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\begin{document}

\textbf{ECON 4113. HOMEWORK 5. 100 POINTS. DUE APRIL 20.}


\begin{enumerate}
\item (35 points) Draw the direction fields ($t$ on the $x$ axis) for the following differential equations:
\begin{enumerate}
\item[a)] $\dot y = \frac{ty}{1+t^2}$
\item[b)] $\dot y = y(y-2)$
\item[c)] $\dot y = t^2-y^2$
\end{enumerate}

\item (40 points) Consider a Cournot duopoly with heterogeneous firms. Suppose that the inverse demand is given by $P(Q) = a - Q$. Firm 1 has cost $c_1$ to produce one unit of output and firm 2 has cost $c_2\neq c_1$.
\begin{enumerate}
\item[a)] Find the unique Nash Equilibrium of the Cournot duopoly game. Under which conditions on $c_1$ and $c_2$ do both firms produce? Under which conditions on $c_1$ and $c_2$ does only one firm produce?
\item[b)] Suppose the firms change the quantities $q_1$ and $q_2$ at each time $t\in[0,\infty)$. Consider the following dynamics:
\begin{eqnarray*}
\dot q_1 &=& BR(q_2) - q_1\\
\dot q_2 &=& BR(q_1) - q_2\\
\end{eqnarray*}
Where $BR(q_2)$ stands for the amount of $q_1$ that is optimal for firm 1 given that firm 2 produces $q_2$. Suppose that $c_1$ and $c_2$ are such that only one firm is producing in Nash Equilibrium. Draw the phase portrait of this system and find the explicit solution. Determine the stability properties of the steady state.
\end{enumerate}
\item (25 points) Find the explicit solution of the following system of linear differential equations:
\begin{eqnarray*}
\dot y_1 &=& y_1 - 5y_2\\
\dot y_2 &=& 2y_1 - 5y_2\\
\end{eqnarray*}
Determine the stability properties of the steady state $y_1=0;\ y_2=0$.
\end{enumerate}
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