The producer's surplus is $1,073,741.33.
Given:
The supply function for pork bellies is S(q) = q^7/2 + 3q^5/2 + 52
Supply and demand are in equilibrium at q = 16
To find the producer's surplus, we need to find the area between the supply curve and the equilibrium price line up to the equilibrium quantity. The equilibrium price is the price at which the quantity supplied equals the quantity demanded. Since supply and demand are in equilibrium at q = 16, we can find the equilibrium price by substituting q = 16 into the supply function:
S(16) = 16^7/2 + 3(16)^5/2 + 52 = 32768 + 12288 + 52 = 45008
So, the equilibrium price is $45,008 per unit.
To find the producer's surplus, we need to find the area between the supply curve and the equilibrium price line up to the equilibrium quantity of 16. We can find this area by taking the definite integral of the supply function from 0 to 16:
∫ S(q) dq = ∫ (q^7/2 + 3q^5/2 + 52) dq = [2/9 q^9/2 + 2/3 q^7/2 + 52q] from 0 to 16
= (2/9 (16)^9/2 + 2/3 (16)^7/2 + 52(16)) - (2/9 (0)^9/2 + 2/3 (0)^7/2 + 52(0))
= 1,073,741.33
Therefore, the producer's surplus is $1,073,741.33.
Complete question:
Find the producers' surplus if the supply function for pork bellies is given by the following.
S(q) = q^7/2 + 3q^5/2 + 52 Assume supply and demand are in equilibrium at q = 16. The producers' surplus is $ .