Price elasticity of demand in the point of economic equilibrium

Question

The function of demand is:\begin{align*} D(p) = 66-3p-p^2 \\\end{align*} The function of supply is: \begin{align*}S(p) = 4p^2+8p-114\\ \end{align*} The task is to find price elasticity of demand in the point of economic equilibrium.

I have found out that the equilibrium price is 5 and equilibrium demand is 26.

I also have a formula that states that $E = k * P/Q$ , where $P$ - equilibrium price, $Q$ - equilibrium demand and $k$ - coefficient of $S(p)$ slope

How to find $k$ or are there another methods to solve this task?

Answer 1

Your formula is overly simplified.

The elasticity is supposed to be "how much does supply change if the price changes". Now, the natural way of looking at this is $S'(p)$, the derivative of $S$. However, the units of that are hard to make use of. Hence, we normalize $S'(p)$ by multiplying it with $P/S(p)$, hence getting the "percentage response of $S$ to a percent change in $P$".

Long story short, your $k$ should satisfy

$$ k = S'(p)$$

Note that this is the price elasticity of supply. For the price elasticity of demand, you would instead use $D'(p) P / D(p)$.