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\textbf{\Large Homework 2}
\textbf{due 2/10/2017}
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\bigskip\noindent\textbf{Problem 1 } Senator Iacovelli and
Senator Jacoban are bargaining over which policy should be implemented out of
the set $\{X,Y,Z\}$. The game they play is as follows:
\begin{itemize}
\item First, Sen. Iacovelli vetoes one of the three policies
\item Second, after observing Sen. Iacovelli's choice, Sen. Jacoban vetoes one
of the remaining policies
\item The policy that has not been vetoed at this point is implemented
\end{itemize}
Sen. Iacovelli's preferences are $X\succ Y\succ Z$, while Sen. Jacoban's
are given by $Z\succ Y\succ X$.
\medskip\noindent\textbf{a. } Solve for the game's subgame perfect Nash
equilibria. Be precise.
\medskip\noindent\textbf{b. } Now suppose the game is changed so that Sen.
Jacoban moves first, followed by Sen. Iacovelli. Solve for the modified game's
subgame perfect Nash equilibrium. Be precise.
\newpage\noindent\textbf{Problem 2} Consider the
sequential move game below. Each set of payoffs is ordered $u_1, u_2, u_3$, where $u_i$ is player i's utility.
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For all possible values of $x$, $y$, and $z$, What is the subgame perfect equilibrium of this game? Remember to describe players' choices at all nodes, including those that are unreached.
\bigskip\noindent\textbf{Problem 3} Consider a market with inverse
market demand given by $P=10-\frac{1}{100}Q$. Firm A is a monopoly producer,
with marginal cost equal to \$2.
\medskip\noindent\textbf{a. } Calculate Firm A's optimal quantity, and its profit as a monopolist.
\bigskip\noindent Now, suppose that Firm A has discovered a new technology that
will allow it to produce at a marginal cost of \$0. Implementing the new
technology will cost firm A to incur a fixed cost of \$1,000.
\medskip\noindent\textbf{b. } Is it profitable for Firm A to implement the new
technology?
\bigskip\noindent Now, suppose that Firm A learns that Firm B is considering
entering the market to compete with Firm A. To enter, Firm B would have to
construct a factory at a cost of \$500, and then Firm A and Firm B would
compete in Cournot oligopoly.\footnote{So that inverse market demand is given
by $P=10-\frac1{100}(q_1+q_2)$, where $q_i$ is firm $i$'s quantity.} If firm B
entered, its marginal cost would also equal \$2.
\medskip\noindent\textbf{c. } Calculate the market price, firm A's profit, and
firm B's profit under Cournot competition. Would firm B profitably enter the
market? (Assume for part c. that Firm A has not implemented the new
technology.)
\medskip\noindent\textbf{d. } Consider an extensive form game with two rounds.
In round 1, Firm A decides whether or not to implement the new technology. In
round 2, Firm B decides whether or not to enter the market. Using your answers
above (and possibly new calculations), determine the subgame perfect
equilibrium of this game.
\medskip\noindent\textbf{e. } Policymakers sometimes worry that monopolists are
less likely to innovate than firms in a competitive market.\footnote{See e.g.
``Enhanced market power can also be manifested in... diminished innovation.'',
page 2 of \textit{Horizonal Merger Guidelines}, 2010, U.S. Department of
Justice and Federal Trade Commission.} Do your answers above suggest any
caveats to this view?
\bigskip\noindent\textbf{Problem 4 } Gibbons, problem 2.5
\bigskip\noindent\textbf{Problem 5 } Gibbons, problem 2.7
\bigskip\noindent\textbf{Problem 6 }
Elroy and Morgan compete in a race. At the start of the race, both players are 6 steps away from the finish line. Who gets the first turn is determined by a toss of a fair coin; the players then alternate turns, with the results of all previous turns being observed before the current turn occurs.
During a turn, a player chooses from these four options:
\begin{itemize}
\item Do nothing at cost 0;
\item Advance 1 step at cost 2;
\item Advance 2 steps at cost 7;
\item Advance 3 steps of at cost 15.
\end{itemize}
The race ends when the first player crosses the finish line.
The winner of the race receives a payoff of 20, while the loser
gets nothing. Assume there is no discounting, but that all else equal each
player prefers to finish the game more quickly.
Find the subgame perfect equilibria of this game.\footnote{Hint: In the SPE, a player's choice at
a decision node only depends on the number of steps he has left
and on the number of steps his opponent has left. Make a table with columns and rows numbered from 1-6, representing how many steps each player has left to finish. Solve for what one player will do at each possible state. Since the game is symmetric, solving for what one player will do at each point in your table is sufficient to solve the game.}
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