For A Consumer With Demand Function Q=1005p^{1/2}, Find: A) Consumer Surplus(CS), At Price P₀=9 B) CS,
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Introduction
Understanding consumer surplus is fundamental in microeconomics, as it measures the benefit consumers receive when they purchase a product at a price lower than the maximum they are willing to pay. In this article, we analyze a specific demand function, Q = 1005p^{1/2}, and calculate the consumer surplus at a given market price P₀=9. We will also explore the derivation of the demand function, the concept of consumer surplus, and the implications of the results.
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Understanding the Demand Function Q = 1005p^{1/2}
Interpreting the Demand Function
The demand function provided, Q = 1005p^{1/2}, expresses the relationship between the quantity demanded (Q) and the price (p). It indicates that:
- The quantity demanded varies directly with the square root of the price.
- As price increases, demand increases, but at a decreasing rate—reflecting typical demand behavior.
Implications of the Demand Function
- The demand function suggests a nonlinear relationship between price and quantity.
- The demand is positively related to the square root of price, which implies diminishing marginal demand as price increases.
- Consumers' maximum willingness to pay at different quantities can be derived from the inverse demand function.
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Deriving the Inverse Demand Function
To analyze consumer surplus, it is essential to express the maximum willingness to pay for each quantity demanded—that is, the inverse demand function p = f(Q).
Step-by-Step Derivation
Given:
\[
Q = 1005 p^{1/2}
\]
Solve for p:
- Divide both sides by 1005:
\[
\frac{Q}{1005} = p^{1/2}
\]
- Square both sides to eliminate the square root:
\[
p = \left(\frac{Q}{1005}\right)^2
\]
Thus, the inverse demand function is:
\[
p(Q) = \left(\frac{Q}{1005}\right)^2
\]
This function indicates the maximum price a consumer is willing to pay for a given quantity Q.
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Calculating Consumer Surplus (CS)
Consumer surplus represents the difference between what consumers are willing to pay and what they actually pay.
General Formula for Consumer Surplus
\[
CS = \int{0}^{Q0} p(Q) \, dQ - P0 \times Q0
\]
Where:
- \(Q0\) = quantity demanded at the market price \(P0\).
- \(p(Q)\) = inverse demand function.
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Part A: Consumer Surplus at Price P₀=9
Step 1: Find the Quantity Demanded \(Q0\) at Price \(P0=9\)
Using the original demand function:
\[
Q = 1005 p^{1/2}
\]
Plug in \(p = 9\):
\[
Q_0 = 1005 \times \sqrt{9} = 1005 \times 3 = 3015
\]
So, at price 9, the consumer demands 3015 units.
Step 2: Compute the Consumer Surplus
Using the inverse demand function:
\[
p(Q) = \left(\frac{Q}{1005}\right)^2
\]
The consumer surplus is:
\[
CS = \int{0}^{Q0} p(Q) \, dQ - P0 \times Q0
\]
Calculate the integral:
\[
CS = \int_{0}^{3015} \left(\frac{Q}{1005}\right)^2 dQ - 9 \times 3015
\]
Simplify the integral:
\[
CS = \frac{1}{(1005)^2} \int_{0}^{3015} Q^2 dQ - 9 \times 3015
\]
Calculate the integral:
\[
\int{0}^{Q0} Q^2 dQ = \frac{Q_0^3}{3}
\]
Plugging in values:
\[
CS = \frac{1}{(1005)^2} \times \frac{(3015)^3}{3} - 9 \times 3015
\]
Calculate numerator:
\[
(3015)^3 = 3015 \times 3015 \times 3015
\]
Since \(3015 = 3 \times 1005\), then:
\[
(3015)^3 = (3 \times 1005)^3 = 3^3 \times 1005^3 = 27 \times 1005^3
\]
Now, substitute back:
\[
CS = \frac{1}{1005^2} \times \frac{27 \times 1005^3}{3} - 9 \times 3015
\]
Simplify numerator:
\[
\frac{27 \times 1005^3}{3} = 9 \times 1005^3
\]
Thus,
\[
CS = \frac{1}{1005^2} \times 9 \times 1005^3 - 9 \times 3015
\]
Simplify:
\[
CS = 9 \times 1005^3 / 1005^2 - 9 \times 3015
\]
\[
CS = 9 \times 1005 - 9 \times 3015
\]
Calculate:
\[
9 \times 1005 = 9045
\]
\[
9 \times 3015 = 27135
\]
Finally,
\[
CS = 9045 - 27135 = -18090
\]
Since consumer surplus cannot be negative, check the calculations for consistency.
Note: The negative result indicates an error in the calculation—likely due to misinterpretation of the integral limits or the substitution process. Let's revisit the integral calculation carefully.
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Revised Calculation of Consumer Surplus
Integral:
\[
CS = \int{0}^{Q0} p(Q) \, dQ - P0 \times Q0
\]
with:
\[
p(Q) = \left(\frac{Q}{1005}\right)^2
\]
So,
\[
CS = \int{0}^{Q0} \left(\frac{Q}{1005}\right)^2 dQ - P0 \times Q0
\]
Expressed explicitly:
\[
CS = \frac{1}{(1005)^2} \int{0}^{Q0} Q^2 dQ - P0 \times Q0
\]
Calculate the integral:
\[
\int{0}^{Q0} Q^2 dQ = \frac{Q_0^3}{3}
\]
Plug in \(Q_0=3015\):
\[
CS = \frac{1}{(1005)^2} \times \frac{(3015)^3}{3} - 9 \times 3015
\]
Express \( (3015)^3 \):
\[
(3015)^3 = (3 \times 1005)^3 = 27 \times 1005^3
\]
Substitute:
\[
CS = \frac{1}{1005^2} \times \frac{27 \times 1005^3}{3} - 9 \times 3015
\]
Simplify numerator:
\[
\frac{27 \times 1005^3}{3} = 9 \times 1005^3
\]
So,
\[
CS = \frac{1}{1005^2} \times 9 \times 1005^3 - 9 \times 3015
\]
Simplify the fraction:
\[
\frac{9 \times 1005^3}{1005^2} = 9 \times 1005
\]
Therefore:
\[
CS = 9 \times 1005 - 9 \times 3015
\]
Calculate:
\[
9 \times 1005 = 9045
\]
\[
9 \times 3015 = 27135
\]
Finally:
\[
CS = 9045 - 27135 = -18090
\]
Again, negative consumer surplus indicates an inconsistency, because consumer surplus should be positive.
Correcting the approach:
The error stems from the incorrect interpretation of the integral bounds. Consumer surplus is the area between the demand curve and the price line, from zero quantity up to \(Q_0\).
But because the demand function is positive and decreasing in price, the integral should be:
\[
CS = \int{p=0}^{p=P0} Q(p) \, dp
\]
Alternatively, the consumer surplus can be calculated as:
\[
CS = \int{p=P0}^{p_{max}} Q(p) \, dp
\]
