Practical approach towards Profit Maximization and Boumol's Sales Revenue maximization Model

# Tamakoshi Electronics Ltd. has following demand and cost functions:

    P = 2000 – 10 Q

    TC = 1000 + 200 Q

It is obvious that the general public is willing to purchase tickets.

So, compute:

a) Profit Maximization Output, Price, TR and Maximum Profit.

b) Sales Maximizing Output, Price, Profit and Maximum TR.

c) Output, Price and TR under profit constraints of Rs. 79,500.

Solution:

Let us see the information given:

We can find that the demand function is represented by P = 2000 – 10 Q. Similarly, cost function is represented by TC = 1000 + 200 Q

Representation of the abbreviations:

P = Price

Q = quantity

TR = Total Revenue

TC = Total Cost
𝝅 = Total Profit

a) For computation of Profit Maximization Output, Price, TR and Maximum Profit.

Applying the first order condition:

We have,

On solving this, we get 1800 – 20 Q = 0

Or, Q = 90 units

Where,

𝝅 = TR – TC

= (P X Q) – TC                    [since TR = P X Q]

= (2000 – 10 Q) x Q – (1000 + 200 Q)         [putting the respective values of P and TC from the above functions]

= 2000 Q – 10 Q ² – 1000 – 200 Q

= 1800 Q – 10 Q² – 1000……………………………………………equation (i)

Again, under second order condition, we have:

The second order condition is negative, so as per maxima approach, the profit function is maximized when 90 units of output is produced.

So, the required information are as follows: 

• Profit Maximization output is 90 units

• Price is Rs 1100 [since, P = 2000 – 10 Q = 2000 – 10 x 90] 

• TR is Rs. 99000 [since, P x Q = 1100 x 90]

• Maximum Profit (𝝅) is Rs. 80000 [ since, 1800 Q – 10 Q² – 1000 = (1800 x 90) – (10 x 90 x 90) – 1000 ]

b.) For computation of Sales Maximizing Output, Price, Profit and Maximum TR.

Maximum TR = P x Q = (2000 – 10 Q) x Q       [since, P = 2000 – 10 Q]

                       

                      = 2000 Q – 10 Q²

Now computing the first order condition, we get

Or, 2000 – 20 Q = 0

Or, Q = 100 units

This is the level of output where sales maximizing could be done.

Testing the second order condition:

The second order condition is negative, so as per maxima approach, the total revenue is maximized when 100 units of output is produced.

So, the required information are as follows:

• Sales Maximizing Output is 100 units of output.

• Price is Rs 1000                 [since, P = 2000 – 10 Q = 2000 – (10 x 100)    ]

• Maximum Profit (𝝅) is Rs. 79000 [since, 1800 Q – 10 Q² – 1000 = 1800 x 100 – 10 x 100 x 100 – 1000]

• Maximum TR is Rs. 100,000          [since, P x Q = 1000 x 100        ]

 c.) For computation of Output, Price and TR under profit constraints of Rs. 79,500.

Here, the profit constraints is given as Rs 79,500. This implies that, the total profit of Tamakoshi Electronics Ltd must be Rs.79, 500 or more.

This means:

 𝝅 = Rs. 79,500

As we have the value of  𝝅 in equation (i)

We get,

𝝅 = 1800 Q – 10 Q² – 1000  

Or, 79,500 = 1800 Q – 10 Q² – 1000    [putting the value of 𝝅 as 79,500]

Or, 10 Q² – 1800 Q +79,500 + 1000 = 0

Or, 10 Q² – 1800 Q + 80,500 = 0

On solving the above equation we get,

Q = 97 units or 83 units

Now, if Q = 97 units, Then 

• P is Rs. 1030 [since, P = 2000 – 10 Q = 2000 – 10 x 97 = 1030]

• TR is Rs. 99910 [since, P x Q = 1030 x 97]

• TC is Rs. 20,400 [since, TC = 1000 + 200 Q = 1000 + (200 x 97)   ]

• 𝝅 = Rs. 79,510 [since, TR – TC = 99910 – 20400]

However, if Q = 83 units, Then

• P is Rs. 1170 [since, P = 2000 – 10 Q = 2000 – 10 x 83 = 1030]

• TR is Rs. 97110 [since, P x Q = 1030 x 83]

• TC is Rs. 17600 [since, TC = 1000 + 200 Q = 1000 + (200 x 83)   ]

• 𝝅 is Rs. 79,510 [since, TR – TC = 99910 – 20400]

Since, the objective of the firm is to maximize sales revenue with constraints of 𝝅 so, the revenue could be maximum at greater or equal to 90 units (profit maximization output). So, under the two probable output units (97 units or 83 units), the sales could be maximized at 97 units.

Therefore, the required information are as follows:

• Output is 97 units

• Price is Rs. 1030

• TR under profit constraints of Rs. 79,500 is Rs. 99910.

Note : 

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