```{r data generation, 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) ``` Question ======== Consider the following inverse demand function: $p(x) = `r a` - `r b` \cdot x$ for the price $p$ given the demanded quantity $x$. What is the price elastiticy of demand at a price of $p = `r p`$? ```{r questionlist, echo = FALSE, results = "asis"} answerlist(sc$questions, markup = "markdown") ``` Solution ======== First, we obtain the demand function by inverting the inverse demand function: $x = D(p) = (`r a` - p)/`r b` = `r fmt(a/b, 6)` - `r fmt(1/b, 6)` \cdot p$. Then, at $p = 5$ the price elasticity of demand is $$ \frac{D'(p)}{D(p)} p = \frac{-`r fmt(1/b, 6)`}{`r x`} `r p` = `r fmt(sol, 6)`. $$ ```{r solutionlist, echo = FALSE, results = "asis"} answerlist(ifelse(sc$solutions, "True", "False"), markup = "markdown") ``` Meta-information ================ extype: schoice exsolution: `r mchoice2string(sc$solutions)` exname: price elasticity of demand