Homework 3 - William Candido
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Objective: To give the student practice working with different assumptions/types of preferences. Demonstrate how modeling preferences can be applied to a policy discussion. Homework 3 The environment continues to be a major issue in United States policy discussions. In this homework imagine you are working for the environmental protection agency (EPA), who is concerned by the consumption of Styrofoam (which is effectively non-biodegradable). Specifically, the EPA wishes to understand the behavior of consumers who chose between eating at two fictional restaurants: Enviro-
taco, which provides recyclable containers for their tacos, and Polluto-burger who insists on serving low quality hamburgers in Styrofoam containers. Formally, let T represent enviro-tacos and B represent polluto-burgers. In all of these problems, when graphing place tacos on the X-axis and burgers on the Y-
axis. Further, assume each burger is 1/4th of a pound and each taco is 1/8
th
of a pound. 1.
Patrick has rational and well behaved (or "standard") preferences for tacos and burgers (𝑇, 𝐵)
. It is known that he prefers the bundle A=(8,6) to the bundle B=(10,4). It is also known that he prefers the bundle C=(10,3) to D=(11,2). For each of the following, if possible, determine which bundle is preferred. State which assumptions on preferences you used to reach the conclusion. Use the appropriate preference relation to denote whether Patrick weakly prefers a bundle, strongly prefers a bundle, or is indifferent between the two bundles. a.
Which bundle does Patrick prefer, B=(10,4) or C=(10,3)? B___C by the assumption of _____________ b.
Which bundle does Patrick prefer E=(9,5) or B=(10,4)? E___B by the assumption of _____________ c.
Which bundle does Patrick prefer E=(9,5) or D=(11,2)? Make a formal argument below your answer. 2.
In these problems, we will better understand the axioms of consumer theory by seeing ways in which they can fail: a.
Suppose Patrick from above was blind folded and given two meals A and B. Patrick stated that he strictly preferred meal A to B. It was then revealed that the meals consisted of the following combinations of tacos and hamburgers: A=(2,2) and B=(2,2). Which fundamental assumption on preferences were violated by Patrick’s choice?
b.
The relation “is preferred to” between commodity bundles is just one example of a binary relation. Another example is the relation “is a full brother of” defined over the set of all human beings. Let xRy mean person x is a full brother of person y. Apply the properties of completeness, reflexivity, and transitivty to xRy. Is this relation complete, transitive, and reflexive? For each either prove the property or provide a counter example.
c.
Both c.1 and c.2 are required for MA STUDENTS ONLY: c.1: let each stake be a bundle of goods ranked over three characteristics: (size, juciness, and seasoning). Ex: (2,1,0) is a large staeak, somewhat jucy, and not seasoned. (1,1,1) is a medium steak, somewhat juicy with some seasonings. I have very, very specific preferences over steaks. I prefer a large and juicy steak to a juicy and seasoned steak. I prefer juicy and seasoned to large and seasoned. BUT: I prefer large and seasoned to large and juicy. Are my preferences transitive? Justify your answer by showing they are always transitive or showing a case of three bundles where transitivity fails. c.2. Coach likes his players to be big, fast, and obedient. If player A is better than player B in two of these three characteristics, then Coach prefers A to B, but if B is better than A in two of these three characteristics, then Coach prefers B to A. Otherwise, the Coach is indifferent between them. Wilbur Westinghouse weighs 340 pounds, runs very slowly, and is fairly obedient. Harold Hotpoint weighs 240 pounds, runs very fast, and is very disobedient. Jacuzzi weighs 150 pounds, runs at average speed, and is extremely obedient. Are coaches preferences transitive? Justify your answer. 3.
Assume Jenkins only cares about the total pounds of fast food that he consumes. The more pounds of fast food he consumes, that happier he is. Remember each burger is 1/4th of a pound and each taco is 1/8
th
of a pound. a.
Graph the indifference curve that passes through the point (4,2). If the indifference curve intercepts the axes, label where they intercept. Graph several other representative indifference curves as well. Draw an arrow showing the direction in which the indifference curves are increasing. What is Jenkin’s marginal rate of substitution between tacos and hamburgers? b.
Alternatively, instead of caring about the total pounds of food he eats, Jenkin only cares about the total number of burgers and tacos that he consumesThe greater the total number of tacos and burgers he consumes the happier he is. Graph the indifference curve that passes through the point (4,2). Label the slope of the indifference curve. If the indifference curve intercepts the axes, label where they intercept. Graph several other representative indifference curves as well. Draw an arrow showing the direction in which the indifference curves are increasing. c.
For both of the preferences above, what type of preferences are these? What wording in the description of Jenkin tipped you off? Are these preferences monotone? Are these preferences strictly convex?
d.
Suppose Jenkin’s
preferences match b. above. Also suppose he was currently consuming all burgers and no tacos. The EPA is debating between two policies to reduce pollution. One is to provide a per-unit subsidy on enviro-tacos and the other is to give an in-kind transfer in the form of a voucher for enviro-tacos. Which will be generally more effective at reducing the consumption of polluto burgers? Explain, if necessary explain by graphing how the budget constraint is changing under the two different policies. 4.
There is a new fast food lover in town. Call her Ms. X. Ms. X always consumes tacos and hamburgers together. Specifically she consumes one taco with one hamburger and has no use for any additional tacos/hamburgers she is given. a.
Graph the indifference curve that passes through the point (3,3). If the indifference curve intercepts the axes, label where they intercept. Graph several other representative indifference curves as well. Draw an arrow showing the direction in which the indifference curves are increasing. b.
Instead suppose Ms. X consumes two tacos with every hamburger. Graph the indifference curve that passes through the point (3,3). If the indifference curve intercepts the axes, label where they intercept. Graph several other representative indifference curves as well. Draw an arrow showing the direction in which the indifference curves are increasing. c.
For both of the cases above, what type of preferences does Ms X have? What language in the question tipped you off? Are her preferences monotone? Are her preferences strictly convex? d.
Suppose the EPA wants to reduce pollution given a consumer has Ms X’
s preferences. What do you think will be generally more effective at reducing pollution, applying a per unit tax to enviro-tacos or a per unit tax to polluto burgers? 5.
Finally suppose Mr. Z starts eating fast food. Mr. Z is happiest when eating a combination of tacos and hamburgers. Specifically, we know an equation for the set of all tacos and burgers such that Mr. Z is indifferent between eating 5 of each. Mathematically: ?ℎ? ??? ?? 𝑎𝑙𝑙 ??𝑖??? ??𝑐ℎ ?ℎ𝑎?: (𝑇, 𝐵)~ (5,5) 𝑖? 𝐵 =
25
𝑇
. Where 𝑇
is the number of tacos, 𝐵
is the number of burgers and ~
symbolizes indifference. a.
Assume Mr. Z. is consuming on the indifference curve that passes through (5,5). Find the values for the number of burgers when he is eating 1, 2, 6, and 10 tacos. b.
Graph the combination of tacos and burgers above and approximate Mr. Z’s indifference curve. c.
True or false: (5,5) ~ (15, 1 2/3). d.
Ture or false: (5,5) ~ (6, 4). e.
Are Mr. Z’
s preferences convex? Are they montone?
6.
Jason also chooses between consuming tacos and burgers. However, after eating 5 burgers, Jason gets tired of burgers and consuming each additional burger makes him more and more sick. Jason is always happy to eat more tacos. a.
graph what one of Jason’s
typical indifference curves could look like: b.
Jason still chooses between tacos and burgers. Eating more than 5 burgers still makes Jason sick. However, the government provides no-charge waste disposal facilities such that if he ends up with more than five burgers he can just throw them away without being charged. Now it is essentially possible for Jason to consume more than 5 burgers without getting sick. Graph what a typical indifference curve of Jason’s now looks like:
c.
Let Jason’s preferences match the graph in b. above. Assume Jason is currently consuming at the point (3,10). Jason is not bothered by the pollution caused by consuming polluto-burgers. However, the rest of society actively dislikes pollution. Suppose the government reallocated resources such that Jason consumes at the point (3,5) instead of (3,10). Does Jason weakly prefer, strictly prefer, or is indifferent between (3,5) and (3,10)? Is this reallocation a pareto improvement? Explain why or why not. 7.
Mary has unusual preferences in which she only cares about winning an eating contest. The rules of the contest are such that the judges note how many tacos and burgers each contestant eats and then ranks the contestant based on the highest number of either tacos or burgers. That is, if the contestant eats more tacos than burgers their rank in the contest is based on the number of tacos they have eaten. If the contestant has eaten more burgers than tacos their rank in the contest is based on the number of burgers they have eaten. Mary only cares about getting the highest possible rank in the contest. Assume she is currently consuming at (4,2). a.
Graph a typical indifference curve that passes through the point (4,2). Draw an arrow showing the direction the indifference curves increase. b.
Are Mary’s preferences montone? Are they convex? Are they weakly convex? c.
Suppose Mary is at the point (4,2) would she trade one burger for one taco? Would she trade one taco for one burger? What are the exchanges that would leave marry indifferent between the new consumption bundle and her current level of consumption?
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Year
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2007
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5,000
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1.00
1.25
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2.50
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A
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