The demand equation for a product is:
q=60/p + ln(65-p^3)
A) Determine the point of elasticity of demand when p=4, and classify the demand as elastic, inelastic, or of unit elasticity at this price level.
B) If the price is lowered by 2% (from $4.00 to $3.92), use the answer in part (a) to estimate the corresponding percentage change in quantity sold.
C) Will the changes in part (b) result in an increase or decrease in revenue? Explain.
Can someone tell me if this is right? Pretty please?
I think:
A)
((ΔQ/Average of Q)*100) /((ΔP/Average of P)*100)
If P= 2 Q= 34.04
If P= 4 Q= 15
(19.04/24.52)/(2/3) = .7765/.667 =1.164 >1 = elastic
B) Q= 16.8671
16.8671-15= 1.867/15= .1244= 12%
C)
4*15=60
3.92*16.8671=66.11
Decrease in price = increase in demand = increase in profit
...anyone?
Yes, your approach is correct, except watch for the following.
(1) In principle, part (A) should be done using calculus. If you have not done calculus before, what you did is appropriate, EXCEPT that the interval for p between 4 and 2 is way too large because q(p) varies rapidly between p=2 and p=4.
The theoretical result using calculus should give E(4)=-13.8
Remember that elasticity is always negative, although we only use the absolute (positive) value.
(2) You have given the exact answer correctly, but not using the result of part (A) as requested.
Using part (A),
E(p)=E(4)=-13.8
Δp=-0.02
so
Δq=E(p)*Δp=-13.8*(-.02)=0.276, or 27.6%
(compared with the correct value of 12%).
The enormous error is due to the rapidity of change of E(p) at this point. If we had used a change of 0.1%, it would have been 1.38% vs 1.27% and the approximation would have been acceptable.
(3)
Also, you need to watch the use of equal sign.
Equal sign means just that, values on each side are equal.
In your statement:
16.8671-15= 1.867/15= .1244= 12%
you have added the division of 15 on the second expression, which tips the equality. The proper way to present it is as follows:
(16.8671-15)/15= 1.867/15= .1244= 12%
This ensures that equality is true throughout.
(4) Part (C)
Your calculation of the revenues is correct. The increase is 10.2%.
A) To determine the point of elasticity of demand, we need to find the value of q when p = 4, and then calculate the corresponding value of the absolute derivative of q with respect to p at that point.
Step 1: Substitute p = 4 into the demand equation:
q = 60/4 + ln(65-4^3)
q = 15 + ln(65-64)
q = 15 + ln(1)
q = 15
Step 2: Now, take the first derivative of q with respect to p:
dq/dp = -60/p^2 + 3p^2/(65-p^3)
Step 3: Substitute p = 4 into the derivative:
dq/dp = -60/4^2 + 3(4^2)/(65-4^3)
dq/dp = -3.75 + 3(16)/(65-64)
dq/dp = -3.75 + 48/1
dq/dp = -3.75 + 48
dq/dp = 44.25
The absolute value of the derivative at p = 4 is 44.25.
Step 4: Determine the point of elasticity:
To find the point of elasticity, we need to set the derivative equal to zero and solve for p:
-60/p^2 + 3p^2/(65-p^3) = 0
Simplifying the equation:
-60(65-p^3) + 3p^4 = 0
-3900 + 60p^3 + 3p^4 = 0
3p^4 + 60p^3 - 3900 = 0
You can use numerical methods or calculus techniques to approximate the value of p that satisfies this equation: p ≈ 3.9
Step 5: Classify the demand:
Since the point of elasticity lies to the right of p = 4, which is the price level we are interested in, we can classify the demand as elastic at this price level.
B) To estimate the percentage change in quantity sold when the price is lowered by 2%, we can use the concept of price elasticity of demand at the point of elasticity found above.
Step 1: Calculate the percentage change in price:
Percentage change in price = (New price - Old price) / Old price * 100%
= (3.92 - 4) / 4 * 100%
≈ -2%
Step 2: Use the point of elasticity value obtained in part (a) to estimate the percentage change in quantity sold:
Percentage change in quantity sold = Price elasticity of demand * Percentage change in price
≈ 44.25 * (-2%)
≈ -88.5%
Therefore, the estimated percentage change in quantity sold when the price is lowered by 2% is approximately -88.5%.
C) To determine if the changes in part (b) will result in an increase or decrease in revenue, we need to consider the price and quantity relationship.
Since elastic demand indicates that quantity changes more significantly than price, a decrease in price will generally result in an increase in quantity sold. In this case, the estimated percentage change in quantity sold is -88.5%, indicating a decrease in quantity.
Assuming no other factors affecting revenue, the change in revenue can be estimated by multiplying the percentage change in quantity sold by the percentage change in price:
Change in revenue ≈ Percentage change in quantity sold * Percentage change in price
≈ -88.5% * -2%
≈ 1.77%
The positive value suggests that there will be a slight increase in revenue due to the edge case of price elasticity at p ≈ 3.9 being very close to the elastic range, indicating a decrease in quantity sold outweighing the decrease in price, resulting in higher revenue overall.