Problem 2 Suppose that the demand curve is... [FREE SOLUTION]

Chapter 14: Problem 2

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

Expert verified

The gross benefit from consuming 6 units is 39.

Step by step solution

Interpret the Demand Function

The demand function is given as \(D(p) = 10 - p\). This equation describes the relationship between the price \(p\) of a good and the quantity \(D(p)\) demanded by consumers. Notice that this is a linear demand curve, with the y-intercept at 10 (when \(p = 0\)) and a slope of -1.

Find the Price at Each Unit of Demand

To find the price consumers are willing to pay for each unit, set \(D(p) = q\), where \(q\) is the quantity. For each unit \(q\) from 1 to 6, solve the equation \(q = 10 - p\) to find the price \(p\):- At \(q=1\), \(10 - 1 = 9\)- At \(q=2\), \(10 - 2 = 8\)- At \(q=3\), \(10 - 3 = 7\)- At \(q=4\), \(10 - 4 = 6\)- At \(q=5\), \(10 - 5 = 5\)- At \(q=6\), \(10 - 6 = 4\)

Calculate the Gross Benefit

The gross benefit of consuming a certain quantity is the sum of the prices consumers are willing to pay for each unit up to that quantity. Add up the prices for units 1 to 6:- \(9 + 8 + 7 + 6 + 5 + 4\).Calculate the total: \(9 + 8 = 17\), \(17 + 7 = 24\), \(24 + 6 = 30\), \(30 + 5 = 35\), \(35 + 4 = 39\).

Confirm the Calculated Gross Benefit

Review the calculations of the price willingness for each unit and the accumulation to ensure accuracy. The gross benefit for consuming 6 units indeed sums to 39, confirming the calculations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Demand Curve

The demand curve is a fundamental concept in microeconomics that visualizes how much of a good consumers are willing to purchase at various prices. A typical demand curve slopes downward from left to right, illustrating that as prices decrease, the quantity demanded generally increases.
For the given demand function, \( D(p) = 10 - p \), we see it's a linear curve with an intercept of 10, which represents the maximum quantity demand when the price is zero. The slope of -1 tells us that for every unit increase in price, the quantity demanded decreases by one.

This linear relation makes it easy to predict consumer behavior for different price levels.
Understanding the demand curve helps businesses set prices to maximize revenue while keeping consumer interest alive.

Gross Benefit

Gross benefit refers to the total value obtained by consumers from purchasing a certain quantity of a good. It is essentially the sum of what consumers are willing to pay for each unit up to a given quantity.
For six units of the good, the gross benefit can be calculated by adding the price consumers are willing to pay for each unit:

Unit 1: \(9\)
Unit 2: \(8\)
Unit 3: \(7\)
Unit 4: \(6\)
Unit 5: \(5\)
Unit 6: \(4\)

Adding them up gives a gross benefit of \(39\).
Gross benefit is a crucial measure in understanding consumer behavior and the perceived value of goods in a market.

Consumer Willingness to Pay

Consumer willingness to pay is the maximum amount a consumer is ready to spend to acquire a unit of a good. It reflects the personal value consumers place on a product. In our exercise, we find this by solving the demand equation \(q = 10 - p\) for each quantity level, giving us the price for individual units.

For the first unit, consumers are willing to pay \(9\).
For the second unit, they are willing to pay \(8\), and so on.

This pattern of declining prices indicates diminishing marginal utility; as consumers acquire more units, their willingness to pay decreases.
Understanding this concept helps businesses in price-setting strategies and enhancing consumer satisfaction by delivering values perceived at various price points.

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