Calculating Consumer Surplus from a Demand Function

The question asks us to find the consumer's surplus for a given demand function at a specific quantity demanded. Consumer surplus is a measure of the economic benefit consumers receive when they pay a price lower than the maximum price they are willing to pay for a good or service.

The demand function is given by $p = 27 - 3x - x^2$, where $p$ is the price and $x$ is the quantity demanded. We need to calculate the consumer's surplus at $x = 3$.

Steps to Calculate Consumer Surplus

To calculate the consumer surplus, we follow these steps:

1. Determine the market price at the given quantity demanded.
2. Calculate the total amount consumers are willing to pay for the quantity demanded. This is represented by the area under the demand curve from $x=0$ to the specific quantity. In calculus terms, this is the definite integral of the demand function over this range.
3. Calculate the total amount consumers actually pay for the quantity demanded. This is the market price multiplied by the quantity.
4. Subtract the total amount consumers actually pay from the total amount they are willing to pay. The result is the consumer surplus.

Step 1: Find the Market Price at x = 3

Substitute $x = 3$ into the demand function:

$\qquad p = 27 - 3(3) - (3)^2$

$\qquad p = 27 - 9 - 9$

$\qquad p = 9$

So, the market price at a quantity of 3 units is 9.

Step 2: Calculate Total Willingness to Pay

The total willingness to pay is the integral of the demand function $p(x) = 27 - 3x - x^2$ from $x = 0$ to $x = 3$.

$\qquad \text{Total Willingness to Pay} = \int_{0}^{3} (27 - 3x - x^2) dx$

Integrate the function:

$\qquad \int (27 - 3x - x^2) dx = 27x - \frac{3x^2}{2} - \frac{x^3}{3} + C$

Now, evaluate the definite integral from 0 to 3:

$\qquad \left[27x - \frac{3x^2}{2} - \frac{x^3}{3}\right]_{0}^{3}$

Evaluate at the upper limit ($x=3$):

$\qquad \left(27(3) - \frac{3(3)^2}{2} - \frac{(3)^3}{3}\right) = \left(81 - \frac{3 \times 9}{2} - \frac{27}{3}\right) = \left(81 - \frac{27}{2} - 9\right) = (81 - 13.5 - 9) = 58.5$

Evaluate at the lower limit ($x=0$):

$\qquad \left(27(0) - \frac{3(0)^2}{2} - \frac{(0)^3}{3}\right) = (0 - 0 - 0) = 0$

Total Willingness to Pay = Value at upper limit - Value at lower limit = $58.5 - 0 = 58.5$.

Step 3: Calculate Total Expenditure

The total amount consumers actually pay is the market price multiplied by the quantity demanded.

$\qquad \text{Total Expenditure} = \text{Price} \times \text{Quantity}$

$\qquad \text{Total Expenditure} = 9 \times 3 = 27$

Step 4: Calculate Consumer Surplus

Consumer Surplus is the difference between the total willingness to pay and the total expenditure.

$\qquad \text{Consumer Surplus} = \text{Total Willingness to Pay} - \text{Total Expenditure}$

$\qquad \text{Consumer Surplus} = 58.5 - 27$

$\qquad \text{Consumer Surplus} = 31.5$

The consumer's surplus at $x=3$ is 31.5.

Revision Table: Key Concepts in Consumer Surplus

Additional Information on Consumer Surplus Calculation

Consumer surplus is a fundamental concept in welfare economics. It measures the benefit that buyers receive from participating in a market. A higher consumer surplus generally indicates that consumers are getting a good deal, paying less than their full valuation of the product.

Calculating consumer surplus using integration is necessary when the demand function is a continuous curve, as in this problem $p = 27 - 3x - x^2$. The integral $\int_{0}^{x_0} p(x) dx$ gives the total willingness to pay for $x_0$ units. The total expenditure is simply $p_0 \times x_0$, where $p_0$ is the price corresponding to $x_0$.

For a linear demand function, the area under the curve above the price is a triangle, and its area can be calculated using the formula for the area of a triangle ($0.5 \times \text{base} \times \text{height}$). However, for non-linear demand functions like the one given ($p = 27 - 3x - x^2$, which is a quadratic function), integration is typically required to find the exact area under the curve.

Understanding consumer surplus helps analyze the impact of various market changes, such as price controls, taxes, or subsidies, on consumer welfare.