The market for 40 lb. bags of pine bark mulch (sold at John's shop) is described by the following functions:
p = -.3q + 66
p = 1.3q - 30
where p is the price per unit and q is the number of bags bought and sold.
a) Which of the functions above is the demand function? Explain. b) Find the equilibrium price and quantity in this market. q = ______________
p = ______________
c) Rewrite the demand function with quantity as the dependent variable.
d) If the price is $40, is there a surplus or a shortage? How much? Explain.
a) To identify the demand function, we need to look for the function that represents the relationship between the price (p) and the quantity demanded (q). In this case, the demand function is given by:
p = 1.3q - 30
The reason this is the demand function is because it represents the relationship between the price and the quantity demanded by consumers. As the price increases, the quantity demanded decreases, and vice versa.
b) To find the equilibrium price and quantity, we need to set the demand function equal to the supply function. The supply function is given by:
p = -.3q + 66
By setting the demand and supply functions equal to each other, we can find the equilibrium price and quantity:
1.3q - 30 = -.3q + 66
Combine like terms:
1.3q + .3q = 66 + 30
1.6q = 96
Divide both sides by 1.6:
q = 96/1.6
q = 60
Now that we have the equilibrium quantity, we can substitute this value back into either the demand or supply function to find the equilibrium price. Let's use the demand function:
p = 1.3(60) - 30
p = 78 - 30
p = 48
Therefore, the equilibrium quantity is 60 bags and the equilibrium price is $48.
c) To rewrite the demand function with quantity as the dependent variable, we need to isolate q on one side. Let's rearrange the demand function:
p = 1.3q - 30
Add 30 to both sides:
p + 30 = 1.3q
Divide both sides by 1.3:
(p + 30) / 1.3 = q
Therefore, the demand function with quantity as the dependent variable is q = (p + 30) / 1.3.
d) If the price is $40, we can use the demand function to determine whether there is a surplus or a shortage. If the quantity demanded is greater than the quantity supplied, there is a shortage. If the quantity supplied is greater than the quantity demanded, there is a surplus.
Let's substitute the price of $40 into the demand function:
q = (40 + 30) / 1.3
q = 70 / 1.3
q ≈ 53.85
The quantity demanded at a price of $40 is approximately 53.85 bags. Now, let's substitute this value into the supply function to determine the quantity supplied:
p = -.3(53.85) + 66
p ≈ 50.24
The quantity supplied at a price of $40 is approximately 50.24 bags.
Since the quantity demanded (53.85 bags) is greater than the quantity supplied (50.24 bags), there is a shortage. The shortage is calculated by subtracting the quantity supplied from the quantity demanded:
Shortage = Quantity Demanded - Quantity Supplied
= 53.85 - 50.24
≈ 3.61 bags
Therefore, there is a shortage of approximately 3.61 bags if the price is $40.