Problem 45 - 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. \(D(q)=10 e^{-0.1 q} ; q_{0}=4\) units

Step-by-Step Solution

Answer

Consumers' surplus is approximately 6.158.

Step 1: Find the price at the given quantity

First, substitute the given quantity into the demand curve equation to find the price. The demand function is given as \[D(q) = 10 e^{-0.1q}\] Substitute \(q_0 = 4\) into the equation: \[p_0 = D(4) = 10 e^{-0.1 \times 4}\]Simplify the exponent: \[p_0 = 10 e^{-0.4}\]Now, calculate the value of \(e^{-0.4}\): \[p_0 \approx 10 \times 0.6703 = 6.703\]Therefore, the price \(p_0\) when \(q_0 = 4\) units is approximately \(6.703\).

Step 2: Set up the integral for consumers' surplus

Consumers' surplus is the area between the demand curve and the horizontal line at the market price from quantity 0 to \(q_0\). The formula for consumers' surplus (CS) is: \[\text{CS} = \int_{0}^{q_0} (D(q) - p_0) \, dq\] Substitute the given demand function and the calculated price \(p_0\) into the integral: \[\text{CS} = \int_{0}^{4} (10 e^{-0.1q} - 6.703) \, dq\]

Step 3: Integrate the demand function

To solve the integral, integrate the function term by term. First, integrate \[\int_{0}^{4} 10 e^{-0.1q} \, dq\]Let \(u = -0.1q\) which means \(du = -0.1 dq \Rightarrow dq = -10 du\) Now substitute and change the limits accordingly. When \( q = 0, u = 0\) and when \(q= 4, u = -0.4\).So the integral becomes:\[\int_{0}^{-0.4} 10 e^{u}(-10 du) = -100 \int_{0}^{-0.4} e^{u} du= -100[e^{u}]_{0}^{-0.4} = -100(e^{-0.4}-1)\]Substitute back the original limits:\[-100 (0.6703-1)\approx 32.97\]

Step 4: Integrate the constant price

Next, integrate the constant price term: \[\int_{0}^{4} 6.703 \, dq = 6.703 q|_0^4= 6.703 \times 4 = 26.812\]

Step 5: Compute the consumers' surplus

Combine the results of the integrals to find the consumers' surplus: \[\text{CS} = 32.97 -26.812 = 6.158\]Therefore, the consumers' surplus when \(q_0 = 4\) units is approximately \(6.158\).

Key Concepts

Demand Curve

The demand curve shows the relationship between the price of a good and the quantity demanded.
It typically slopes downward, indicating that as the price falls, the quantity demanded increases.
In our exercise, the demand curve is given as the function: \[ D(q) = 10 e^{-0.1q} \] This means the price per unit decreases exponentially as more units are demanded.
Understanding the demand curve is crucial in determining prices and quantities in various market scenarios.
When we plug in a specific quantity ( q_0 , which is 4 in our example) into the demand equation, we find the corresponding price ( p_0 ).
By substituting item 4 in the demand equation, D(4) = 10 e^{-0.1 \times 4} = 6.703. This calculation helps us understand the price consumers are willing to pay at a specific quantity of goods.

Integral Calculus

Integral calculus is a major branch of math that deals with integrals and their properties.
In economics, integrals are used to calculate areas under curves, which represent accumulated quantities.
For consumers' surplus, the integral helps us find the area between the demand curve and the price level.
The formula we use is: \[ \text{CS} = \int_{0}^{q_0} (D(q) - p_0) \, dq \]
By integrating this area, we can understand how much more consumers are willing to pay as opposed to the market price.
In our example, it involves integrating 10 e^{-0.1q} from q=0 to q=4, and then subtracting the area at the constant price level, which shows the true economic benefit or surplus.

Consumer Economics

Consumer economics studies how individuals make decisions to allocate their resources.
Consumers' surplus is an essential concept that measures economic welfare from consumption.
It indicates the difference between what consumers are willing to pay and what they actually pay.
In our exercise, consumers' surplus is calculated as the area between the demand curve and the actual price line.
Calculating this involves understanding the demand function, and how much more consumers' value the goods beyond the price they paid.
This measure helps in understanding the benefits consumers receive from market transactions which, in our case, is approximately 6.158.

Applied Calculus

Applied calculus uses calculus methods to solve real-world problems.
We use it extensively in economics to model and analyze various situations.
In the context of our exercise, applied calculus helps us compute the consumers' surplus.
By setting up integrals and solving them, we can determine economic benefits and other business metrics.
Applied calculus requires translating a problem into mathematical terms, solving it, and interpreting the results.
In this case, it meant setting up the integral for D(q) - p_0 and integrating to find the total surplus, illustrating how calculus aids in practical decision-making processes.