```{r data generation, echo = FALSE, results = "hide"} ## 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 ``` 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 $p = `r p`$? 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 $$ \frac{D'(p)}{D(p)} p = \frac{-`r fmt(1/b, 6)`}{`r x`} `r p` = `r fmt(sol, 6)`. $$ Meta-information ================ extype: num exsolution: `r fmt(sol, 3)` extol: 0.01 exname: price elasticity of demand