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\textbf{Econ 301}

\textbf{Intermediate Microeconomics}

\textbf{Prof. Marek Weretka}

\begin{center}
\textbf{Midterm 1 (A)}
\end{center}

{\small You have 70 minutes to complete the exam. The midterm consists of 4
questions (55+15+15+15=100 points) + bonus (just for fun). Make sure you
answer the first four questions before working on the bonus one! }

\bigskip

\textbf{{\small Problem 1 (55p}) }{\small (Well-behaved preferences)}

{\small Ava is a big fun of delicious gourmet steaks: their robust and
hearty beef flavors is what she likes the most. Her two most favorite steaks
are marinated ribeye }$(x_{1})${\small \ and top sirloin }$(x_{2})${\small . 
}

{\small a) The price of a marinated ribeye steak is }$p_{1}=\$50${\small ,
the price of top sirloin is\ }$p_{2}=\$25${\small ,\ and her income (spent
entirely on steaks) is }$m=\$500$.{\small \ Show geometrically Ava's budget
constraint. Mark the two extreme consumption bundles (give concrete values).
Show on the graph how her budget set would be affected by inflation that
increased the prices of both steaks by }$100\%$,{\small \ leaving }$m$%
{\small \ unchanged (draw a new budget line). }

{\small b) Ava's utility function over both types of steaks is }$U\left(
x_{1},x_{2}\right) =x_{1}^{5}x_{2}^{5}${\small . Find her marginal rate of
substitution (MRS) for any bundle }$\left( x_{1},x_{2}\right) ${\small \
(give a formula for MRS). What is the value of MRS at bundle }$\left(
2,4\right) $ {\small (one number)? \ Which of the two steak types is more
valuable to Ava?}

{\small d) Write down two secrets of happiness, assuming that }$%
p_{1},p_{2},m ${\small \ are parameters (two equations).}

{\small - Provide economic intuition behind the two conditions (two
sentences for each).}

{\small - Derive optimal consumption of }$x_{1}${\small \ and }$x_{2}$%
{\small \ as a function of }$p_{1},p_{2},m${\small \ (show the derivation of
magic formulas).}

{\small - Using magic formulas determine whether the solution is interior
for all values of }$p_{1},p_{2},m${\small \ (one sentence). }

{\small - What fraction of total income is spent on top sirloin?}

{\small e) Find the optimal consumption levels of two types of steak }$%
\left( x_{1},x_{2}\right) ${\small \ for:}

{\small - }$p_{1}=\$50,p_{2}=\$25${\small \ and }$m=\$500${\small \ (give
two numbers).}

{\small and after the price of ribeye decreased:}

{\small - for }$p_{1}=\$25,p_{2}=\$25${\small \ and }$m=\$500${\small \
(give two numbers).}

{\small What is the total change in consumption of marinated ribeye?\ (give
a number). Illustrate the change on the graph. Is marinated ribeye an
ordinary of a Giffen good? (Chose one + one sentence explaining why.)}

{\small f) Decompose the total change in consumption of }$x_{1}${\small \
from e) into a substitution and income effect. (Calculate the two numbers
and show how can you find the effect on the graph.) }

{\small \bigskip }

\textbf{{\small Problem 2 (15p}) }{\small (Perfect complements) }

{\small Alex is a hotdog lover. In each hotdog bun (}$x_{1}${\small ) Alex
always inserts two hotdogs (}$x_{2}${\small ): he hates hotdogs with ratios
of buns to hotdogs different from 1:2. }

{\small a) Propose a utility function that captures Alex's preferences over (%
}$x_{1}${\small ) and (}$x_{2}${\small ) (function }$U\left( {\small x}_{1}%
{\small ,x}_{2}\right) ${\small ).}

{\small b) In the commodity space, carefully depict Alex's indifference
curves (marking the optimal proportion line).}

{\small c) Assume prices }$p_{1}=4${\small \ and }$p_{2}=3${\small \ and
income }$m=40${\small .\ Write down two secrets of happiness (two equations)
and determine the optimal choice (two numbers). Is your solution interior
(yes or no)?}

{\small d) Suppose price of a hotdog bun goes down to }$p_{1}=2,${\small \
while price }$p_{2}=3${\small \ income }$m=40${\small \ are unchanged. Find
optimal choice (two numbers). Without calculations, how much of this change
can be attributed to a substitution effect (number + one sentence explaining
the number).}

{\small \bigskip }

\textbf{{\small Problem 3 (15p}) }{\small (Perfect substitutes)}

{\small Pepsi }$x_{1}${\small \ and Coke }$x_{2}${\small \ both are goods
that are indistinguishable. }

{\small a) Propose a utility function (the \textquotedblleft
craziest\textquotedblright\ function you can imagine), which captures
preferences for such perfect substitutes. }

{\small b) Plot the budget line, assuming }$p_{1}=\$2,${\small \ }$p_{2}=\$3$%
{\small \ and }$m=\$12.$ {\small In the same graph plot indifference curves.}

{\small c) Find the optimal bundle for }$U\left( {\small x}_{1}{\small ,x}%
_{2}\right) =x_{1}+x_{2}${\small .\ (give two numbers). Is your solution
interior or corner? (one sentence)}

{\small d) Plot income offer curve and Engel curve given fixed prices }$%
p_{1}=\$2,p_{2}=\$3.${\small \ (two graphs). Is Pepsi normal or inferior
(choose one + one sentence)}

\bigskip

\textbf{{\small Problem 4 (15p}) }{\small (Quasilinear Preferences, Labor
Supply)}

{\small Your best friend Aiden asked you to help him to determine how much
time he should spend at work. His total available time (each day) is 24h and
his only source of income is labor income given wage rate }${\small w=\$10}$%
{\small . His spending on consumption good }${\small C}${\small \ is equal
to what he earns. The price of such good is }$p_{c}=2${\small \ }

{\small a) What is his real wage rate? (number+interpretation, one sentence )%
}

{\small b) Plot his budget set on the graph.}

{\small c) Aiden has quasilinear utility function }${\small U}\left( {\small %
R,C}\right) {\small =}\ln {\small R+C}${\small \ where }${\small R}${\small %
\ is leisure (or relaxation time) and }${\small C}${\small \ is consumption
of other goods. Write down two secrets of happiness that determine optimal
choice of }${\small R,C}${\small \ for arbitrary }${\small w,p}_{c}${\small %
. Find optimal }$R${\small \ as a function }$w,p_{c}${\small \ (you can
assume interior solution)}.{\small \ }

{\small d) What is Aiden's labor supply }${\small L}${\small \ if }${\small %
w=\$1}${\small \ and }${\small p}_{c}{\small =\$10}${\small \ (one number)}.%
{\small \ How about }${\small w=\$2}${\small \ and }${\small p}_{c}{\small %
=\$10}${\small \ (one number). Is labor supply increasing in real wage rate,
when preferences are quasilinear? (yes no answer)}

{\small \bigskip }

\textbf{\small Bonus question }{\small (Just for fun)}

{\small Let }$MRS^{V}${\small \ and }$MRS^{U}${\small \ be marginal rates of
substitution of utility functions }${\small V(x}_{1}{\small ,x}_{2}{\small )}
${\small \ and }${\small U(x}_{1}{\small ,x}_{2}{\small )}${\small \
respectively. Moreover, assume that }${\small V(x}_{1}{\small ,x}_{2}{\small %
)=f}\left[ {\small U(x}_{1}{\small ,x}_{2}{\small )}\right] ${\small \ where 
}${\small f}\left[ \cdot \right] ${\small \ is a strictly monotone
transformation. Show that }${\small MRS}^{V}{\small =MRS}^{U}$. {\small %
(Hint: Use formula for the derivative of a composite function)}

{\small \pagebreak }

\textbf{Econ 301}

\textbf{Intermediate Microeconomics}

\textbf{Prof. Marek Weretka}

\begin{center}
\textbf{Midterm 1 (B)}
\end{center}

{\small You have 70 minutes to complete the exam. The midterm consists of 4
questions (55+15+15+15=100 points) + bonus (just for fun). Make sure you
answer the first four questions before working on the bonus one! }

\bigskip

\textbf{{\small Problem 1 (55p}) }{\small (Well-behaved preferences)}

{\small Ava is a big fun of delicious gourmet steaks: their robust and
hearty beef flavors is what she likes the most. Her two most favorite steaks
are marinated ribeye }$(x_{1})${\small \ and top sirloin }$(x_{2})${\small . 
}

{\small a) The price of a marinated ribeye steak is }$p_{1}=\$10${\small ,
the price of top sirloin is\ }$p_{2}=\$5${\small ,\ and her income (spent
entirely on steaks) is }$m=\$100$.{\small \ Show geometrically Ava's budget
constraint. Mark the two extreme consumption bundles (give concrete values).
Show on the graph how her budget set would be affected by inflation that
increased the prices of both steaks by }$200\%$,{\small \ leaving }$m$%
{\small \ unchanged (draw a new budget line). }

{\small b) Ava's utility function over both types of steaks is }$U\left(
x_{1},x_{2}\right) =x_{1}^{3}x_{2}^{3}${\small . Find her marginal rate of
substitution (MRS) for any bundle }$\left( x_{1},x_{2}\right) ${\small \
(give a formula for MRS). What is the value of MRS at bundle }$\left(
3,6\right) $ {\small (one number)? \ Which of the two steak types is more
valuable to Ava?}

{\small d) Write down two secrets of happiness, assuming that }$%
p_{1},p_{2},m ${\small \ are parameters (two equations).}

{\small - Provide economic intuition behind the two conditions (two
sentences for each).}

{\small - Derive optimal consumption of }$x_{1}${\small \ and }$x_{2}$%
{\small \ as a function of }$p_{1},p_{2},m${\small \ (show the derivation of
magic formulas).}

{\small - Using magic formulas determine whether the solution is interior
for all values of }$p_{1},p_{2},m${\small \ (one sentence). }

{\small - What fraction of total income is spent on top sirloin?}

{\small e) Find the optimal consumption levels of two types of steak }$%
\left( x_{1},x_{2}\right) ${\small \ for:}

{\small - }$p_{1}=\$10,p_{2}=\$5${\small \ and }$m=\$100${\small \ (give two
numbers).}

{\small and after the price of ribeye decreased:}

{\small - for }$p_{1}=\$5,p_{2}=\$5${\small \ and }$m=\$100${\small \ (give
two numbers).}

{\small What is the total change in consumption of marinated ribeye?\ (give
a number). Illustrate the change on the graph. Is marinated ribeye an
ordinary of a Giffen good? (Chose one + one sentence explaining why.)}

{\small f) Decompose the total change in consumption of }$x_{1}${\small \
from e) into a substitution and income effect. (Calculate the two numbers
and show how can you find the effect on the graph.) }

{\small \bigskip }

\textbf{{\small Problem 2 (15p}) }{\small (Perfect complements) }

{\small Alex is a hotdog lover. In each hotdog bun (}$x_{1}${\small ) Alex
always inserts three hotdogs (}$x_{2}${\small ): he hates hotdogs with
ratios of buns to hotdogs different from 1:3. }

{\small a) Propose a utility function that captures Alex's preferences over (%
}$x_{1}${\small ) and (}$x_{2}${\small ) (function }$U\left( {\small x}_{1}%
{\small ,x}_{2}\right) ${\small ).}

{\small b) In the commodity space, carefully depict Alex's indifference
curves (marking the optimal proportion line).}

{\small c) Assume prices }$p_{1}=4${\small \ and }$p_{2}=2${\small \ and
income }$m=40${\small .\ Write down two secrets of happiness (two equations)
and determine the optimal choice (two numbers). Is your solution interior
(yes or no)?}

{\small d) Suppose price of a hotdog bun goes down to }$p_{1}=2,${\small \
while price }$p_{2}=2${\small \ income }$m=40${\small \ are unchanged. Find
optimal choice (two numbers). Without calculations, how much of this change
can be attributed to a substitution effect (number + one sentence explaining
the number).}

{\small \bigskip\ }

\textbf{{\small Problem 3 (15p}) }{\small (Perfect substitutes)}

{\small Pepsi }$x_{1}${\small \ and Coke }$x_{2}${\small \ both are goods
that are indistinguishable. }

{\small a) Propose a utility function (the \textquotedblleft
craziest\textquotedblright\ function you can imagine), which captures
preferences for such perfect substitutes. }

{\small b) Plot the budget line, assuming }$p_{1}=\$4,${\small \ }$p_{2}=\$2$%
{\small \ and }$m=\$12.$ {\small In the same graph plot indifference curves.}

{\small c) Find the optimal bundle for }$U\left( {\small x}_{1}{\small ,x}%
_{2}\right) =x_{1}+x_{2}${\small .\ (give two numbers). Is your solution
interior or corner? (one sentence)}

{\small d) Plot income offer curve and Engel curve given fixed prices }$%
p_{1}=\$4,p_{2}=\$2.${\small \ (two graphs). Is Pepsi normal or inferior
(choose one + one sentence)}

\bigskip

\textbf{{\small Problem 4 (15p}) }{\small (Quasilinear Preferences, Labor
Supply)}

{\small Your best friend Aiden asked you to help him to determine how much
time he should spend at work. His total available time (each day) is 24h and
his only source of income is labor income given wage rate }${\small w=\$10}$%
{\small . His spending on consumption good }${\small C}${\small \ is equal
to what he earns. The price of such good is }$p_{c}=2${\small \ }

{\small a) What is his real wage rate? (number+interpretation, one sentence )%
}

{\small b) Plot his budget set on the graph.}

{\small c) Aiden has quasilinear utility function }${\small U}\left( {\small %
R,C}\right) {\small =}\ln {\small R+C}${\small \ where }${\small R}${\small %
\ is leisure (or relaxation time) and }${\small C}${\small \ is consumption
of other goods. Write down two secrets of happiness that determine optimal
choice of }${\small R,C}${\small \ for arbitrary }${\small w,p}_{c}${\small %
. Find optimal }$R${\small \ as a function }$w,p_{c}${\small \ (you can
assume interior solution)}.{\small \ }

{\small d) What is Aiden's labor supply }${\small L}${\small \ if }${\small %
w=\$1}${\small \ and }${\small p}_{c}{\small =\$10}${\small \ (one number)}.%
{\small \ How about }${\small w=\$2}${\small \ and }${\small p}_{c}{\small %
=\$10}${\small \ (one number). Is labor supply increasing in real wage rate,
when preferences are quasilinear? (yes no answer)}

{\small \bigskip }

\textbf{\small Bonus question }{\small (Just for fun)}

{\small Let }$MRS^{V}${\small \ and }$MRS^{U}${\small \ be marginal rates of
substitution of utility functions }${\small V(x}_{1}{\small ,x}_{2}{\small )}
${\small \ and }${\small U(x}_{1}{\small ,x}_{2}{\small )}${\small \
respectively. Moreover, assume that }${\small V(x}_{1}{\small ,x}_{2}{\small %
)=f}\left[ {\small U(x}_{1}{\small ,x}_{2}{\small )}\right] ${\small \ where 
}${\small f}\left[ \cdot \right] ${\small \ is a strictly monotone
transformation. Show that }${\small MRS}^{V}{\small =MRS}^{U}$. {\small %
(Hint: Use formula for the derivative of a composite function)}

{\small \pagebreak }

\textbf{Econ 301}

\textbf{Intermediate Microeconomics}

\textbf{Prof. Marek Weretka}

\begin{center}
\textbf{Midterm 1 (C)}
\end{center}

{\small You have 70 minutes to complete the exam. The midterm consists of 4
questions (55+15+15+15=100 points) + bonus (just for fun). Make sure you
answer the first four questions before working on the bonus one! }

\bigskip

\textbf{{\small Problem 1 (55p}) }{\small (Well-behaved preferences)}

{\small Ava is a big fun of delicious gourmet steaks: their robust and
hearty beef flavors is what she likes the most. Her two most favorite steaks
are marinated ribeye }$(x_{1})${\small \ and top sirloin }$(x_{2})${\small . 
}

{\small a) The price of a marinated ribeye steak is }$p_{1}=\$10${\small ,
the price of top sirloin is\ }$p_{2}=\$5${\small ,\ and her income (spent
entirely on steaks) is }$m=\$200$.{\small \ Show geometrically Ava's budget
constraint. Mark the two extreme consumption bundles (give concrete values).
Show on the graph how her budget set would be affected by inflation that
increased the prices of both steaks by }$100\%$,{\small \ leaving }$m$%
{\small \ unchanged (draw a new budget line). }

{\small b) Ava's utility function over both types of steaks is }$U\left(
x_{1},x_{2}\right) =x_{1}^{\frac{1}{2}}x_{2}^{\frac{1}{2}}${\small . Find
her marginal rate of substitution (MRS) for any bundle }$\left(
x_{1},x_{2}\right) ${\small \ (give a formula for MRS). What is the value of
MRS at bundle }$\left( 3,6\right) $ {\small (one number)? \ Which of the two
steak types is more valuable to Ava?}

{\small d) Write down two secrets of happiness, assuming that }$%
p_{1},p_{2},m ${\small \ are parameters (two equations).}

{\small - Provide economic intuition behind the two conditions (two
sentences for each).}

{\small - Derive optimal consumption of }$x_{1}${\small \ and }$x_{2}$%
{\small \ as a function of }$p_{1},p_{2},m${\small \ (show the derivation of
magic formulas).}

{\small - Using magic formulas determine whether the solution is interior
for all values of }$p_{1},p_{2},m${\small \ (one sentence). }

{\small - What fraction of total income is spent on top sirloin?}

{\small e) Find the optimal consumption levels of two types of steak }$%
\left( x_{1},x_{2}\right) ${\small \ for:}

{\small - }$p_{1}=\$10,p_{2}=\$5${\small \ and }$m=\$200${\small \ (give two
numbers).}

{\small and after the price of ribeye decreased:}

{\small - for }$p_{1}=\$5,p_{2}=\$5${\small \ and }$m=\$200${\small \ (give
two numbers).}

{\small What is the total change in consumption of marinated ribeye?\ (give
a number). Illustrate the change on the graph. Is marinated ribeye an
ordinary of a Giffen good? (Chose one + one sentence explaining why.)}

{\small f) Decompose the total change in consumption of }$x_{1}${\small \
from e) into a substitution and income effect. (Calculate the two numbers
and show how can you find the effect on the graph.) }

{\small \bigskip }

\textbf{{\small Problem 2 (15p}) }{\small (Perfect complements) }

{\small Alex is a hotdog lover. In each hotdog bun (}$x_{1}${\small ) Alex
always inserts four hotdogs (}$x_{2}${\small ): he hates hotdogs with ratios
of buns to hotdogs different from 1:4. }

{\small a) Propose a utility function that captures Alex's preferences over (%
}$x_{1}${\small ) and (}$x_{2}${\small ) (function }$U\left( {\small x}_{1}%
{\small ,x}_{2}\right) ${\small ).}

{\small b) In the commodity space, carefully depict Alex's indifference
curves (marking the optimal proportion line).}

{\small c) Assume prices }$p_{1}=6${\small \ and }$p_{2}=1${\small \ and
income }$m=40${\small .\ Write down two secrets of happiness (two equations)
and determine the optimal choice (two numbers). Is your solution interior
(yes or no)?}

{\small d) Suppose price of a hotdog bun goes down to }$p_{1}=4,${\small \
while price }$p_{2}=1${\small \ income }$m=40${\small \ are unchanged. Find
optimal choice (two numbers). Without calculations, how much of this change
can be attributed to a substitution effect (number + one sentence explaining
the number).}

{\small \bigskip\ }

\textbf{{\small Problem 3 (15p}) }{\small (Perfect substitutes)}

{\small Pepsi }$x_{1}${\small \ and Coke }$x_{2}${\small \ both are goods
that are indistinguishable. }

{\small a) Propose a utility function (the \textquotedblleft
craziest\textquotedblright\ function you can imagine), which captures
preferences for such perfect substitutes. }

{\small b) Plot the budget line, assuming }$p_{1}=\$6,${\small \ }$p_{2}=\$2$%
{\small \ and }$m=\$12.$ {\small In the same graph plot indifference curves.}

{\small c) Find the optimal bundle for }$U\left( {\small x}_{1}{\small ,x}%
_{2}\right) =x_{1}+x_{2}${\small .\ (give two numbers). Is your solution
interior or corner? (one sentence)}

{\small d) Plot income offer curve and Engel curve given fixed prices }$%
p_{1}=\$6,p_{2}=\$2.${\small \ (two graphs). Is Pepsi normal or inferior
(choose one + one sentence)}

\bigskip

\textbf{{\small Problem 4 (15p}) }{\small (Quasilinear Preferences, Labor
Supply)}

{\small Your best friend Aiden asked you to help him to determine how much
time he should spend at work. His total available time (each day) is 24h and
his only source of income is labor income given wage rate }${\small w=\$10}$%
{\small . His spending on consumption good }${\small C}${\small \ is equal
to what he earns. The price of such good is }$p_{c}=2${\small \ }

{\small a) What is his real wage rate? (number+interpretation, one sentence )%
}

{\small b) Plot his budget set on the graph.}

{\small c) Aiden has quasilinear utility function }${\small U}\left( {\small %
R,C}\right) {\small =}\ln {\small R+C}${\small \ where }${\small R}${\small %
\ is leisure (or relaxation time) and }${\small C}${\small \ is consumption
of other goods. Write down two secrets of happiness that determine optimal
choice of }${\small R,C}${\small \ for arbitrary }${\small w,p}_{c}${\small %
. Find optimal }$R${\small \ as a function }$w,p_{c}${\small \ (you can
assume interior solution)}.{\small \ }

{\small d) What is Aiden's labor supply }${\small L}${\small \ if }${\small %
w=\$1}${\small \ and }${\small p}_{c}{\small =\$10}${\small \ (one number)}.%
{\small \ How about }${\small w=\$2}${\small \ and }${\small p}_{c}{\small %
=\$10}${\small \ (one number). Is labor supply increasing in real wage rate,
when preferences are quasilinear? (yes no answer)}

{\small \bigskip }

\textbf{\small Bonus question }{\small (Just for fun)}

{\small Let }$MRS^{V}${\small \ and }$MRS^{U}${\small \ be marginal rates of
substitution of utility functions }${\small V(x}_{1}{\small ,x}_{2}{\small )}
${\small \ and }${\small U(x}_{1}{\small ,x}_{2}{\small )}${\small \
respectively. Moreover, assume that }${\small V(x}_{1}{\small ,x}_{2}{\small %
)=f}\left[ {\small U(x}_{1}{\small ,x}_{2}{\small )}\right] ${\small \ where 
}${\small f}\left[ \cdot \right] ${\small \ is a strictly monotone
transformation. Show that }${\small MRS}^{V}{\small =MRS}^{U}$. {\small %
(Hint: Use formula for the derivative of a composite function)}

{\small \pagebreak }

\textbf{Econ 301}

\textbf{Intermediate Microeconomics}

\textbf{Prof. Marek Weretka}

\begin{center}
\textbf{Midterm 1 (D)}
\end{center}

{\small You have 70 minutes to complete the exam. The midterm consists of 4
questions (55+15+15+15=100 points) + bonus (just for fun). Make sure you
answer the first four questions before working on the bonus one! }

\bigskip

\textbf{{\small Problem 1 (55p}) }{\small (Well-behaved preferences)}

{\small Ava is a big fun of delicious gourmet steaks: their robust and
hearty beef flavors is what she likes the most. Her two most favorite steaks
are marinated ribeye }$(x_{1})${\small \ and top sirloin }$(x_{2})${\small . 
}

{\small a) The price of a marinated ribeye steak is }$p_{1}=\$4${\small ,
the price of top sirloin is\ }$p_{2}=\$2${\small ,\ and her income (spent
entirely on steaks) is }$m=\$80$.{\small \ Show geometrically Ava's budget
constraint. Mark the two extreme consumption bundles (give concrete values).
Show on the graph how her budget set would be affected by inflation that
increased the prices of both steaks by }$100\%$,{\small \ leaving }$m$%
{\small \ unchanged (draw a new budget line). }

{\small b) Ava's utility function over both types of steaks is }$U\left(
x_{1},x_{2}\right) =x_{1}^{\frac{1}{3}}x_{2}^{\frac{1}{3}}${\small . Find
her marginal rate of substitution (MRS) for any bundle }$\left(
x_{1},x_{2}\right) ${\small \ (give a formula for MRS). What is the value of
MRS at bundle }$\left( 3,6\right) $ {\small (one number)? \ Which of the two
steak types is more valuable to Ava?}

{\small d) Write down two secrets of happiness, assuming that }$%
p_{1},p_{2},m ${\small \ are parameters (two equations).}

{\small - Provide economic intuition behind the two conditions (two
sentences for each).}

{\small - Derive optimal consumption of }$x_{1}${\small \ and }$x_{2}$%
{\small \ as a function of }$p_{1},p_{2},m${\small \ (show the derivation of
magic formulas).}

{\small - Using magic formulas determine whether the solution is interior
for all values of }$p_{1},p_{2},m${\small \ (one sentence). }

{\small - What fraction of total income is spent on top sirloin?}

{\small e) Find the optimal consumption levels of two types of steak }$%
\left( x_{1},x_{2}\right) ${\small \ for:}

{\small - }$p_{1}=\$4,p_{2}=\$2${\small \ and }$m=\$80${\small \ (give two
numbers).}

{\small and after the price of ribeye decreased:}

{\small - for }$p_{1}=\$2,p_{2}=\$2${\small \ and }$m=\$80${\small \ (give
two numbers).}

{\small What is the total change in consumption of marinated ribeye?\ (give
a number). Illustrate the change on the graph. Is marinated ribeye an
ordinary of a Giffen good? (Chose one + one sentence explaining why.)}

{\small f) Decompose the total change in consumption of }$x_{1}${\small \
from e) into a substitution and income effect. (Calculate the two numbers
and show how can you find the effect on the graph.) }

{\small \bigskip }

\textbf{{\small Problem 2 (15p}) }{\small (Perfect complements) }

{\small Alex is a hotdog lover. In each hotdog bun (}$x_{1}${\small ) Alex
always inserts two hotdogs (}$x_{2}${\small ): he hates hotdogs with ratios
of buns to hotdogs different from 1:2. }

{\small a) Propose a utility function that captures Alex's preferences over (%
}$x_{1}${\small ) and (}$x_{2}${\small ) (function }$U\left( {\small x}_{1}%
{\small ,x}_{2}\right) ${\small ).}

{\small b) In the commodity space, carefully depict Alex's indifference
curves (marking the optimal proportion line).}

{\small c) Assume prices }$p_{1}=2${\small \ and }$p_{2}=2${\small \ and
income }$m=30${\small .\ Write down two secrets of happiness (two equations)
and determine the optimal choice (two numbers). Is your solution interior
(yes or no)?}

{\small d) Suppose price of a hotdog bun goes down to }$p_{1}=1,${\small \
while price }$p_{2}=2${\small \ income }$m=30${\small \ are unchanged. Find
optimal choice (two numbers). Without calculations, how much of this change
can be attributed to a substitution effect (number + one sentence explaining
the number).}

{\small \bigskip\ }

\textbf{{\small Problem 3 (15p}) }{\small (Perfect substitutes)}

{\small Pepsi }$x_{1}${\small \ and Coke }$x_{2}${\small \ both are goods
that are indistinguishable. }

{\small a) Propose a utility function (the \textquotedblleft
craziest\textquotedblright\ function you can imagine), which captures
preferences for such perfect substitutes. }

{\small b) Plot the budget line, assuming }$p_{1}=\$2,${\small \ }$p_{2}=\$6$%
{\small \ and }$m=\$12.$ {\small In the same graph plot indifference curves.}

{\small c) Find the optimal bundle for }$U\left( {\small x}_{1}{\small ,x}%
_{2}\right) =x_{1}+x_{2}${\small .\ (give two numbers). Is your solution
interior or corner? (one sentence)}

{\small d) Plot income offer curve and Engel curve given fixed prices }$%
p_{1}=\$2,p_{2}=\$6.${\small \ (two graphs). Is Pepsi normal or inferior
(choose one + one sentence)}

\bigskip

\textbf{{\small Problem 4 (15p}) }{\small (Quasilinear Preferences, Labor
Supply)}

{\small Your best friend Aiden asked you to help him to determine how much
time he should spend at work. His total available time (each day) is 24h and
his only source of income is labor income given wage rate }${\small w=\$10}$%
{\small . His spending on consumption good }${\small C}${\small \ is equal
to what he earns. The price of such good is }$p_{c}=2${\small \ }

{\small a) What is his real wage rate? (number+interpretation, one sentence )%
}

{\small b) Plot his budget set on the graph.}

{\small c) Aiden has quasilinear utility function }${\small U}\left( {\small %
R,C}\right) {\small =}\ln {\small R+C}${\small \ where }${\small R}${\small %
\ is leisure (or relaxation time) and }${\small C}${\small \ is consumption
of other goods. Write down two secrets of happiness that determine optimal
choice of }${\small R,C}${\small \ for arbitrary }${\small w,p}_{c}${\small %
. Find optimal }$R${\small \ as a function }$w,p_{c}${\small \ (you can
assume interior solution)}.{\small \ }

{\small d) What is Aiden's labor supply }${\small L}${\small \ if }${\small %
w=\$1}${\small \ and }${\small p}_{c}{\small =\$10}${\small \ (one number)}.%
{\small \ How about }${\small w=\$2}${\small \ and }${\small p}_{c}{\small %
=\$10}${\small \ (one number). Is labor supply increasing in real wage rate,
when preferences are quasilinear? (yes no answer)}

{\small \bigskip }

\textbf{\small Bonus question }{\small (Just for fun)}

{\small Let }$MRS^{V}${\small \ and }$MRS^{U}${\small \ be marginal rates of
substitution of utility functions }${\small V(x}_{1}{\small ,x}_{2}{\small )}
${\small \ and }${\small U(x}_{1}{\small ,x}_{2}{\small )}${\small \
respectively. Moreover, assume that }${\small V(x}_{1}{\small ,x}_{2}{\small %
)=f}\left[ {\small U(x}_{1}{\small ,x}_{2}{\small )}\right] ${\small \ where 
}${\small f}\left[ \cdot \right] ${\small \ is a strictly monotone
transformation. Show that }${\small MRS}^{V}{\small =MRS}^{U}$. {\small %
(Hint: Use formula for the derivative of a composite function)}

{\small \pagebreak }

\end{document}
