earns a supernormal profit.
2. Normal Profit
   If the total revenue equals the total cost, then the difference between \(TR\) and \(TC\) becomes zero. Such a situation is known as normal profit or no profit no loss.

\(T R-T C=0\)
\(T R=T C\)
\(\frac{T R}{Q}=\frac{T C}{Q}\)
\(\Rightarrow A R=A C\)

Example:

If a firm A sells \(10\) units of goods for \(R s .10\) each. Its total cost of production is \(Rs .90\). Does supernormal profit exist, and how much?

Sol. Given: Firm sells \(10\) units of a goods for \(Rs .10\)

Therefore, total revenue \(=TR=10 \times 10=100\)

Total production cost, \(TC=90\)

Now, \(TR-TC=100-90\)

\(=R s .10\)

Since \(10>0\) i.e. \(TR-TC\) is positive, there is a supernormal profit, and it is equal to \(Rs. 10\).

Loss

When total cost exceeds a firm’s total revenue, it is said to have incurred a loss.

\(TC>TR\) incurs loss for the firm.

For a unit level, the loss implies average revenue \((AR)\) or price \((P)\) is less than average \(\operatorname{cost}(AC)\).

\(A R<A C\)

In the situation of loss the firm is unable to recover its cost of production after selling the product.

TR and TC Approach to Maximisation Profit

We know that the profit is nothing but the difference between total revenue and the total cost. Profit maximisation through this approach states that the firm should produce that quantity of output such that the difference between the total revenue \((TR)\) and the total cost \((TC)\) should be maximum.

i.e., \(TR-TC\) is maximum

Revenue vs Profit Maximisation

Following is the comparison between Revenue and Profit Maximisation.

Revenue Maximisation: It is when a business keeps selling its products until the marginal revenue does not fall negative.

Profit Maximisation: A point at which its marginal cost is less than the marginal revenue.

Revenue Maximisation	Profit Maximisation
Quantity sold to a point at which \(MR=0\), where \(MR\) is marginal revenue.	Quantity sold to a point at which \(MR=MC\), where \(MR\) is marginal revenue and \(MC\) is the marginal cost
The main aim is to increase the customer base and capture a high market share.	Aim to increase the profitability of any firm or business.
It is suitable for a fresher in the market or for expanding an existing business into a new product line.	It is suitable for an existing business that has maintained its reputation in the market and has a well-established client base and has maintained its quality without cutting the price down.
It is a long term objective.	It is a short term objective.
The businesses having this type of strategy can bring any new opportunity in the market and take up their maximum advantage.	Business becomes rigid and loses a few customers in the non-flexibility of cutting down prices.

Solved Examples on Maximisation of Total Revenue and Profit

Below are a few solved examples that can help in getting a better idea:

Q.1. Given the total cost function \(C=5 q+\frac{q^{2}}{50}\) and the demand function \(q=400-20 p\). Then find the total revenue function and maximise the total revenue function.

Sol: Total Revenue, \(T R=p \cdot q\)

Let us first rearrange the demand function as

\(p=20-\frac{q}{20}\)

Then, \(T R=\left(20-\frac{q}{20}\right) \cdot q\)

\(\Rightarrow T R=\left(20 q-\frac{q^{2}}{20}\right)\)

To maximise the total revenue function, find the critical value(s) by setting the first derivative of the TR function equal to zero.

\(\frac{d(T R)}{d q}=20-\frac{q}{10}=0\)

\(\Rightarrow q=200\)

At critical value, total revenue function is maximised provided the sufficient condition is satisfied

That is \(\frac{d^{2}(T R)}{d q^{2}}<0\)

\(\frac{d^{2}(T R)}{d q^{2}}=-\frac{1}{10}<0\)

Thus, total revenue is maximised at \(q=200\), and the maximum total revenue is

\(T R=\left(20 q-\frac{q^{2}}{20}\right)\)

\(=\left(20(200)-\frac{200^{2}}{20}\right)\)

\(\therefore T R=2000\)

Q.2. Given the total cost function \(T C=5 q+\frac{q^{2}}{50}\) and the demand function \(q=400-20 p\). Then maximise profit function.

Sol: We know that Total Revenue \((T R)=p.q\)

Let us first rearrange the demand function as

\(p=20-\frac{q}{20}\)

Then, \(T R=\left(20-\frac{q}{20}\right) \cdot q\)

\(\Rightarrow T R=\left(20 q-\frac{q^{2}}{20}\right)\)

The profit function is \(\pi=T R-T C\)

\(\pi=\left(20 q-\frac{q^{2}}{20}\right)-5 q-\frac{q^{2}}{50}\)

\(\therefore \pi=15 q-\frac{7 q^{2}}{100}\)

The first-order condition for profit maximisation is

\(\frac{d \pi}{d q}=\frac{d T R}{d q}-\frac{d T C}{d q}=0\)

\(\frac{d \pi}{d q}=\left(20-\frac{q}{10}\right)-\left(5+\frac{q}{25}\right)=0\)
\(\Rightarrow q \approx 107\)

If profit is maximised at \(q=107\), we need to ensure that the sufficient condition is satisfied

That is \(\frac{d^{2} \pi}{d q^{2}}<0\)

\(\frac{d^{2} \pi}{d q^{2}}=-\frac{1}{10}-\frac{1}{25}\)

\(\frac{d^{2} \pi}{d q^{2}}=-\frac{7}{50}<0\)

Thus, profit is maximised at \(q=107\), and the maximum profit is \(\pi=803\).

Q.3. Given the total cost function \(C=15 q+\frac{q^{2}}{20}\) and the demand function \(q=600-30 p\).
Find the total revenue function and maximise the total revenue function.

Sol: Total revenue, \(T R=p . q\)

Let us first rearrange the demand function as

\(p=20-\frac{q}{30}\)

Then, \(T R=\left(20-\frac{q}{30}\right) \cdot q\)

\(\Rightarrow T R=\left(20 q-\frac{q^{2}}{30}\right)\)

To maximise the total revenue function, find the critical value(s) by setting the first derivative of the TR function equal to zero.

\(\frac{d(T R)}{d q}=20-\frac{q}{15}=0\)

\(\Rightarrow q=300\)

At critical value, total revenue function is maximised provided the sufficient condition is satisfied

That is \(\frac{d^{2}(T R)}{d q^{2}}<0\)

\(\frac{d^{2}(T R)}{d q^{2}}=-\frac{1}{15}<0\)

Thus, total revenue is maximised at \(q=300\) and the maximum total revenue is

\(T R=\left(20 q-\frac{q^{2}}{30}\right)\)

\(=\left(20(300)-\frac{300^{2}}{30}\right)\)

\(\therefore T R=3000\)

Q.4. Given the total cost function \(TC=15 q+\frac{q^{2}}{20}\) and the demand function \(q=600-30 p\). Maximise profit function.

Sol: We know that Total Revenue \((TR)=p \cdot q\)

Let us first rearrange the demand function as

\(p=20-\frac{q}{30}\)

Then, \(T R=\left(20-\frac{q}{30}\right) \cdot q\)

\(\Rightarrow T R=\left(20 q-\frac{q^{2}}{30}\right)\)

The profit function is \(\pi=T R-T C\)

\(\pi=\left(20 q-\frac{q^{2}}{30}\right)-15 q-\frac{q^{2}}{20}\)

\(=5 q-\frac{q^{2}}{12}\)

The first-order condition for profit maximisation is

\(\frac{d \pi}{d q}=\frac{d T R}{d q}-\frac{d T C}{d q}=0\)

\(\frac{d \pi}{d q}=\left(20-\frac{q}{15}\right)-\left(15+\frac{q}{10}\right)=0\)

\(\Rightarrow q \approx 30\)

Whether profit is maximised at \(q=30\), we need to ensure that the sufficient condition is satisfied

That is \(\frac{d^{2} \pi}{d q^{2}}<0\)

\(\frac{d^{2} \pi}{d q^{2}}=-\frac{1}{15}-\frac{1}{10}=-\frac{1}{6}<0\)

Thus, profit is maximised at \(q=30\), and the maximum profit is \(\pi=75\).

Q.5. Given the total cost function \(T C=25 q+\frac{q^{2}}{10}\) and the total revenue function is \(T R=\left(40 q-\frac{q^{2}}{10}\right)\). Then maximise profit function.

Sol: The profit function is \(\pi=T R-T C\)

\(\pi=\left(40 q-\frac{q^{2}}{10}\right)-\left(25 q+\frac{q^{2}}{10}\right)\)

The first-order condition for profit maximisation is

\(\frac{d \pi}{d q}=\frac{d T R}{d q}-\frac{d T C}{d q}=0\)

\(\frac{d \pi}{d q}=\left(40-\frac{q}{5}\right)-\left(25+\frac{q}{5}\right)=0\)

\(\Rightarrow q=\frac{75}{2}\) approximately

If profit is maximised at \(q=\frac{75}{2}\), we need to ensure that the sufficient condition is satisfied

That is \(\frac{d^{2} \pi}{d q^{2}}<0\)

\(\frac{d^{2} \pi}{d q^{2}}=-\frac{1}{5}-\frac{1}{5}=-\frac{2}{5}<0\)

Thus, profit is maximised at \(q=\frac{75}{2}\), and the maximum profit is \(\pi=\frac{1125}{4}\).

Summary

Revenue is the total income obtained by the sale of goods and services related to the primary operations of the business. The total revenue is defined as the total sales proceeds in the market. It is calculated by multiplying price by quantity.

The average revenue is the ratio of total revenue to the quantity of the product. The marginal revenue is defined as the increase in the total revenue from selling an extra unit of the commodity by the firm. AR and MR are always equal.

Among MR and TR, MR is known as the rate of change of the total revenue(TR). The profit of any company is defined as the difference of total revenue (TR) over the total cost (TC).

Frequently Asked Questions (FAQs)

Students might be having many questions regarding the Maximisation of Total Revenue and Profit. Here are a few commonly asked questions and answers.

Q.2. What is the formula for profit maximisation?
Ans: Profit maximisation states that the firm should produce that quantity of output such that the difference between the total revenue \(TR\) and the total cost \(TC\) should be maximum.
\(TR-TC\) is maximum

Q.3. What is the difference between revenue and profit maximisation?
Ans: The difference between them is listed below in the table:

Revenue Maximisation	Profit Maximisation
Quantity sold to a point at which \(MR=0\), where \(MR\) is marginal revenue.	Quantity sold to a point at which \(MR=MC\), where \(MR\) is marginal revenue and \(MC\) is the marginal cost
The main aim is to increase the customer base and capture a high market share.	Aim to increase the profitability of any firm or business.
It is suitable for a fresher in the market or for expanding an existing business into a new product line.	It is suitable for an existing business that has maintained its reputation in the market and has a well-established client base and has maintained its quality without cutting the price down.
It is a long term objective.	It is a short term objective.
The businesses having this type of strategy can bring any new opportunity in the market and take up their maximum advantage.	Business becomes rigid and loses a few customers in the non-flexibility of cutting down prices.

Q.4. Why is profit maximisation good?
Ans:

1. To increase the wealth of shareholders.
2. To pay more wages to the employees if some parts of their wages are linked with the profits of the business.
3. To increase the capital investment and spend on research and development.

Q.5. What is total revenue in business?
Ans: The total revenue is the total sales proceeds in the market. The firm sells different product quantities to its customers at the prevailing market price. So the total revenue can be calculated by multiplying price by quantity.
\(T R=P \times Q\)
where
\(T R=\) Total revenue
\(P=\) Price
\(Q=\) Quantity

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