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\begin{document}
\textbf{Econ 301}
\textbf{Intermediate Microeconomics}
\textbf{Prof. Marek Weretka}
\begin{center}
\textbf{Final Exam (Group A)}
\end{center}
{\small You have 2h to complete the exam. The final consists of 6 questions
(25+20+20+15+10+10=100). }
\bigskip
\textbf{Problem 1. (Quasilinaer income effect)}
{\small Mirabella consumes chocolate candy bars }$x_{1}${\small \ and fruits
}$x_{2}$.{\small \ The prices of the two goods are }${\small p}_{1}{\small %
=4,p}_{2}{\small =4},${\small \ respectively and Mirabella's income is }$%
{\small m=20}${\small . Her utility function is }%
\[
{\small U(x}_{1}{\small ,x}_{2}{\small )=2\ln x}_{1}{\small +x}_{2}
\]
{\small a) In the commodity space plot Mirabella's budget set. Find the
slope of budget line (one number). Provide the economic interpretation of
the slope (one sentence). }
{\small b) Find analytically formula that gives Mirabella's }${\small MRS}$%
{\small \ for any bundle }$\left( {\small x}_{1}{\small ,x}_{2}\right) $%
{\small \ (a function). Give the economic and the geometric interpretation
of }${\small MRS}${\small \ (two sentences). Find the value of }${\small MRS}
${\small \ at bundle }${\small (x}_{1},{\small x}_{2}{\small )=(4,4)}$
{\small (one number). At this bundle, which of the two commodities is
(locally) more valuable? (chose one)}
{\small c) Write down two secrets of happiness that determine Mirabellas's
optimal choice (two equation). Provide the geometric interpretation of the
conditions in the commodity space. }
{\small d) Find Mirabella's optimal choice (two numbers). Is solution
interior (yes-no answer).}
{\small e) Suppose the price of a chocolate candy bar goes down to }${\small %
p}_{1}{\small =2},$ {\small while other price }${\small p}_{2}{\small =4}$
{\small and income }${\small m=20}$ {\small are unchanged. Find the new
optimal choice (two numbers). Is a chocolate candy bar an ordinary or Giffen
good (pick one)?}
{\small f) Decompose the change in demand for }${\small x}_{1}$ {\small in
points d) and e) into a substitution and income effect.}
\bigskip
\textbf{Problem 2. (Equilibrium)}
{\small Consider an economy with two consumers, Adalia and Briana and two
goods: bicycles }$x_{1}${\small \ and flowers }$x_{2}${\small . Adalia
initial endowment of the commodities is }${\small \omega }^{A}{\small %
=(40,60)}${\small \ and Briana endowment is }${\small \omega }^{B}{\small %
=(60,40)}${\small . Adalia and Briana utility functions are given by, }$%
{\small i=A,B}${\small \ }%
\[
{\small U}^{i}{\small (x}_{1}{\small ,x}_{2}{\small )=4\ln x}_{1}{\small %
+4\ln x}_{2}
\]
{\small a) Plot an Edgeworth box and mark the point that corresponds to
initial endowments. }
{\small b) Give a definition of a Pareto efficient allocation (one sentence).%
}
{\small c)} {\small Give a (general) equivalent condition for Pareto
efficiency in terms of }${\small MRS.}$\ {\small Provide geometric arguments
that demonstrate the necessity and sufficiency of }${\small MRS}${\small \
condition for Pareto efficiency.}
{\small d) Find competitive equilibrium (six numbers). Depict the obtained
equilibrium in the Edgeworth box. Using }${\small MRS}$ {\small condition
verify that the equilibrium is Pareto efficient. }
{\small e) Using (one of) the secrets of happiness prove that a competitive
equilibrium is Pareto efficient in any economy.}
\bigskip
\textbf{Problem 3. (Short questions)}
{\small a) Using }$\lambda $ {\small argument prove that Cobb-Douglass
production function }${\small y=2KL}$ {\small exhibits increasing returns to
scale. Without any calculations, sketch total cost function }${\small c(y)}$%
{\small \ corresponding to the production function.}
{\small b) Now consider a firm (different from point a)) with variable cost }%
${\small c(y)=2y}^{2}$ {\small and fixed cost }${\small F=2}$. {\small Find }%
${\small ATC}^{MES}${\small \ and }${\small y}^{MES}${\small \ (two
numbers).\ In a long-run equilibrium with free entry how many firms should
be expect in the industry if inverse demand is }${\small D(p)=10-p?}$
{\small c) Suppose a Bernoulli utility function is }${\small u(x)=x}^{2}$
{\small and two states are equally likely (probability }$\frac{1}{2}$).
{\small Write down the corresponding von Neuman-Morgenstern utility
function. Find the certainty equivalent and the expected value of lottery }$%
{\small (0,2)}$ {\small (two numbers). Which of the two is bigger and why?
(two numbers and one sentence.)}
{\small d) Find Herfindahl--Hirschman Index (HHI) for industry with }$%
{\small N=50}${\small \ identical firms (one number). Is the industry
concentrated?}
{\small e) Derive formula for the present value of perpetuity.}
\newpage
\textbf{Problem 4. (Market Power)}
{\small Consider an industry with inverse demand }${\small p}\left( y\right)
{\small =8-y},${\small \ and a monopoly with cost function }${\small TC}%
\left( y\right) {\small =0}$ {\small who cannot discriminate.}
{\small a) What are the total gains-to-trade (or potential total surplus) in
this industry? (give one number)}
{\small b) Write down monopoly's profit function. Derive the condition on }$%
{\small MR}${\small \ and }${\small MC}$ {\small that gives profit
maximizing level of production. Provide economic interpretation of this
condition. }
{\small c) Find the level of production, the price, the deadweight loss and
the elasticity of the demand at optimum (four numbers). Illustrate the
choice in a graph. }
{\small d) Assuming the same demand function find the individual and the
aggregate level of production and the price in the Cournot-Nash equilibrium
with }${\small N=3}${\small \ identical firms (give three numbers). Show the
deadweight loss in the graph.}
\bigskip
\textbf{Problem 5.(Externality)}
{\small Lucy is addicted to nicotine. Her utility from smoking }$c$ {\small %
cigarettes (net of their cost) is given by }%
\[
{\small U}^{L}\left( {\small c}\right) {\small =2\ln c-c}
\]
{\small Her sister Taja prefers healthy lifestyle, her favorite commodity is
orange juice, }$j${\small . The two sisters live together and Taja is
exposed to second-hand smoke and hence her utility is adversely affected by
Lucy consumption of cigarettes }$c${\small . In particular, her utility
function (net of cost of orange juice) is given by }%
\[
{\small U}^{T}\left( j,{\small c}\right) {\small =\ln }\left( {\small j-c}%
\right) {\small -j.}
\]
{\small a) Market outcome: Find consumption of cigarettes }$c${\small \ that
maximizes the utility of Lucy and the amount of orange juice chosen by Taja
(assuming }$c$ {\small is optimal for Lucy) (two numbers)}
{\small b) Find the Pareto efficient level of }$c${\small \ and }$j.${\small %
\ Is the value of }$c${\small \ higher or smaller than in a)? Why? (two
numbers + one sentence) Hint: Derivative of }${\small \ln }\left( {\small j-c%
}\right) $ {\small with respect to }$c$ is $-\frac{1}{j-c}.$
\bigskip
\textbf{Problem 6. (Asymmetric information)}
{\small In Shorewood Hills area there are two types of homes: lemons (bad
quality homes) and plums (good quality ones). The fraction of lemons is
equal to }$\frac{1}{2}.$ {\small The value of a home for the two parties
depends on its type and is given by }
\begin{center}
\[
\begin{tabular}{lll}
& {\small Lemon} & {\small Plum} \\
{\small Seller} & ${\small 0}$ & ${\small 12}$ \\
{\small Buyer} & ${\small 10}$ & ${\small 18}$%
\end{tabular}%
\]
\end{center}
{\small Both parties agree on the price that is in between the value of a
buyer and a seller. }
{\small a) Buyers and sellers can perfectly determine the quality of a house
before transaction takes place \ What is expected total, buyers and sellers
surplus (three numbers)}
{\small b) Now assume that the buyers are not able to determine quality of a
house. Find the price of a house, and the expected buyers and sellers
surplus (three numbers). Is a pooling equilibrium sustainable, or will this
market result in a separating equilibrium? Is outcome Pareto efficient (why
or why not)? }\newpage
\textbf{Econ 301}
\textbf{Intermediate Microeconomics}
\textbf{Prof. Marek Weretka}
\begin{center}
\textbf{Final Exam (Group B)}
\end{center}
{\small You have 2h to complete the exam. The final consists of 6 questions
(25+20+20+15+10+10=100). }
\bigskip
\textbf{Problem 1. (Quasilinaer income effect)}
{\small Mirabella consumes chocolate candy bars }$x_{1}${\small \ and fruits
}$x_{2}$.{\small \ The prices of the two goods are }${\small p}_{1}{\small %
=2,p}_{2}{\small =2},${\small \ respectively and Mirabella's income is }$%
{\small m=20}${\small . Her utility function is }%
\[
{\small U(x}_{1}{\small ,x}_{2}{\small )=2\ln x}_{1}{\small +x}_{2}
\]
{\small a) In the commodity space plot Mirabella's budget set. Find the
slope of budget line (one number). Provide the economic interpretation of
the slope (one sentence). }
{\small b) Find analytically formula that gives Mirabella's }${\small MRS}$%
{\small \ for any bundle }$\left( {\small x}_{1}{\small ,x}_{2}\right) $%
{\small \ (a function). Give the economic and the geometric interpretation
of }${\small MRS}${\small \ (two sentences). Find the value of }${\small MRS}
${\small \ at bundle }${\small (x}_{1},{\small x}_{2}{\small )=(8,8)}$
{\small (one number). At this bundle, which of the two commodities is
(locally) more valuable? (chose one)}
{\small c) Write down two secrets of happiness that determine Mirabellas's
optimal choice (two equation). Provide the geometric interpretation of the
conditions in the commodity space. }
{\small d) Find Mirabella's optimal choice (two numbers). Is solution
interior (yes-no answer).}
{\small e) Suppose the price of a chocolate candy bar goes down to }${\small %
p}_{1}{\small =1},$ {\small while other price }${\small p}_{2}{\small =2}$
{\small and income }${\small m=20}$ {\small are unchanged. Find the new
optimal choice (two numbers). Is a chocolate candy bar an ordinary or Giffen
good (pick one)?}
{\small f) Decompose the change in demand for }$x_{1}$ {\small in points d)
and e) into a substitution and income effect.}
\bigskip
\textbf{Problem 2. (Equilibrium)}
{\small Consider an economy with two consumers, Adalia and Briana and two
goods: bicycles }$x_{1}${\small \ and flowers }$x_{2}${\small . Adalia
initial endowment of the commodities is }${\small \omega }^{A}{\small %
=(50,100)}${\small \ and Briana endowment is }${\small \omega }^{B}{\small %
=(100,50)}${\small . Adalia and Briana utility functions are given by, }$%
{\small i=A,B}${\small \ }%
\[
{\small U}^{i}{\small (x}_{1}{\small ,x}_{2}{\small )=2\ln x}_{1}{\small %
+2\ln x}_{2}
\]
{\small a) Plot an Edgeworth box and mark the point that corresponds to
initial endowments. }
{\small b) Give a definition of a Pareto efficient allocation (one sentence).%
}
{\small c)} {\small Give a (general) equivalent condition for Pareto
efficiency in terms of }${\small MRS.}$\ {\small Provide geometric arguments
that demonstrate the necessity and sufficiency of }${\small MRS}${\small \
condition for Pareto efficiency.}
{\small d) Find competitive equilibrium (six numbers). Depict the obtained
equilibrium in the Edgeworth box. Using }${\small MRS}$ {\small condition
verify that the equilibrium is Pareto efficient. }
{\small e) Using (one of) the secrets of happiness prove that a competitive
equilibrium is Pareto efficient in any economy.}
\bigskip
\textbf{Problem 3. (Short questions)}
{\small a) Using }$\lambda $ {\small argument prove that Cobb-Douglass
production function }${\small y=2KL}$ {\small exhibits increasing returns to
scale. Without any calculations, sketch total cost function }${\small c(y)}$%
{\small \ corresponding to the production function.}
{\small b) Now consider a firm (different from point a)) with variable cost }%
${\small c(y)=4y}^{2}$ {\small and fixed cost }${\small F=4}$. {\small Find }%
${\small ATC}^{MES}${\small \ and }${\small y}^{MES}${\small \ (two
numbers).\ In a long-run equilibrium with free entry how many firms should
be expect in the industry if inverse demand is }${\small D(p)=16-p?}$
{\small c) Suppose a Bernoulli utility function is }${\small u(x)=x}^{2}$
{\small and two states are equally likely (probability }$\frac{1}{2}$).
{\small Write down the corresponding von Neuman-Morgenstern utility
function. Find the certainty equivalent and the expected value of lottery }$%
{\small (0,2)}$ {\small (two numbers). Which of the two is bigger and why?
(two numbers and one sentence.)}
{\small d) Find Herfindahl--Hirschman Index (HHI) for industry with }$%
{\small N=100}${\small \ identical firms (one number). Is the industry
concentrated?}
{\small e) Derive formula for the present value of perpetuity}
\bigskip \newpage
\textbf{Problem 4. (Market Power)}
{\small Consider an industry with inverse demand }${\small p}\left( y\right)
{\small =12-y},${\small \ and a monopoly with cost function }${\small TC}%
\left( y\right) {\small =0}$ {\small who cannot discriminate.}
{\small a) What are the total gains-to-trade (or potential total surplus) in
this industry? (give one number)}
{\small b) Write down monopoly's profit function. Derive the condition on }$%
{\small MR}${\small \ and }${\small MC}$ {\small that gives profit
maximizing level of production. Provide economic interpretation of this
condition. }
{\small c) Find the level of production, the price, the deadweight loss and
the elasticity of the demand at optimum (four numbers). Illustrate the
choice in a graph. }
{\small d) Assuming the same demand function, find the individual and the
aggregate level of production and the price in the Cournot-Nash equilibrium
with }${\small N=3}${\small \ identical firms (give three numbers). Show the
deadweight loss in the graph.}
\bigskip
\textbf{Problem 5.(Externality)}
{\small Lucy is addicted to nicotine. Her utility from smoking }$c$ {\small %
cigarettes (net of their cost) is given by }%
\[
{\small U}^{L}\left( {\small c}\right) {\small =2\ln c-c}
\]
{\small Her sister Taja prefers healthy lifestyle, her favorite commodity is
orange juice, }$j${\small . The two sisters live together and Taja is
exposed to second-hand smoke and hence her utility is adversely affected by
Lucy consumption of cigarettes }$c${\small . In particular, her utility
function (net of cost of orange juice) is given by }%
\[
{\small U}^{T}\left( j,{\small c}\right) {\small =\ln }\left( {\small j-c}%
\right) {\small -j.}
\]
{\small a) Market outcome: Find consumption of cigarettes }$c${\small \ that
maximizes the utility of Lucy and the amount of orange juice chosen by Taja
(assuming }$c$ {\small is optimal for Lucy) (two numbers)}
{\small b) Find the Pareto efficient level of }$c${\small \ and }$j.${\small %
\ Is the value of }$c${\small \ higher or smaller than in a)? Why? (two
numbers + one sentence) Hint: Derivative of }${\small \ln }\left( {\small j-c%
}\right) $ {\small with respect to }$c$ is $-\frac{1}{j-c}.$
\bigskip
\textbf{Problem 6. (Asymmetric information)}
{\small In Shorewood Hills area there are two types of homes: lemons (bad
quality homes) and plums (good quality ones). The fraction of lemons is
equal to }$\frac{1}{2}.$ {\small The value of a home for the two parties
depends on its type and is given by }
\begin{center}
\[
\begin{tabular}{lll}
& {\small Lemon} & {\small Plum} \\
{\small Seller} & ${\small 6}$ & ${\small 14}$ \\
{\small Buyer} & ${\small 10}$ & ${\small 22}$%
\end{tabular}%
\]
\end{center}
{\small Both parties agree on the price that is in between the value of a
buyer and a seller. }
{\small a) Buyers and sellers can perfectly determine the quality of a house
before transaction takes place \ What is expected total, buyers and sellers
surplus (three numbers)}
{\small b) Now assume that the buyers are not able to determine quality of a
house. Find the price of a house, and the expected buyers and sellers
surplus (three numbers). Is a pooling equilibrium sustainable, or will this
market result in a separating equilibrium? Is outcome Pareto efficient (why
or why not)? }
\end{document}