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Sagot :
Certainly! Let's address each part of the question step-by-step.
### Part (a): Calculate the market equilibrium price and quantity
The market equilibrium occurs where the quantity demanded equals the quantity supplied, i.e., [tex]\( Q^d = Q^s \)[/tex].
Given the market demand equation:
[tex]\[ Q^d = 100 - 2P \][/tex]
And the market supply equation:
[tex]\[ P = 10 + \frac{Q^s}{2} \][/tex]
First, let's manipulate the supply equation to express [tex]\( Q^s \)[/tex] in terms of [tex]\( P \)[/tex]:
[tex]\[ P = 10 + \frac{Q^s}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 2P = 20 + Q^s \][/tex]
Then, isolate [tex]\( Q^s \)[/tex]:
[tex]\[ Q^s = 2P - 20 \][/tex]
Now, set [tex]\( Q^d \)[/tex] equal to [tex]\( Q^s \)[/tex]:
[tex]\[ 100 - 2P = 2P - 20 \][/tex]
Solve for [tex]\( P \)[/tex]:
[tex]\[ 100 + 20 = 2P + 2P \][/tex]
[tex]\[ 120 = 4P \][/tex]
[tex]\[ P = 30 \][/tex]
So, the equilibrium price [tex]\( P \)[/tex] is [tex]\( 30 \)[/tex].
Next, calculate the equilibrium quantity [tex]\( Q \)[/tex]:
Using the demand equation:
[tex]\[ Q = 100 - 2P \][/tex]
[tex]\[ Q = 100 - 2(30) \][/tex]
[tex]\[ Q = 100 - 60 \][/tex]
[tex]\[ Q = 40 \][/tex]
So, the equilibrium quantity [tex]\( Q \)[/tex] is [tex]\( 40 \)[/tex].
### Part (b): Determine whether there is surplus or shortage at [tex]\( P = 25 \)[/tex] and [tex]\( P = 35 \)[/tex]
Let's analyze both scenarios separately.
At [tex]\( P = 25 \)[/tex]:
Calculate the quantity demanded ([tex]\( Q^d \)[/tex]):
[tex]\[ Q^d = 100 - 2P \][/tex]
[tex]\[ Q^d = 100 - 2(25) \][/tex]
[tex]\[ Q^d = 100 - 50 \][/tex]
[tex]\[ Q^d = 50 \][/tex]
Calculate the quantity supplied ([tex]\( Q^s \)[/tex]) using the supply equation rearranged earlier:
[tex]\[ Q^s = 2P - 20 \][/tex]
[tex]\[ Q^s = 2(25) - 20 \][/tex]
[tex]\[ Q^s = 50 - 20 \][/tex]
[tex]\[ Q^s = 30 \][/tex]
The difference between quantity supplied and quantity demanded:
[tex]\[ Q^s - Q^d = 30 - 50 \][/tex]
[tex]\[ Q^s - Q^d = -20 \][/tex]
A negative value indicates a shortage. So, at [tex]\( P = 25 \)[/tex], there is a shortage of 20 units.
At [tex]\( P = 35 \)[/tex]:
Calculate the quantity demanded ([tex]\( Q^d \)[/tex]):
[tex]\[ Q^d = 100 - 2P \][/tex]
[tex]\[ Q^d = 100 - 2(35) \][/tex]
[tex]\[ Q^d = 100 - 70 \][/tex]
[tex]\[ Q^d = 30 \][/tex]
Calculate the quantity supplied ([tex]\( Q^s \)[/tex]):
[tex]\[ Q^s = 2P - 20 \][/tex]
[tex]\[ Q^s = 2(35) - 20 \][/tex]
[tex]\[ Q^s = 70 - 20 \][/tex]
[tex]\[ Q^s = 50 \][/tex]
The difference between quantity supplied and quantity demanded:
[tex]\[ Q^s - Q^d = 50 - 30 \][/tex]
[tex]\[ Q^s - Q^d = 20 \][/tex]
A positive value indicates a surplus. So, at [tex]\( P = 35 \)[/tex], there is a surplus of 20 units.
### Summary
a)
- Equilibrium price: [tex]\( P = 30 \)[/tex]
- Equilibrium quantity: [tex]\( Q = 40 \)[/tex]
b)
- At [tex]\( P = 25 \)[/tex]: Shortage of 20 units
- At [tex]\( P = 35 \)[/tex]: Surplus of 20 units
### Part (a): Calculate the market equilibrium price and quantity
The market equilibrium occurs where the quantity demanded equals the quantity supplied, i.e., [tex]\( Q^d = Q^s \)[/tex].
Given the market demand equation:
[tex]\[ Q^d = 100 - 2P \][/tex]
And the market supply equation:
[tex]\[ P = 10 + \frac{Q^s}{2} \][/tex]
First, let's manipulate the supply equation to express [tex]\( Q^s \)[/tex] in terms of [tex]\( P \)[/tex]:
[tex]\[ P = 10 + \frac{Q^s}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 2P = 20 + Q^s \][/tex]
Then, isolate [tex]\( Q^s \)[/tex]:
[tex]\[ Q^s = 2P - 20 \][/tex]
Now, set [tex]\( Q^d \)[/tex] equal to [tex]\( Q^s \)[/tex]:
[tex]\[ 100 - 2P = 2P - 20 \][/tex]
Solve for [tex]\( P \)[/tex]:
[tex]\[ 100 + 20 = 2P + 2P \][/tex]
[tex]\[ 120 = 4P \][/tex]
[tex]\[ P = 30 \][/tex]
So, the equilibrium price [tex]\( P \)[/tex] is [tex]\( 30 \)[/tex].
Next, calculate the equilibrium quantity [tex]\( Q \)[/tex]:
Using the demand equation:
[tex]\[ Q = 100 - 2P \][/tex]
[tex]\[ Q = 100 - 2(30) \][/tex]
[tex]\[ Q = 100 - 60 \][/tex]
[tex]\[ Q = 40 \][/tex]
So, the equilibrium quantity [tex]\( Q \)[/tex] is [tex]\( 40 \)[/tex].
### Part (b): Determine whether there is surplus or shortage at [tex]\( P = 25 \)[/tex] and [tex]\( P = 35 \)[/tex]
Let's analyze both scenarios separately.
At [tex]\( P = 25 \)[/tex]:
Calculate the quantity demanded ([tex]\( Q^d \)[/tex]):
[tex]\[ Q^d = 100 - 2P \][/tex]
[tex]\[ Q^d = 100 - 2(25) \][/tex]
[tex]\[ Q^d = 100 - 50 \][/tex]
[tex]\[ Q^d = 50 \][/tex]
Calculate the quantity supplied ([tex]\( Q^s \)[/tex]) using the supply equation rearranged earlier:
[tex]\[ Q^s = 2P - 20 \][/tex]
[tex]\[ Q^s = 2(25) - 20 \][/tex]
[tex]\[ Q^s = 50 - 20 \][/tex]
[tex]\[ Q^s = 30 \][/tex]
The difference between quantity supplied and quantity demanded:
[tex]\[ Q^s - Q^d = 30 - 50 \][/tex]
[tex]\[ Q^s - Q^d = -20 \][/tex]
A negative value indicates a shortage. So, at [tex]\( P = 25 \)[/tex], there is a shortage of 20 units.
At [tex]\( P = 35 \)[/tex]:
Calculate the quantity demanded ([tex]\( Q^d \)[/tex]):
[tex]\[ Q^d = 100 - 2P \][/tex]
[tex]\[ Q^d = 100 - 2(35) \][/tex]
[tex]\[ Q^d = 100 - 70 \][/tex]
[tex]\[ Q^d = 30 \][/tex]
Calculate the quantity supplied ([tex]\( Q^s \)[/tex]):
[tex]\[ Q^s = 2P - 20 \][/tex]
[tex]\[ Q^s = 2(35) - 20 \][/tex]
[tex]\[ Q^s = 70 - 20 \][/tex]
[tex]\[ Q^s = 50 \][/tex]
The difference between quantity supplied and quantity demanded:
[tex]\[ Q^s - Q^d = 50 - 30 \][/tex]
[tex]\[ Q^s - Q^d = 20 \][/tex]
A positive value indicates a surplus. So, at [tex]\( P = 35 \)[/tex], there is a surplus of 20 units.
### Summary
a)
- Equilibrium price: [tex]\( P = 30 \)[/tex]
- Equilibrium quantity: [tex]\( Q = 40 \)[/tex]
b)
- At [tex]\( P = 25 \)[/tex]: Shortage of 20 units
- At [tex]\( P = 35 \)[/tex]: Surplus of 20 units
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