Given The Supply And Demand Functions:${ \begin{align*} P &= Q_s^2 + Q_s + 32 \ P &= -Q_d^2 - 4Q_d + 200 \end{align*} }$Calculate The Equilibrium Price And Quantity.

Equilibrium Price and Quantity: A Mathematical Analysis

1. Introduction

In economics, the concept of equilibrium price and quantity is crucial in understanding the behavior of supply and demand in a market. The equilibrium price is the price at which the quantity supplied equals the quantity demanded, resulting in a stable market condition. In this article, we will calculate the equilibrium price and quantity using the given supply and demand functions.

2. Supply and Demand Functions

The supply function is given by $P = Q_s^2 + Q_s + 32$, where $P$ is the price and $Q_s$ is the quantity supplied. The demand function is given by $P = -Q_d^2 - 4Q_d + 200$, where $P$ is the price and $Q_d$ is the quantity demanded.

3. Equilibrium Condition

To find the equilibrium price and quantity, we need to set the supply and demand functions equal to each other and solve for $Q_s$ and $P$. This is because at equilibrium, the quantity supplied equals the quantity demanded.

\begin{align*} Q_s^2 + Q_s + 32 &= -Q_d^2 - 4Q_d + 200 \\ Q_s^2 + Q_s + Q_d^2 + 4Q_d - 168 &= 0 \end{align*}

Since $Q_s = Q_d$ at equilibrium, we can substitute $Q_d$ for $Q_s$ in the above equation.

\begin{align*} Q_d^2 + Q_d + Q_d^2 + 4Q_d - 168 &= 0 \\ 2Q_d^2 + 5Q_d - 168 &= 0 \end{align*}

4. Solving the Quadratic Equation

To solve the quadratic equation $2Q_d^2 + 5Q_d - 168 = 0$, we can use the quadratic formula:

$Q_d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 2$, $b = 5$, and $c = -168$. Plugging these values into the formula, we get:

$Q_d = \frac{-5 \pm \sqrt{5^2 - 4(2)(-168)}}{2(2)}$

Simplifying the expression, we get:

$Q_d = \frac{-5 \pm \sqrt{25 + 1344}}{4}$

$Q_d = \frac{-5 \pm \sqrt{1369}}{4}$

$Q_d = \frac{-5 \pm 37}{4}$

Therefore, we have two possible values for $Q_d$:

$Q_d = \frac{-5 + 37}{4} = \frac{32}{4} = 8$

$Q_d = \frac{-5 - 37}{4} = \frac{-42}{4} = -10.5$

Since the quantity demanded cannot be negative, we discard the second solution and conclude that $Q_d = 8$.

5. Equilibrium Price

Now that we have found the equilibrium quantity, we can substitute this value into either the supply or demand function to find the equilibrium price. We will use the demand function:

$P = -Q_d^2 - 4Q_d + 200$

Substituting $Q_d = 8$ into the demand function, we get:

$P = -(8)^2 - 4(8) + 200$

$P = -64 - 32 + 200$

$P = 104$

Therefore, the equilibrium price is $P = 104$.

6. Conclusion

In this article, we calculated the equilibrium price and quantity using the given supply and demand functions. We found that the equilibrium quantity is $Q_d = 8$ and the equilibrium price is $P = 104$. This result demonstrates the importance of understanding the behavior of supply and demand in a market and how it affects the equilibrium price and quantity.
Equilibrium Price and Quantity: A Q&A Guide

1. Introduction

In our previous article, we calculated the equilibrium price and quantity using the given supply and demand functions. In this article, we will address some common questions and concerns related to equilibrium price and quantity.

2. Q&A

Q: What is the difference between supply and demand?

A: The supply function represents the quantity of a good or service that producers are willing to sell at a given price, while the demand function represents the quantity of a good or service that consumers are willing to buy at a given price.

Q: How do you determine the equilibrium price and quantity?

A: To determine the equilibrium price and quantity, you need to set the supply and demand functions equal to each other and solve for the quantity. This is because at equilibrium, the quantity supplied equals the quantity demanded.

Q: What is the significance of the equilibrium price and quantity?

A: The equilibrium price and quantity represent the stable market condition where the quantity supplied equals the quantity demanded. This is the price and quantity at which the market is in balance.

Q: Can the equilibrium price and quantity change over time?

A: Yes, the equilibrium price and quantity can change over time due to changes in supply and demand. For example, if there is an increase in demand, the equilibrium price may rise, and the equilibrium quantity may increase.

Q: How do external factors affect the equilibrium price and quantity?

A: External factors such as changes in government policies, technological advancements, and weather conditions can affect the equilibrium price and quantity. For example, a change in government policy may affect the supply of a good or service, leading to a change in the equilibrium price and quantity.

Q: What is the relationship between the supply and demand curves?

A: The supply curve is upward-sloping, meaning that as the price increases, the quantity supplied also increases. The demand curve is downward-sloping, meaning that as the price increases, the quantity demanded decreases.

Q: Can the supply and demand curves intersect at more than one point?

A: No, the supply and demand curves can only intersect at one point, which represents the equilibrium price and quantity.

Q: What is the concept of elasticity of demand?

A: The elasticity of demand refers to the responsiveness of the quantity demanded to changes in the price of a good or service. If the demand curve is steep, it means that a small change in price will lead to a large change in quantity demanded.

Q: How do you calculate the elasticity of demand?

A: The elasticity of demand can be calculated using the following formula:

Elasticity of demand = (Percentage change in quantity demanded) / (Percentage change in price)

Q: What is the significance of the elasticity of demand?

A: The elasticity of demand is important because it helps businesses and policymakers understand how changes in price will affect the quantity demanded of a good or service.

3. Conclusion

In this article, we addressed some common questions and concerns related to equilibrium price and quantity. We hope that this Q&A guide has provided you with a better understanding of the concept of equilibrium price and quantity and how it relates to supply and demand.

4. Additional Resources

For further reading on equilibrium price and quantity, we recommend the following resources:

• [1] "Microeconomics" by Gregory Mankiw
• [2] "Principles of Economics" by Gregory Mankiw
• [3] "Economics" by Samuelson and Nordhaus

5. References

[1] Mankiw, G. (2017). Microeconomics. Cengage Learning.

[2] Mankiw, G. (2017). Principles of Economics. Cengage Learning.

[3] Samuelson, P. A., & Nordhaus, W. D. (2010). Economics. McGraw-Hill Education.