Problem Set 1: Modeling Preferences with Multivariable Calculus
Due Saturday, September 27 at 11pm on Gradescope
Note: while for some assignments you are welcome to skip a question or two, for this one I would encourage you to do all five questions. However, read carefully: for the first three, you don’t need to do all the utility functions if you don’t want to!
Preferences and Utility
Suppose $A = (4,16)$ and $B = (16,4)$. 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): \(u(x_1,x_2) = \min\{x_1,4x_2\} = \begin{cases}x_1 & \text{ if }x_1 \le 4x_2 \\ 4x_2 & \text{ if }x_1 \ge 4x_2\end{cases}\)
For (g), also evaluate the utility at point $C = (16,1)$. How does this utility compare to the other two bundles?
Marginal Utility and the MRS
For each of the functions you chose in the first question, find:
- the expression for the marginal utility of each good at an arbitrary point; that is, $MU_1(x_1,x_2)$ and $MU_2(x_1,x_2)$
- the expression for the marginal rate of substitution at an arbitrary point; that is, $MRS(x_1,x_2)$
- the values of $MU_1$, $MU_2$, and $MRS$ at points $A = (4,16)$ and $B = (16,4)$. If you do (g), also evaluate these at point $C = (16,1)$…and note that for the purposes of this class, $1/0 = \infty$, while $0/0$ is undefined…
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. At points $A$ and $B$, indicate the MRS by drawing a tangent line with the precise slope you calculated in the last question. (For example, if you found that the MRS was 2 at point $A$, you should draw a line with slope $-2$ passing through $A$; i.e., from $(3,11)$ to $(5,7)$).
Here’s a template to use if you need one:
Using the MRS to sketch an indifference curve
Let’s now try plotting a slightly more difficult indifference curve. Consider the utility function \(u(x_1,x_2) = 16 \ln x_1 + x_2\) Repeat exercises 1.1-1.3 for this function.
- To answer 1.1, feel free to use a calculator to evaluate the utilities.
- To answer 1.2, use the fact that if $f(x) = \ln(x)$, then $f^\prime(x) = 1/x$.
- To answer 1.3, first plot the tangent lines at bundles $A$ and $B$; this tells you what the slope of the indifference curves at those points must be. Then, use the expression for the MRS you found in 1.2 to think about what the shape of the indifference curve must be. Does the slope get flatter or steeper as you move along an indifference curve to the right? Finally, bearing in mind your answer to 1.1 (so you know which bundle is preferred), sketch the two indifference curves.
More Fruit
Allana is choosing between two baskets of fruit. Each contains some apples (good 1) and some bananas (good 2). The first has more total fruit than the second; however, the first has more apples than bananas, and the second has more bananas than apples. Allana’s preferences may be represented by the utility function \(u(x_1,x_2) = x_1^ax_2^b\) where $a$ and $b$ are both strictly greater than zero.
State whether each of the following statements is true or false. If true, prove why. If false, provide a counterexample consistent with the story above. (That is, make up some numbers for how much fruit is in each basket, as well as values for $a$ and $b$.)
Allana must prefer the first basket, because it has more total fruit.
Allana’s MRS at the first bundle must be lower than her MRS at the second bundle.
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: