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\title{General Mathematics and Computational Science I}
\author{Exercise 18}
\date{November 20, 2007}
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\begin{enumerate}


\item Solve the difference equation
\[
  x_{n+1} = x_n + h \, x_n \,, \qquad x_0 = 1 \,.
\]
Then show that
\[
  \lim_{h\to 0} x_{t/h} = \mathrm{e}^t \,,
\]
where the limit is taken such that $t/h$ is always integer.

\emph{Hint:} Recall that
\[
  \lim_{n \to \infty} \biggl(1 + \frac1n \biggr)^n = \mathrm{e} \,.
\]

\item Reconsider the argument leading to the ``cobweb theorem of
economics'' from class.
\begin{enumerate}
\item Describe the relative size of the coefficients $m_{\mathrm d}$,
$m_{\mathrm s}$, $b_{\mathrm d}$, and $m_{\mathrm s}$ that lead to a
negative stable equilibrium price $p^*$.

\item Can you think of an economic context where a negative stable
equilibrium price might occur?
\end{enumerate}



\end{enumerate}
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