Problem 46 - 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. \(D(q)=5+3 e^{-0.2 q} ; q_{0}=10\) units

Step-by-Step Solution

Answer

Consumers' surplus is approximately 8.87 dollars.

Step 1: Calculate Demand at Given Level of Production

First, substitute the given level of production, 10 units, into the demand function to find the price per unit.\( p_0 = D(q_0) = 5 + 3e^{-0.2 \times 10} \)

Step 2: Simplify the Exponential Term

Calculate the value of the exponential term.\( -0.2 \times 10 = -2 \)Thus, \( e^{-2} \approx 0.1353 \)

Step 3: Solve for the Price per Unit

Now, substitute the value of the exponential term back into the demand function.\( p_0 = 5 + 3 \times 0.1353 \approx 5 + 0.4059 = 5.4059 \)Therefore, \( p_0 \approx 5.41 \) dollars per unit.

Step 4: Set Up the Consumers' Surplus Integral

Consumers' surplus is given by the integral of the difference between the demand function and the price per unit from 0 to the given level of production. \( CS = \int_{0}^{q_0} [D(q) - p_0] \, dq \)Using \( p_0 = 5.41 \), we have:\( CS = \int_{0}^{10} [5 + 3e^{-0.2q} - 5.41] \, dq \)

Step 5: Simplify the Integrand

Combine the constants inside the integral:\( CS = \int_{0}^{10} [-0.41 + 3e^{-0.2q}] \, dq \)

Step 6: Integrate the Function

Integrate the function term by term.\( \int -0.41 \, dq = -0.41q \)\( \int 3e^{-0.2q} \, dq = -15e^{-0.2q} \)Thus, the integral becomes:\( CS = \left[-0.41q - 15e^{-0.2q}\right]_{0}^{10} \)

Step 7: Evaluate the Definite Integral

Evaluate at the limits and subtract.\( [-0.41(10) - 15e^{-0.2(10)}] - [-0.41(0) - 15e^{-0}] \)\( = [-4.1 - 15(0.1353)] - [0 - 15(1)] \)\( = [-4.1 - 2.0295] - [-15] \)\( = -6.1295 + 15 \)\( = 8.8705 \)

Key Concepts

demand curve

The demand curve is an essential concept in economics and represents the relationship between the price of a commodity and the quantity demanded. When we say 'demand curve,' we are usually referring to a graph that shows how many units of a product consumers will buy at different prices. In this exercise, the demand curve is given by a function: \(D(q) = 5 + 3 e^{-0.2q}\). Here, 'q' stands for the quantity of the commodity, and \(D(q)\) gives us the price per unit. Understanding the demand curve helps in determining price points and the overall market behavior.

exponential decay

Exponential decay describes a process where the value of a quantity decreases exponentially over time. In our demand function \(D(q) = 5 + 3 e^{-0.2q}\), the term \(3 e^{-0.2q}\) represents exponential decay. The constant \(-0.2\) is the decay rate, which slows down how quickly the value changes. As 'q' (quantity) increases, the exponential term \(e^{-0.2q}\) gets smaller. This means that the additional price effect contributed by this term becomes less significant as more units are produced, indicating how prices might stabilize or decrease as production ramps up.

integral calculus

Integral calculus is a critical tool for solving problems involving areas and accumulations. One of its primary uses is to find the area under a curve, which can represent a variety of real-world quantities. In this exercise, we use integral calculus to calculate consumer surplus, which measures the difference between what consumers are willing to pay and what they actually pay. Using the integral \(\int_{0}^{10} [D(q) - p] \, dq\), we sum up the tiny differences (areas) between the demand curve and the actual price over a given range of quantities.

definite integral evaluation

Evaluating a definite integral means finding the exact value of an integral over a specific interval. In this problem, we evaluate the integral \(\int_{0}^{10} [5 + 3e^{-0.2q} - 5.41] \, dq\) from 0 to 10. The goal is to compute the consumer surplus, a measure of the economic benefit consumers receive. We split the integral into simpler parts, integrate each term, and then apply the limits. The final step involves calculating these integrals at specific points. The result, in this case, shows how much benefit consumers get from buying at the market price versus what they are willing to pay.