The supply and demand functions for a certain commodity are given by: supply: p=20+1.5x, and demand: p=40-x. Then producers' surplus and consumers' surplus is?
To calculate the producer's surplus and consumer's surplus, we need to first determine the equilibrium price and quantity of the commodity by equating the supply and demand functions.
Setting the supply and demand functions equal to each other:
20 + 1.5x = 40 - x
Combining like terms:
2.5x = 20
x = 8
Substituting the value of x back into either the supply or demand function, let's use the demand function:
p = 40 - x
p = 40 - 8
p = 32
Therefore, the equilibrium price is 32 and the equilibrium quantity is 8.
Now we can calculate the producer's surplus and consumer's surplus.
Producer's Surplus:
The producer's surplus represents the difference between the price at which producers are willing to sell a certain quantity and the price they actually receive.
To calculate the producer's surplus, we need to find the area above the supply curve and below the equilibrium price.
Using the supply function: p = 20 + 1.5x
We need to find the area under the curve from x=0 to x=8 and above p=32.
First, let's calculate the price at x=0:
p = 20 + 1.5(0) = 20
The producer's surplus is the area of the trapezoid formed by the supply function and the equilibrium price:
(1/2)(base1 + base2)(height)
= (1/2)(32 + 20)(8 - 0)
= (1/2)(52)(8)
= 208
Therefore, the producer's surplus is 208.
Consumer's Surplus:
The consumer's surplus represents the difference between the price consumers are willing to pay for a certain quantity and the price they actually pay.
To calculate the consumer's surplus, we need to find the area below the demand curve and above the equilibrium price.
Using the demand function: p = 40 - x
We need to find the area under the curve from x=0 to x=8 and above p=32.
First, let's calculate the price at x=0:
p = 40 - 0 = 40
The consumer's surplus is the area of the triangle formed by the demand function and the equilibrium price:
(1/2)(base)(height)
= (1/2)(8)(40 - 32)
= (1/2)(8)(8)
= 32
Therefore, the consumer's surplus is 32.
In summary:
Producer's surplus = 208
Consumer's surplus = 32
To find the producer's surplus and consumer's surplus, we first need to find the equilibrium point where supply equals demand.
Setting the supply and demand functions equal to each other, we get:
20 + 1.5x = 40 - x
Combining like terms and solving for x, we find:
2.5x = 20
x = 8
Substituting this value of x back into either the supply or demand function, we can find the equilibrium price:
p = 40 - x
= 40 - 8
= 32
So, the equilibrium price is $32.
Next, let's calculate the producer's surplus and consumer's surplus.
Producer's surplus is the area above the supply curve but below the equilibrium price. To find this, we need to find the area of the triangle bounded by the supply curve, the equilibrium price, and the x-axis. The formula for the area of a triangle is (base * height) / 2.
The base of the triangle is 8 units (x), and the height of the triangle can be found by substituting x = 8 into the supply function:
Supply: p = 20 + 1.5x
= 20 + 1.5(8)
= 32
The height of the triangle is 32 units (p).
Now, we can calculate the producer's surplus:
Producer's surplus = (base * height) / 2
= (8 * 32) / 2
= 128
Therefore, the producer's surplus is $128.
Consumer's surplus is the area below the demand curve but above the equilibrium price. To find this, we need to find the area of the triangle bounded by the demand curve, the equilibrium price, and the x-axis.
Similarly, the base of the triangle is 8 units (x), and the height can be found by substituting x = 8 into the demand function:
Demand: p = 40 - x
= 40 - 8
= 32
The height of the triangle is 32 units (p).
Now, we can calculate the consumer's surplus:
Consumer's surplus = (base * height) / 2
= (8 * 32) / 2
= 128
Therefore, the consumer's surplus is $128.
In summary:
- The producer's surplus is $128.
- The consumer's surplus is $128.