Problem 64 - Precalculus

Question

For Exercises \(63-66,\) employ the following supply and demanc equations: $$\begin{array}{ll}\text { Demand: } & P=80-0.01 x \\\\\text { Supply: } & P=20+0.02 x\end{array}$$ where \(P\) is the price in dollars when \(x\) units are produced. Write a system of linear inequalities corresponding to the producer surplus.

Step-by-Step Solution

Answer

The system of inequalities is \(\begin{cases} P > 20 + 0.02x \\ P < 60 \\ x \leq 2000 \end{cases}\).

Step 1: Identify Equilibrium Point

To find the producer surplus, first identify the equilibrium point where the supply equals the demand. Equate the demand equation and the supply equation: \(80 - 0.01x = 20 + 0.02x\).

Step 2: Solve for \(x\) at Equilibrium

Rearrange the equation from Step 1 to solve for \(x\): \(80 - 20 = 0.02x + 0.01x\). Simplify to: \(60 = 0.03x\). Divide by 0.03: \(x = 2000\).

Step 3: Find \(P\) at Equilibrium

Substitute \(x = 2000\) back into either the supply or demand equation to find \(P\). Using the demand equation, \(P = 80 - 0.01 \times 2000 = 80 - 20 = 60\). Thus, at equilibrium, \(P = 60\).

Step 4: Set up Supply Inequality

For the producer surplus, set up the inequality for prices below the equilibrium price using the supply equation: \(P > 20 + 0.02x\).

Step 5: Set up Demand Inequality

Ensure that the quantities produced \((x)\) are within the range up to equilibrium using the demand function: \(P < 80 - 0.01x\).

Step 6: Define Producer Surplus System of Inequalities

The system of inequalities representing the producer surplus is: \(\begin{cases} P > 20 + 0.02x \ P < 60 \ x \leq 2000 \end{cases}\). These inequalities ensure the price is above the supply curve and below equilibrium, with quantities within the range.

Key Concepts

Equilibrium Point

The concept of the equilibrium point in economics represents the situation where supply matches demand. This means that the quantity of products that suppliers want to sell is exactly the same as the quantity consumers want to buy. In mathematical terms, this is found where the supply and demand equations intersect.

To find this point in our given problem, we start by equating the supply and demand equations:

• Demand: \( P = 80 - 0.01x \)
• Supply: \( P = 20 + 0.02x \)

Setting them equal, \( 80 - 0.01x = 20 + 0.02x \), allows us to solve for \( x \), called the equilibrium quantity. The result, \( x = 2000 \), tells us that 2000 units are the equilibrium quantity where supply equals demand. Next, substituting \( x = 2000 \) back into either equation provides the equilibrium price, \( P = 60 \). This indicates that at the price of 60 dollars per unit, supply and demand are balanced.

Supply and Demand Equations

Supply and demand equations are mathematical formulations used to describe how the price \( P \) of a good is influenced by the quantity \( x \) produced and purchased. The demand equation shows how much consumers are willing to buy at different prices, typically demonstrating that as price decreases, demand increases.

In this exercise:

• The demand equation is \( P = 80 - 0.01x \). This indicates that for every unit increase in quantity, the price decreases by 0.01 dollars.

The supply equation, on the other hand, illustrates how much producers are willing to sell at varying prices. Often, as price increases, the supply increases, showcasing producers’ interest in maximizing profits:

• The supply equation is \( P = 20 + 0.02x \). Here, as the quantity increases by one unit, the price goes up by 0.02 dollars, showing a direct relationship between price and supply.

These equations together offer a system to understand market behaviors, predict price changes, and determine the effects of external factors on supply and demand.

System of Linear Inequalities

To find the producer surplus, we often create a system of linear inequalities that defines conditions for price and quantity. Producer surplus represents the difference between what producers are willing to accept for a good versus what they receive. It is typically the area above the supply curve and below the equilibrium price level.

In this case, setting up a system of inequalities helps to visualize this surplus:

• The supply inequality, \( P > 20 + 0.02x \), ensures the price is always above the supply curve for producers to gain surplus.
• The demand inequality, \( P < 80 - 0.01x \), ensures the price is still within consumer demand limits but below the equilibrium point.
• \( x \leq 2000 \) ensures that production does not exceed the equilibrium quantity.

These inequalities together constitute the producer surplus apparatus, describing conditions where producers earn more than the minimum price they would accept while still staying within market limits.