Problem Set 1: Tell Me What You Want
Due Saturday, September 26 at 11pm on Gradescope
Note: for most assignments, each question (e.g. Exercise 1.1) is worth 3 points. Because this is an abbreviated week, exercise 1.1 and 1.2 are worth 6 points each!
Preferences and Utility
Suppose $A = (9,4)$ and $B = (4,9)$. For at least three of the following functions, determine if someone whose preferences could be represented by that utility function prefers $A$ to $B$, prefers $B$ to $A$, or is indifferent between the two. You’re welcome to do all of them, but three is sufficient for full credit. Students in 50Q must do the challenge function (g) as one of their functions. I would recommend doing at least one of a/b, one of c/d, and one of e/f; as you can see, each of these pairs shares a general functional form, but differs in the exponent or coefficient on good 1.
$u(x_1,x_2) = x_1x_2$
$u(x_1,x_2) = x_1x_2^2$
$u(x_1,x_2) = x_1 + x_2$
$u(x_1,x_2) = 2x_1 + x_2$
$u(x_1,x_2) = x_1^{1 \over 2} + x_2^{1 \over 2}$
$u(x_1,x_2) = x_1^{1 \over 2} + 2x_2^{1 \over 2}$
Challenge (required for 50Q, optional for everyone else): \(u(x_1,x_2) = \min\{ x_1,2x_2\} = \begin{cases}x_1 & \text{ if }x_1 \le 2x_2 \\ 2x_2 & \text{ if }x_1 \ge 2x_2\end{cases}\)
For (g), also evaluate the utility at point $C = (4 , 2)$. How does this utility compare to the other two bundles?
Indifference Curves
For each of the functions you chose in the first question, derive expressions for the indifference curves passing through bundles $A$ and $B$. Then, plot them as precisely as possible using graph paper.
Here’s a template to use if you need one:
Representing Preferences with Utility Functions
This question is only required for 50Q students.
For each of the following scenarios, write a utility function which represents the listed preferences, or explain why such a utility function cannot exist: