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\begin{document}
\bigskip \textbf{Econ 301}
\textbf{Intermediate Microeconomics}
\textbf{Prof. Marek Weretka}
\begin{center}
\textbf{Final}
\end{center}
{\small You have 2h to complete the exam. The final consists of 6 questions
(10+10+15+25+25+15=100). }
\bigskip
\textbf{Problem 1.}
{\small Ace consumes bananas }$x_{1}${\small \ and kiwis }$x_{2}$.{\small \
The prices of both goods are }$p_{1}=4,p_{2}=10${\small \ and Ace's income
is }$m=120${\small . His utility function is }%
\[
{\small U}\left( x_{1},x_{2}\right) {\small =}\left( x_{1}\right)
^{20}\left( x_{2}\right) ^{40}
\]
{\small a) Find analytically Ace's }${\small MRS}${\small \ as a function of
}$\left( {\small x}_{1}{\small ,x}_{2}\right) ${\small \ (give a function)
and find its value for the consumption bundle }$\left( {\small x}_{1}{\small %
,x}_{2}\right) {\small =}\left( {\small 20,20}\right) $. {\small Give its
economic and geometric interpretation (one sentence and find }${\small MRS}$%
{\small \ on the graph) }
{\small b) Give two secrets of happiness that determine Ace's optimal choice
of fruits (give two equation). Explain why violation of any of them implies
that the bundle is not optimal (one sentence for each condition).}
{\small c) Using magic forumula find the optimal bundle of Ace (two
numbers), and show geometrically the .}
\bigskip
\textbf{Problem 2.}
{\small Adria collects two types of rare coins: Jefferson Nickels }$x_{1}$%
{\small and Seated Half Dimes }$x_{2}$.{\small \ Her utility from a
collection }$({\small x}_{1}{\small ,x}_{2})${\small \ is}%
\[
{\small U}\left( x_{1},x_{2}\right) {\small =x}_{1}+{\small x}_{2}
\]
{\small a) Propose a utility function that gives a higher level of utility
for any }$(x_{1},x_{2}{\small )}${\small , but represents the same
preferences (give utility function).}
{\small b) Suppose the prices of the two types of coins are }${\small p}_{1}%
{\small =4}${\small \ and }${\small p}_{2}{\small =2}${\small \ for }$%
x_{1},x_{2}${\small \ respectively} {\small and the Adria's income is }$%
{\small m=\$20}${\small . Plot her budget set and find the optimal
collection }$\left( x_{1},x_{2}\right) ${\small \ and mark it in your graph
(give two numbers)}
{\small c) Are the coins Giffen goods (yes or no and one sentence explaining
why)?}
{\small d) Harder:\ Suppose Adria's provider of coins currently has only six
Seated Half Dimes }$x_{2}${\small \ in stock (hence }$x_{2}\leq 6${\small ).
Plot a \ budget set with the extra constraint and find (geometrically) an
optimal collection given the constraint.}
\bigskip
\textbf{Problem 3. (Equilibrium)}
{\small There are two commodities traded on the market: umbrellas }$x_{1}$%
{\small \ and swimming suits }$x_{2}${\small . Abigail has ten umbrellas and
twenty swimming suits (}${\small \omega }^{A}{\small =(10,20)}${\small \ )}.
{\small Gabriel has forty umbrellas and twenty swimming suits (}${\small %
\omega }^{G}{\small =(40,20)}${\small ). Abigail and Gabriel have identical
utility functions given by }%
\[
{\small U}^{i}\left( x_{1},x_{2}\right) {\small =}\frac{{\small 1}}{{\small 2%
}}\ln \left( x_{1}\right) {\small +}\frac{{\small 1}}{{\small 2}}\ln \left(
x_{2}\right)
\]
{\small a) Plot an Edgeworth box and mark the point corresponding to
endowments of Abigail and Gabriel. }
{\small b) Give a definition of a Pareto efficient allocation (one sentence)
and the equivalent condition in terms of }${\small MRS}${\small \
(equation). Verify whether endowment is Pareto efficient (two numbers+one
sentence).}
{\small c) Find prices and an allocation of umbrellas and swimming suits in
a competitive equilibrium and mark it in your graph.}
{\small d) Harder: Plot a contract curve in the Edgeworth box assuming
utilities for two agents }$U^{i}\left( x_{1},x_{2}\right) =\min (x_{1},x_{2})
${\small \ }.
\bigskip
\textbf{Problem 4.(Short questions)}
{\small a) You are going to pay taxes of }$\$200$ {\small every year,
forever. Find the Present Value of your taxes if the yearly interest rate is
}$r=10\%${\small . }
{\small b) Consider a lottery that pays }$0${\small \ with probability }$%
\frac{1}{2}${\small \ and }$16${\small \ with probability }$\frac{1}{2}$
{\small and a Bernoulli utility function is }$u\left( x\right) =\sqrt{x}$%
{\small . Give a corresponding von Neuman-Morgenstern utility function. Find
the certainty equivalent of the lottery. Is it bigger or smaller than the
expected value of the lottery? Why? (give a utility function, two numbers
and one sentence.)}
{\small c) Give an example of a Cobb-Douglass production function that is
associated with increasing returns to scale, decreasing MPK and decreasing
MPL (give a function). Without any calculations, sketch the average total
cost function (}$ATC${\small ) associated with your production function.}
{\small d) Let the variable cost be }$c\left( y\right) =y^{2}$ {\small and
fixed cost }$F=4$. {\small Find }${\small ATC}^{MES}${\small \ and }${\small %
y}^{MES}${\small \ (two numbers).\ Given demand }${\small D}\left( p\right)
{\small =8-p\ }${\small determine a number of firms in the industry assuming
free entry (and price taking). Is the industry monopolistic, duopolistic,
oligopolistic or perfectly competitive? Find Herfindahl--Hirschman Index
(HHI) of this industry (one number).}
{\small e) In a market for second-hand vehicles two types of cars can be
traded: lemons (bad quality cars) and plums (good quality ones). The value
of a car depends on its type and is given by }
\begin{center}
\[
\begin{tabular}{lll}
& {\small Lemon} & {\small Plum} \\
{\small Seller} & ${\small 0}$ & ${\small 20}$ \\
{\small Buyer} & ${\small 10}$ & ${\small 26}$%
\end{tabular}%
\]
\end{center}
{\small Will we observe plums traded on the market if the probability of a
lemon is equal to }$\frac{1}{2}?${\small \ (compare two relevant numbers).
Is the equilibrium outcome Pareto efficient (yes-no answer+ one sentence)?
Give a threshold probability for which we might observe pooling equilibrium
(number).}
\bigskip
\textbf{Problem 5.(Market Power)}
{\small Consider an industry with the inverse demand equal to }${\small p}%
\left( y\right) {\small =6-y},${\small \ and suppose that the total cost
function is }${\small TC=2y}${\small .}
{\small a) What are the total gains to trade in this industry? (give one
number)}
{\small b) Find the level of production and the price if there is only one
firm in the industry (i.e., we have a monopoly) charging a uniform price
(give two numbers). Find demand elasticity at optimum. (give on number)
Illustrate the choice using a graph. Mark a DWL.}
{\small c) Find the profit of the monopoly and a DWL given that monopoly
uses the first degree price discrimination.}
{\small d) Find the individual and aggregate production and the price in a
Cournot-Nash equilibrium given that there are two firms (give three
numbers). Show DWL in the graph.}
{\small e) In which of the three cases, (b,c or d) the outcome is Pareto
efficient? (chose one+ one sentence)}
\bigskip
\textbf{Problem 6.(Externality)}
{\small A bee keeper chooses the number of hives }$h${\small . Each hive
produces one pound of honey which sells at the price of }$\$10${\small \ per
pound. The cost of holding }$h${\small \ hives is }$TC\left( h\right) =\frac{%
1}{2}h^{2}.${\small \ Consequently the profit of bee keeper is equal to}%
\[
{\small \pi }_{h}\left( {\small h}\right) {\small =10h-}\frac{{\small 1}}{%
{\small 2}}{\small h}^{2}
\]%
{\small The hives are located next to an apple tree orchard. The bees
pollinate the trees and hence the total production of apples }$y=h+t$
{\small is increasing in number of trees and bees. Apples sell for }${\small %
\$3}${\small \ and the cost of }$t${\small \ trees is }${\small TC}%
_{t}\left( t\right) =\frac{1}{2}t^{2}.${\small \ Therefore the profit of an
orchard grower is }%
\[
\pi _{t}\left( t\right) =3\left( {\small t+h}\right) -\frac{{\small 1}}{%
{\small 2}}{\small t}^{2}
\]
{\small a) Market outcome: Find the level of hives }$h${\small \ that
maximizes the profit of a beekeeper and the number of trees that maximizes
the profit of an orchard owner (assuming }$h$ {\small optimal for a bee
keeper) (two numbers)}
{\small b) Find the Pareto efficient level of }$h${\small \ and }$t.${\small %
\ Are the two values higher or smaller then the ones in a)? Why? (two
numbers + one sentence)}
\end{document}