Question

Using the elasticity formulas.] Answer the following questions: a) The initial price for an item is $6.00, and the quantity demanded is 550 units. When the price is raised to $6.50, the quantity demanded falls to 520 units. What is the point elasticity of demand? How would we categorize this elasticity (elastic, inelastic, unit elastic,...)? b) A product’s point price elasticity has been estimated at -1.85 (different from part a). At the initial price of $30, the quantity demanded was 30 units. If the firm cuts the price to $27.25, how much will quantity demanded and sold increase? c) A firm’s demand curve is estimated to be Q = 450 - 2.5P, where Q is quantity and P is the price of the good. At P = $40, what is the point elasticity of demand?

Answer 1

Answer on Question #81445, Economics / Microeconomics

a) $P_0 = 6$; $Q_0 = 550$

$$P_1 = 6.5; \quad Q_1 = 520$$

% change of price is

$$\frac{\Delta P}{P_1} = \frac{0.5}{6.5} \approx 0.08$$

It’s 8%.

% change of quantity is

$$\frac{\Delta Q}{Q_0} = \frac{30}{550} \approx 0.05$$

It’s 5%.

$$PED = \frac{\% Q_{change}}{\% P_{change}} = \frac{5}{8} = 0.625$$

PED<1 - inelastic.

b) $P_0 = 30$; $Q_0 = 30$

$$P_1 = 27,25; \quad Q_1 - ?$$ $$PED = \frac{\% Q_{change}}{\% P_{change}} = \frac{\frac{\Delta Q}{Q_0}}{\frac{\Delta P}{P_1}} = \frac{\Delta Q \cdot P_1}{Q_0 \cdot \Delta P}$$ $$PED = -1.85$$ $$\frac{\Delta Q \cdot P_1}{Q_0 \cdot \Delta P} = -1.85$$ $$\Delta Q \cdot P_1 = -1.85 \cdot Q_0 \cdot \Delta P$$ $$\Delta Q = \frac{-1.85 \cdot Q_0 \cdot \Delta P}{P_1}$$ $$\Delta Q = \frac{-1.85 \cdot 30 \cdot (-2.75)}{27.25} \approx \frac{152.625}{27.25} \approx 6$$ $$\Delta Q = Q_1 - Q_0$$ $$Q_1 = \Delta Q + Q_0$$ $$Q_1 = 6 + 30 = 36$$ $$\text{c)} Q = 450 - 2.5P$$

First we need to obtain the derivative of the demand function when it's expressed with $Q$ as a function of $P$. Since quantity $(Q)$ goes down by 2.5 each time price $(P)$ goes up by 1,

$$\frac{\Delta Q}{\Delta P} = -2.5$$

Next we need to find the quantity demanded at each associated price and pair it together with the price: $(40; 350^*)$.

$$(* Q = 450 - 2.5P, P = 40 : Q = 450 - 100 = 350)$$

Then we plug those values into our point elasticity of demand formula to obtain the following:

$$e = \frac{\Delta Q}{\Delta P} \cdot \left(\frac{P}{Q}\right)$$ $$e = \frac{\Delta Q}{\Delta P} \cdot \left(\frac{P}{Q}\right) = -2.5 \cdot \frac{40}{350} = -0.29.$$

Answer provided by https://www.AssignmentExpert.com