<<echo=FALSE, results=hide>>=
sol <- 0
while(sol > -0.11) {
## p = a - b * x
p <- sample(5:15, 1)
fctr <- sample(c(2, 4, 5, 10), 1)
x <- sample(5:15, 1) * fctr
b <- sample(1:5, 1) / fctr
a <- p + b * x

## elasticity
sol <- -1/b * p/x
}

## single-choice incl. typical errors
err <- c(1/sol, sol/p, p/sol)
err <- err[(err > -5) & (err < -0.2) & abs(err - sol) > 0.01]
rng <- c(min(1.5 * sol, -1), -0.01)
sc <- num_to_schoice(sol, wrong = err, range = rng,
  delta = 0.017, method = "delta", digits = 3)
@


\begin{question}
Consider the following inverse demand function:
$p(x) = \Sexpr{a} - \Sexpr{b} \cdot x$
for the price $p$ given the demanded quantity $x$.
What is the price elastiticy of demand at a price of
$p = \Sexpr{p}$?

<<echo=FALSE, results=tex>>=
answerlist(sc$questions)
@
\end{question}


\begin{solution}
First, we obtain the demand function by inverting the
inverse demand function:
$x = D(p) = (\Sexpr{a} - p)/\Sexpr{b} = \Sexpr{fmt(a/b, 6)} - \Sexpr{fmt(1/b, 6)} \cdot p$.

Then, at $p = 5$ the price elasticity of demand is
$$
\frac{D'(p)}{D(p)} p = \frac{-\Sexpr{fmt(1/b, 6)}}{\Sexpr{x}} \Sexpr{p} = \Sexpr{fmt(sol, 6)}.
$$

<<echo=FALSE, results=tex>>=
answerlist(ifelse(sc$solutions, "True", "False"))
@
\end{solution}


%% \extype{schoice}
%% \exsolution{\Sexpr{mchoice2string(sc$solutions)}}
%% \exname{price elasticity of demand}