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Prof.
 Marek Weretka's
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Econ 301 Intermediate Microeconomics 
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Problem Set 3
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\begin_inset Formula $\vphantom{}$
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\begin_layout Subsubsection*
Problem 1 (Cobb Douglas Utility function) 
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Tony likes nuts 
\begin_inset Formula $x_{1}$
\end_inset

 and berries 
\begin_inset Formula $x{}_{2}$
\end_inset

, and his preferences are described by the following utility function 
\begin_inset Formula $U(x_{1},x{}_{2})=x_{1}^{a}x{}_{2}^{b}$
\end_inset

.
 Find the following variables: 
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\begin_inset Formula $\vphantom{}$
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1) the optimal fraction (percentage) of income spent on berries; 
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2) the optimal amount of total cash (dollars) spent on berries; 
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3) the optimal quantity of nuts consumed; 
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4) the slope of the indifference curve at the optimal bundle for the following
 values of parameters: 
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\begin_inset Formula $\vphantom{}$
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a) 
\begin_inset Formula $a=4,b=8,p_{1}=5,p_{2}=10,m=60$
\end_inset


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b) 
\begin_inset Formula $a=(1/3),b=(1/3),p_{1}=4,p_{2}=1,m=12$
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c) 
\begin_inset Formula $a=(1/2),b=(3/2),p_{1}=5,p_{2}=1,m=20$
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Hint: Instead of calculating optimal choices using "two secrets of happiness",
 take advantage of the demand formulas for Cobb Douglas utility that we
 derived in the class.
 
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\begin_inset Formula $\vphantom{}$
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\begin_layout Subsubsection*
Problem 2 
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Benjamin spends his time either watching movies (
\begin_inset Formula $x{}_{1}$
\end_inset

) (he uses "on demand" option, cable TV) or listening to songs - MP3 downloaded
 from the Internet (
\begin_inset Formula $x{}_{2}$
\end_inset

) .
 His preferences are described by
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\begin_inset Formula $\vphantom{}$
\end_inset


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\begin_layout Standard
\begin_inset Formula $U(x_{1},x_{2})=ln(x_{1})+ln(x_{2})$
\end_inset


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\begin_inset Formula $\vphantom{}$
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Answer the following questions: 
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\begin_inset Formula $\vphantom{}$
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a) Derive Benjamin's demand for movies and MP3 files as a function of prices
 
\begin_inset Formula $p_{1},p_{2}$
\end_inset

 and his income 
\begin_inset Formula $m$
\end_inset

.
 (do not use Cobb Douglas formula but rather derive demand using "two secrets
 of happiness").
 
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b) Fix the price of MP3 at 
\begin_inset Formula $p_{2}=1$
\end_inset

, and income on 
\begin_inset Formula $m=10$
\end_inset

.
 Find the price offer curve (give an exact formula 
\begin_inset Formula $x{}_{2}=f(x_{1})$
\end_inset

) and plot it in the commodity space.
 Find the demand curve 
\begin_inset Formula $x_{1}=f(p_{1})$
\end_inset

 and plot it in the graph (with 
\begin_inset Formula $p_{1}$
\end_inset

 on vertical axis and 
\begin_inset Formula $x_{1}$
\end_inset

 on horizontal axis).
 
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c) Is 
\begin_inset Formula $x_{1}$
\end_inset

 an ordinary good or a Giffen good? Explain.
 
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d) Now fix 
\begin_inset Formula $p_{1}=1$
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 and 
\begin_inset Formula $p_{2}=1$
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.
 In the commodity space, plot the income offer curve.
 In addition, in two separate graphs, plot Engel curves for both movies
 and MP3 files.
 Argue that the two commodities are normal (not inferior).
 
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e) For the demand functions from point a), determine whether the two goods
 are gross complements, substitutes or neither.
 
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\begin_inset Formula $\vphantom{}$
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\begin_layout Subsubsection*
Problem 3 (Perfect Complements) 
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Consider Trevor from our previous problem set who begins his day with strawberry
 milkshake.
 To prepare it, he mixes milk, 
\begin_inset Formula $x_{1}$
\end_inset

, strawberries 
\begin_inset Formula $x{}_{2}$
\end_inset

 and does so always in his favorite: proportion 1 glass of milk to 2 strawberrie
s.
 What is his utility function? Answer all the questions from Problem 2,
 from a) to e) using these preferenes.
 
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\begin_inset Formula $\vphantom{}$
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Problem 4 (Perfect Substitutes) 
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Kate's preferences are defined over consumption of two types of apples:
 Red Delicious (
\begin_inset Formula $x_{1}$
\end_inset

) and Jonagold (
\begin_inset Formula $x{}_{2}$
\end_inset

)
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\begin_inset Formula $\vphantom{}$
\end_inset


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\begin_inset Formula $U(x_{1},x{}_{2})=2x_{1}+x{}_{2}.$
\end_inset


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\begin_inset Formula $\vphantom{}$
\end_inset


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Answer all the questions from Problem 2, starting from a) to e) using these
 preferenes.
 
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\begin_inset Formula $\vphantom{}$
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Problem 5 (Quasilinear preferences) 
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George is a stamp (
\begin_inset Formula $x_{1}$
\end_inset

) collector, but he also likes fancy clothes (
\begin_inset Formula $x{}_{2}$
\end_inset

).
 His utility function is given by
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\begin_inset Formula $\vphantom{}$
\end_inset


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\begin_layout Standard
\begin_inset Formula $U(x_{1},x{}_{2})=x_{1}+10x{}_{2}-(1/2)x{}_{2}².$
\end_inset


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\begin_inset Formula $\vphantom{}$
\end_inset


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Each stamp costs 
\begin_inset Formula $p_{1}=1$
\end_inset

 and a piece of his favorite clothing costs 
\begin_inset Formula $p_{2}=2$
\end_inset

 
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a) Assuming that his total income is given by m=$10, find his optimal choice
 of 
\begin_inset Formula $x_{1}$
\end_inset

 and 
\begin_inset Formula $x{}_{2}.$
\end_inset

 (Is it interior?) 
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b) Suppose next year George's salary doubles, resulting in his higher income
 m=$20.
 Find his new demanded quantities of stamps and clothes.
 (Is it interior?).
 
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c) Harder: In point a) and b) what is the marginal utility from one dollar
 invested in stamps, and in clothing (at the optimal demand).
 Are they equal? Hint: Unlike in a Cobb-Douglas utility function, with quasiline
ar preferences we might have corners!
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