Problem 1 - Applied Calculus

Question

1-8. For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=4000-12 x, \quad x=100 $$

Step-by-Step Solution

Answer

The consumer surplus is $60,000.

Step 1: Understanding Consumer Surplus

Consumer surplus is the difference between what consumers are willing to pay and what they actually pay. It is represented by the area between the demand curve and the price level up to the given quantity, x.

Step 2: Determine the Actual Price

First, we need to find the price at the given demand level. Substitute the given value of \(x\) into the demand function: \(d(100) = 4000 - 12 \, \times \, 100\).

Step 3: Calculate the Actual Price

Perform the calculation: \(d(100) = 4000 - 1200 = 2800\). Thus, the price when \(x = 100\) is $2800.

Step 4: Find the Maximum Price Willing to Pay

At \(x = 0\), the price consumers are willing to pay is the intercept of the demand function on the y-axis, i.e., \(d(0) = 4000\).

Step 5: Find Consumer Surplus Formula

The consumer surplus can be calculated using the area of the triangle formed between the demand curve and the equilibrium price line. The formula for this triangle is: \[CS = \frac{1}{2} (\text{Base} \times \text{Height})\] where Base is \(x\) and Height is \(d(0) - d(x)\).

Step 6: Substitute Values Into the Consumer Surplus Formula

Substitute \(x = 100\), \(d(0) = 4000\), and \(d(100) = 2800\) into the formula: \[CS = \frac{1}{2} \, \times \, 100 \, \times \, (4000 - 2800) = \frac{1}{2} \, \times \, 100 \, \times \, 1200\]

Step 7: Calculate the Consumer Surplus

Perform the calculation: \[CS = \frac{1}{2} \, \times \, 100 \, \times \, 1200 = 60000\]So, the consumer surplus is $60,000.

Key Concepts

Understanding the Demand Function

The demand function is a powerful tool to understand consumer behavior. It represents the relationship between the price of a good and the quantity demanded by consumers. Think of it as a formula that tells you how many units of a product people want, based on its price at a certain point in time. In mathematical terms, it's often expressed as a linear equation of the form \(d(x) = a - bx\), where \(a\) is the maximum price consumers are willing to pay when no units are demanded (intercept), and \(b\) represents the change in demand for each unit increase in price (slope).

• The intercept \(d(0) = 4000\) shows the highest price consumers are willing to pay when zero units are demanded.
• The negative slope \(-12\) indicates that as the price decreases by 1, the demand increases by approximately 12 units.

When you substitute \(x = 100\) in the demand function \(d(x) = 4000 - 12x\), you can find the price consumers are willing to pay for 100 units. This step is crucial in determining consumer surplus and analyzing consumer behavior in economic terms.

Deciphering the Price Level

The price level is essential for calculating consumer surplus. It indicates the actual amount consumers pay in the market. In the context of the problem, the price level for 100 units is found by substituting into the demand function: \(d(100) = 4000 - 12 \times 100\). By performing this calculation, we get a price level of \(2800.
This means that when the company sells 100 units, the price per unit is \)2800. The ability to calculate the price level using the demand function allows businesses to determine how changes in demand affect pricing strategy.

• Price level helps in setting realistic pricing policies that attract consumer interest.
• Analyzing the price level can assist in adjusting production to match consumer demand.

Understanding the price consumers pay versus what they are willing to pay sheds light on consumer spending and saving habits.

Consumer Behavior Analysis

Consumer behavior in economics focuses on how individuals or groups choose to acquire, use, and dispose of products or services. This behavior can be quantified using aspects like consumer surplus. In our example, consumer surplus is the difference between the maximum price consumers are willing to pay at zero demand (\(d(0) = 4000\)) and the actual price paid (\(d(100) = 2800\)), multiplied by the number of units (100).

Calculating Consumer Surplus

By using the formula, consumer surplus is calculated as the area of the triangle: \(CS = \frac{1}{2} \times 100 \times (4000 - 2800)\). The resulting value is $60,000.
Understanding this surplus provides insights into consumer satisfaction and market efficiency, showcasing both the value consumers place on a product and their willingness to purchase it beyond the actual price.

• Consumer surplus is an indicator of consumer welfare and economic efficiency.
• It influences policy-making, pricing strategies, and market assessments.

By dissecting such aspects of consumer behavior, businesses can better tailor their offerings to meet market needs while maximizing their revenue.