A monopolist faces the following demand curve:

P=50-0.5Q

where Q is the quantity demanded and P is the price.

The monopolist's total cost function is given by:

TC=50+4Q where TC is the total cost and Q is the quantity produced.

What is the profit maximizing quantity?

Question

A monopolist faces the following demand curve:

P=50-0.5Q

where Q is the quantity demanded and P is the price.

The monopolist's total cost function is given by:

TC=50+4Q where TC is the total cost and Q is the quantity produced.

What is the profit maximizing quantity?

BlackTom AI · Accepted Answer

First, I will restate the problem in my own words to ensure understanding: a monopolist faces the demand P = 50 - 0.5Q and has total cost TC = 50 + 4Q. We need to find the quantity Q that maximizes profit.

Step 1: Compute total revenue (TR) as a function of quantity Q.
TR = P × Q = (50 - 0.5Q)Q = 50Q - 0.5Q^2.

Step 2: Write the profit function, π(Q).
π(Q) = TR - TC = (50Q - 0.5Q^2) - (50 + 4Q) = -0.5Q^2 + 46Q - 50.

Step 3: Use marginal analysis to find the profit-maximizing quantity.
- Marginal revenue (MR) is the derivative of TR with respect to Q: MR = d(TR)/dQ = 50 - Q.
- Marginal cost (MC) is the derivative of TC with respect to Q: MC = d(TC)/dQ = 4.
Set MR = MC for profit maximization: 50 - Q = 4.
Solving gives Q = 46.

Step 4: (Optional consistency check) compute price and profit at Q = 46.
P = 50 - 0.5 × 46 = 50 - 23 = 27.
TR = 27 × 46 = 1242.
TC = 50 + 4 × 46 = 50 + 184 = 234.
π = TR - TC = 1242 - 234 = 1008.

Conclusion (from the calculation): the profit-maximizing quantity is 46.

ECON_104B_001_25S Lecture Quiz #5

A monopolist faces the following demand curve: P=50-0.5Q where Q is the quantity demanded and P is the price. The monopolist's total cost function is given by: TC=50+4Q where TC is the total cost and Q is the quantity produced. What is the profit maximizing quantity?

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Figure 16-4Refer to Figure 16-4. How much output will the monopolist produce in order to maximize profit?

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Figure 16-5 The following graph depicts the market situation for a monopoly pastry shop called Bearclaws. Refer to Figure 16-5. Based upon the information shown, how many units will Bearclaws produce to maximize profits?

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ECON_104B_001_25S Lecture Quiz #5

A monopolist faces the following demand curve: P=50-0.5Q where Q is the quantity demanded and P is the price. The monopolist's total cost function is given by: TC=50+4Q where TC is the total cost and Q is the quantity produced. What is the profit maximizing quantity?

View Explanation

Log in for full answers

Similar Questions

Use the model above to answer the following question. The profit maximizing quantity for the monopoly is?

Figure 16-4Refer to Figure 16-4. How much output will the monopolist produce in order to maximize profit?

Figure 16-4Refer to Figure 16-4. How much output will the monopolist produce in order to maximize profit?

Figure 16-4Refer to Figure 16-4. How much output will the monopolist produce in order to maximize profit?

Figure 16-4Refer to Figure 16-4. How much output will the monopolist produce in order to maximize profit?

Figure 16-4Refer to Figure 16-4. How much output will the monopolist produce in order to maximize profit?

Figure 16-5 The following graph depicts the market situation for a monopoly pastry shop called Bearclaws. ​ ​ ​ Refer to Figure 16-5. Based upon the information shown, how many units will Bearclaws produce to maximize profits?

More Practical Tools for Students Powered by AI Study Helper

Making Your Study Simpler

Figure 16-5 The following graph depicts the market situation for a monopoly pastry shop called Bearclaws. Refer to Figure 16-5. Based upon the information shown, how many units will Bearclaws produce to maximize profits?