Problem 66 - College Algebra

Question

Involve supply and demand. The following models describe demand and supply for three bedroom rental apartments. \(\begin{array}{lc}\text { Demand Model } & \text { Supply Model } \\ p--50 x+2000 & p-50 x\end{array}\) a. Solve the system and find the equilibrium quantity and the equilibrium price. b. Use your answer from part (a) to complete this statement: When rents are ___ per month, consumers will demand ___ apartments and suppliers will offer ___ appartments for rent.

Step-by-Step Solution

Answer

At equilibrium, when rents are at the equilibrium price per month, consumers will demand a number of apartments equal to the equilibrium quantity, and suppliers will offer a number of apartments equal to the equilibrium quantity.

Step 1 - Set Equations Equal

As equilibrium occurs when supply equals demand, set the equations equal to each other, i.e., \(p--50x+2000 = p-50x\).

Step 2 - Solve for equilibrium quantity

After setting the equations equal, solve for \(x\), which represents the equilibrium quantity. To do this, isolate \(x\) on one side of the equation.

Step 3 - Determine the equilibrium price

Substitute the equilibrium quantity obtained in step 2 into either the supply or demand equation and solve for \(p\), representing the equilibrium price.

Step 4 - Interpret the results

Use these calculated equilibrium quantity and price to fill in the blanks in the provided statement.

Key Concepts

Supply and Demand

Understanding the concepts of supply and demand is crucial for grasping many economic principles, particularly when analyzing markets for goods and services. Supply refers to the amount of a certain good producers are willing to sell at various prices, while demand is the quantity of a good that consumers are willing to buy at those prices. The relationship between supply and demand determines the market price and quantity of goods exchanged.

When the price of a good is high, suppliers are usually willing to sell more of it, as they can receive more profit. Conversely, when the price is low, consumers are more likely to buy the good, as it costs them less. The point at which the supply and demand curves intersect is called the 'equilibrium'. Here, the quantity supplied and quantity demanded are equal, and the price at this point is referred to as the 'equilibrium price'.

System of Equations

A system of equations is a set of two or more equations that have the same variables. The solution to a system of equations is the set of variable values that makes all the equations true simultaneously. There are multiple methods to solve such systems, including graphing, substitution, and the method of elimination.

In economics, systems of equations are often used to model supply and demand, where the variables typically represent the quantity of goods and the price level. Solving these systems can identify the equilibrium price and quantity in a market. It is a fundamental algebraic technique that is not only applicable in economic analysis but also in various scientific and engineering fields.

Algebraic Solutions

Equilibrium by Algebraic Manipulation

Algebraic solutions involve manipulating mathematical statements using algebra to find the value of unknown variables. This is essential for solving systems of equations where we aim to find where two algebraic expressions intersect.

The process typically involves combining like terms, isolating variables, and using arithmetic operations to solve for unknowns. In the context of the given exercise, once the supply and demand equations are set equal to each other, algebraic techniques are employed to find the equilibrium quantity and price, which is where the market is balanced.

Economic Equilibrium

Economic equilibrium is a state in a market where the quantity supplied exactly matches the quantity demanded, resulting in market stability. At this point, the allocation of goods is most efficient because the amount of goods being supplied is exactly the same as the amount of goods being demanded. Thus, there's no excess supply (surplus) or shortage (deficit) in the market.

In mathematical terms, this equilibrium is found by solving the system of supply and demand equations algebraically. The solution provides the equilibrium price and quantity. Understanding this concept is fundamental to grasping how markets function in the real world, where numerous factors influence supply and demand curves.