# Mathematical Economics-Price and output determination

1. Assume you are the manager of a firm that holds a patent that makes it the exclusive manufacturer of a unique SD card. Based on the estimates provided by a consultant, you know that the relevant demand and cost functions for this SD card are Q = 120 - 2P; C = 20 + 2.5Q2.

What are the levels of quantity and price when you are maximizing profits? Show your work.

What are the levels of quantity and price when you are maximizing profits? Show your work.

2. Continuing from the previous question where you are a manager of a firm that holds a patent that makes it the exclusive manufacturer of a unique SD card with demand and cost functions for this SD card that are Q = 120 - 2P; C = 20 + 2.5Q2.

Calculate the deadweight loss caused by the firm in this market. (Give the number only with no $, you will show your work on the next page.)

3. Continuing from the previous question where you are a manager of a firm that holds a patent that makes it the exclusive manufacturer of a unique SD card with demand and cost functions for this SD card that are Q = 120 - 2P; C = 20 + 2.5Q2.

Calculate the deadweight loss caused by the firm in this market.

(On the previous page you input the numerical answer. Show your work here the best you can. No need to draw a graph, just give the key equations.)

4. Continuing from the previous questions where you are a manager of a firm that holds a patent that makes it the exclusive manufacturer of a unique SD card with demand and cost functions for this SD card that are Q = 120 - 2P; C = 20 + 2.5Q2.

Now assume that the firm incurs a one time cost of $10 for a zoning violation. What is the price that the manager should now charge to maximize profits? (Just give the number, no need to include "$")

5. Assume a firm selling wine faces the following inverse demand curve: P = 30 − 2Q. The total cost function is TC= 10 + 1.5Q2. A local regulation states that wine must be sold for at least $20 per bottle. The firm plans to start a wine club and utilize 2-part pricing. Calculate the membership fee the firm will charge in order to maximize profits and comply with the regulation.

(Give only the numerical answer with no dollar sign. You will show your work on the next page.)

6. Assume a firm selling wine faces the following inverse demand curve: P = 30 − 2Q. The total cost function is TC= 10 + 1.5Q2. A local regulation states that wine must be sold for at least $20 per bottle. The firm plans to start a wine club and utilize 2-part pricing. Calculate the membership fee the firm will charge in order to maximize profits and comply with the regulation.

(You already input in answer on the previous page. Show your work here the best you can. There is no need to draw a graph, just give your logic and key equations.)

7. In the previous question the local regulation dictated that wine had to be sold for $20 per bottle. Explain how much surplus a typical consumer lost because of the regulation as compared to the situation where price is not regulated.

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#### Solution Preview

Answer:

Given that,

Demand function:

Q=120-2P

Or,

P=(120-Q)/2

Cost function:

C=20+2.5Q^2

Now,

Revenue=P*Q

Or,

Revenue=(120-Q)/2*Q=60Q-Q^2/2

Therefore,

MR=d(60Q-Q^2/2)/dQ=60-Q

Similarly,

MC=dC/dQ=d(20+2.5Q^2 )/dQ

Or,

MC=5Q

For Profit maximization,

MR=MC

Or,

60-Q=5Q

Or,

Q=10

Putting ...

#### Solution Summary

This answer helps to determine equilibrium price and output in different market structure.