Microeconomics Tutors at Yale

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Intermediate Microeconomics tutors for Yale

Problem Submitted by Intermediate Microeconomics Tutor at Yale

Consider the real-valued function ƒ defined over the interval -3, 3 by:

fx=4-(x+2)2if x≤-1x2(2-x)if x>-1

Sketch a rough graph of the function

Find all critical points (i.e., points with zero first derivative) of the function. Identify the interior local maxima and minima.
Is the function discontinuous anywhere? Is the function non-differentiable anywhere? If your answer to either of these questions is yes, is there a local maximum or minimum at these points?
What is the function‘s global maximum and minimum. [Hint: Check the function values at all critical points and endpoints.]

Problem Submitted by Intermediate Microeconomics Tutor at Yale

You have two midterm exams coming up, and have to decide how to allocate your study time. After eating, sleeping, exercising, and maintaining some human contact, you will have 10 hours each day in which to study for your exams. You have figured out that your GPA (G) from your two courses, Economics and Sociology, takes the form

G=47(2 E+S)

where E is the number of hours per day spent studying Economics and S is the number of hours per day spent studying Sociology. You only care about your GPA.

What is your optimal allocation of study time?
If you follow this strategy, what will your GPA be?
What will the shadow value, measured in GPA units, of the study time? [Hint: In a constrained optimization problem, the optimal value of the Lagrange multiplier is the shadow value of the constraint.]

Problem Submitted by Intermediate Microeconomics Tutor at New York University

John is a stamp (x1) collector, but he also likes fancy clothes (x2). His utility function is

Ux1,x2=x1+10x2-12x22

Each stamp costs p1=1 and a piece of his favorite clothing costs p2=2.

Assuming that his total income is $10, find his optimal x1 and x2. Is this solution interior?
Suppose now George‘s salary doubles, resulting in his higher income $20. Find his new demanded quantities of stamps and clothes. Is this solution interior?
In cases (a) and (b), what is the marginal utility from one dollar invested in stamps, MU1p1, and in clothing (at the optimal demand), MU2p2, . Are they equal? Discuss.

Problem Submitted by Intermediate Microeconomics Tutor at New York University

Mark wants to buy T-shirts (good x), whose (per unit) price is px, and notebooks (good y), whose (per unit) price is py. His utility function is Cobb-Douglas of the form ux,y=x2y3 and his income is I.

Find the marginal utility of each good. Are the preferences monotone?
Define an indifference curve and show that (for some positive level of utility) is decreasing and convex.
Write down the budget constraint of the consumer.
Write the Lagrangian for the utility maximization problem.
Solve the first-order conditions to find the demand functions for both x and y. [Hint: Your results should be functions of px,py and I].
Check the second order conditions for maximum.
In the optimal consumption bundle, how much money is spent on x and how much on y? Express your answer in proportion to the total income I.
Suppose now that the price of notebooks doubles and Mark has to pay 2py per unit. How does the optimal consumption bundle change? Does the spending proportion change?

Problem Submitted by Intermediate Microeconomics Tutor at New York University

John (from last week‘s problem set) is wearing his fancy clothes and is going to a cinema to watch "Destroyer" (Nicole Kidman in the leading role) with some friends. Once he is arriving at the cinema, he wants to buy something to eat. In the cinema, he can buy chocolate bars, z, and popcorn, y (measured in cups). He loves chocolate and popcorn. His utility function is uz,u=z2+y2. The budget constraint ispzz+pyy=I.

Define an indifference curve. Write down the function for the indifference curve for some positive level of utility, u-. Draw the indifference curve.
Solve the first-order conditions of the utility maximization problem and find the critical (not necessarily optimal!) quantities for chocolate bars, z*, and cups of popcorn, y*.
Derive the Hessian matrix of the utility function. Then, write down the bordered Hessian matrices, and calculate their determinants. (Hint: It is almost always simpler to calculate if you put the Lagrange multiplier as the first variable in the Hessian matrices.)
Check the second order conditions. Is your answer to part (b) a constrained maximum?
Go back to your graph of the indifference curve and use intuition to explain where one might find the true optimal consumption bundles. Consider each of the following three cases:i pz<py ii pz>py iii pz=py.

Problem Submitted by Intermediate Microeconomics Tutor at Columbia University

Sophia likes donuts, x, and milkshakes, y . The prices are px and py, respectively. Her utility function ux,y=x12+y.

Derive the marginal rate of substitution at an arbitrary point x-, y-, where x-, y->0.
Suppose that Sophia‘s income is I. Use the Lagrangian method to obtain the optimal consumption bundle of the consumer (as functions of I,px,py). [Hint: Make sure you cover cases in which I is small, as well as cases in which I is high; consumptions cannot be negative!]
Suppose Sophia’s total income is I=$10 , the price of a donut is px=$0.05, while the price of a milkshake is py=$1. What is Sophia‘s optimal consumption bundle? Is this an interior solution?
Now suppose Sophia’s total income triples, so that I=$30. What is her optimal consumption? Is this new demand interior?

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