1. Point Elasticity of Demand

The demand function is given by:

$x = 48 - 3p^2$

To find the point elasticity of demand at $p = 3$, we use the formula for point elasticity of demand:

$E(p) = \frac{d\ln(x)}{d\ln(p)} = \frac{p}{x} \cdot \frac{dx}{dp}$

Step 1: Compute $\frac{dx}{dp}$

First, differentiate the demand function $x = 48 - 3p^2$ with respect to $p$:

$\frac{dx}{dp} = -6p$

Step 2: Evaluate $\frac{dx}{dp}$ at $p = 3$

Substituting $p = 3$:

$\frac{dx}{dp} = -6 \times 3 = -18$

Step 3: Compute $x$ at $p = 3$

Substitute $p = 3$ into the demand function:

$x = 48 - 3(3^2) = 48 - 27 = 21$

Step 4: Calculate the Point Elasticity

Now, using the point elasticity formula:

$E(3) = \frac{3}{21} \cdot (-18) = \frac{3 \times (-18)}{21} = -\frac{54}{21} = -\frac{18}{7} \approx -2.57$

So, the point elasticity of demand at $p = 3$ is approximately $-2.57$.

2. Relative Increase in Demand Due to a 4% Decrease in Price

Step 1: Calculate the new price after a 4% decrease

The price is decreased by 4%:

$\text{New price} = 3 \times (1 - 0.04) = 3 \times 0.96 = 2.88$

Step 2: Compute the new demand at $p = 2.88$

Substitute $p = 2.88$ into the demand function:

$x_{new} = 48 - 3(2.88^2) = 48 - 3(8.2944) = 48 - 24.8832 = 23.1168$

Step 3: Calculate the relative increase in demand

The original demand at $p = 3$ was $x = 21$. The relative increase in demand is:

$\text{Relative increase} = \frac{x_{new} - x_{old}}{x_{old}} \times 100 = \frac{23.1168 - 21}{21} \times 100 \approx \frac{2.1168}{21} \times 100 \approx 10.08\%$

So, the relative increase in demand is approximately 10.08%.

3. Approximation to the Elasticity of Demand

Using the formula for elasticity approximation:

$E \approx \frac{\text{Percentage change in demand}}{\text{Percentage change in price}} = \frac{10.08\%}{4\%} \approx 2.52$

So, the approximate elasticity of demand is $-2.52$.

Summary

• Point Elasticity at $p = 3$: $-2.57$
• Relative increase in demand: $10.08\%$
• Approximate Elasticity: $-2.52$

Would you like more details or have any questions?

5 Related Questions:

1. How does elasticity of demand change with different price levels?
2. What are the factors that influence the elasticity of a commodity?
3. How can businesses use elasticity of demand in pricing strategies?
4. What are the implications of a negative elasticity value?
5. How does elasticity affect total revenue?

Tip:

Elasticity of demand helps predict how changes in price affect the quantity demanded, crucial for setting prices strategically.