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


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:

Hint … start of answer:

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