Problem 4 - Applied Calculus

Question

Given the demand curve \(p=35-q^{2}\) and the supply curve \(p=3+q^{2},\) find the producer surplus when the market is in equilibrium.

Step-by-Step Solution

Answer

Producer surplus is 32 when the market is in equilibrium.

Step 1: Find the Equilibrium Quantity

To find the equilibrium quantity, equate the demand curve to the supply curve: \(35-q^2 = 3+q^2\). Solving for \(q\), we get \(35 - 3 = q^2 + q^2\). This simplifies to \(32 = 2q^2\), and dividing by 2 gives \(q^2 = 16\). Taking the square root, we find \(q = 4\).

Step 2: Find the Equilibrium Price

Substitute \(q = 4\) into either the demand or supply function to find the equilibrium price. Using the supply curve: \(p = 3 + q^2 = 3 + 4^2 = 3 + 16 = 19\). Thus, the equilibrium price is \(p = 19\).

Step 3: Calculate Producer Surplus

Producer surplus is the area above the supply curve and below the price line up to the equilibrium quantity. The formula for producer surplus is \( \frac{1}{2} \times \text{base} \times \text{height} \). Here, the base is the equilibrium quantity \(q = 4\), and the height is the price difference \(19 - 3\) because \(p_{min} = 3\). Therefore, the producer surplus is \( \frac{1}{2} \times 4 \times (19 - 3) = \frac{1}{2} \times 4 \times 16 = 32\).

Key Concepts

Equilibrium Quantity

Finding the equilibrium quantity is essential in understanding how much of a product will be exchanged in a market when supply and demand are balanced. In this case, you start by setting the demand curve equal to the supply curve:

• Demand curve: \( p = 35 - q^2 \)
• Supply curve: \( p = 3 + q^2 \)

Equating them gives us \( 35 - q^2 = 3 + q^2 \). Solving this equation will allow us to find the equilibrium quantity \( q \).
Rearranging the terms gives us \( 32 = 2q^2 \). When we divide by 2, we simplify it to \( q^2 = 16 \). Taking the square root of both sides results in \( q = 4 \).
Thus, the equilibrium quantity in this market, where the demand equals supply, is 4 units.

Demand and Supply Curves

Understanding demand and supply curves allows you to see how market conditions will interact. These curves graphically represent the relationship between price and quantity for both consumers and producers.

• The demand curve \( p = 35 - q^2 \) shows how the quantity demanded decreases as price increases. It is downward sloping because as prices rise, fewer people are willing or able to buy a product.
• The supply curve \( p = 3 + q^2 \) indicates how the quantity supplied increases with price. This upward slope exists because higher prices cover the costs of production for more units, motivating producers to supply more.

The meeting point of these curves is crucial because it represents market equilibrium
where the quantity of goods consumers want is exactly what producers are willing to supply.

Equilibrium Price

The equilibrium price is where the market settles, balancing the quantity supplied with the quantity demanded. To find this price, substitute the equilibrium quantity \( q = 4 \) back into either the demand or supply equation. Doing this allows us to calculate the price at which the market clears.
Using the supply curve \( p = 3 + q^2 \):

• Plugging in \( q = 4 \), the equation becomes \( p = 3 + 4^2 \).
• This simplifies to \( p = 3 + 16 \), resulting in an equilibrium price \( p = 19 \).

Thus, the equilibrium price is 19 units, meaning consumers are willing to buy, and producers are willing to sell, at this price. It reflects an efficient market state where there is no excess supply or demand, ensuring optimal distribution of goods.