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.
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:
|
30 -| /
| /
| /
| /
| /
| /
| /
| /
| /
| /
| /
----------------------------
0 5 10 15 20 25 30
Case 2:
|
30 -| /
| /
| /
| /
| /
| /
| /
| /
| /
20 -| /
| /
----------------------------
0 5 10 15 20 25 30
Case 3:
|
40 -|
| /
| /
| /
| /
| /
| /
| /
20 -| /
| /
| /
| /
| /
----------------------------
0 5 10 15 20 25 30
To calculate the market demand, we need to add up the individual demand quantities of all consumer types.
Let's denote the quantity demanded as Qd and the price as p.
For the first type of consumer with demand Qd = 10 - 2p, the market demand is the sum of all individual demands of this type for different price levels. We can represent this mathematically as:
Market demand for the first type = Σ(10 - 2p) = n * (10 - 2p)
where n is the number of consumers belonging to this type.
Similarly, for the second type of consumer with demand Qd = 10 - 5p, the market demand is:
Market demand for the second type = Σ(10 - 5p) = m * (10 - 5p)
where m is the number of consumers belonging to this type.
Now, let's consider the three cases mentioned:
Case 1: Both types of consumers have the same number of individuals, n = m. Thus, the market demand would be:
Market demand = n * (10 - 2p) + m * (10 - 5p)
= n * (10 - 2p) + n * (10 - 5p)
= n * (20 - 7p)
Case 2: The first type of consumer has twice the number of individuals compared to the second type, n = 2m. In this case, the market demand would be:
Market demand = n * (10 - 2p) + m * (10 - 5p)
= 2m * (10 - 2p) + m * (10 - 5p)
= 20m - 4np + 10m - 5mp
= 30m - 9np
Case 3: The first type of consumer has half the number of individuals compared to the second type, n = 0.5m. In this case, the market demand would be:
Market demand = n * (10 - 2p) + m * (10 - 5p)
= 0.5m * (10 - 2p) + m * (10 - 5p)
= 5m - mp + 10m - 5mp
= 15m - 6mp
Now we can sketch the market demand function based on the above calculations.
To calculate the market demand, we need to sum up the demand of all individual consumers. In this case, we have 3 types of consumers with demand 𝑄𝑑 = 10 − 2𝑝 and 5 types of consumers with demand 𝑄𝑑 = 10 − 5𝑝.
Case 1: When all consumers of both types are active in the market
The market demand can be calculated as follows:
Market demand = (Demand of 1st consumer type + Demand of 2nd consumer type) × Number of consumers of the respective type
For the 1st type of consumer:
Consumer 1 demand = 10 - 2𝑝
Number of consumers of 1st type = 3
For the 2nd type of consumer:
Consumer 2 demand = 10 - 5𝑝
Number of consumers of 2nd type = 5
The market demand in this case can be calculated as:
Market demand = (10 - 2𝑝) × 3 + (10 - 5𝑝) × 5
Case 2: When only the first type of consumers are active in the market
In this case, the market demand is simply the demand of the first type of consumers, as there are no consumers of the second type.
Market demand = (10 - 2𝑝) × 3
Case 3: When only the second type of consumers are active in the market
Similar to case 2, in this case, the market demand is simply the demand of the second type of consumers, as there are no consumers of the first type.
Market demand = (10 - 5𝑝) × 5
To sketch the market demand function, plot the different market demand equations for different cases on a graph. Represent price (𝑝) on the x-axis and market demand (𝑄𝑑) on the y-axis.