ECO3011 Intermediate Microeconomic Theory: Assignment 1
Total points: 100
Due Date: February 17 (Monday) 11:59pm
Please show detailed steps for ALL the questions.
1. (7 pts) Professor Goodheart gives 3 midterm exams. He drops the lowest score and gives each student her average score on the other two exams. Polly Sigh is taking his course and has a 60 on her first exam. Let x2 be her score on the second exam and x3 be her score on the third exam. If we draw her indifference curves for scores on the second and third exams with x2 represented by the horizontal axis and x3 represented by the vertical axis, explain how you determine the shape of her indifference curve through the point (x2, x3) = (50, 70) and display in a plot. In your graph, label the numerical values of important points that characterize the indifference curve.
2. (14 pts) Sammy and Jimmy are twin brothers. Each gets a weekly allowance of ✩2. Sammy’s preferences for baseball cards (quantity denoted as x) and “famous economists” cards (quantity denoted as y) can be represented by the utility function u(x, y) = xy. Suppose that both goods are ✩1 per unit, and x and y can be any non-negative real numbers.
(a) (2 pts) Solve for Sammy’s optimal consumption bundle.
(b) (3 pts) Suppose the price for baseball cards, px, rises to ✩2. What is Sammy’s new optimal consumption bundle?
(c) (5 pts) How much would his parents have to increase his allowance in order to leave him exactly as well off as he was originally? Save your final result to 2 decimal point.
(d) (4 pts) Jimmy’s preferences are represented by v(x, y) = ln(x) + ln(y). Answer questions (a), (b) and (c) for Jimmy. Compare the result with Sammy. Explain why you have such result.
3. (23 pts) People buy all sorts of different cars depending on their income levels as well as their tastes. Industrial organization economists who study product characteristic choices (and advise firms like car manufacturers) often model consumer tastes as tastes over product characteristics (rather than as tastes over different types of products). We explore this concept below. Suppose people cared about only two different aspects of cars: the size of the interior passenger cabin and the quality of handling of the car on the road. Compare three car models: a Chevrolet Minivan, a Porsche 944, and a Toyota Camry. Porsche’s do not have much space in the interior but they handle well at high speeds. Minivans have tons of interior space but don’t handle that well at high speeds. And Toyota Camrys are somewhere in between — with more space than Prosche’s but not as much as minivans, and with better handling at high speeds than minivans but not as good as Porsches.
(a) (3 pts) Putting x1 = “cubic feet of interior space” on the horizontal axis and x2 = “speed at which the car can handle a curved mountain road” on the vertical, based on the above information, draw the three types of cars in a plot assuming that they will fall on one line.
(b) (6 pts) Suppose we considered three different individuals whose preferences (over space and ma-neuverability) satisfy basic assumptions and are monotonic and strictly convex, and suppose each person owns different one of the three types of cars. Suppose further that each indifference curve from one person’s indifference map crosses any indifference curve from another person’s indiffer-ence map at most once. (When two indifference maps satisfy this condition, we often say that they satisfy the single crossing property.) Consider the case that the indifference curves through Toyota Camry for the three individuals all intersect at the Toyota Camry. Now suppose you know person A’s MRS at the Toyota Camry is larger (in absolute value) than person B’s, and person B’s MRS at the Toyota Camry is larger (in absolute value) than person C’s. Who owns which car? Draw a plot to show their indifference curves through the Toyota Camry and explain your result.
(c) (4 pts) Suppose we had not assumed the “single crossing property” in part (b). Would you have been able to answer the question “Who owns which car” assuming everything else remained the same? Explain with a plot.
(d) (2 pts) Suppose you are currently person B and you just found out that your uncle has passed away and bequeathed to you his 3 children,aged 4, 6 and 8 (and nothing else). This results in a change in how you value space and maneuverability. Is your new MRS at the Toyota Camry now larger or smaller (in absolute value)?
(e) (3 pts) Suppose that the tastes of persons A, B and C can be represented by the utility functions uA(x1, x2) = x
α
1 x2, uB(x1, x2) = xβ1x2, u
C (x1, x2) = xγ1x2 respectively. Calculate the MRS for each person.
(f) (5 pts) In the above utility functions, assuming α, β and γ take on different values, is the “single crossing property” defined in part (b) satisfied? Given the description of the three persons’ preferences in part (b), what is the relationship between the values of α, β and γ? Explain all of your results in details.
4. (16 pts) Dudley’s utility function is U(C, R) = C − (12 − R)2, where R is the amount of leisure he has per day and C is the quantity of all consumption goods. He has 16 hours a day to divide between work and leisure. He has an income of ✩20 a day from nonlabor sources. The price of consumption goods is $1 per unit.
(a) (2 pts) If Dudley can work as many hours a day as he likes but gets zero wages for his labor, how many hours of leisure will he choose?
(b) (4 pts) If Dudley can work as many hours a day as he wishes for a wage rate of ✩10 an hour, how many hours will he choose to work? Write down his budget constraint.
(c) (6 pts) If Dudley’s nonlabor income decreased to $5 a day and wage rate is still ✩10 per hour, how many hours would he choose to work? Explain your result. Draw in a graph the old budget constraint, the new budget constraint, and indifference curves. Also display Dudley’s optimal choices under the old and new nonlabor income. Label important points in your graph.
(d) (4 pts) Suppose that Dudley has to pay an income tax of 20 percent on all of his income (labor and nonlabor), and suppose that his before-tax wage remained at $10 an hour and his before-tax nonlabor income was $20 per day. How many hours would he choose to work?
5. (14 pts) There are two goods in the world, pumpkins (x1) and apple cider (x2). Pumpkins are $2 each. Cider is $7 per gallon for the first two gallons. After the second gallon, the price of cider drops to $4 per gallon.
(a) (4 pts) Peter’s income is $54. Draw his budget constraint. Clearly show in the plot the intercepts on the axes, the kink in the budget line and the slope of each segment of the budget constraint.
(b) (3 pts) Peter’s utility function is u(x1, x2) = x1 + 3x2. Sketch some indifference curves in your graph. Find Peter’s optimal consumption bundle (x*1
, x*2).
(c) (4 pts) Paul’s income is $22. Draw his budget constraint in a new graph. Clearly show in the new plot the intercepts on the axes, the kink in the budget line and the slope of each segment of the budget constraint.
(d) (3 pts) Paul’s utility function is u(x1, x2) = min{3x1, 2x2}. Sketch some indifference curves in your graph for Paul. Find Paul’s optimal consumption bundle (x*1
, x*2).
6. (26 pts) I have two 5-year old girls — Ellie and Jenny — at home. Suppose I begin the day by giving each girl 10 toy cars and 10 princess toys. I then ask them to plot their indifference curves that contain these endowment bundles on a graph with cars on the horizontal and princess toys on the vertical axis. Assume they are both rational.
A. Ellie’s indifference curve appears to have a marginal rate of substitution of -1 at her endowment bundle, while Jenny’s appears to have a marginal rate of substitution of -2 at the same bundle.
(a) (2 pts) Can you propose a trade that would make both girls better off?
(b) (2 pts) Suppose the girls cannot figure out a trade on their own. So I open a store where they can buy and sell any toy for ✩1. Illustrate the budget constraint for each girl in two separated plots.
(c) (6 pts) Will either of the girls shop at my store? If so, what will they buy? Explain your result with indifference curves displayed in the plots you drew for part (b).
(d) (4 pts) Suppose I do not actually have any toys in my store and simply want my store to help the girls make trades among themselves. Suppose I fix the price at which princess toys are bought and sold to $1. Without being specific about what the price of toy cars would have to be, illustrate, using final indifference curves for both girls on the same graph, a situation where the prices in my store result in an efficient allocation of toys.
(e) (1 pts) What values might the price for toy cars take to achieve the efficient trades you described in your answer to (d)? Explain.
B. Now suppose that my girls’ tastes could be described by the utility function u(x1, x2) = xα1x12
−α
, where x1 represents toy cars, x2 represents princess toys and 0 < α < 1.
(a) (2 pts) What must be the value of α for Ellie (given the information in part A)? What must the value be for Jenny?
(b) (4 pts) When I set all toy prices to $1, what exactly will Ellie do? What will Jenny do? Assume that it is possible to trade fractions of toys.
(c) (1 pt) Given that I am fixing the price of princess toys at $1, do I have to raise or lower the price of car toys in order for me to operate a store in which I don’t keep inventory but simply facilitate trades between the girls?
(d) (4 pts) Suppose I raise the price of car toys to $1.40, and assume that it is possible to sell fractions of toys. Have I found a set of prices that allow me to keep no inventory?