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Microeconomics, a synthesis of modern and neoclassical theory

R. Robert Russell, Maurice Wilkinson

Chapter 3

UTILITY THEORY - all with Video Answers

Educators

Chapter Questions

01:29

Problem 1

Consider the utility functions defined by
$$
\begin{aligned}
& U(x)=x_1 \cdot x_2 . \\
& O(x)=\log x_1+\log x_2 .
\end{aligned}
$$
and
$$
O(x)=\left(x_1\right)^2 \cdot\left(x_2\right)^2 .
$$

Do they imply different consumer preference orderings? Demonstrate.

Heather Zimmers
Heather Zimmers
Numerade Educator
00:01

Problem 2

Global nonsatiation is defined as follows: for all $x$ in $X$, there exists an $\vec{x}$ in $X$ such that $\dot{x} \succ x$.
a. Show that global nonsatiation, together with strict convexity (assumption 3.2), implies local nonsatiation: for all $x$ in $X$, there is no neighborhood of $x$ such that $x \geqslant \bar{x}$ for all $\bar{x}$ in the neighborhood. ${ }^{17}$ (Note: local nonsatiation rules out "thick" indifference surfaces-i.e." "indifference bands" in two-space. See figure 17.8.)
b. Show that local nonsatiation implies that the consumer spends all of his income (so that the budget constraint is satisfied as an equality).

Oluwadamilola Ameobi
Oluwadamilola Ameobi
Numerade Educator
01:09

Problem 3

Show that the sign but not the magnitude of $\partial u(x) / \partial x_i$ is invariant under a monotonic transformation of $U$.

Adriano Chikande
Adriano Chikande
Numerade Educator
04:18

Problem 4

Prove that the assumption of diminishing and independent marginal utilities,
$$
\frac{\partial U^2(x)}{\partial x_i^2}<0, \quad i=1, \ldots, n
$$
and
$$
\frac{\partial^2 u(x)}{\partial x_i \partial x_j}=0, \quad i \neq i, \quad i, j=1, \ldots, n,
$$
is stronger than the assumption of diminishing marginal rates of substitution.

Akash M
Akash M
Numerade Educator
04:42

Problem 5

Draw the general shape of the indifference curves associated with each of the following situations. Also, put in a budget constraint and identify the optimum.
a. Nickels and dimes are the two commodities.
b. Right and left shoes are the two commodities.
c. The more salted peanuts you eat the more you want (use money as the other commodity).
d. You would have to be paid to eat jalapeño peppers (use money as the other commodity).
e. You can't tell the difference between margarine and "the high-priced spread."

Rashmi Sinha
Rashmi Sinha
Numerade Educator

Problem 6

Derive the first-order conditions for maximizing
$$
U(x)=\sum_{i=1}^n \beta_i \cdot \log \left(x_i-\gamma_i\right) .
$$
where $\beta_i$, and $\gamma_i, i=1, \ldots, n$, respectively, are positive and nonpositive parameters, subject to a budget constraint.

Check back soon!
01:41

Problem 7

What is the reason for the restrictions on the parameters, $\beta_i$ and $\gamma_i$, $i=1, \ldots, n$, in exercise 3.6 ? (This one is tricky.)

Linh Vu
Linh Vu
Numerade Educator