Find the elasticity of demand at P = 6 and P = 20 if the demand function is given by:
Qd = 40p^-1
anyone know where i could start by anwsering this question?
For $\displaystyle p = 6$, $\displaystyle q = \frac{20}{3}$.
For $\displaystyle p = 20$, $\displaystyle q = 2$.
Elasticity of demand is given as negative the change in quantity demanding divided by change in price. ($\displaystyle E_d = -\frac{\Delta Q_d}{\Delta p} $)
Change in quantity demanded = $\displaystyle \Delta Q_d = 2 - \frac{20}{3} = \frac{-14}{3}$
Change in price = $\displaystyle \Delta p= 20 - 6 = 14$
Negative change in quantity demanded dividided by change in price = $\displaystyle E_d= -\frac{\Delta Q_d}{\Delta p} =-\frac{\frac{-14}{3}}{14} = -\frac{-14}{3}.\frac{1}{14} = \frac{1}{3} $.
I believe that elasticity is defined as sensitivity of quantity to price changes, which is the percent change in quantity divided by the percent change in price. Thus, elasticity of demand is defined informally as:
$\displaystyle \epsilon_D = - \frac{\frac{\Delta Q}{Q}}{\frac{\Delta P}{P}}$
With some algebra, we can write this expression in an alternative form as:
$\displaystyle \epsilon_D = - \frac{P}{Q} \frac{\Delta Q}{\Delta P}$
Since the demand function is given to you in a nonlinear form, we need to use calculus to find a more accurate answer:
$\displaystyle \epsilon_D = \lim_{\Delta P \to 0} - \frac{P}{Q} \frac{\Delta Q}{\Delta P} = - \frac{P}{Q} \frac{d Q}{d P}$
In your case, because the demand function is $\displaystyle Q_D = \frac{40}{P}$, we differentiate with respect to P (price) to get:
$\displaystyle \frac{d Q_D}{d P} = - \frac{40}{P^2}$
As a result, your final expression for elasticity is:
$\displaystyle \epsilon_D = - \frac{P}{Q} \frac{d Q}{d P} = - \frac{P}{Q} (- \frac{40}{P^2}) = \frac{40}{PQ}$
Plug in the respective values for P and Q at that point and you have your answer.
Note that elasticity varies not only depending on the slope of the function, but also the values of price and quantity themselves. You may have heard your economics professor stress that a linear demand function, while clearly having a constant slope, does NOT have a constant elasticity. Intuitively, it makes sense too: how much price changes impact you depends on the quantity of the product.
Moreover, I know you meant to say Q for one of these: P = 6 and P = 20. When posing a question on these forums, please take the time to make sure your question is typed correctly, as it helps us when we know what to answer. Thanks!