Step 1: Understand the formula for point elasticity of demand

The point elasticity of demand ($E_d$) is given by: $E_d = \frac{p}{q} \cdot \frac{dq}{dp}$ Where:

• $p$ is the price
• $q$ is the quantity demanded
• $\frac{dq}{dp}$ is the derivative of the demand function with respect to price

Step 2: Differentiate the demand equation

The demand equation is: $q = 400 - 35p + p^2$ Differentiate $q$ with respect to $p$: $\frac{dq}{dp} = -35 + 2p$

Step 3: Calculate $q$ and $\frac{dq}{dp}$ at $p = 13$

Substitute $p = 13$ into the demand equation to find $q$: $q = 400 - 35(13) + (13)^2$ $q = 400 - 455 + 169 = 114 \, \text{(in thousands)}$

Now substitute $p = 13$ into $\frac{dq}{dp}$: $\frac{dq}{dp} = -35 + 2(13) = -35 + 26 = -9$

Step 4: Compute point elasticity

Substitute $p = 13$, $q = 114$, and $\frac{dq}{dp} = -9$ into the elasticity formula: $E_d = \frac{13}{114} \cdot (-9)$ $E_d = \frac{13 \cdot -9}{114} = \frac{-117}{114} \approx -1.03$

Step 5: Interpret the elasticity

The point elasticity of demand is approximately $-1.03$. Since $|E_d| > 1$, demand is elastic, meaning that changes in price will have a greater-than-proportional effect on quantity demanded.

Step 6: Calculate the approximate change in demand

If the price increases by $1\%$, the approximate change in demand ($\%\Delta q$) can be found using: $\%\Delta q \approx E_d \cdot \%\Delta p$ Here, $\%\Delta p = 1\%$, and $E_d = -1.03$: $\%\Delta q \approx -1.03 \cdot 1 = -1.03\%$

Thus, the demand will decrease by approximately 1.03% if the price increases by 1%.

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Summary:

• Point Elasticity of Demand: $-1.03$
• Approximate Change in Demand: $-1.03\%$

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Would you like further details or assistance with another related topic?

Related Questions:

1. How do you derive the formula for elasticity in general?
2. What happens to elasticity when demand is perfectly inelastic?
3. Can elasticity help determine optimal pricing strategies?
4. How would a $2\%$ price change affect demand in this scenario?
5. What are the implications of elastic demand for revenue management?

Tip:

Elasticity can be a powerful tool for predicting consumer behavior. Always consider both the magnitude and the sign of the elasticity when interpreting results.