Question

Suppose the demand schedule in the motorcycle market is given by the equation \(\mathrm{P}_{\mathrm{D}}=100-5 \mathrm{Q}_{\mathrm{D}}\), where P represents price, and \(Q\) represents quantity. If the supply schedule is given by the equation \(\mathrm{P}_{\mathrm{s}}=40+10 \mathrm{Q}_{\mathrm{s}}\), what is the equilibrium price and quantity in the motorcycle market?

Short answer

The equilibrium price and quantity in the motorcycle market are \(P = 80\) and \(Q = 4\), respectively.

Step-by-step solution

Set demand and supply equations equal

First, we need to set the demand and supply equations equal to each other. The demand equation is \(P_D = 100 - 5Q_D\) and the supply equation is \(P_S = 40 + 10Q_S\). To find the equilibrium point, we set the two equations equal to each other. Since \( P_D = P_S \), we have \(100 - 5Q_D = 40 + 10Q_S\).

Solve for the equilibrium quantity

Next, we'll solve this equation for the equilibrium quantity, which we'll denote as Q. Because the equilibrium quantity is the same for both supply and demand, we can set \(Q_D = Q_S = Q\). Then the equation becomes \(100 - 5Q = 40 + 10Q\). To solve for Q, first combine the terms involving Q on one side of the equation: \(100 -5Q - 10Q = 40\) Now, simplify the left side: \(-15Q = 40 - 100\) Divide both sides by -15: \(Q = \frac{40 - 100}{-15} = \frac{-60}{-15} = 4\) So, the equilibrium quantity \(Q\) is 4.

Solve for the equilibrium price

Now that we know the equilibrium quantity, we can find the equilibrium price by plugging the value of Q into either the demand or supply equation. Let's use the demand equation: \(P_D = 100 - 5Q_D = 100 - 5(4) = 100 - 20 = 80\) Thus, the equilibrium price \(P\) is 80. In conclusion, the equilibrium price and quantity in the motorcycle market are 80 and 4, respectively.

Key concepts

Demand and Supply

The fundamental concepts of demand and supply describe the relationship between how much of a product people want and how much is available. In this example, the motorcycle market provides an interesting case study.

• **Demand**: Represented as \( P_D = 100 - 5Q_D \), this equation shows that as the quantity demanded (\( Q_D \)) increases, the price people are willing to pay decreases. This negative relationship is typical in economics and is known as the law of demand.
• **Supply**: The equation \( P_S = 40 + 10Q_S \) describes supply. Here, as the quantity supplied (\( Q_S \)) increases, so does the price. This positive relationship represents the law of supply, which indicates that producers are willing to supply more at higher prices.

Understanding demand and supply is key to finding the equilibrium point. It's about balancing what customers want with what is available from producers.

Equilibrium Price

The equilibrium price is where demand meets supply, resulting in a balance in the market. It's the price at which the quantity of the product consumers want to buy equals the quantity sellers want to sell.
In our motorcycle market, setting the demand equation equal to the supply equation allows us to find this equilibrium.

• Solving the Equations: By equating \( 100 - 5Q_D = 40 + 10Q_S \), we solve for the quantity to find where both lines on a graph would intersect.
• Substitution to Find Price: Once we find the equilibrium quantity \( Q = 4 \), substituting back into either the demand or supply equation helps us discover the equilibrium price. In this case, it gives us an equilibrium price of 80.

This equilibrium price ensures that the market is in balance, preventing excess supply or unmet demand at this level.

Equilibrium Quantity

The equilibrium quantity is the amount of the product that is both supplied and demanded at the equilibrium price. It's the perfect harmony of how much people want to buy and how much sellers want to sell.

• **Balancing Act**: In the motorcycle market scenario, after solving the demand and supply equations, we find that the equilibrium quantity \( Q \) is 4. This means that at the price of 80, four motorcycles will be sold.
• **Market Stability**: At this point, there is no pressure to change the price or quantity, as both sides of the market are satisfied. If the price were higher, there would be excess supply. If lower, excess demand.

Identifying the equilibrium quantity helps businesses and economists predict how markets behave, ensuring efficient operation.