Problem Set #2.
Due at 11:59 pm, Tuesday, September 22, 2020.
Solutions
Problem 1. Consider the reference dependent utility function for mugs in
... [Show More] problem 1 of
PS#1. Imagine 2 subjects, each of whom has this utility function. Person 1 is initially
endowed with 10 mugs and no money, while person 2 is initially endowed with $15 and
no mugs. Suppose we used the methodology of Knetsch (1992) to map out the indifference
curves of the two individuals respectively, and then used the methodology we used in lecture
to map out their demand curves.
(a) What shape would the demand curves for the two individuals be between zero and ten
mugs, and why? Make sure to explain how you arrived at your answer, and show any
math or diagrams you used in the process of answering the question. Note, the demand
curve for person 2 will be very different after ten mugs. Make sure you understand
why.
[Note: there is no right answer to the question, “How much math do I need to show?”
The answer is, “As much as you need to show how you answered the question.”]
A: Horizontal, because the value function is piecewise linear.
B: Kinked, because the value function is kinked.
C: Downward sloping, because the indifference curves are downward sloping.
D: A and B.
E: B and C.
(b) What would be the height of each demand curve?
A: $6 for person 1 and $1.50 for person 2.
B: $1.50 for person 1 and $6 for person 2.
C: $6 up to the kink, and $1.50 after the kink.
D: It would depend on the quantity being bought or sold.
The answer to (a) is A. The first key to understanding this is to see that
the endowments are on the axes of the diagram of indifference curves, which
means that person 1 is only able to sell, and person 2 is only able to buy.
(This is because consumption cannot be negative. That would mean you
were producing mugs, or printing money.) Thus, we only have the selling
“leg” of person 1’s IC, and we only have the buying leg of person 2’s IC.
(These IC’s are just like the ones we saw in PS#1, except the endowment
Problem Set #2
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points are shifted to the axes, so you only see one leg of each IC.) The
next key is to realize that the IC’s for the value function we’ve been using
are linear except at the kink. (Because the value function is linear in both
gains and losses, which is a simplification we’ve been using for mathematical tractability.) Once you’ve grasped all of this, you will see that what
we’ve got is a couple of IC’s that look a whole lot like the IC’s for perfect
substitutes. (Indeed, except for at the kink, mugs and money are perfect
substitutes, again because of the piecewise linearity of the value function.1
Now you go back to intermediate micro and dig up what you know about
perfect substitutes. For any price greater than the slope of the IC the consumer will choose zero mugs, and for any price less than the slope of the
IC the consumer will choose zero money. (These are corner solutions in the
LaGrangian maximization problem.) At the price that is exactly equal to
the slope of the IC the consumer is exactly indifferent between any combinations of mugs and money, so their utility will be maximized at any point
on their IC. If you map out these consumption choices as a function of price,
you get a horizontal demand curve with height equal to the slope of the IC,
which is Ps for person 1 and Pb for person 2. You’ve already computed those
prices in PS#1; they are 6 and 1.5 respectively, so the answer to (b) is A.
A couple of additional points [Show Less]