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Question 21 1 pts Suppose the demand for oranges is given by $Q_{d}=A p^{-5} m^...
May 15, 2024
Solution by Steps
step 1
The demand function for oranges is given by Qd=Ap−5m5Q_d = A p^{-5} m^5
step 2
To find the partial elasticity of demand with respect to price, we use the formula for elasticity: Ep=∣∂Qd∂p⋅pQd∣E_p = \left| \frac{\partial Q_d}{\partial p} \cdot \frac{p}{Q_d} \right|
step 3
First, compute the partial derivative of QdQ_d with respect to pp: ∂Qd∂p=A⋅(−5)⋅p−6⋅m5=−5Ap−6m5\frac{\partial Q_d}{\partial p} = A \cdot (-5) \cdot p^{-6} \cdot m^5 = -5A p^{-6} m^5
step 4
Substitute ∂Qd∂p\frac{\partial Q_d}{\partial p} and QdQ_d into the elasticity formula: Ep=∣−5Ap−6m5⋅pAp−5m5∣E_p = \left| \frac{-5A p^{-6} m^5 \cdot p}{A p^{-5} m^5} \right|
step 5
Simplify the expression: Ep=∣−5Ap−6m5⋅pAp−5m5∣=∣−5Ap−5m5Ap−5m5∣=∣−5∣=5E_p = \left| \frac{-5A p^{-6} m^5 \cdot p}{A p^{-5} m^5} \right| = \left| \frac{-5A p^{-5} m^5}{A p^{-5} m^5} \right| = \left| -5 \right| = 5
Answer
The partial elasticity of demand with respect to price is 5.
Key Concept
Partial Elasticity of Demand
Explanation
The partial elasticity of demand with respect to price measures the responsiveness of the quantity demanded to a change in price, holding other factors constant. In this case, the elasticity is 5, indicating a highly elastic demand.
Question 22
step 1
The demand function for apples is given by Qd=Ap−5m5Q_d = A p^{-5} m^5
step 2
To find the partial elasticity of demand with respect to income, we use the formula for elasticity: Em=∣∂Qd∂m⋅mQd∣E_m = \left| \frac{\partial Q_d}{\partial m} \cdot \frac{m}{Q_d} \right|
step 3
First, compute the partial derivative of QdQ_d with respect to mm: ∂Qd∂m=A⋅p−5⋅5m4=5Ap−5m4\frac{\partial Q_d}{\partial m} = A \cdot p^{-5} \cdot 5m^4 = 5A p^{-5} m^4
step 4
Substitute ∂Qd∂m\frac{\partial Q_d}{\partial m} and QdQ_d into the elasticity formula: Em=∣5Ap−5m4⋅mAp−5m5∣E_m = \left| \frac{5A p^{-5} m^4 \cdot m}{A p^{-5} m^5} \right|
step 5
Simplify the expression: Em=∣5Ap−5m4⋅mAp−5m5∣=∣5Ap−5m5Ap−5m5∣=∣5∣=5E_m = \left| \frac{5A p^{-5} m^4 \cdot m}{A p^{-5} m^5} \right| = \left| \frac{5A p^{-5} m^5}{A p^{-5} m^5} \right| = \left| 5 \right| = 5
Answer
The partial elasticity of demand with respect to income is 5.
Key Concept
Partial Elasticity of Demand
Explanation
The partial elasticity of demand with respect to income measures the responsiveness of the quantity demanded to a change in income, holding other factors constant. In this case, the elasticity is 5, indicating a highly elastic demand.
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