Question

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

The gross benefit from consuming 6 units is 39.

Step-by-step solution

Understanding the Problem

The demand curve is given by \(D(p) = 10 - p\). This represents the relationship between price \(p\) and quantity demanded. We need to find the consumers' gross benefit from consuming 6 units of the good.

Determine the Price for Each Unit

To find the price consumers are willing to pay for each unit, substitute \(D(p)\) values from 1 to 6 into the demand curve equation to get corresponding prices. For each unit \(i\), calculate \(p_i = 10 - i\).

Price for First 6 Units

Calculate the price for each of the 6 units based on \(D(p) = 10 - p\):- For the first unit: \(p_1 = 10 - 1 = 9\)- For the second unit: \(p_2 = 10 - 2 = 8\)- For the third unit: \(p_3 = 10 - 3 = 7\)- For the fourth unit: \(p_4 = 10 - 4 = 6\)- For the fifth unit: \(p_5 = 10 - 5 = 5\)- For the sixth unit: \(p_6 = 10 - 6 = 4\)

Calculate Gross Benefit

The gross benefit is the sum of the prices consumers are willing to pay for each of the 6 units. Add the calculated prices: \(Gross = 9 + 8 + 7 + 6 + 5 + 4 = 39\).

Conclusion

The gross benefit from consuming 6 units of the good is 39. This represents the total willing-to-pay amount for these units based on the demand curve.

Key concepts

Demand Curve

A demand curve is a graphical representation of the relationship between the price of a good and the quantity demanded by consumers. In simpler terms, it shows us how much of a good people want to buy at different price levels. The demand curve, typically downward sloping, reflects an inverse relationship between price and quantity. This means that as the price decreases, the quantity demanded tends to increase, and vice versa.

In our exercise, the demand curve is given by the equation \(D(p) = 10 - p\). Here, \(p\) represents the price, and \(D(p)\) stands for the quantity demanded. The expression \(10 - p\) means that for every unit increase in price, the quantity demanded decreases by one unit. Understanding this relationship is crucial as it allows us to interpret consumer behavior in regard to changes in price.

Price Determination

Price determination is a key concept in economics, as it involves finding the price at which consumers are willing to purchase a certain quantity of goods. The law of demand tells us that prices and quantity demanded are inversely related. By knowing the demand curve, one can determine prices for different quantities.

In the given problem, the task is to determine the price for each of the first six units of the good using the equation \(p = 10 - D(p)\). You simply substitute the unit values, from 1 to 6, into the demand curve equation:

• For the first unit, consumers pay \(p_1 = 9\).
• For the second unit, consumers pay \(p_2 = 8\).
• For the third unit, consumers pay \(p_3 = 7\).
• For the fourth unit, consumers pay \(p_4 = 6\).
• For the fifth unit, consumers pay \(p_5 = 5\).
• For the sixth unit, consumers pay \(p_6 = 4\).

This step-by-step calculation allows us to see how price changes affect consumer decisions.

Gross Benefit Calculation

The gross benefit calculation sums up the values consumers are willing to pay for each unit consumed. This calculation helps measure the total benefit or value that consumers derive from consuming a certain quantity of goods.

To calculate the gross benefit from consuming six units, add up the prices consumers are willing to pay for each unit:

• The first unit is valued at 9.
• The second unit is valued at 8.
• The third unit is valued at 7.
• The fourth unit is valued at 6.
• The fifth unit is valued at 5.
• The sixth unit is valued at 4.

By adding these amounts together: \[ Gross ext{ }Benefit = 9 + 8 + 7 + 6 + 5 + 4 = 39. \]This sum represents the total amount consumers are willing to pay for six units based on the provided demand curve. Knowing the gross benefit is useful for understanding consumer surplus and the overall value creation in the market.