Question

what are the respective price elasticities of supply at a and b on the supply curve shown in the figure below? supply instruction: simplify and enter your responses as ratios. for instance, a ratio of 12/20 should be changed to 3/5. elasticity of supply at point a: / . elasticity of supply at point b: / .

Explanation:

Step1: Recall the formula for price - elasticity of supply

The formula for price - elasticity of supply ($E_s$) using the point - method is $E_s=\frac{\text{d}Q}{\text{d}P}\times\frac{P}{Q}$. For a linear supply curve $Q = a + bP$, $\frac{\text{d}Q}{\text{d}P}=b$. The slope of the supply curve $S$ is constant. The slope of the supply curve $m=\frac{\Delta P}{\Delta Q}=\frac{6 - 4}{12 - 9}=\frac{2}{3}$, so $\frac{\text{d}Q}{\text{d}P}=\frac{3}{2}$.

Step2: Calculate elasticity at point A

At point $A$, $P = 4$ and $Q = 9$. Using the formula $E_s=\frac{\text{d}Q}{\text{d}P}\times\frac{P}{Q}$, we substitute $\frac{\text{d}Q}{\text{d}P}=\frac{3}{2}$, $P = 4$, and $Q = 9$. Then $E_s=\frac{3}{2}\times\frac{4}{9}=\frac{2}{3}$.

Step3: Calculate elasticity at point B

At point $B$, $P = 6$ and $Q = 12$. Using the formula $E_s=\frac{\text{d}Q}{\text{d}P}\times\frac{P}{Q}$, we substitute $\frac{\text{d}Q}{\text{d}P}=\frac{3}{2}$, $P = 6$, and $Q = 12$. Then $E_s=\frac{3}{2}\times\frac{6}{12}=\frac{3}{4}$.

Answer:

Elasticity of supply at point A: $2/3$
Elasticity of supply at point B: $3/4$