# 5 Problems

1. (a) Using calculus, derive the relationship between a monopolist's marginal revenue, the monopolists' price, and the price elasticity of demand. (b) Consider a monopolist who produces output at a constant marginal and average cost of $12. The price elasticity for the monopolist's product is 3. Use your answer to (a) to find the monopolist's profit maximizing price.

2. Bob's pharmaceutical company has developed a dew drug, Econosil, which helps students relax during economics exams.* A paten was obtained which entitles Bob to be the sold producer of Econosil. Bob's cost function for producing this new drug is given by C(Q) = 20Q, where Q is the number of capsules produced (in thousands). After extensive market research, Bob has determined the demand for his product is very high, and is given by the demand function P(Q) = 100 - Q. *Side effects may include nausea, hallucinations of enjoying economics, or premature death. (a) Find Bob's marginal revenue and marginal cost functions and plot them on the same graph, along with the demand function. (b) Assuming Bob is a profit-maximizing monopolist, how many capsules will Bob produce? What price will he charge? Indicate the profit-maximizing quantity (and price) on your graph. (c) Calculate Bob's profit as this level of output. Show your work. (d) If Econosil were instead a generic drug produced in perfectly competitive market, how many capsules would be produced, and what price would they be sold (assume the same cost and demand functions apply) (e) Finally, calculate the deadweight loss associated with monopoly production of Econosil and indicate this area on your graph. Hint: the area of a triangle is (1/2)base*height.

3. Use the demand and cost functions from Question 1 to answer the following questions, but here assume Bob practice first-degree price discrimination. (a) In this case, how many capsules would Bob produce? Show the profit on the diagram. Again, calculate the profit he would earn. (b) Of these two cases (standard monopoly pricing and fist degree price discrimination), which would result in the highest total surplus? Which would result in the highest consumer surplus? Explain.

4. Acme Widgets is a monopolist in the widget market. The widget market consists of 75 A's and 25 B's. The A's are willing to pay $90 for a high quality widget; the B's are willing to pay $200 for a high quality widget. Each consumer will either buy 1 high quality widget or no widgets. Acme can produce a high quality widget at zero cost. (a) Suppose initially Acme cannot engage in any form of price discrimination. Find Acme's profit maximizing price. How much profit will Acme earn? (b) Now suppose Acme can identify who is an A and who is a B and is able to engage in third degree price discrimination. How much profit will Acme earn?

5. College Park Airlines sells a single daily flight from College Park to Long Island to two types of passengers-business people, who have daily demand of QA = 260 - 0.4PA and University of Maryland students, who have daily demand of QB = 260 - 0.6 PB. The airline can easily distinguish students from business people, so they decide to charge them different prices (they also have to show a student ID to get the special rate-business passengers cannot pose as students). The cost of running each flight is $30,000 plus $100 per passenger. Answer the following questions, and show your work. (a) Find the marginal revenue function for each of College Park Airline's submarkets. (b) What price will the airline charge the students? What price will they charge business people? (c) How many of each type of customer will take the flight? (d) Under this pricing strategy, what will be College Park Airline's per-flight profit? (e) If third degree price discrimination is not allowed and only one price can be charged, what price will the airline charge? How many customers will take the flight altogether? What will be the College Park Airline's per-flight profit?

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Different Problems

Week 4/2

Write three quadratic equations, with a, b, and c (coefficients of x2, x, and the constant) as:

1. Integers

2. Rational numbers

3. Irrational numbers

Week 4/ 3

1. How many solutions exist for a quadratic equation? How do we determine algebraically whether the solutions are real or complex?

2. What three techniques can be used to solve a quadratic equation? Demonstrate these techniques on the equation "x2 - 10x - 39 = 0".

3.

Look at the graph above and comment on the sign of D or the discriminant. Form the quadratic equation based on the information provided and find its solution.

4. Translate the following into a quadratic equation, and solve it: The length of a rectangular garden is four times its width; if the area of the garden is 196 square meters, what are its dimensions?

Week 5/1

1. Do exponential functions only model phenomena that grow, or can they also model phenomena that decay? Explain what is different in the form of the function in each case.

2. True or false: The function "f(x) = 4x" grows four times faster than the function "g(x) = x". Explain.

3. What are the asymptotes of the functions "f(x) = 4x" and "g(x) = log5x"?

4. A cell divides into two identical copies every 4 minutes. How many cells will exist after 5 hours?

5. The level of thorium in a sample decreases by a factor of one-half every 2 million years. A meteorite is discovered to have only 8.6% of its original thorium remaining. How old is the meteorite?

Week 5/3

Refer to the graph given below and identify the graph that represents the corresponding function. Justify your answer.

y = 4x

y = log4x

Plot the graphs of the following functions. Scan the graphs and post them to the Facilitator along with your response.

1. f(x) = 5x

2. f(x) = 4x+2

3. f(x) = (1/3)x

4. f(x) = log5x

Week 6/1

1. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph of both the functions and post to the discussion forum. Discuss these functions and their graphs with your classmates.

2. Form each of the following:

? A linear equation in one variable

? A linear equation in two variables

? A quadratic equation

? A polynomial of three terms

? An exponential function

? A logarithmic function

3. Plot the graph of the above equations formed in question 2, and post your response to the discussion forum.

4. Derive the quadratic and linear equations from the corresponding graphs of a classmate.

Week 6/2

Answer the following question:

For the exponential function ex and logarithmic function log x, graphically show the effect if x is halved. Include a table of values for all four functions.

Week 6/3

Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Rank in order from fastest to slowest the rate at which each function changes with increasing values of x.

1. f(x) = 2x - 5

2. f(x) = x2 - 4x + 3

3. f(x) = x3 + 4x2 - 2x - 3

4. f(x) = 7x

5. f(x) = log x