Math Problem Statement
A company estimates that the weekly sales q of its product is related to the product's price by the function q=13900/p^12/5 where p is in dollars. Currently, each unit of the product is selling for $3. Determine the point elasticity of demand of this product. (Do not round e; answer exactly.) The point elasticity of demand is e=. Round these next two parts to the nearest whole number. Determine the effect on revenue if the price were decreased by 1%. If the price were decreased by 1%, the revenue will (increase/decrease) Select an answer by $.
Solution
To solve this problem, we'll break it down into parts:
1. Point Elasticity of Demand (e)
The point elasticity of demand, , is given by the formula: Where:
- (the sales function)
- is the price of the product
Step 1: Differentiate with respect to
To find , we differentiate with respect to :
= -\frac{166800}{5}p^{-\frac{17}{5}}$$ #### Step 2: Substitute $$p = 3$$ and $$q = \frac{13900}{3^{12/5}}$$ Now substitute $$p = 3$$ into both $$\frac{dq}{dp}$$ and $$q$$. - $$q = \frac{13900}{3^{12/5}}$$ - $$\frac{dq}{dp} = -\frac{166800}{5} \cdot 3^{-17/5}$$ #### Step 3: Calculate the point elasticity $$e$$ Using the formula for $$e$$: $$e = \frac{\frac{dq}{dp} \cdot p}{q} = \frac{-\frac{166800}{5} \cdot 3^{-17/5} \cdot 3}{\frac{13900}{3^{12/5}}} = -\frac{500400 \cdot 3^{-17/5}}{5 \cdot 13900 \cdot 3^{-12/5}} = -\frac{500400}{69500} = -7.2$$ Thus, the point elasticity of demand is: $$e = -7.2$$ ### 2. Effect on Revenue When Price Decreases by 1% Revenue, $$R$$, is given by: $$R = p \cdot q = p \cdot \frac{13900}{p^{12/5}} = 13900 \cdot p^{-7/5}$$ To determine the effect of a 1% price decrease on revenue, we use the elasticity of demand: - $$\Delta R \approx e \cdot \Delta p \cdot R$$ Since $$e = -7.2$$, and a 1% price decrease means $$\Delta p = -0.01$$, the change in revenue is: $$\Delta R \approx -7.2 \cdot (-0.01) \cdot R = 0.072 \cdot R$$ So, revenue will increase by approximately 7.2%. Finally, if the price were decreased by 1%, revenue will **increase** by approximately 7%. Would you like more details on any step? Here are some related questions: 1. What is the formula for revenue in terms of price and sales? 2. How does elasticity affect revenue when prices change? 3. Can you explain how differentiation helps find elasticity? 4. What happens if elasticity is greater than -1 (inelastic demand)? 5. How does elasticity vary between different types of goods? **Tip:** The point elasticity helps businesses understand how sensitive their sales are to price changes—critical for pricing strategies!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Elasticity of Demand
Differentiation
Revenue Analysis
Formulas
Point Elasticity of Demand: e = (dq/dp) * (p/q)
Revenue: R = p * q
Effect on Revenue: ΔR ≈ e * Δp * R
Theorems
-
Suitable Grade Level
Undergraduate Economics or Business Math
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