#the demand function of 3 commodity: A, B and C are given as: DA=20-3PA+PB+PC, DB=30+PA+5PB , DC=15+PA-3PC and the corresponding supply function are SA=9A, SB=30PB-3 , SC=3PC-1 , then find the equilibrium of price and quantity of each commodity
To find the equilibrium price and quantity for each commodity, we need to equate the demand and supply functions for each commodity.
Let's start by setting the demand and supply functions equal to each other for commodity A:
Demand for A (DA) = Supply for A (SA)
20 - 3PA + PB + PC = 9A
Simplifying the equation, we get:
-3PA + PB + PC = 9A - 20
Now, let's equate the demand and supply functions for commodity B:
Demand for B (DB) = Supply for B (SB)
30 + PA + 5PB = 30PB - 3
Rearranging the equation:
PA - 29PB = -33
Finally, let's equate the demand and supply functions for commodity C:
Demand for C (DC) = Supply for C (SC)
15 + PA - 3PC = 3PC - 1
Simplifying the equation:
PA - 6PC = -16
Now we have a system of equations:
-3PA + PB + PC = 9A - 20 [Equation 1]
PA - 29PB = -33 [Equation 2]
PA - 6PC = -16 [Equation 3]
To solve this system of equations, we can use any method, such as substitution or elimination. Let's use the method of substitution.
From Equation 3, we can express PA in terms of PC:
PA = 6PC - 16
Substituting this value into Equation 2, we get:
6PC - 16 - 29PB = -33
Rearranging the equation:
29PB - 6PC = -17 [Equation 4]
Now we have two equations:
-3PA + PB + PC = 9A - 20 [Equation 1]
29PB - 6PC = -17 [Equation 4]
To solve this system of equations, you can use a tool like a graphing calculator or matrix operations. You can also substitute the value of PA into Equation 1 or Equation 4 to solve for PB and PC. Once you have the values of PB and PC, you can substitute them back into the supply functions to find the equilibrium quantity for each commodity.