Deriving indifference curves

Question

I have been trying to work this out for quite a while now, but I can't seem to understand how to solve these kind of questions. Any help (or hint) would be highly appreciated.

Professor Goodheart's colleague Dr. Kremepu gives 3 midterm exams. He drops the lowest and gives each student her average score on the other two exams. Polly Sigh is taking his course and has a 60 on her first exam. Let $x_2$ be her score on the second exam and $x_3$ be her score on the third exam. If we draw her indifference curves for scores on the second and third exams with $x_2$ represented by the horizontal axis and $x_3$ represented by the vertical axis, then her indifference curve through the point ($x_2; x_3$) = (50; 70) is:

1. L-shaped with a kink where $x_2 = x_3$.
2. three line segments, one vertical, one horizontal, and one running from (70; 60) to (60; 70).
3. a straight line, running from (0; 120) to (120; 0).
4. three line segments, one vertical, one horizontal, and one running from (70; 50) to (50; 70).
5. a V-shaped curve with its point at (50; 70).

Answer 1

The Utility function here is the average score that Polly gets i.e.

$U(x_2,x_3)=\frac{max\{x_2,x_3,60\}+max^2\{x_2,x_3,60\}}{2}\\$

Where: $max^2\{.\}$ stands for second highest number. Now this utility function can be rewritten as:

$U(x_2,x_3)= \begin{cases}\frac{x_2+x_3}{2}, & \text{ if } x_2,x_3\geq 60,\\ \frac{60+x_3}{2}, & \text{ if } x_2<60, x_3>x_2,\\ \frac{60+x_2}{2}, & \text{ if } x_3<60,x_3<x_2,\\ \frac{60+k}{2}, & \text{ if } k<60,x_3=x_2=k.\\ \end{cases}$

Plotting its level curve i.e. the IC for utility level $c$ would be a horizontal line from point $(0,2c-60)$ to $(60,2c-60)$, a vertical line from point $(2c-60,0)$ to $(60,2c-60)$ and a line from $(60,2c-60)$ to $(2c-60,60)$.

So if $(x_2,x_3)=(50,70)$ option 2 satisfies the IC's description.

Answer 2

Another way to represent the preference is:

\begin{eqnarray*} u(x_1, x_2, x_3) & = & x_1 + x_2 + x_3 - \min(x_1, x_2, x_3) \\ & = & \max(x_2+x_3, x_1+x_3, x_1+x_2)\end{eqnarray*} You can divide the above by $2$ to write the utility in terms of average.

Given that $x_1=60$, \begin{eqnarray*} u(60, x_2, x_3) & = & 60 + x_2 + x_3 - \min(60, x_2, x_3) \\ & = & \max(x_2+x_3, 60+x_3, 60+x_2)\end{eqnarray*}

Now $u(60, 50, 70) = 130$ (sum of the top two scores). So the indifference curve where $u = 130$ is the set: $\{(x_1, x_2, x_3)\in\mathbb{R}^3_+| \max(x_2+x_3, x_1+x_3, x_1+x_2) = 130\}$

And indifference curve holding $x_1 =60$ fixed is: $\{(x_2, x_3)\in\mathbb{R}^2_+| \max(x_2+x_3, 60+x_3, 60+x_2) = 130\}$

Observe that $\{(x_2, x_3)\in\mathbb{R}^2_+| \max(x_2+x_3, 60+x_3, 60+x_2) = 130\}$ $\subset \{(x_2, x_3)\in\mathbb{R}^2_+| x_2+x_3 = 130 \vee x_3 = 70 \vee x_2 = 70\}$

So we can first plot the lines $x_2+x_3 = 130$, $x_3 = 70$, $x_2 = 70$ and get:

Now we can eliminate the part of these lines where $\max(x_2+x_3, 60+x_3, 60+x_2) > 130$ and get the indifference curve: