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Prof.
 Marek Weretka's
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Econ 301 Intermediate Microeconomics 
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Problem Set 12
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\begin_layout Subsubsection*
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Problem 1 (Nonexlcudable and Non-Rival Goods) 
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a) Explain what a non-exlcudable and non-rival good is 
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b) Give an example of:
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\begin_inset Formula $\hphantom{}$
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\begin_inset Formula $\hphantom{}$
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\begin_inset Formula $\qquad$
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- excludable and rival good 
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\begin_inset Formula $\hphantom{}$
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\begin_inset Formula $\hphantom{}$
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\begin_inset Formula $\qquad$
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- nonexcludable and rival good 
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\begin_inset Formula $\hphantom{}$
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\begin_inset Formula $\hphantom{}$
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\begin_inset Formula $\qquad$
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- excludable and nonrival good 
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\begin_inset Formula $\hphantom{}$
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\begin_inset Formula $\hphantom{}$
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\begin_inset Formula $\qquad$
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- non excludable and nonrival good 
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The examples should be different from the ones given in class.
 In each example explain why the good belongs to the given category 
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c) Which of the four categories is called a pure public good? 
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\begin_inset Formula $\vphantom{}$
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\begin_layout Subsubsection*
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Problem 2 (Provision of Public Good) 
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There are two major owners of real estate in Shorewood Hills area in Madison,
 WI, called Alfon Inc (A) and Beton Inc.
 (B).
 Both firms specialize in renting their apartments to UW faculty.
 Each year firm A and firm B decides how much to spend on the common areas
 in Shorewood Hills, such us playgrounds and bike paths.
 
\begin_inset Formula $x{}_{1}^{A}$
\end_inset

 and 
\begin_inset Formula $x{}_{1}^{B}$
\end_inset

 are spendings for Alfon and Beton respectively and 
\begin_inset Formula $x_{1}=x{}_{1}^{A}+x{}_{1}^{B}$
\end_inset

 is the total spending for both firms.
 The second type of cost is the maintainance of their individual properties
 (
\begin_inset Formula $x{}_{2}^{A}$
\end_inset

 and 
\begin_inset Formula $x{}_{2}^{B}$
\end_inset

).
 The common area is used by all members of the community, and hence 
\begin_inset Formula $x_{1}$
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 is non-excludable.
 The market rent received by firm A depends on the quality of the property
 and its surrounding common area.
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\begin_inset Formula $v^{A}(x_{1},x{}_{2}^{A})=ln(x{}_{1}^{A}+x{}_{1}^{B})+ln(x{}_{2}^{A})$
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The profit then is given by:
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\begin_inset Formula $π^{A}=v^{A}(x_{1},x{}_{2}^{A})-x{}_{1}^{A}-x{}_{2}^{A}$
\end_inset


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The customers of firm B value the common area more hence the value property
 is given by:
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\begin_inset Formula $v^{B}(x_{1},x{}_{2}^{B})=2ln(x{}_{1}^{A}+x{}_{1}^{B})+ln(x{}_{2}^{B})$
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The profit of firm B is them given by:
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\begin_inset Formula $π^{B}=v^{B}(x_{1},x{}_{2}^{B})-x{}_{1}^{B}-x{}_{2}^{B}$
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a) suppose 
\begin_inset Formula $x{}_{1}^{B}=0.5$
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.
 Find the optimal level of investment in a common area by firm A, 
\begin_inset Formula $x{}_{1}^{A}$
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.
 Mark this point in space (
\begin_inset Formula $x{}_{1}^{A},x{}_{1}^{B}$
\end_inset

).
 
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b) Find analytically the best response function for firm A (an optimal investmen
t 
\begin_inset Formula $x{}_{1}^{A}$
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 as a function of 
\begin_inset Formula $x{}_{1}^{B}$
\end_inset

) and plot the entire function in your graph from a) Make sure you also
 show the optimal 
\begin_inset Formula $x{}_{1}^{A}$
\end_inset

 for 
\begin_inset Formula $x{}_{1}^{B}>1$
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.
 
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c) Find the best response function for firm B.
 Add it to the graph.
 
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d) Find in your graph the (Nash) equilibrium for two firms (
\begin_inset Formula $x{}_{1}^{∗A},x{}_{1}^{∗B}$
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).
 What is the total amount of money invested in common area? 
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e) Is the predicted outcome associated with free riding? If yes by which
 firm? Explain why this firm is free riding? 
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f) What is the Pareto efficient joint level of investment in the common
 area? Is it greater, smaller or equal to the one observed in the market
 (point d)? Explain intuitively why is it so? 
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\begin_inset Formula $\vphantom{}$
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\begin_layout Subsubsection*
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Problem 3 (Adverse Selection) 
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Consider a second-hand car market.
 For each trader the value of a car depends on whether a trader is a buyer
 or a seller, and also on whether a car is a lemon or a plum.
 The values of cars are given by
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Lemon
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Plum
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Seller
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0
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100
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Buyer
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30
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120
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a) What are the total gains-to-trade in the market for second-hand cars?
 
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b) Assuming perfect information and equal splitting of gains-to-trade between
 buyers and sellers on each segment of the market, give the two prices for
 lemons and plums.
 Is an allocation 
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pareto efficient
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 if information is perfect? 
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Suppose the buyers cannot tell a lemon from a plum now.
 (asymmetric information).
 
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c) What is the expected value of a car for a buyer if 
\begin_inset Formula $(1/3)$
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 of the cars are lemons? What is the maximal price that a buyer might pay?
 
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d) Shall we observe a pooling or separating equilibrium if 
\begin_inset Formula $(1/3)$
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? Are plums traded in equilibrium? 
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e) Is the outcome of market interactions in a separating equilibirum 
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pareto efficient
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 ? Why or why not? 
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f) Find a treshold probability of a lemon for which we might observe a polling
 equilibrium.
 
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g) Is the allocation in pooling equilibrium Pareto efficient? Why or why
 not? 
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h) Propose a signal by which plum owners can differentiate themselves from
 the lemon owners.
 
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\begin_layout Subsubsection*
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Problem 4 (Signalling) 
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Consider a signalling model presented in the class.
 Suppose GMC is looking for new workers to its new factory in China (the
 same we considered in PS10).
 The pool of potential workers consists of two types workers: workaholics
 
\begin_inset Formula $(w)$
\end_inset

 and lazybones 
\begin_inset Formula $(l)$
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.
 The productivity of a workaholic is 10 cars, while a lazybones produces
 only 4 cars.
 Labor market is competitive - the wage is equal to the expected productivity.of
 a worker (and one car costs one $) 
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a) Find a pooling equilibrium in which workers cannot credibly signal their
 true type.
 
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Suppose now that before being hired by GMC workers can take a number of
 skill tests (the number of passed tests is 
\begin_inset Formula $e$
\end_inset

) that prove their abilities.
 The attempt to pass a test costs $1.
 If they are workaholics, they pass each test in the first approach and
 hence e passed tests cost them 
\begin_inset Formula $c^{w}(e)=e$
\end_inset

 dollars.
 If they are lazybones, then it takes two approaches to pass one test.
 Consequently e passed tests cost 
\begin_inset Formula $c^{l}(e)=2e$
\end_inset

 
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b) Are two passed tests a credible signal that a worker is a workaholic?
 Why or why not? 
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c) What is the minimal number of tests that constitutes a credible signal
 for the employer? 
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d) Is signalling if form of taking tests efficient form the point of view
 of the society?
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