%% LyX 2.3.3 created this file. For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\setlength{\parskip}{\smallskipamount}
\setlength{\parindent}{0pt}
\usepackage{calc}
\usepackage{amstext}
\usepackage{amssymb}
\usepackage{babel}
\begin{document}
\title{8100 Problem Set 5.}
\maketitle
1. Find the Marshallian demand for the utility function: (assume $\alpha,\beta,\gamma>0$
and $a,b,c\geq0.$) \emph{Mind the corners.}
\[
\left(x_{1}+a\right)^{\alpha}\left(x_{2}+b\right)^{\beta}\left(x_{3}+c\right)^{\gamma}
\]
\\
2. Consider an environment of choice under uncertainty. There are
finite outcomes $A=\left\{ a_{1},a_{2},...,a_{n}\right\} $ and you
can assume $a_{i}\succ a_{j}$ for $i0$.
A consumer's preferences over compound gambles are such that $g\succ g'$
if and only if $b\left(g\right)\succ b\left(g'\right)$ or $b\left(g\right)\sim b\left(g'\right)$
and $p\left(b\left(g\right)\right)>p\left(b\left(g'\right)\right)$.
\\
\fbox{\begin{minipage}[t]{1\columnwidth - 2\fboxsep - 2\fboxrule}%
Let $\succsim$ be the preference relation on $\mathcal{G}$ (the
set of compound gambles).
Axiom 1. \textbf{\emph{Complete: }}\emph{$\succsim$ is complete.}
Axiom 2. \textbf{\emph{Transitive: }}\emph{$\succsim$ is transitive.}\\
Axiom 3. \textbf{\emph{Monotonic:}}\emph{ }For all $\left(\alpha\circ a_{1},\left(1-\alpha\right)\circ a_{n}\right)\succsim\left(\beta\circ a_{1},\left(1-\beta\right)\circ a_{n}\right)$
iff $\alpha\geq\beta$,
Axiom 4. \textbf{\emph{Continuous:}}\emph{ }For all $g$ $\exists p\in\left[0,1\right]$
such that $g\sim\left(p\circ a_{1},\left(1-p\right)\circ a_{n}\right)$
Axiom 5. \textbf{\emph{Substitution:}}\emph{ }If $g=\left(p_{1}\circ g_{1},...,p_{k}\circ g_{k}\right)$
and $h=\left(p_{1}\circ h_{1},...,p_{k}\circ h_{k}\right)$ and if
$g_{i}\sim h_{i}$ for all $i\in\left\{ 1,...,k\right\} $ then $g\sim h$.
Axiom 6. \textbf{\emph{Reduction:}} For any gamble $g$ and the simple
gamble it induces $g_{s}$, $g\sim g_{s}$. \\
%
\end{minipage}}
A) Among \textbf{completeness, transitivity, monotonicity, continuity,
substitution, reduction}. Which assumptions are met by these preferences?
B) Can you construct a utility function that represents these preferences?\\
3. A consumer is an expected utility maximizer and has a utility function
for wealth equal to $v\left(w\right)=\sqrt{w}$.
A) If the consumer starts with $\$0$, what is their certainty equivelent
for game that pays $\$x$ with $p=0.5$ and $\$0$ with $p=0.5$.
B) If the consumer starts with $w_{0}$, what is their certainty equivelent
for the same gamble?
C) As the consumer becomes more wealthy ($w$ increases) how does
their certainty equivalent for this gamble compare to the certainty
equivalent for a risk-neutral consumer?\\
4. Consider the production function:
\[
f\left(x_{1},x_{2}\right)=\left(x_{1}^{r}+x_{2}^{r}\right)^{\frac{1}{2r}}
\]
A) Find the conditional factor demands.
B) What is the cost function?
C) Show the cost function can be decomposed into the cost of producing
one unit and a power function of output $y$.
D) What is the profit function when output price of $y$ is $p$?\\
5. Consider the production function:
\[
f\left(x_{1},x_{2}\right)=\left(x_{1}+x_{2}\right)^{\frac{1}{4}}+x_{3}^{\frac{1}{2}}
\]
A) Show that the ratio of marginal products of $x_{1}$ and $x_{2}$
do not depend on the level of $x_{3}$.
B) What is the cost of producing $y_{1}$ units of output using only
$x_{1}$ and $x_{2}$.
C) What is the cost of producing $y_{2}$ units of output using only
$x_{3}$?
D) What is the cost of producing $y$ units of output from $x_{1},x_{2},x_{3}$
when $w_{1}=w_{2}=w_{3}=1$?
E) What is firms profit when output price of $y$ is $p$ and $w_{1}=w_{2}=w_{3}=1$?\\
\end{document}