Question
Suppose there are 3 types of consumers with demand: 𝑄𝑑 = 10 − 2𝑝 and 5 types of consumers with demand 𝑄𝑑 = 10 − 5𝑝.
a. Calculate market demand (make sure to consider 3 cases). Sketch the market demand function.
a. Calculate market demand (make sure to consider 3 cases). Sketch the market demand function.
Answers
GPT 3.5
To calculate the market demand, we need to sum up the individual demand functions for each type of consumer.
For the first type of consumer with demand 𝑄𝑑 = 10 − 2𝑝, let's denote the number of consumers of this type as 𝑛1. Therefore, their total demand is 𝑛1(10 - 2𝑝).
For the second type of consumer with demand 𝑄𝑑 = 10 − 5𝑝, let's denote the number of consumers of this type as 𝑛2. Therefore, their total demand is 𝑛2(10 - 5𝑝).
Now, let's consider the three cases:
Case 1: 𝑛1 = 1 and 𝑛2 = 1
The market demand in this case is 𝑛1(10 - 2𝑝) + 𝑛2(10 - 5𝑝) = 1(10 - 2𝑝) + 1(10 - 5𝑝) = 20 - 2𝑝 + 10 - 5𝑝 = 30 - 7𝑝.
Case 2: 𝑛1 = 2 and 𝑛2 = 1
The market demand in this case is 𝑛1(10 - 2𝑝) + 𝑛2(10 - 5𝑝) = 2(10 - 2𝑝) + 1(10 - 5𝑝) = 20 - 4𝑝 + 10 - 5𝑝 = 30 - 9𝑝.
Case 3: 𝑛1 = 1 and 𝑛2 = 2
The market demand in this case is 𝑛1(10 - 2𝑝) + 𝑛2(10 - 5𝑝) = 1(10 - 2𝑝) + 2(10 - 5𝑝) = 20 - 2𝑝 + 20 - 10𝑝 = 40 - 12𝑝.
Now, let's sketch the market demand function for each case:
Case 1: Market Demand = 30 - 7𝑝
This is a linear function with a negative slope of -7.
Case 2: Market Demand = 30 - 9𝑝
This is also a linear function with a negative slope of -9.
Case 3: Market Demand = 40 - 12𝑝
This is a linear function with a negative slope of -12.
To sketch these functions, we can use a graph with 𝑝 (price) on the x-axis and 𝑄𝑑 (quantity demanded) on the y-axis. The slope of the line represents the coefficient of 𝑝 in the demand function.
Here is a visual representation of the market demand functions for the three cases (assuming price is measured on the x-axis and quantity demanded is measured on the y-axis):
Case 1:
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0 5 10 15 20 25 30
Case 2:
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20 -| /
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Case 3:
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0 5 10 15 20 25 30
For the first type of consumer with demand 𝑄𝑑 = 10 − 2𝑝, let's denote the number of consumers of this type as 𝑛1. Therefore, their total demand is 𝑛1(10 - 2𝑝).
For the second type of consumer with demand 𝑄𝑑 = 10 − 5𝑝, let's denote the number of consumers of this type as 𝑛2. Therefore, their total demand is 𝑛2(10 - 5𝑝).
Now, let's consider the three cases:
Case 1: 𝑛1 = 1 and 𝑛2 = 1
The market demand in this case is 𝑛1(10 - 2𝑝) + 𝑛2(10 - 5𝑝) = 1(10 - 2𝑝) + 1(10 - 5𝑝) = 20 - 2𝑝 + 10 - 5𝑝 = 30 - 7𝑝.
Case 2: 𝑛1 = 2 and 𝑛2 = 1
The market demand in this case is 𝑛1(10 - 2𝑝) + 𝑛2(10 - 5𝑝) = 2(10 - 2𝑝) + 1(10 - 5𝑝) = 20 - 4𝑝 + 10 - 5𝑝 = 30 - 9𝑝.
Case 3: 𝑛1 = 1 and 𝑛2 = 2
The market demand in this case is 𝑛1(10 - 2𝑝) + 𝑛2(10 - 5𝑝) = 1(10 - 2𝑝) + 2(10 - 5𝑝) = 20 - 2𝑝 + 20 - 10𝑝 = 40 - 12𝑝.
Now, let's sketch the market demand function for each case:
Case 1: Market Demand = 30 - 7𝑝
This is a linear function with a negative slope of -7.
Case 2: Market Demand = 30 - 9𝑝
This is also a linear function with a negative slope of -9.
Case 3: Market Demand = 40 - 12𝑝
This is a linear function with a negative slope of -12.
To sketch these functions, we can use a graph with 𝑝 (price) on the x-axis and 𝑄𝑑 (quantity demanded) on the y-axis. The slope of the line represents the coefficient of 𝑝 in the demand function.
Here is a visual representation of the market demand functions for the three cases (assuming price is measured on the x-axis and quantity demanded is measured on the y-axis):
Case 1:
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30 -| /
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0 5 10 15 20 25 30
Case 2:
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30 -| /
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20 -| /
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0 5 10 15 20 25 30
Case 3:
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40 -|
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20 -| /
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0 5 10 15 20 25 30