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EDPE 4056 – Microeconomic Theory: Applications to Education

Homework 2

Question 1 (12 points total):

For the following utility functions (1 pt per task, per utility function):

Task 1: Solve for the marginal utility of good x and good y at the point (7, 3). Show your work!

Task 2: Pick any positive point. Determine whether the marginal utility decreases as consumption of each good increases at this point (i.e., does the utility function exhibit diminishing marginal utility in each good?).

Task 3: Find the marginal rate of substitution of x for y at the point (2, 2).

Task 4: Graph the indifference curve for when utility is equal to 10; label at least three points on the curve.

a. U(X,Y) = 10X + 20Y

b. U(X,Y) = (X^1/3 )*(Y^2/3)

c. U(X,Y) = 5*(X^1/2)*Y

Question 2:

In this question, we study the consumption behavior of Apoorva, a student in EDPE 4056. She views studying from the Goolsbee et. al textbook and the Pindyck and Rubinfeld textbook as perfect substitutes for each other.

a. (2 pts) Draw a set of indifference curves that plot her preferences over studying from these two books, where the price is the number of hours per chapter. Note: you will have to assume some ratio of prices.

b. (3 pts) Now assume that, while the two are perfect substitutes, the Goolsbee book is written less clearly than the Pindyck one, so that it takes two hours to read a chapter in Pindyck and three hours to read a chapter in Goolsbee. Assume additionally the chapters contain the same content. She has 30 hours of studying to allocate between the two books in preparation for the midterm. Given the information we know, how will she allocate her time? Show your answer graphically with a budget line (over number of chapters in each book on the x and y axes, respectively) and several indifference curves using the process of utility maximization.

c. (3 pts) Take the relative prices and budget curve from part b, and add the assumption that she must read at least 6 chapters of Goolsbee to pass the exam. Under these constraints, how will she allocate her time? Show your answer graphically with several budget lines and an indifference curve using the process of cost minimization.

Question 3:

Henry is a student leaving high school considering what to do with his life. If he attends college next year, he will graduate with a degree with probability 0.9. With probability 0.1, he will attend for four years but not get his degree. The cost of attending college is $400,000. If he attends college, Henry will work for 50 years and then retire. If he earns a college degree, he will make $120,000 per year. If he attends college but does not earn a degree, he will make $50,000 per year. If he does not attend college, he will work for 54 years, earning $100,000 per year, and then retire.

a. (2 pts) Write out the equation for the net present value of Henry attending college. Note that you do not have enough information yet to solve it for a number.

b. (2 pts) Write a mathematical statement (i.e., an equation or inequality, with words as needed) to express under what circumstances Henry will choose to attend college.

c. (3 pts) Now imagine he does not know the probability of graduating with a degree, if he chooses to attend college. Write a mathematical statement expressing the probability of graduating with a degree, conditional on attending college, at which Henry is indifferent between attending and not attending college.

Now let’s set the interest rate at 3%.

d. (2 pts) What is the present discounted value of the income Henry will make if he does not attend college? Show your work – write down the mathematical formula you use to generate this and the work you used to come to your final answer. You can use excel / google sheets to come up with the final numerical answer.

e. (3 pts) What is the present discounted value of the income Henry will make if he does attend college? Show your work. Write down the mathematical formula you use to generate this and the work you used to come to your final answer. You can use excel / google sheets to come up with the final numerical answer.

Question 4:

Shanshan is considering what major to study in college. Her utility function is based on the income she earns, and is defined by U(I) = I0.8 . If she majors in metalworking, she will earn $145,000 per year with probability 1. If she majors in finance, she will earn $300,000 per year with probability 0.6 (assuming the market goes well) and $30,000 with probability 0.4 (if the market tanks and she has to go move in with her parents and work at Burger King).

a. (2 pts) Is she risk averse, risk neutral, or risk loving? Explain.

b. (2 pts) Write out the equation for her expected utility for each major

c. (3 pts) Which major will she pick? Show your work.

d. (3 pts) Suppose someone offers her insurance for the possibility that the market tanks. This insurance will provide her an amount of income in addition to the burger king wages that makes her indifferent between metalworking and finance. What is this amount, and what is the cost of the insurance? (note: many possible answers)

Question 5:

Peter loves spicy food. He consumes both Laoganma chili crisp and Texas Pete hot sauce. His annual budget curve maps from (20 bottles of Laoganma, 0 cups of Texas Pete) to (0 bottles of Laoganma, 24 bottles of Texas Pete).

a. (2 pts) Draw his budget constraint. Express the price of Texas Pete, in terms relative to the price of Laoganma.

b. (1.5 pts per answer) Given this set up, would Peter ever consume at the following points of (#bottles of Laoganma, #bottles of Texas Pete)? Why/why not? If we would consume at this point, draw an indifference curve that would justify this decision.

i. (12,12)                                                                    ii. (6,15)

iii. (6,6)                                                                      iv. (12,10)

c. (1 pt) Imagine that Peter consumes 4 bottles of Laoganma. How many bottles of Texas Pete will he consume?

d. (1 pt) Now imagine that the price of Texas Pete doubles. Draw the new budget constraint in Laoganma / Texas Pete space.

e. (3 pts) Assume Peter goes from the consumption point found in part c of this question to a new point where he consumes 5 bottles of Laoganma to some new consumption point defined by the budget constraint in part d. Graphically depict his new consumption bundle and the income and substitution effect in his coming to his new consumption bundle (this does not have to be precise – just graph it showing you understand the concepts).