Question

Read the following and answer this question based on the same:

The demand for a product (Q) is related to the price (P) of the product as follows: \(Q=100-2P\).

The cost (C) of manufacturing the product is related to the quantity produced in the following manner: \(C=Q^2-16Q+2000\).

As of now the corporate profit tax rate is zero. But the Government of India is thinking of imposing 25% tax on the profit of the company.

If the government imposes the 25% corporate profit tax, then what will be the profit maximizing output?

• A. 16.5
• B. 16.125
• C. 15
• D. None of the above

Hint:

A flat percentage tax on profit scales the whole profit curve by a constant but never shifts its peak, so check whether the pre-tax optimum (22) appears among the options.

Answer 1

The Correct Option is D

Step 1: Recall the profit function before tax.
As found from the same demand and cost equations, the pre-tax profit is
\[ \pi(Q)=66Q-\frac{3}{2}Q^2-2000 \]
and this was maximized at \(Q=22\).

Step 2: Write the after-tax profit.
A 25% tax on profit means the company keeps 75% of whatever profit it makes.
After-tax profit is:
\[ \pi_{\text{after}}(Q)=0.75\times\pi(Q)=0.75\left(66Q-\frac{3}{2}Q^2-2000\right) \]

Step 3: Use the key idea about scaling by a constant.
Multiplying a function by a fixed positive number, here 0.75, stretches or squashes its graph vertically but does not move it left or right.
So the value of Q where \(\pi_{\text{after}}(Q)\) is largest is exactly the same Q where \(\pi(Q)\) is largest.
This can be confirmed by differentiating directly:
\[ \frac{d}{dQ}\pi_{\text{after}}(Q)=0.75\times\frac{d\pi}{dQ}=0.75\times(66-3Q) \]
Setting this to zero gives \(66-3Q=0\), the exact same equation as before, so \(Q=22\) again.

Step 4: Conclude what the tax actually does.
A flat percentage tax on profit reduces how much profit the firm keeps, but it does not change which output level gives the largest profit.
So the profit-maximizing output after the 25% tax is still 22 units, exactly the same as before the tax.

Step 5: Compare with the given options.
The listed numbers are 16.5, 16.125 and 15, and none of these equals 22.
Since the true profit-maximizing output (22) is not one of the specific numbers offered, none of options (A), (B) or (C) can be correct.

Final Answer:
The profit-maximizing output does not change after the tax. It stays at 22, which is not listed, so the answer is "None of the above".
\[ \boxed{\text{None of the above (still 22 units)}} \]