HW3 (1)

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin Assignment Overview: This assignment is designed to enhance your understanding of constrained optimization, using a graphical approach or a calculus-based approach. Required Research: Microeconomics , Chapter 3; Perloff Class Notes/Slides Assumptions and Agreements: 1. See Canvas/Assignments for the due date and time. 2. You will enter your multiple-choice answers in Canvas under Assignments/ Homework 3 MC. Long-form answers will also be submitted in Canvas, but under Assignments/ Homework 3 LF. No hard copies will be accepted. The assignment will be graded automatically in Canvas. Please plan accordingly and submit your assignment early. If you experience severe technological difficulties using Canvas, you may e-mail me your solutions in a PDF file BEFORE the due date/time but try to work the problem out with Canvas first. See syllabus for rules regarding late submissions. 3. Note the directions for long-form submissions in Section 4 below. 4. You may re-submit your assignment as many times as you would like. Canvas will grade your last submission. 5. Individuals can discuss the assignment with any other member(s) of the class. 6. For all problems, assume that it is possible to purchase fractional quantities of the goods. 7. For all problems, assume only non-negative quantities of the goods can be purchased. 8. Funny-looking answers don’t necessarily mean you did something wrong. In some cases, the answer is completely right. In other cases (for instance, if you find the optimal bundle has negative units of one of the goods), it might mean an interior solution does not exist. 9. Unless explicitly stated, assume preferences are rational, monotonic, continuous, and strictly convex.

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin Section 1: Budget Lines (2 points each) For questions 1-4, consider the market for coffee and donuts. Coffee costs $3, while a donut costs $2. Assume you have $18 to spend on coffee and donuts. 1. (coffee/donuts) Which of the following accurately portrays your budget line?

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin 4. (coffee/donuts) Now assume the particular brand of coffee you enjoy implements a new pricing policy. In particular, if you want to purchase any of their coffee you must pay $3 to become a “Premiere Delectable Coffee Member”. Once you pay the $3, you can purchase as much coffee as you would like for $2 per cup. If you do not purchase the membership, you cannot purchase their coffee. Donuts still cost $2 and you still have $18. Which of the following accurately portrays your budget line?

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin 5. Oftentimes, we reduce a complex economy to a two-good economy by specifying one particular good and combining all other goods into a “numeraire” good that can be interpreted as “everything else” on which the individual spends money. Let us consider the market for Netflix streaming-movies and “everything else”. Netflix costs a fixed $8 per month after which an individual can stream as many movies as she would like for free. Assume a unit of “everything else” costs $4 and an individual has $24 to spend on Netflix movies and everything else. Which of the following accurately portrays her budget line if she signs up for Netflix?

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin 7. You turn on the TV and a very excitable fellow yells “YOU SHOULD BUY A SHAMWOW! THEY ARE ONLY $6 EACH!! YOU SHOULD ALSO BUY A SLAP CHOP! THEY ARE ONLY $2 EACH! AND IF YOU CALL IN THE NEXT 30 MINUTES, FOR EVERY SHAMWOW YOU BUY, YOU’LL GET A SLAP CHOP FOR FREE!” Let’s assume you have $18 to spend on ShamWows and Slap Chops. Let’s also assume you call in the next 30 minutes. Finally, let’s assume that it is impossible to resell Slap Chops (after all, who would want to buy a used Slap Chop?). Which of the following accurately portrays your budget line?

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin 8. Which of the following scenarios could generate the following budget line? A. Sweatshirts cost $30, Dinners cost $30, and you have an income of $300. B. Dinners cost $5, Sweatshirts cost $10, you have an income of $25, and you get a free dinner (that you can resell) with every sweatshirt that you buy. C. Sweatshirts cost $10, Dinners cost $10, you have an income of $40, and you’re given a tax rebate of $20. D. All of the above. E. None of the above.

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin 11. By some strange phenomenon, you wake up from a nap and see the exact same Slap Chop/ShamWow commercial on TV from Question 7. The pricing is identical, as is your income. You also still cannot resell your Slap Chops. Being a self-reflective ND student, you ponder your preferences for these two goods. Surprisingly, you realize that you view these two goods as perfect complements. Specifically, you realize that you need ShamWows and Slap Chops in a 1:1 ratio. In other words, you need 1 ShamWow for every 1 Slap Chop to get any enjoyment from either. After all, how can you enjoy a Slap Chop if you have nothing to clean up your mess? And of course, how can you enjoy a ShamWow without anything that makes a mess? What is your optimal constrained bundle? A. 1 Slap Chop, 4 ShamWows B. 2 Slap Chops, 3.5 ShamWows C. 3 Slap Chops, 3 ShamWows D. 4 Slap Chops, 2.5 ShamWows E. None of the above

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin 12. Let us explore the policy of Food Stamps. Note we will explore this further in the long-form section. Oftentimes, we reduce a complex economy to a two-good economy by specifying one particular good and combining all other goods into a “numeraire” good that can be interpreted as “everything else” on which the individual spends money. Let us consider the preferences of an individual over “food” and “everything else”. To keep our analysis simple, let us assume the individual originally has $30 to spend, a unit of food costs $10, and a unit of “everything else” also costs $10. Assume preferences are of the form: Compare the optimal bundle under this baseline scenario to the scenario in which prices are identical, the individual still has $30, but the individual also has $20 worth of food stamps. Note food stamps can ONLY be spent on food and cannot be resold. With food stamps (compared to without), the individual A. Will purchase exactly $20 worth of extra food. B. Will purchase less than $20 worth of extra food. C. Will purchase more than $20 worth of extra food. D. None of the above.

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin 16. I have $20 to spend on pizza and view slices of pizza and a whole pizza as perfect substitutes. In particular, I am indifferent between 8 slices of pizza and 1 whole pizza. As such, my utility function can be represented as , where is a slice of pizza and is a whole pizza. A slice of pizza costs $2. A whole pizza costs $14. What is my optimal bundle of slices of pizza and whole pizzas? A. B. C. D. No optimal bundle exists 17. Consider an individual with quasilinear preferences . The individual has $200 to spend on the two goods. The price of one unit of good 1 is $60. The price of one unit of good 2 is $1. As always, the individual can purchase fractions of a unit if she is so inclined, but cannot purchase negative amounts of the good. What is the individual’s optimal bundle? A. B. C. D. E. None of the above 18. Consider an individual with preferences over two goods, and represented by the Constant Elasticity of Substitution utility function . The individual has $20 to spend on the two goods. The price of one unit of Good 1 is $1. The price of one unit of Good 2 is . Solve for the optimal bundle in terms of . In other words, set up the Lagrangian and solve the First Order Conditions in the manner in which you are accustomed. The only difference is that instead of a specific value of , it will be a constant variable that appears in your final solutions. A. B. C. D. ( , ) 8 U S P S P = + S P 10 0, 7 S P = = 10, 0 S P = = 8 2, 7 S P = = = - + x x U x x 1 2 1 2 1/2 , ) 2 ( 2 = = x x 1 2 10 , 0 3 = = x x 1 2 0, 200 = = x x 1 2 5 , 225 12 = = x x 1 2 0, 0 1 x 2 x 1/3 1/3 1 2 1 3 2 , ) ( ) ( x x x u x = + 2 p 2 p 2 p 1/2 1/2 2 2 1 2 3/2 2 1 1 , 20 20 p x p x p - - + + = = 1 2 1/2 1/2 2 2 2 20 , 1 1 ) 20 ( x p x p p - = + = + 1 2 10, 0 x x = = 1 2 2 10 0, x x p = =

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin Section 4: Long-Form Questions (10 points each) Your long-form answers should be typed or neatly hand-written/drawn. Please place your name and netid on each page. You will submit a single .pdf file, containing clearly labeled answers (and associated work) to both LF questions . It is important that you use an actual scanner or mobile pdf scanner (e.g., Adobe Scan) to scan your work. If you simply take a photo using your phone, I will deduct points. Your long-form answers should be in your words. You may work through ideas with your classmates, but I should not receive overly similar answers in different handwriting. Be concise with your responses. A long answer does not equate a good answer. 19. Let us return to Question 5. Assume that the budget constraint is as specified, but we also know the preferences of the individual are quasilinear and have the following shape. Recall from HW2 that quasilinear indifference curves are parallel shifts of each other. In this case, the parallel shift is horizontal. A. Assume it cost Netflix $0.10 per movie that it streams to its customers. Moreover, as in question 5, assume that average individual income is $24, a monthly subscription to Netflix is $8, and the price of everything else is $4. Using what we know about graphical constrained optimization, make an argument as to whether Netflix will stay in business much longer. Specifically, make an argument as to how many movies an individual with these preferences will consume. Hint: Recall that the full indifference map will have many more of these indifference curves, all of which are a horizontal shift of the curves you see. In other words, there are many more indifference curves to the right of I 3 .

ECON 30010: Intermediate Microeconomic Theory HW3, Cronin 20. Freddie has preferences over gym equipment represented by the Cobb-Douglas utility function, where G is gym equipment and E is everything else. Assume Freddie has an income of $200, the price of gym equipment is $100 and the price of a unit of “everything else” is $50. A. Set up Freddie’s Lagrangian function, take the first order conditions, and solve the system of equations to find Freddie’s optimal combination of gym equipment and everything else. B. With gym equipment on the x-axis and everything else on the y-axis, draw Freddie’s budget constraint. On the same graph, draw the indifference curve passing through the optimal point you found in part A (plot at least 5 points for each curve). Hint: where are the optimal values found in part A. Are your results consistent? C. Now assume that Freddie’s gym offers a new promotion whereby gym equipment is buy 1, get 1 free. While Freddie can resell his goods, he cannot take the free gym equipment out of the packaging and try to sell that. If he wants to resell a piece of gym equipment, he must sell it with the free piece with it. Draw this new budget line. Without solving for the new optimal bundle, can you unequivocally state whether Freddie will be better off? Hint: Are the preferences represented by the Cobb-Dougas utility function monotonic? = U G E G E 1/5 4/5 ( , ) = U G E 1/5 4/5 * * *, * G E