Pathways in Economics – Microeconomics Problem Set
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Pathways in Economics – Microeconomics Problem Set
Victor Lima's Materials –Utility Maximization Problem
Problem 1. Chef Vera buys pounds of pepper(x),and salt(y)for her restaurant,according to her utility function U(x,y)=xy.Pepper costs $2 per pound and salt costs $1,and she has $4 to spend.
1.Write Vera's budget constraint.In other words,what is the inequality which states how many apples and bananas can Vera can buy?Would Vera ever spend less than all of her income?
(Bonus:what feature of the utility function makes this true?)
2.What is the marginal utility of consuming an additional pound of pepper?How about an additional pound of salt?Recall that Marginal Utility of x is defined by the partial derivative:
3.You'll notice that MUaincreases as y increases,and likewise that MUy increases as z increases.What is the economic intuition behind this?
Hint:No amount of pepper is gonna save an unsalted dish
4.Draw the indifference curve corresponding the Vera's maximized utility level,and label the point at where her utility is maximized.
5.Write down the utility maximization problem using the following structure:
6.Solve the UMP using the method of substitution(or if you feel like it,the method of Lagrange multi- pliers).Recall that the solution involves finding an optimum bundle(x*,y*)that maximizes the utility function under the constraint.
7.Let's go back to the marginal utilities that you found.We define the Marginal Rate of Substitution between x and y(MRS)by 
Solve for MRSzy in Vera's utility function.What does MRSry equal at(x*,y*)?
8.Note that the ratio of the price of salt to paper is 2.If you did the previous problem correctly,you should also get that MRSa,y=2.Therefore,the ratio of marginal utilities of x and y equal the ratio of their prices when utility is maximized.What is the economic intuition behind this?Remember that MRSry is just a ratio of marginal utilities!
Victor Lima's Material-Matching Problem
Short Answers. Please answer the following questions in 5 to 10 sentences.If a picture helps you,feel free to provide one.Points will be assigned for clarity.
1.The cost to a student of a year of college is the cost of books,tuition,room,and board.
2.If the price of first class-postage increases from 20 cents to 30 cents and the price of overnight delivery increases from 1 dollar to 1.10 dollars,then the proportion in which the two types of mail are consumed will not change.
3.Education is currently quite expensive.If education were to be free,however,then people would definitely go to school for more years.
Question 2: Dating Norms and The Gale-Shapley Algorithm.
Consider a dating market with four females and four males with the following prefer- ences over potential partners(the males are labeled using lower cases and the females are labeled using upper cases):
1.In current dating markets,the norm is for males to ask females out on dates.Under this assumption,what is the stable matching?
2.Suppose,now,that in a different society the norm is for females to ask the males out. Under this assumption,what is the stable matching?
3.Do the males(females)prefer to ask the females(out)or do they prefer to be asked out?Is you are answer true for every male(female)or just for some of them?
4.Can you propose another stable matching system for this dating market?
5.How does it compare to the male optimal one?The female optimal one?
Min Sok Lee's Material-Experimental Economics
1.Causality and Empiricism I[25 POINTS]
Ivan and Min are engaged in the debate on the return to attending college.They reason that attending college must lead to higher salaries.In fact,they collected the following data:average annual salary of 30-year-old high school graduates and college graduates in the US.On average, annual salaries of college graduates were 50%higher than those of high school graduates.
a.Min concludes that high school students who want to maximize salaries should attend college.Do you agree with him?Fully justify your answer. [7 points]
Broadly speaking,economists have offered two mechanisms for this relationship:(1)signaling: more productive individuals signal their higher productivity (thus,higher salaries)by going to college,(2)human capital:colleges teach their students skills that lead to higher productivity.
b.Assume you had data on annual salaries of individuals with 2,3 and 4 years of college education.How would you use these data to shed light on the two mechanisms?[9 points]
C.Ivan argues that the signaling mechanism does not make any sense because then
anyone can go to college to signal that he/she is a high productivity individual.Based on your intuition,what must be true for the signaling mechanism to work?Fully justify your answer. [9 points]
2.Causality and Empiricism IⅡ[25 POINTS]
The public K-12 education in the US is in a crisis.Despite the increase in per student spending over time,high school graduation rate has been flat or declining.As a potential solution, education reformers have proposed charter schools(e.g.,schools that receive public funding but have more flexibility in terms of curriculum and employing non-unionized teachers).
Usually,parents of students who want to attend charter schools must apply to them and if demand is higher than seats available,they use a lottery to decide who gets the available seats.
a.High school graduation rate of students who attend charter schools is higher than the graduation rate of those attending traditional public schools.Education reformers have used these data to support for more charter schools.Do you agree with this interpretation of the data?Fully justify your answer. [10 points]
b.Additionally,education reformers have performed the following analysis.They focused on charter schools where demand was higher than supply.Then,they compared the graduation rate of students who went to charter schools by lottery to that of students who were left out by the lottery.They found that the graduation rate of the former group was higher than that of the latter group.Thus,they concluded that if all the traditional public schools are turned into charter schools,high school graduation rates would increase.Do you agree with this interpretation of the data?Fully justify your answer. [15 points]
2025-06-23