Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the point of equilibrium of the demand and supply equations. The point of equilibrium is the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations. Demand \(\quad\) Supply \(p=400-0.0002 x \quad p=225+0.0005 x\)

Step-by-Step Solution

Answer

The point of equilibrium of the demand and supply equations is at the price p = 350 and the number of units x = 250000.

Step 1: Set two equations equal

To find the equilibrium point, it is needed to set the demand and supply equations equal to each other. Hence, \(400 - 0.0002x = 225 + 0.0005x\)

Step 2: Simplify the equation

Consolidate similar terms. Then the equation becomes \(400 - 225 = 0.0005x + 0.0002x\), which simplifies to \(175 = 0.0007x\)

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by 0.0007. This gives, \(x = 175 / 0.0007 = 250000\) units.

Step 4: Find the price at this point

Substitute \(x = 250000\) into either the supply or demand equation to find the price at that point. Using the demand equation yields \(p = 400 - 0.0002*250000 = 350\)

Step 5: State the Equilibrium Point

The equilibrium point is where the quantity demanded is equal to the supply. So the equilibrium point (p, x) is (350, 250000)

Key Concepts

Demand Equation

In economics, the demand equation is an essential concept used to describe how much of a product or service consumers are willing to purchase at different prices. Specifically, the demand equation reflects the relationship between price and quantity demanded. For the given exercise, the demand equation is specified as:

• \( p = 400 - 0.0002x \)

Here, the price \( p \) depends inversely on the number of units \( x \). This means that as the price decreases, the quantity demanded increases, which is a typical behavior in most markets. The specific coefficients in this equation help in determining the slope of the demand curve. A smaller absolute value of the coefficient \(-0.0002\) indicates a gradual slope, meaning the demand is somewhat elastic to changes in price. Understanding this equation is key to predicting consumer behavior and setting prices effectively in a competitive market.

Supply Equation

In contrast to the demand equation, the supply equation represents the relationship between the price of a good and the quantity supplied by producers. It shows how much suppliers are willing to produce at different prices. For this exercise, the supply equation is defined as:

• \( p = 225 + 0.0005x \)

In this equation, the price \( p \) is directly related to the number of units \( x \). This implies that as the supply of a product increases, the price tends to rise, assuming other factors are constant. The positive coefficient \(0.0005\) highlights that the supply curve is upward sloping, indicating that higher prices incentivize producers to supply more. This relationship is crucial for understanding how suppliers respond to changes in market conditions and determining optimal production levels.

Solve for x

To find the equilibrium point where both demand and supply are equal, we need to solve for \( x \). Setting the demand equation equal to the supply equation allows us to find this point:

• \( 400 - 0.0002x = 225 + 0.0005x \)

Solving for \( x \) involves isolating \( x \) on one side of the equation and moving constants to the other. Here is how it's done:

• Start with: \\( 400 - 225 = 0.0005x + 0.0002x \)
• Simplify to: \\( 175 = 0.0007x \)

Finally, divide both sides by \(0.0007\) to calculate \( x \):

• \( x = \frac{175}{0.0007} = 250,000 \)

The value \( x = 250,000 \) signifies the quantity at which demand equals supply, known as the equilibrium quantity.

Simplify Equations

Simplifying equations is a critical step toward solving them effectively and accurately. In this context, simplifying refers to combining like terms and facilitating easier manipulation of terms. When finding an equilibrium point, it's necessary to first equalize the supply and demand equations and then simplify:

• From: \\( 400 - 0.0002x = 225 + 0.0005x \)
• Subtract \( 225 \) from \( 400 \), and bring the \( x \)-related terms on one side: \\( 175 = 0.0005x + 0.0002x \)
• Combine the \( x \) terms: \\( 175 = 0.0007x \)

This process involves consolidation, where terms on both sides are simplified into a form that is easier to handle. Simplifying makes it easier to track changes in variable values and confirms that the equation correctly represents the economic relationship. It ensures that the solution, when calculated, is both logical and mathematically robust, leading to sound economic analysis.