consider the following functions;
Q1=36-1/3p
Q2=-9+1/2p
1. Identify the demand and the supply functions with reasons.
2. Determine the equilibrium price and equilibrium quantity.
since Q1 declines as price increases, it must be the demand function
set Q1=Q2 to find equilibrium p
To identify the demand and supply functions, we will look at the given equations:
Q1 = 36 - (1/3)p
Q2 = -9 + (1/2)p
1. Demand Function: The demand function represents the relationship between the price of a good and the quantity demanded at that price. In this case, Q1 represents the quantity demanded. As we can see, Q1 decreases as the price (p) increases. Therefore, Q1 is the demand function, and we can write it as:
Qd = 36 - (1/3)p
Supply Function: The supply function represents the relationship between the price of a good and the quantity supplied at that price. In this case, Q2 represents the quantity supplied. As we can see, Q2 increases as the price (p) increases. Therefore, Q2 is the supply function, and we can write it as:
Qs = -9 + (1/2)p
2. Equilibrium Price and Quantity: At equilibrium, the quantity demanded is equal to the quantity supplied. Thus, we can set Qd equal to Qs and solve for the equilibrium price and quantity.
36 - (1/3)p = -9 + (1/2)p
To solve this equation, we can get rid of the fractions by multiplying both sides by the least common multiple (LCM) of 3 and 2, which is 6:
6(36 - (1/3)p) = 6(-9 + (1/2)p)
216 - 2p = -54 + 3p
5p = 270
p = 54
So, the equilibrium price (p) is 54. To find the equilibrium quantity, we can substitute the equilibrium price into either the demand or supply function:
Qd = 36 - (1/3)(54)
Qd = 36 - 18
Qd = 18
Therefore, the equilibrium quantity (Q) is 18.
In conclusion, the demand function is Qd = 36 - (1/3)p, the supply function is Qs = -9 + (1/2)p, the equilibrium price is p = 54, and the equilibrium quantity is Q = 18.
To identify the demand and supply functions, we need to understand the characteristics of each function.
1. Demand Function:
The demand function represents the relationship between the price of a product and the quantity demanded by consumers. Typically, as the price of a product decreases, the quantity demanded increases (law of demand).
In the given functions Q1 and Q2, we can observe that:
- Q1 = 36 - (1/3)p
- Q2 = -9 + (1/2)p
Q1 represents the quantity demanded, whereas Q2 represents the quantity supplied. Since the demand function represents the quantity demanded by consumers, we can conclude that Q1 is the demand function. This is because Q1 decreases as the price increases, indicating the inverse relationship between price and quantity demanded.
Therefore, Q1 = 36 - (1/3)p is the demand function.
2. Supply Function:
The supply function represents the relationship between the price of a product and the quantity supplied by producers. Typically, as the price of a product increases, the quantity supplied increases (law of supply).
Based on the given functions Q1 and Q2, we can deduce that:
- Q1 = 36 - (1/3)p
- Q2 = -9 + (1/2)p
Q2 represents the quantity supplied. As Q2 increases with the increase in price, we can conclude that Q2 is the supply function.
Therefore, Q2 = -9 + (1/2)p is the supply function.
To determine the equilibrium price and equilibrium quantity, we need to find the point where the demand and supply functions intersect. This intersection represents the market equilibrium.
Equating the demand and supply functions, we can write:
36 - (1/3)p = -9 + (1/2)p
To simplify this equation, we can multiply both sides by 6:
216 - 2p = -54 + 3p
Combining like terms:
5p = 270
Dividing both sides by 5:
p = 54
Substituting the value of p back into either the demand or supply function, we can find the equilibrium quantity.
Let's use the demand function:
Q1 = 36 - (1/3)p
Q1 = 36 - (1/3)(54)
Q1 = 36 - 18
Q1 = 18
Therefore, the equilibrium price is 54 and the equilibrium quantity is 18.