Question

Suppose that the demand curve is given by D(p) = 10 ? p. What is the gross benefit from consuming 6 units of the good?

Short answer

The gross benefit from consuming 6 units is 42.

Step-by-step solution

Understanding the Demand Curve

We are given the demand curve as \( D(p) = 10 - p \). The demand function expresses the maximum unit price \( p \) that consumers are willing to pay for \( D(p) \) units of a good. To find the gross benefit from consuming 6 units, we need to calculate the consumer's total willingness to pay for those 6 units.

Setting Up the Integral

The gross benefit from consumption up to a quantity \( Q \) is the area under the demand curve from \( 0 \) to \( Q \). For the given demand curve \( D(p) = 10 - p \), we want to find the integral of this function from \( 0 \) to \( 6 \), which represents the quantities of the good.

Calculating the Integral

To calculate the area under the curve from 0 to 6 units, we compute the definite integral:\[ \int_{0}^{6} (10 - p) \, dp.\]

Evaluating the Integral

First, find the antiderivative of \( 10 - p \), which is \( 10p - \frac{p^2}{2} \). Then, evaluate from 0 to 6:\[ [10p - \frac{p^2}{2}]_0^6 = (10 \times 6 - \frac{6^2}{2}) - (10 \times 0 - \frac{0^2}{2})\]\[ = (60 - 18) - 0 = 42\]

Summarizing the Result

The result of the integration shows the gross benefit from consuming 6 units. This value, 42, represents the total willingness to pay for 6 units, capturing the entire area under the demand curve from 0 to 6 units.

Key concepts

Demand Curve

The demand curve is a fundamental concept in economics that helps us understand consumer purchasing behavior. It graphically represents the relationship between the price of a good and the quantity demanded by consumers. In a typical demand curve, as the price decreases, the quantity demanded increases, which is shown as a downward sloping line on the graph.
For the equation given in this exercise, the demand curve is represented by \( D(p) = 10 - p \). This means that the maximum price consumers are willing to pay for any quantity decreases by one unit of currency for each additional unit of the good.
It's beneficial to think of this curve as capturing consumer preferences—more specifically, the maximum amount consumers are willing to pay for different quantities of a good. This is essential for calculating important economic measures like consumer surplus and gross benefit.

Definite Integral

The concept of a definite integral is crucial when calculating areas under curves, which in economics translates to measuring total benefits or costs. In this context, we use a definite integral to compute the gross benefit of consuming a certain number of units under a demand curve.
Mathematically, a definite integral sums all small areas under the curve from one point to another, effectively calculating the total area under the curve. In this exercise, the definite integral \( \int_{0}^{6} (10 - p) \, dp \) measures the gross benefit of purchasing 6 units.
The process involves finding the antiderivative of the demand function, which is \( 10p - \frac{p^2}{2} \), and evaluating it from 0 to 6. This gives us the measure of the total willingness to pay for these units, a fundamental application of integrals in economics.

Willingness to Pay

Willingness to pay (WTP) is a key economic concept that describes the maximum amount a consumer is willing to spend on a good or service. It reflects the value that consumers place on a given quantity of a good.
In this exercise, the willingness to pay is illustrated by the area under the demand curve up to the consumption of 6 units. Hence, the sum of these values from zero to six reflects the overall willingness to pay for those units, totaling to 42 in the solution.
Understanding WTP is important because it helps businesses set prices strategically. It also allows policymakers to gauge consumer preferences and the perceived value of goods, which can affect economic policy decisions. Calculating the total WTP as seen here can provide insights into consumer behavior and market dynamics.