1.) Find the producers' surplus if the supply function is: S(q) = q^7/2+3q^5/2 + 54. Assume the supply and demand are in equilibrium at q= 25.
2.) S(q) = q^2 + 12q and D(q) = 900 - 18q - q^2
The point at which the supply and demand are equilibrium is (15, 405).
a.) Find the consumers' surplus
b.) Find the producers' surplus
Thank you for any help!
1.) Find the producers' surplus if the supply function is: S(q) = q^7/2+3q^5/2 + 54. Assume the supply and demand are in equilibrium at q= 25.
2.) S(q) = q^2 + 12q and D(q) = 900 - 18q - q^2
The point at which the supply and demand are equilibrium is (15, 405).
a.) Find the consumers' surplus
b.) Find the producers' surplus
To find the producers' surplus in these scenarios, we need to understand what it represents and how to calculate it.
1.) Producers' Surplus for Supply Function S(q) = q^7/2 + 3q^5/2 + 54
Producers' surplus represents the difference between the price that producers are willing to supply a good and the actual price they receive. It is the area between the supply curve and the equilibrium price (price at the point of equilibrium). Mathematically, it can be calculated as follows:
Step 1: Find the equilibrium price by substituting q = 25 into the supply function:
S(25) = (25)^7/2 + 3(25)^5/2 + 54
Step 2: Calculate the producers' surplus by finding the integral of the supply function from equilibrium quantity (q) to positive infinity:
Producers' Surplus = ∫[25 to ∞] S(q) - Equilibrium Price dq
To find the exact value of the integral, you can use calculus software or numerical methods. Additionally, you could approximate the integral by dividing the range into smaller intervals and calculating the sum of areas.
2.) Producers' Surplus for Supply Function S(q) = q^2 + 12q and Demand Function D(q) = 900 - 18q - q^2
Note: In this case, we already have the equilibrium point (15, 405), so we can directly calculate the producers' surplus.
Step 1: Find the equilibrium price by substituting q = 15 into either the supply or demand function, as they should give the same value:
S(15) = (15)^2 + 12(15)
D(15) = 900 - 18(15) - (15)^2
Either S(15) or D(15) will give the equilibrium price.
Step 2: Calculate the producers' surplus by finding the integral of the supply function from equilibrium quantity (q) to positive infinity:
Producers' Surplus = ∫[15 to ∞] S(q) - Equilibrium Price dq
Similar to the previous case, you can find the exact value using calculus software, numerical methods, or approximate it using intervals.
I hope this explanation helps you understand how to find the producers' surplus in these situations.