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Short Assignment Requirements
Monday 2nd April, hard copy handed in at the end of class, electronic submission before class on NYU classes.
This is a take-home exam. You may not discuss the exam with anyone except the instructors of the class. You are allowed to consult your notes, the textbook and other sources, except other people. If anything is unclear, please email the instructor. The instructor will remove your name from the email, and then your questions along with the response will be sent to all students in the class. The instructor will not discuss the exam in person with any student.
Your solutions to the exam must be written-up clearly. All solutions, including workings, arguments, diagrams, and reasoning, must be clearly written-up in black ink. Everything must be legible, graphs should be labeled clearly, your arguments should be easy to follow, and your final solutions should be easy to identify. The clarity of solutions will be a factor in the grading of the exam. Please make sure that your name and a page number is on every page of your solutions before you hand them in.
There are two problems on the exam. All problems will be graded, and each will be worth one half of the grade for this exam. The grading on the exam will be stricter than on problem sets, so please make sure that you answer the questions carefully.
Late submission will not be accepted. If you submit the exam after the deadline, you will receive a grade of 0.
By signing the declaration below, you affirm that you have understood the rules of the exam and that the solutions you have submitted are your own work (without any outside assistance). You should submit the signed declaration along with your solutions.
Academic Integrity Declaration:
I, , hereby declare that
I have understood and followed all rules of the exam, and that any solutions submitted by me represent my own work.
Problem 1: Non-linear pricing
Suppose that there two goods X and Y , available in arbitrary nonnegative quantities (so the the consumption set is). The consumer has preferences over consumption bundles that are strongly monotone, strictly convex, and represented by the following (differentiable) utility function:
√ u(x,y) = y + 2α x,
where x is the quantity of good X, and y is the quantity of good Y , and α ≥ 0 is a utility parameter.
The consumer has strictly positive wealth w > 0. The price of good Y is pY = 1. However, the price of good X depends on the quantity of good X that the consumer purchases. In particular, , where x is the quantity of good X the consumer purchases. (Note that pX(x) is the price per unit when the consumer purchases x units).
(1) In an appropriate diagram, illustrate (i) the indifference map for the consumer, and (ii) the consumer’s budget set. Make sure you label diagrams clearly, and include as part of your answer any calculations about the slopes and intercepts of the indifference curves and the budget line.
(2) Formulate and solve the consumer’s utility maximization problem, and find the demand and value functions. Your demand and value functions should be functions of the parameters w and α.
(3) Suppose w = 10. In an appropriate diagram, illustrate the demand function for
good X, x(α|w = 10), and for good Y , y(α|w = 10).
Problem 2: Endowment effect
Suppose that there two goods X and Y , available in arbitrary nonnegative quantities (so the the consumption set is
Instead of being endowed with a fixed amount of wealth w, the consumer has an initial endowment of the two goods, ¯x > 0 and ¯y > 0, where ¯x is the quantity of good X the consumer owns and ¯y is the quantity of good Y the consumer owns. The consumer has no additional wealth, but the consumer can buy and sell goods X and Y at the fixed prices pX > 0 (for good X) and pY > 0 (for good Y ).
(1) In an appropriate diagram, illustrate the consumer’s budget set. Make sure you label the diagrams clearly, and include as part of your answer, the slope and intercepts of the budget line.
The consumer has preferences over consumption bundles that are strongly monotone and strictly convex. However, the consumer’s preferences depend on their initial endowment of the good. (Preferences with this property are called endowment-dependent preferences and have been studied widely by economists in recent years). In particular, the consumer’s preferences can be represented by the following “endowmentdependent” utility function:
where x is the quantity of good X, y is the quantity of good Y , and (¯x,y¯) >> 0 is the consumer’s initial endowment of the two goods. (Note that the utility function is not differentiable, but the function is differentiable for any parameters (p,x,¯ y¯) >> 0).
(2) In an appropriate diagram, illustrate the indifference map for the consumer. Make sure you label the diagram clearly.
(3) Formulate and solve the consumer’s utility maximization problem, and find the demand functions. Note that the demand functions will depend on the parameters (pX,pY,x,¯ y¯) >> 0.
(4) For what prices (pX,pY) does the consumer optimally decide to (i) consume their initial endowment of the goods, (ii) sell some units of good Y to buy more units of good X, or (iii) sell some units of good X to buy more units of good Y ?