Question

For a​ monopolist's product, the demand function is p equals 75 minus 0.05 q and the cost function is c equals 700 plus 25 q​, where q is number of​ units, and both p and c are expressed in dollars per unit. At what level of output will profit be​ maximized? At what price does this​ occur, and what is the​ profit?

Answer 1

Answer:

The correct answer is at 500 units offering at $50, the monopolist will have a profit equal to $11,800.

Step-by-step explanation:

For a​ monopolist's product, where q is number of​ units, and both p and c are expressed in dollars per unit,

Demand function is p = 75 - 0.05 × q

Revenue function is R = Demand function × Quantity = 75 × q - 0.05 × $q^{2}$

Cost function is c = 700 + 25 × q​

Profit function is given by π = R - c = 75 × q - 0.05 × $q^{2}$ - 700 - 25 × q​

⇒ π = -0.05 ×$q^{2}$ + 50 × q - 700

Now for profit to be​ maximum ⇒ $\frac{d}{dq}$π = 0

⇒ -0.1 ×q + 50 = 0

⇒ 0.1 × q = 50

⇒ q = 500.

We can easily see that the second order derivative is negative hence telling us that the profit is maximum.

There at 500 units of the product the monopolist will have his maximum output.

Price at which the profit occurs = 75 - 0.05 × 500 = $50

Profit = -0.05 × 500 × 500 + 50 × 500 - 700 = -12500 + 25000 -700 = $11800