Problem 48 - Precalculus with Limits

Question

Finding the Equilibrium Point In Exercises \(45-48\) , find the equilibrium point of the demand and supply equations. $$ \begin{array}{ll}{\text { Demand }} & {\text { Supply }} \\ {p=400-0.0002 x} & {p=225+0.0005 x}\end{array} $$

Step-by-Step Solution

Answer

The equilibrium point is \(p = 350\), \(x = 250000\). This means that the market price at which the quantity demanded equals the quantity supplied is $350, and the quantity at which this occurs is 250,000 units.

Step 1: Set the two equations equal to each other

Set the demand equation equal to the supply equation: \(400 - 0.0002x = 225 + 0.0005x\)

Step 2: Solve for x

Rearrange the equation to isolate x on one side: \(400 - 225 = 0.0005x + 0.0002x\), which simplifies to \(175 = 0.0007x\). Then divide both sides of the equation by 0.0007 to solve for x, yielding \(x = 175/0.0007 = 250000\)

Step 3: Substitute x into one of the equations to solve for p

Substitute \(x = 250000\) into the demand equation \(p=400-0.0002x\) to solve for p. This gives \(p=400-0.0002*250000 = 350\)

Key Concepts

Understanding the Demand Equation

The demand equation in economic terms describes how much of a product consumers are willing to buy at different prices. For our exercise, the demand equation is given by\[ p = 400 - 0.0002x \].
Here:

• \( p \) is the price consumers are willing to pay.
• \( x \) is the quantity of items demanded.

The equation shows that as the quantity \( x \) increases, the price \( p \) consumers are willing to pay decreases. This reflects the typical behavior in a market where higher availability of a product often leads to lower prices. The negative coefficient \( -0.0002 \) indicates this inverse relationship.
The intercept point, \( 400 \), highlights the maximum price consumers would be willing to pay when no product is available.

Deciphering the Supply Equation

The supply equation represents how much of a product suppliers are willing to produce at different prices. For our scenario, the supply equation is described as\[ p = 225 + 0.0005x \].
In this equation:

• \( p \) represents the price at which producers are willing to sell.
• \( x \) signifies the quantity of items supplied.

Unlike the demand equation, this equation shows a direct relationship between price \( p \) and quantity \( x \). As the quantity supplied \( x \) increases, producers are willing to accept a lower price. The coefficient \( 0.0005 \) indicates the positive relationship between price and quantity.
The intercept \( 225 \) represents the minimum price at which producers would begin to offer their products, even when no product is supplied.

Solving Equations to Find the Equilibrium

Solving equations involves finding the values of variables that satisfy the given equations. In finding the equilibrium point, we set the demand and supply equations equal.
This step reflects when market demand matches supply, aligning with \[ 400 - 0.0002x = 225 + 0.0005x \].To solve:

• Subtract 225 from 400 to simplify the left side: \( 175 = 0.0007x \).
• Isolate \( x \) by dividing both sides by 0.0007: \( x = 250000 \).

This \( x \) value represents the quantity where the demand and supply intersect.Finally, substitute \( x = 250000 \) into either equation to find \( p \). Here, using the demand equation gives \( p = 350 \). Thus, the equilibrium point is \( (250000, 350) \), indicating the price and quantity where the market balances.

Grasping Economic Equilibrium

Economic equilibrium occurs when market demand equals market supply, leading to a stable market condition. At this point, neither excess demand nor excess supply exists.
Equilibrium ensures:

• Exact quantity demanded is produced and sold.
• Stable prices without external disruptions.

Our exercise reveals that at a quantity \( x = 250000 \) and a price \( p = 350 \), the market reaches equilibrium. This balance results from the intersection of the demand and supply curves, representing optimal allocation of resources.
Understanding economic equilibrium is crucial, as it aids businesses and policymakers in predicting consumer behavior, optimizing production levels, and setting fair prices.