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\textbf{Econ 301}

\textbf{Intermediate Microeconomics}

\textbf{Prof. Marek Weretka}

\begin{center}
\textbf{Midterm 2 (Group A)}
\end{center}

{\small You have 70 minutes to complete the exam. The midterm consists of 4
questions (25, 20, 30, 25 points)+Just For Fun question. }

{\small \bigskip }

{\small \textbf{Problem 1. (25p) (Intertemporal Choice) } }

{\small Joseph earns }${\small m}_{1}{\small =\$10}${\small \ when young and 
}${\small m}_{2}{\small =\$90}${\small \ when old.}

{\small a) Write down Joseph's budget constraint (one inequality) and plot
his budget set given interest rate }${\small r=200\%}${\small \ in the
graph. Find PV and FV of income and mark it in your graph (give two numbers).%
}

{\small b) Joseph's utility is given by }${\small U}\left(
C_{1},C_{2}\right) {\small =}\ln \left( c_{1}\right) {\small +}\frac{1}{%
1+\delta }\ln \left( c_{2}\right) ${\small \ where discount rate is }$%
{\small \delta =2.}${\small \ Using magic formulas, find the optimal
consumption plan and the optimal saving strategy (give three numbers }$%
{\small C}_{1}{\small ,C}_{2}{\small ,S}${\small ). }

{\small c) (Annuity) Joseph is contemplating leasing a car. Leasing would
require three annual payments, each }${\small \$1600,}${\small \ starting
next year (after which you can keep your car). Alternatively, he can buy a
car for }${\small \$1,500}${\small \ (payment this year). Which of the two
alternatives should he choose if annual interest is }${\small r=100\%}?$ 
{\small Why? (calculate PV of the two alternatives)}

{\small d) (Perpetuity) Derive the (general) formula for PV of perpetuity.}

\bigskip

{\small \textbf{Problem 2. (20p) (Uncertainty and Insurance)} }

{\small Trevor's motorbike is worth }${\small 16}$ {\small (thousand }$%
{\small \$}${\small ).}$\ ${\small In an event (state of the world) of a
crash, its value drops to }${\small 0}$ {\small (and, hence, the bike is a
lottery with }${\small w=(w}_{1},{\small w}_{2})={\small (16,0)).}$ {\small %
Assume that the probability of a crash is equal to }$\frac{1}{2}.${\small \ }

{\small a) Find the expected value of the \textquotedblleft
bike\textquotedblright\ lottery }${\small (16,0)}$ {\small (one number) and
its certainty equivalent }${\small CE}$,{\small \ assuming Bernoulli utility
function }${\small u(y)=}\sqrt{y}$ {\small (one number). Is the certainty
equivalent bigger or smaller than the expected value? Why? (one sentence) . }

{\small b) Write down the Von Neumann-Morgenstern (expected) utility
function over lotteries }${\small U(C}_{1}{\small ;C}_{2}{\small )}${\small %
\ for Bernoulli utility function }${\small u(y)=}\ln {\small y.}$ {\small Is
Trevor risk averse, risk neutral or risk loving? (chose one+ one sentence)}

{\small c) Assume insurance premium }$\gamma =\frac{1}{2}${\small \ and the
utility function as in point b), find the optimal consumption }$\left( 
{\small C}_{1},{\small C}_{2}\right) $ {\small and insurance coverage }$x$ 
{\small (give three numbers, you can use Magic Formulas). Is Trevor fully
insured? Why? (one sentence Hint: is insurance fair?)}

{\small \bigskip }

{\small \textbf{Problem 3. (30p) (Edgeworth box and equilibrium)} }

{\small Jayden and Olivia are consume two types of commodities, ice cream $%
x_{1}$ and burgers $x_{2}$. Jayden\ is initially endowed with $\omega
^{J}=(9,1)$ of the two goods and Olivia's endowment is $\omega ^{O}=(1,9)$.
The utility function of Jayden and Olivia is the same and is given by 
\[
U\left( x_{1},x_{2}\right) =\frac{1}{3}\ln x_{1}+\frac{1}{3}\ln x_{2}. 
\]%
}

{\small a) Plot an Edgeworth box and mark the initial endowments. }

{\small b) Find the competitive equilibrium (give six numbers) and depict
the equilibrium in the Edgeworth box. }

{\small c) Verify that the allocation in the competitive equilibrium is
Pareto efficient (compare two numbers). }

{\small d) Find analytically the contract curve (write down the condition
for Pareto efficiency and solve for a contract curve line). Are the initial
endowments located on this curve? (a yes or no answer) }

{\small \bigskip }

{\small \textbf{Problem 4. (25p) (Producers)} }

{\small A producer has the following technology 
\[
y=K^{\frac{1}{2}}L^{\frac{1}{2}} 
\]%
}

{\small a) Is MPL increasing, decreasing or constant (choose one) What are
the returns to scale? (choose: IRS, DRS or CRS) }

{\small b) Given short-run level of capital $\bar{K}=1$ derive the labor
demand (formula) of a competitive firm. Find the equilibrium real wage rate
if labor supply is given by $L^{s}=9$ (one number). }

{\small c) Find the unemployment rate if the minimal (real) wage is }$%
{\small w}^{min}{\small /p=}\frac{1}{2}${\small \ (one number+graph). }

{\small d) Find analytically a cost function if $w_{K}=w_{L}=2$ (formula).
Plot the cost function in the graph. }

\bigskip

{\small \textbf{Just For Fun} }

{\small Give a definition of the Pareto efficient allocation. Prove that
allocation is Pareto efficient if and only if "MRS" of consumers are equal
(show the necessary and sufficient condition).}

\bigskip

\pagebreak \textbf{Econ 301}

\textbf{Intermediate Microeconomics}

\textbf{Prof. Marek Weretka}

\begin{center}
\textbf{Midterm 2 (Group B)}
\end{center}

{\small You have 70 minutes to complete the exam. The midterm consists of 4
questions (25, 20, 30, 25 points)+Just For Fun question. }

{\small \bigskip }

{\small \textbf{Problem 1. (25p) (Intertemporal Choice) } }

{\small Joseph earns }${\small m}_{1}{\small =\$20}${\small \ when young and 
}${\small m}_{2}{\small =\$180}${\small \ when old.}

{\small a) Write down Joseph's budget constraint (one inequality) and plot
his budget set given interest rate }${\small r=200\%}${\small \ in the
graph. Find PV and FV of income and mark it in your graph (give two numbers).%
}

{\small b) Joseph's utility is given by }${\small U}\left(
C_{1},C_{2}\right) {\small =}\ln \left( c_{1}\right) {\small +}\frac{1}{%
1+\delta }\ln \left( c_{2}\right) ${\small \ where discount rate is }$%
{\small \delta =2.}${\small \ Using magic formulas, find the optimal
consumption plan and the optimal saving strategy (give three numbers }$%
{\small C}_{1}{\small ,C}_{2}{\small ,S}${\small ). }

{\small c) (Annuity) Joseph is contemplating leasing a car. Leasing would
require three annual payments, each }${\small \$8,000,}${\small \ starting
next year (after which you can keep your car). Alternatively, he can buy a
car for }${\small \$7,500}${\small \ (payment this year). Which of the two
alternatives should he choose if annual interest is }${\small r=100\%}?$ 
{\small Why? (calculate PV of the two alternatives)}

{\small d) (Perpetuity) Derive the (general) formula for PV of perpetuity.}

\bigskip

{\small \textbf{Problem 2. (20p) (Uncertainty and Insurance)} }

{\small Trevor's motorbike is worth }${\small 36}$ {\small (thousand }$%
{\small \$}${\small ).}$\ ${\small In an event (state of the world) of a
crash, its value drops to }${\small 0}$ {\small (and, hence, the bike is a
lottery with }${\small w=(w}_{1},{\small w}_{2})={\small (36,0)).}$ {\small %
Assume that the probability of a crash is equal to }$\frac{1}{2}.${\small \ }

{\small a) Find the expected value of the \textquotedblleft
bike\textquotedblright\ lottery }${\small (36,0)}$ {\small (one number) and
its certainty equivalent }${\small CE}$,{\small \ assuming Bernoulli utility
function }${\small u(y)=}\sqrt{y}$ {\small (one number). Is the certainty
equivalent bigger or smaller than the expected value? Why? (one sentence). }

{\small b) Write down the Von Neumann-Morgenstern (expected) utility
function over lotteries }${\small U(C}_{1}{\small ;C}_{2}{\small )}${\small %
\ for Bernoulli utility function }${\small u(y)=}\ln {\small y.}$ {\small Is
Trevor risk averse, risk neutral or risk loving? (chose one+ one sentence)}

{\small c) Assume insurance premium }$\gamma =\frac{1}{2}${\small \ and the
utility function as in point b), find the optimal consumption }$\left( 
{\small C}_{1},{\small C}_{2}\right) $ {\small and insurance coverage }$x$ 
{\small (give three numbers, you can use Magic Formulas). Is Trevor fully
insured? Why? (one sentence Hint: is insurance fair?)}

{\small \bigskip }

{\small \textbf{Problem 3. (30p) (Edgeworth box and equilibrium)} }

{\small Jayden and Olivia are consume two types of commodities, ice cream $%
x_{1}$ and burgers $x_{2}$. Jayden\ is initially endowed with $\omega
^{J}=(4,6)$ of the two goods and Olivia's endowment is $\omega ^{O}=(6,4)$.
The utility function of Jayden and Olivia is the same and is given by 
\[
U\left( x_{1},x_{2}\right) =\frac{1}{5}\ln x_{1}+\frac{1}{5}\ln x_{2}. 
\]%
}

{\small a) Plot an Edgeworth box and mark the initial endowments. }

{\small b) Find the competitive equilibrium (give six numbers) and depict
the equilibrium in the Edgeworth box. }

{\small c) Verify that the allocation in the competitive equilibrium is
Pareto efficient (compare two numbers). }

{\small d) Find analytically the contract curve (write down the condition
for Pareto efficiency and solve for a contract curve line). Are the initial
endowments located on this curve? (a yes or no answer) }

{\small \bigskip }

{\small \textbf{Problem 4. (25p) (Producers)} }

{\small A producer has the following technology 
\[
y=K^{\frac{1}{2}}L^{\frac{1}{2}} 
\]%
}

{\small a) Is MPL increasing, decreasing or constant (choose one) What are
the returns to scale? (choose: IRS, DRS or CRS) }

{\small b) Given short-run level of capital $\bar{K}=1$ derive the labor
demand (formula) of a competitive firm. Find the equilibrium real wage rate
if labor supply is given by $L^{s}=9$ (one number). }

{\small c) Find the unemployment rate if the minimal (real) wage is }$%
{\small w}^{min}{\small /p=}\frac{1}{2}${\small \ (one number+graph). }

{\small d) Find analytically a cost function if $w_{K}=w_{L}=3$ (formula).
Plot the cost function in the graph. }

\bigskip

{\small \textbf{Just For Fun} }

{\small Give a definition of the Pareto efficient allocation. Prove that
allocation is Pareto efficient if and only if "MRS" of consumers are equal
(show the necessary and sufficient condition).}

\bigskip

\pagebreak \pagebreak

\textbf{Econ 301}

\textbf{Intermediate Microeconomics}

\textbf{Prof. Marek Weretka}

\begin{center}
\textbf{Midterm 2 (Group C)}
\end{center}

{\small You have 70 minutes to complete the exam. The midterm consists of 4
questions (25, 20, 30, 25 points)+Just For Fun question. }

{\small \bigskip }

{\small \textbf{Problem 1. (25p) (Intertemporal Choice) } }

{\small Joseph earns }${\small m}_{1}{\small =\$10}${\small \ when young and 
}${\small m}_{2}{\small =\$60}${\small \ when old.}

{\small a) Write down Joseph's budget constraint (one inequality) and plot
his budget set given interest rate }${\small r=100\%}${\small \ in the
graph. Find PV and FV of income and mark it in your graph (give two numbers).%
}

{\small b) Joseph's utility is given by }${\small U}\left(
C_{1},C_{2}\right) {\small =}\ln \left( c_{1}\right) {\small +}\frac{1}{%
1+\delta }\ln \left( c_{2}\right) ${\small \ where discount rate is }$%
{\small \delta =1.}${\small \ Using magic formulas, find the optimal
consumption plan and the optimal saving strategy (give three numbers }$%
{\small C}_{1}{\small ,C}_{2}{\small ,S}${\small ). }

{\small c) (Annuity) Joseph is contemplating leasing a car. Leasing would
require three annual payments, each }${\small \$2400,}${\small \ starting
next year (after which you can keep your car). Alternatively, he can buy a
car for }${\small \$2,300}${\small \ (payment this year). Which of the two
alternatives should he choose if annual interest is }${\small r=100\%}?$ 
{\small Why? (calculate PV of the two alternatives)}

{\small d) (Perpetuity) Derive the (general) formula for PV of perpetuity.}

\bigskip

{\small \textbf{Problem 2. (20p) (Uncertainty and Insurance)} }

{\small Trevor's motorbike is worth }${\small 64}$ {\small (thousand }$%
{\small \$}${\small ).}$\ ${\small In an event (state of the world) of a
crash, its value drops to }${\small 0}$ {\small (and, hence, the bike is a
lottery with }${\small w=(w}_{1},{\small w}_{2})={\small (64,0)).}$ {\small %
Assume that the probability of a crash is equal to }$\frac{1}{2}.${\small \ }

{\small a) Find the expected value of the \textquotedblleft
bike\textquotedblright\ lottery }${\small (64,0)}$ {\small (one number) and
its certainty equivalent }${\small CE}$,{\small \ assuming Bernoulli utility
function }${\small u(y)=}\sqrt{y}$ {\small (one number). Is the certainty
equivalent bigger or smaller than the expected value? Why? (one sentence) . }

{\small b) Write down the Von Neumann-Morgenstern (expected) utility
function over lotteries }${\small U(C}_{1}{\small ;C}_{2}{\small )}${\small %
\ for Bernoulli utility function }${\small u(y)=}\ln {\small y.}$ {\small Is
Trevor risk averse, risk neutral or risk loving? (chose one+ one sentence)}

{\small c) Assume insurance premium }$\gamma =\frac{1}{2}${\small \ and the
utility function as in point b), find the optimal consumption }$\left( 
{\small C}_{1},{\small C}_{2}\right) $ {\small and insurance coverage }$x$ 
{\small (give three numbers, you can use Magic Formulas). Is Trevor fully
insured? Why? (one sentence Hint: is insurance fair?)}

{\small \bigskip }

{\small \textbf{Problem 3. (30p) (Edgeworth box and equilibrium)} }

{\small Jayden and Olivia are consume two types of commodities, ice cream $%
x_{1}$ and burgers $x_{2}$. Jayden\ is initially endowed with $\omega
^{J}=(6,14)$ of the two goods and Olivia's endowment is $\omega ^{O}=(14,6)$%
. The utility function of Jayden and Olivia is the same and is given by 
\[
U\left( x_{1},x_{2}\right) =3\ln x_{1}+3\ln x_{2}. 
\]%
}

{\small a) Plot an Edgeworth box and mark the initial endowments. }

{\small b) Find the competitive equilibrium (give six numbers) and depict
the equilibrium in the Edgeworth box. }

{\small c) Verify that the allocation in the competitive equilibrium is
Pareto efficient (compare two numbers). }

{\small d) Find analytically the contract curve (write down the condition
for Pareto efficiency and solve for a contract curve line). Are the initial
endowments located on this curve? (a yes or no answer) }

{\small \bigskip }

{\small \textbf{Problem 4. (25p) (Producers)} }

{\small A producer has the following technology 
\[
y=K^{\frac{1}{2}}L^{\frac{1}{2}} 
\]%
}

{\small a) Is MPL increasing, decreasing or constant (choose one) What are
the returns to scale? (choose: IRS, DRS or CRS) }

{\small b) Given short-run level of capital $\bar{K}=1$ derive the labor
demand (formula) of a competitive firm. Find the equilibrium real wage rate
if labor supply is given by $L^{s}=9$ (one number). }

{\small c) Find the unemployment rate if the minimal (real) wage is }$%
{\small w}^{min}{\small /p=}\frac{1}{2}${\small \ (one number+graph). }

{\small d) Find analytically a cost function if $w_{K}=w_{L}=5$ (formula).
Plot the cost function in the graph. }

\bigskip

{\small \textbf{Just For Fun} }

{\small Give a definition of the Pareto efficient allocation. Prove that
allocation is Pareto efficient if and only if "MRS" of consumers are equal
(show the necessary and sufficient condition).}

\bigskip

\pagebreak \pagebreak

\textbf{Econ 301}

\textbf{Intermediate Microeconomics}

\textbf{Prof. Marek Weretka}

\begin{center}
\textbf{Midterm 2 (Group D)}
\end{center}

{\small You have 70 minutes to complete the exam. The midterm consists of 4
questions (25, 20, 30, 25 points)+Just For Fun question. }

{\small \bigskip }

{\small \textbf{Problem 1. (25p) (Intertemporal Choice) } }

{\small Joseph earns }${\small m}_{1}{\small =\$10}${\small \ when young and 
}${\small m}_{2}{\small =\$100}${\small \ when old.}

{\small a) Write down Joseph's budget constraint (one inequality) and plot
his budget set given interest rate }${\small r=100\%}${\small \ in the
graph. Find PV and FV of income and mark it in your graph (give two numbers).%
}

{\small b) Joseph's utility is given by }${\small U}\left(
C_{1},C_{2}\right) {\small =}\ln \left( c_{1}\right) {\small +}\frac{1}{%
1+\delta }\ln \left( c_{2}\right) ${\small \ where discount rate is }$%
{\small \delta =1.}${\small \ Using magic formulas, find the optimal
consumption plan and the optimal saving strategy (give three numbers }$%
{\small C}_{1}{\small ,C}_{2}{\small ,S}${\small ). }

{\small c) (Annuity) Joseph is contemplating leasing a car. Leasing would
require three annual payments, each }${\small \$2400,}${\small \ starting
next year (after which you can keep your car). Alternatively, he can buy a
car for }${\small \$2,300}${\small \ (payment this year). Which of the two
alternatives should he choose if annual interest is }${\small r=100\%}?$ 
{\small Why? (calculate PV of the two alternatives)}

{\small d) (Perpetuity) Derive the (general) formula for PV of perpetuity.}

\bigskip

{\small \textbf{Problem 2. (20p) (Uncertainty and Insurance)} }

{\small Trevor's motorbike is worth }${\small 100}$ {\small (thousand }$%
{\small \$}${\small ).}$\ ${\small In an event (state of the world) of a
crash, its value drops to }${\small 0}$ {\small (and, hence, the bike is a
lottery with }${\small w=(w}_{1},{\small w}_{2})={\small (100,0)).}$ {\small %
Assume that the probability of a crash is equal to }$\frac{1}{2}.${\small \ }

{\small a) Find the expected value of the \textquotedblleft
bike\textquotedblright\ lottery }${\small (100,0)}$ {\small (one number) and
its certainty equivalent }${\small CE}$,{\small \ assuming Bernoulli utility
function }${\small u(y)=}\sqrt{y}$ {\small (one number). Is the certainty
equivalent bigger or smaller than the expected value? Why? (one sentence) . }

{\small b) Write down the Von Neumann-Morgenstern (expected) utility
function over lotteries }${\small U(C}_{1}{\small ;C}_{2}{\small )}${\small %
\ for Bernoulli utility function }${\small u(y)=}\ln {\small y.}$ {\small Is
Trevor risk averse, risk neutral or risk loving? (chose one+ one sentence)}

{\small c) Assume insurance premium }$\gamma =\frac{1}{2}${\small \ and the
utility function as in point b), find the optimal consumption }$\left( 
{\small C}_{1},{\small C}_{2}\right) $ {\small and insurance coverage }$x$ 
{\small (give three numbers, you can use Magic Formulas). Is Trevor fully
insured? Why? (one sentence Hint: is insurance fair?)}

{\small \bigskip }

{\small \textbf{Problem 3. (30p) (Edgeworth box and equilibrium)} }

{\small Jayden and Olivia are consume two types of commodities, ice cream $%
x_{1}$ and burgers $x_{2}$. Jayden\ is initially endowed with $\omega
^{J}=(3,17)$ of the two goods and Olivia's endowment is $\omega ^{O}=(17,3)$%
. The utility function of Jayden and Olivia is the same and is given by 
\[
U\left( x_{1},x_{2}\right) =5\ln x_{1}+5\ln x_{2}. 
\]%
}

{\small a) Plot an Edgeworth box and mark the initial endowments. }

{\small b) Find the competitive equilibrium (give six numbers) and depict
the equilibrium in the Edgeworth box. }

{\small c) Verify that the allocation in the competitive equilibrium is
Pareto efficient (compare two numbers). }

{\small d) Find analytically the contract curve (write down the condition
for Pareto efficiency and solve for a contract curve line). Are the initial
endowments located on this curve? (a yes or no answer) }

{\small \bigskip }

{\small \textbf{Problem 4. (25p) (Producers)} }

{\small A producer has the following technology 
\[
y=K^{\frac{1}{2}}L^{\frac{1}{2}} 
\]%
}

{\small a) Is MPL increasing, decreasing or constant (choose one) What are
the returns to scale? (choose: IRS, DRS or CRS) }

{\small b) Given short-run level of capital $\bar{K}=1$ derive the labor
demand (formula) of a competitive firm. Find the equilibrium real wage rate }%
$\frac{w_{L}}{p}$ {\small if labor supply is given by $L^{s}=9$ (one
number). }

{\small c) Find the unemployment rate if the minimal (real) wage is }$%
{\small w}^{min}{\small /p=}\frac{1}{2}${\small \ (one number+graph). }

{\small d) Find analytically a cost function if $w_{K}=w_{L}=8$ (formula).
Plot the cost function in the graph. }

\bigskip

{\small \textbf{Just For Fun} }

{\small Give a definition of the Pareto efficient allocation. Prove that
allocation is Pareto efficient if and only if "MRS" of consumers are equal
(show the necessary and sufficient condition).}

\pagebreak \pagebreak

\end{document}
