Problem 43 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

CONSUMERS' SURPLUS In Exercises 43 through 46, \(p=D(q)\) is the demand curve for a particular commod- ity; that is, \(q\) units of the commodity will be demanded when the price is \(p=D(q)\) dollars per unit. In each case, for the given level of production \(q_{0}\), find \(p_{0}=D\left(q_{0}\right)\) and compute the corresponding consumers' surplus. \(D(q)=4\left(36-q^{2}\right) ; q_{0}=2\) units

Step-by-Step Solution

Answer

The consumers' surplus is 21.33.

Step 1: Identify the given function and quantity

The demand function is given as \[D(q) = 4(36 - q^2)\] and the level of production is \(q_0 = 2\) units.

Step 2: Calculate the price at the given quantity

Substitute \(q_0 = 2\) into the demand function to find the price \(p_0\). \[\begin{align*} p_0 &= D(q_0) \ &= 4(36 - (2)^2) \ &= 4(36 - 4) \ &= 4 \times 32 \ &= 128 \end{align*}\] Therefore, \(p_0 = 128\) dollars per unit.

Step 3: Set up the integral for consumers' surplus

The formula for the consumers' surplus (CS) is: \[CS = \int_{0}^{q_0} D(q) \, dq - p_0 \cdot q_0\] Substituting the given demand function \(D(q)\) and the values \(q_0 = 2\) and \(p_0 = 128\), we have: \[CS = \int_{0}^{2} 4(36 - q^2) \, dq - 128 \times 2\]

Step 4: Evaluate the integral

First, compute the integral: \[\begin{align*} \int_{0}^{2} 4(36 - q^2) \, dq &= 4 \int_{0}^{2} (36 - q^2) \, dq \ &= 4 \left[ 36q - \frac{q^3}{3} \right]_{0}^{2} \ &= 4 \left[ (36 \times 2 - \frac{2^3}{3}) - (36 \times 0 - \frac{0^3}{3}) \right] \ &= 4 \left[ 72 - \frac{8}{3} \right] \ &= 4 \left[ \frac{216 - 8}{3} \right] \ &= 4 \times 69.33 \ &= 277.33 \end{align*}\]

Step 5: Calculate consumers' surplus

Now use the result of the integral to find the consumers' surplus: \[\begin{align*} CS &= 277.33 - 128 \times 2 \ &= 277.33 - 256 \ &= 21.33 \end{align*}\] Therefore, the consumers' surplus is 21.33.

Key Concepts

demand curve

A demand curve shows the relationship between the price of a commodity and the quantity demanded by consumers. In this problem, the demand curve is represented by the equation \(D(q) = 4(36 - q^2)\). This equation tells us how much quantity \(q\) of an item will be purchased at a particular price \(p\). When the price is high, the demand is usually low, and when the price is low, the demand is higher. In step 2, we calculated the price at a given quantity by substituting \(q_0 = 2\) into the demand function, yielding \(p_0 = 128\) dollars per unit. This means at the price of 128 dollars per unit, exactly 2 units are demanded.

integral calculus

Integral calculus deals with the accumulation of quantities, such as areas under a curve. In this context, we use it to find the consumers' surplus. Consumers' surplus is the difference between what consumers are willing to pay and what they actually pay. The formula for consumers' surplus is: \( CS = \int_{0}^{q_0} D(q) \, dq - p_0 \times q_0 \). In step 3, we set up the integral for consumers' surplus by substituting our specific demand function and known values. This integral calculates the area under the demand curve from 0 to \(q_0 = 2\).

consumer economics

Consumer economics examines how individuals decide to allocate their resources (like money) to achieve maximum satisfaction. Consumers' surplus is a vital concept because it measures the economic benefit to consumers for purchasing goods at a market price lower than the highest price they are willing to pay. Here, steps 3 and 4 show how we calculate this benefit by integrating the demand function and subtracting the expenditure \(p_0 \times q_0\). This calculation incorporates both the attitudes toward valuation of goods and the actual spending, yielding a surplus value.

optimization

Optimization deals with finding the best possible outcome under given circumstances. In this exercise, we aim to calculate the consumers' surplus effectively by optimizing our use of calculus. Here, we set up and evaluate an integral to find the exact area under the demand curve, minus the actual cost. Each step in this problem is a way of optimizing the process to ensure the correct result—\(CS = 21.33\). This helps students see how optimization is crucial in economic calculations, ensuring resources are utilized efficiently to reach precise answers.