4. Given market demand: [tex]Q^d = 100 - 2P[/tex], and market supply: [tex]P = 10 + \frac{Q^s}{2}[/tex]

a) Calculate the market equilibrium price and quantity.

b) Determine whether there is a surplus or shortage at [tex]P = 25[/tex] and [tex]P = 35[/tex].



Answer :

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