Consider the following inverse demand and inverse supply curves representing the market for apartments where p represents the rental price of an apartment unit, Qd represents the total number of apartment units demanded and Qs the total number of apartment units supplied in 1000s of units.

Inverse Demand Curve:

Inverse Supply Curve:

**Question 6**.  Compute the equilibrium price, p*

To compute the equilibrium price (\(p^*\)) for the market for apartments, we need the equations for both the inverse demand and inverse supply curves. These curves are typically given in the form \(p = f(Q)\), where \(p\) is the price and \(Q\) is the quantity. 

Let's assume the following forms for the curves based on the usual representation:

1. Inverse Demand Curve: \( p = a - bQ_d \)
2. Inverse Supply Curve: \( p = c + dQ_s \)

Where:
- \(a\), \(b\), \(c\), and \(d\) are constants.
- \(Q_d\) is the quantity demanded in 1000s of units.
- \(Q_s\) is the quantity supplied in 1000s of units.

At equilibrium, the quantity demanded equals the quantity supplied (\(Q_d = Q_s = Q\)), and the prices are equal. So we set the inverse demand equal to the inverse supply to find the equilibrium price \(p^*\).

\[ a - bQ = c + dQ \]

Solving for \(Q\):

\[ a - c = bQ + dQ \]
\[ a - c = (b + d)Q \]
\[ Q = \frac{a - c}{b + d} \]

Substitute \(Q\) back into either the inverse demand or inverse supply curve to find the equilibrium price \(p^*\).

Let's go through a specific example. Suppose the inverse demand and supply curves are as follows:

1. Inverse Demand Curve: \( p = 100 - 2Q \)
2. Inverse Supply Curve: \( p = 20 + 3Q \)

Set them equal to each other to find \(Q\):

\[ 100 - 2Q = 20 + 3Q \]
\[ 100 - 20 = 2Q + 3Q \]
\[ 80 = 5Q \]
\[ Q = \frac{80}{5} = 16 \]

Now substitute \(Q = 16\) back into either curve to find \(p^*\). Using the inverse demand curve:

\[ p = 100 - 2(16) = 100 - 32 = 68 \]

Therefore, the equilibrium price \(p^*\) is 68.

Would you like further details or have any questions?

Here are some related questions to expand on this information:

1. What is the significance of the equilibrium price in a market?
2. How do shifts in the demand curve affect the equilibrium price?
3. How do shifts in the supply curve affect the equilibrium price?
4. What factors can cause the demand curve for apartments to shift?
5. What factors can cause the supply curve for apartments to shift?
6. How would a price ceiling below the equilibrium price affect the market for apartments?
7. How would a price floor above the equilibrium price affect the market for apartments?
8. Can you explain the concept of consumer and producer surplus at equilibrium?

**Tip:** Always double-check your algebra when solving for equilibrium to avoid simple calculation errors.

Learn how to compute the equilibrium price (p*) in the market for apartments using inverse demand and supply curves. Understand the equations involved and the steps to find the equilibrium price. Suitable for advanced undergraduate students and anyone interested in microeconomic analysis.

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