#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass article
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100

\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize a4paper
\use_geometry true
\use_amsmath 1
\use_esint 1
\use_mhchem 1
\use_mathdots 1
\cite_engine basic
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\use_refstyle 1
\index Index
\shortcut idx
\color #008000
\end_index
\paperwidth 15cm
\paperheight 24cm
\leftmargin 2.5cm
\topmargin 2.5cm
\rightmargin 2.5cm
\bottommargin 2.5cm
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header

\begin_body

\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\align center

\family typewriter
\size large
Prof.
 Marek Weretka's
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\noindent
\align center

\series bold
\size large
Econ 301 Intermediate Microeconomics 
\end_layout

\begin_layout Standard
\paragraph_spacing double
\noindent
\align center

\series bold
\size huge
Problem Set 6
\end_layout

\begin_layout Standard
\begin_inset Formula $\vphantom{}$
\end_inset


\end_layout

\begin_layout Subsubsection*
\paragraph_spacing onehalf
Problem 1 (Insurance) 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
Ben is a proud owner of a romantic mansion facing the lake Mendota worth
 $
\begin_inset Formula $500.000.$
\end_inset

 The location attractive as it is, however has one drawback.
 Occasionally in the spring heavy rains raise the water level in the lake,
 flooding the house.
 When this happens, the value of the house drops to $
\begin_inset Formula $50.000$
\end_inset

.
 The flood occurs with probability 
\begin_inset Formula $1/10$
\end_inset

.
 Ben finds the situation too stressful and therefore he is going to sell
 the house in the summer (after the potential flood).
 By 
\begin_inset Formula $c_{F}$
\end_inset

 and 
\begin_inset Formula $c_{NF}$
\end_inset

 denote his wealth when there is, and there is no flood respectively.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
a) In the commodity space (
\begin_inset Formula $c_{F},c_{NF}$
\end_inset

) show the affordable bundle if there is no insurance (the endowment point).
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
b) Ben can buy insurance that pays $x dollars when there is a food, paying
 premium 
\begin_inset Formula $0.1x$
\end_inset

 (he can choose 
\begin_inset Formula $x$
\end_inset

) - find analytically and show on the graph his budget constraint.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
c) Suppose Ben's Von Neumann Morgenstern utility function is given by
\end_layout

\begin_layout Standard
\begin_inset Formula $\vphantom{}$
\end_inset


\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\align center
\begin_inset Formula $U(C_{F},C_{NF})=0.1√(C_{F})+0.9√(C_{NF})$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\vphantom{}$
\end_inset


\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
Is Ben risk averse (argue using a graph of his Bernoulli utility function)?
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
d) What is his MRS at the endowment point (no insurance)? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
e) Find the optimal insurance and wealth levels under two contingencies.
 Does he insure fully? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
f) On the graph show how your answer changes if the insurance premium is
 
\begin_inset Formula $0.2x$
\end_inset

? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\begin_inset Formula $\vphantom{}$
\end_inset


\end_layout

\begin_layout Subsubsection*
\paragraph_spacing onehalf
Problem 2 (Risk aversion and certainty equivalence) 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
Frank McGambler needs our advice.
 He is thinking about a lottery ticket that gives $
\begin_inset Formula $100$
\end_inset

 with probability (
\begin_inset Formula $1/2$
\end_inset

) and zero otherwise (also with probability (
\begin_inset Formula $1/2$
\end_inset

)).
 The gamble's alternative is $
\begin_inset Formula $40$
\end_inset

 for sure.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
a) Plot the Bernoulli utility function, given by:
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\align center
\begin_inset Formula $u(c)=√c$
\end_inset

 on a graph.
 Is Frank risk averse? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
b) What is the expected value of the lottery? (give the number and mark
 it on the graph).
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
c) What is the expected utility from the lottery? (give a number and mark
 it on the graph).
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
d) What is the certainty equivalent of the lottery? Is it greater or smaller
 than the expected value of the lottery? Write an economic interpretation
 of this number.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
e) Which one should Frank chose to maximize his utility ? $
\begin_inset Formula $4$
\end_inset

0 for sure or the lottery? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
f) Give answers to questions b-e, assuming Bernoulli utility function is
 : 
\begin_inset Formula $u(c)=c$
\end_inset

.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
g) Give answers to questions b-e, assuming Bernoulli utility function is
 : 
\begin_inset Formula $u(c)=c²$
\end_inset

.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\begin_inset Formula $\vphantom{}$
\end_inset


\end_layout

\begin_layout Subsubsection*
\paragraph_spacing onehalf
Problem 3 (Standard Edgeworth Box) 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
Elvis and Miriam both love listening to MP3s 
\begin_inset Formula $(x_{1})$
\end_inset

 and watching DVDs 
\begin_inset Formula $(x_{2}).$
\end_inset

 Elvis has initially 10 DVDs and 10 MP3s, (hence 
\begin_inset Formula $ω^{E}=(10,10)$
\end_inset

).
 Miriam has 90 MP3s only, (so 
\begin_inset Formula $ω^{M}=(90,0)$
\end_inset

).Utility functions of Elvis and Miriam are the same and given by:
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\align center
\begin_inset Formula $U^{i}(x_{1},x_{2})=ln(x_{1})+5ln(x_{2})$
\end_inset


\end_layout

\begin_layout Standard

\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
a) What are the total resources in the "economy" with Elvis and Miriam?
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
b) Plot an Edgeworth box and mark the allocation corresponding to initial
 endowments.
 Argue that such an allocation is (or it is not) 
\shape italic
pareto efficient
\shape default
.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
c) Find analytically the contract curve (collection of all 
\shape slanted
pareto efficient
\shape default
 allocations) and plot it on your graph.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
d) Now let Elvis and Miriam trade.
 Find the equilibrium consumption of both commodities for Elvis and Miriam,
 and the equilibrium prices.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
e) Suggest two other prices that support the same allocation as an equilibrium.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
f) Are the markets efficient in allocating the resources? (argue using the
 criterion of pareto efficiency) 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
g) Find geometrically (in the Edgeworth box) the equilibrium consumption
 and prices of both commodities for the new utility (perfect substitutes)
 given by:
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\align center
\begin_inset Formula $U^{i}(x_{1},x_{2})=x_{1}+5x_{2}$
\end_inset


\end_layout

\begin_layout Subsubsection*
\paragraph_spacing onehalf
Problem 4 (Uncertainty and Asset Pricing) 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
John and Benjamin are investors who trade shares of two companies: Rainalot
 Inc.
 
\begin_inset Formula $(x_{1})$
\end_inset

 and HateRain Inc.
\begin_inset Formula $(x_{2}$
\end_inset

).
 There are two equally likely states of the world in the future: rain and
 no rain, and the profits of two companies are risky.
 The dividend from one share of Rainalot Inc is $1 when it rains and $0
 otherwise and dividends of HateRain Inc.
 are $0 when it rains and $1 otherwise.
 John initially has 100 shares of Rainalot Inc and no shares of HateRain
 Inc.
\begin_inset Formula $(ω^{J}=(100,0))$
\end_inset

 and the endowment of Benjamin is 
\begin_inset Formula $ω^{B}=(0,100)$
\end_inset

.
 Both investors maximize Expected utility given by
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\align center
\begin_inset Formula $U^{i}(x_{1},x_{2})=(1/2)ln(x_{1})+(1/2)ln(x_{2})$
\end_inset


\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
a) Plot an Edgeworth box and mark the allocation corresponding to the initial
 endowments.
 Argue that such an allocation is (or it is not) 
\shape italic
pareto efficient
\shape default
.
 Are the endowments risky? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
b) Find the equilibrium prices of the two companies' shares and their allocation
s,.
 Show them on the graph.
 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
c) Is the equilibrium allocation efficient? Is it risky? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\begin_inset Formula $\vphantom{}$
\end_inset


\end_layout

\begin_layout Subsubsection*
\paragraph_spacing onehalf
Problem 5 (Irving Fisher Determination of Interest Rate) 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
Consumption can take place in two periods only: today 
\begin_inset Formula $(C_{1})$
\end_inset

 and tomorrow 
\begin_inset Formula $(C_{2})$
\end_inset

.
 Jane's income is $0 today (she is a student now).
 Tomorrow she is going to be a CEO with income $1000 (hence her endowment
 is 
\begin_inset Formula $ω^{J}=(0,1000)$
\end_inset

).
 William is a sportsman with $
\begin_inset Formula $1000$
\end_inset

 income today, and tomorrow he will get $0 
\begin_inset Formula $(ω^{W}=(1000,0))$
\end_inset

.
 They both have the same utility function:
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
\align center
\begin_inset Formula $U^{i}(C_{1},C_{2})=ln(C_{1})+βln(C_{2})$
\end_inset


\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
where 
\begin_inset Formula $β$
\end_inset

 is a discount factor that tells how impatient they are (the higher the
 
\begin_inset Formula $β$
\end_inset

, the more patient the consumer is since the value of utility tomorrow relative
 to utility today is higher).
 
\end_layout

\begin_layout Standard
\begin_inset Formula $\vphantom{}$
\end_inset


\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
a) Plot an Edgeworth box and mark the allocation corresponding to the initial
 endowments.
 Is the allocation of initial endowments Pareto efficient? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
b) Assume 
\begin_inset Formula $β=(1/2)$
\end_inset

.
 Find the equilibrium interest rate and depict the equilibrium on the Edgeworth
 box.
 (Hint: Instead of working with a harder "intertemporal" model, you can
 first find equilibrium prices 
\begin_inset Formula $p_{1}$
\end_inset

 and 
\begin_inset Formula $p_{2}$
\end_inset

 similar to the apple and orange model and then use: 
\begin_inset Formula $((p_{1})/(p{}_{2}))=(1+r)$
\end_inset

 )
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
c) Is the equilibrium allocation 
\shape slanted
pareto efficient
\shape default
? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
d) Assume now that consumers are more patient and 
\begin_inset Formula $β=1$
\end_inset

.
 Repeat the question in part b).
 How does your new equilibrium interest rate compare to the one in part
 b? Can you give some economic intuition about your result? 
\end_layout

\begin_layout Standard
\paragraph_spacing onehalf
e) Now assume 
\begin_inset Formula $β=0.5$
\end_inset

, and Jane's income tomorrow changes to 
\begin_inset Formula $2000$
\end_inset

 (
\begin_inset Formula $ω^{J}=(0,2000)$
\end_inset

).
 Is the interest rate higher or lower than the one in part b)? Explain.
\end_layout

\end_body
\end_document