SOLUTION: Suppose that the demand equation for a monopolist’s product is P=400-2Q and the average cost function is AC=400/Q+4+0.2Q , where Q is the number of units and p is the price per

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Question 1198755: Suppose that the demand equation for a monopolist’s product is P=400-2Q and the average cost function is AC=400/Q+4+0.2Q , where Q is the number of units and p is the price per unit.
Write down the equations for TC and TR.
Calculate the price and quantity at which profit is maximised. Determine the maximum profit.

Answer by CPhill(1959) (Show Source):
You can put this solution on YOUR website!
**1. Find the Total Cost (TC) Equation**
* **Total Cost (TC) = Average Cost (AC) * Quantity (Q)**
* TC = (400/Q + 4 + 0.2Q) * Q
* TC = 400 + 4Q + 0.2Q²
**2. Find the Total Revenue (TR) Equation**
* **Total Revenue (TR) = Price (P) * Quantity (Q)**
* TR = (400 - 2Q) * Q
* TR = 400Q - 2Q²
**3. Find the Profit (π) Equation**
* **Profit (π) = Total Revenue (TR) - Total Cost (TC)**
* π = (400Q - 2Q²) - (400 + 4Q + 0.2Q²)
* π = 400Q - 2Q² - 400 - 4Q - 0.2Q²
* π = 396Q - 2.2Q² - 400
**4. Find the Quantity that Maximizes Profit**
* To find the quantity that maximizes profit, we need to find the derivative of the profit function with respect to Q and set it to zero.
* dπ/dQ = 396 - 4.4Q = 0
* 4.4Q = 396
* Q = 90
**5. Find the Price at Maximum Profit**
* Substitute the optimal quantity (Q = 90) into the demand equation:
* P = 400 - 2 * 90
* P = 400 - 180
* P = $220
**6. Calculate the Maximum Profit**
* Substitute the optimal quantity (Q = 90) into the profit function:
* π = 396 * 90 - 2.2 * 90² - 400
* π = 35640 - 17820 - 400
* π = $17420
**Therefore:**
* **Quantity that maximizes profit:** 90 units
* **Optimal price:** $220
* **Maximum profit:** $17,420