The producer surplus for the demand and supply functions x+y=4 and -x+y=2 is?
A) 1/2
B) 3/2
C) 5/2
D) 7/2
E) 3
I've looked at the equation: The integral of Price equilibrium-Supply function with limits of 0 to xe but I don't know how to solve for the price equilibrium with the given equations and what variable would represent Price.
To find the producer surplus for the given demand and supply functions, you first need to find the equilibrium price and quantity.
The demand function is x + y = 4, while the supply function is -x + y = 2.
To find the equilibrium price, you need to equate the demand and supply functions and solve for x or y. Let's solve for y:
x + y = 4
-y = -x + 2
y = x - 2
Now substitute this value of y back into the demand function:
x + (x - 2) = 4
2x - 2 = 4
2x = 6
x = 3
Now that you have the equilibrium quantity (x = 3), you can substitute it into either the demand or supply function to find the equilibrium price (p):
x + y = 4
3 + y = 4
y = 1
So, the equilibrium price (p) is 1.
To calculate the producer surplus, you need to integrate the area between the supply curve and the equilibrium price line with the limits of 0 to the equilibrium quantity (xe).
The supply function is -x + y = 2. Rearrange it to solve for y:
y = x + 2
Now, to calculate the producer surplus, you need to integrate the difference between the price equilibrium (p = 1) and the supply function (y = x + 2) with the limits of 0 to xe (3):
Producer Surplus = ∫(p - supply function) dx from 0 to xe
= ∫(1 - x - 2) dx from 0 to 3
= ∫(-x - 1) dx from 0 to 3
Integrating -x - 1 with respect to x, we get:
= -[(x^2)/2 + x] from 0 to 3
= -[(3^2)/2 + 3] - [0/2 + 0]
= -[(9/2) + 3]
= -[9/2 + 6/2]
= -[15/2]
= -7.5
The producer surplus is -7.5. However, in the answer choices you provided, none of them match this value. Therefore, it seems there might be an error in the given problem or the answer choices.