Question

The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is \(800 per month. A market survey suggests that, on average, one additional unit will remain vacant for each \)10 increase in rent. What rent should the manager charge to maximize revenue?

Short answer

The manager should charge $900 rent to maximize the revenue.

Step-by-step solution

Given data

Number of apartments occupied when rent is $800 = 100.

Decrease in apartment occupation when the rent is increase by $10 = 1.

Maximization condition

The condition for maxima of a function is:

\({\frac{{df\left( x \right)}}{{dx}}_{x = {x_{max}}}} = 0\;\;\;\;\;.....\left( 1 \right)\)

Maximum revenue

Let the apartments occupied be x. Decrease in apartment occupation:

\(100 - x\)

Price increase per unit decrease in occupation is:

\(\frac{{\$ 10}}{1} = \$ 10\)

Therefore, the demand function is:

\(\begin{aligned}{c}p\left( x \right) &= 800 + 10\left( {100 - x} \right)\\ &= 800 + 1000 - 10x\\ &= 1800 - 10x\end{aligned}\)

The revenue function is:

\(\begin{aligned}{c}R\left( x \right) &= xp\left( x \right)\\ &= x\left( {1800 - 10x} \right)\\ &= 1800x - 10{x^2}\end{aligned}\)

From equation (1), the revenue is maximum for:

\(\begin{aligned}{c}{\frac{{dR\left( x \right)}}{{dx}}_{x = {x_{\max }}}} &= 0\\1800 - 20{x_{\max }} &= 0\\{x_{\max }} &= 90\end{aligned}\)

The corresponding rent is:

\(\begin{aligned}{c}p\left( {90} \right) &= 1800 - 10 \times 90\\ &= 1800 - 900\\ &= 900\end{aligned}\)

The required rent is $900.