# Exercises: Consumer preferences and behavior/demand (‘problem set’)

Exeter students: more detailed answers to these questions are provided in the MS Teams class notebook.

O-R chapter 4

Q1a:

Describe the following preference relation formally, giving a utility function that represents the preferences if possible.

Draw some representative indifference sets, and determine whether her preferences are monotone, continuous, and convex.

1. The consumer prefers the bundle $(x_1,x_2)$ to the bundle $(y_1, y_2)$ if and only if $(x_1,x_2)$ is further from (0,0) than is (y1, y2), where the distance between the $(z_1, z_2)$ and $(z_1' , z_2')$ is $\sqrt{(z_1 − z_1')^2 + (z_2 − z_2')^2}$ (the ‘Euclidean distance norm’).

Exeter students: I covered this question in a recorded session (towards the end of the session), which you should be able to access HERE

Q2. … as in Q1 but for…

1. $max\{x_1, x_2\}$

(skip b)

1. $log(x_1 + 1) + log(x_2 + 1)$

Q4 Quasilinear preferences.

A preference relation is represented by a utility function of the form $u(x_1,x_2) = x_2 + g(x_1)$, where $g$ is a continuous increasing function.

1. How does each indifference set for this preference relation relate geometrically to the other indifference sets?

O-R chapter 5

Problem 2: Cobb-Douglas preferences

A consumer’s preference relation is represented by the utility function $x_1^\alpha x_2 ^{1−\alpha}$ where $0 < \alpha < 1$.

These preferences are convex and differentiable. Show that for all prices and wealth levels the consumer spends the fraction $\alpha$ of his budget on good 1.

Exeter students: I covered this question in a recorded session you should be able to access HERE

Problem 5 (introduces the ‘expenditure function’; challenging)

A consumer’s preference relation is monotone, continuous, and convex.

Let $x^{\ast} = (x_1^{\ast},x_2^{\ast})$ be a bundle.

For any pair $(p_1, p_2)$ of prices, let $e((p_1, p_2), x^{\ast})$ be the smallest wealth that allows the consumer to purchase a bundle that is at least as good for him as $x^{\ast}$:

$e((p_1, p_2), x^{\ast}) = \min_{(x1,x2)}\{p_1 x_1 + p_2 x_2 : (x_1,x_2) \succsim (x_1^{\ast}, x_2^{\ast})\}$.

(See Figure 5.9 in O-R)

1. Show that the function $e$ is increasing in $p_1$ (and $p_2$).

Unfold a hint:

Let $Y = \{x \in X : x \succsim x^{\ast} \}$. (The set of all bundles that are ‘at least as good’ as $x^{\ast}$).

Let $a$ be a bundle that is least expensive in $Y$ for the price vector $(p_1 +\epsilon , p_2)$, where $\epsilon>0$.

1. Show that for all $\lambda > 0$ and every pair $(p_1, p_2)$ of prices we have

$\Big((\lambda p_1, \lambda p_2), x^{\ast}\Big) = \lambda e\Big((p_1, p_2),x_{\ast}\Big)$.

(In other words, show that if prices of all goods double, the minimum expenditure to maintain the original utility must double.)

• Problems 8-9 (re-)introduce ‘time preferences’

Possible further questions tbd (unfold for possibilities)

• Business case study/investigation: modeling ‘which (sorts of) new product will consumers prefer?’

• Finding departures from the WARP in real data