Problem 66 - 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. Calculate the producer surplus given the supply and demand equations.

Step-by-Step Solution

Answer

The producer surplus is $40,000.

Step 1: Equilibrium Quantity

To find the equilibrium quantity, set the demand equation equal to the supply equation: \[ 80 - 0.01x = 20 + 0.02x \]Solve for \(x\): \[ 60 = 0.03x \] \[ x = \frac{60}{0.03} = 2000 \]Thus, the equilibrium quantity is 2000 units.

Step 2: Equilibrium Price

Plug the equilibrium quantity back into either the demand or supply equation to find the equilibrium price. Using the demand equation: \[ P = 80 - 0.01 \times 2000 \]\[ P = 80 - 20 = 60 \]Thus, the equilibrium price is $60.

Step 3: Calculate Producer Surplus

The producer surplus is the area above the supply curve and below the equilibrium price. First, find the price when \(x = 0\) using the supply equation: \[ P = 20 + 0.02 \times 0 = 20 \]The producer surplus is the area of the triangle with base \(x = 2000\) and height \(60 - 20 = 40\): \[ \text{Producer Surplus} = \frac{1}{2} \times 2000 \times 40 = 40000\]

Step 4: Final Step: Verify Calculations

Double-check each step to ensure all calculations are correct by reconfirming equilibrium quantity, price, and producer surplus.

Key Concepts

Equilibrium Quantity

To understand how the market achieves balance, we must find the equilibrium quantity, which is the quantity at which the amount demanded by consumers equals the amount suppliers want to sell. Imagine this as a harmonious point where desires of both buyers and sellers perfectly intersect. Using the provided demand and supply equations, we set them equal to solve for this quantity:\[ 80 - 0.01x = 20 + 0.02x \]This equation comes from the idea that at equilibrium, the price consumers are willing to pay equals the price suppliers are willing to accept. Simplifying gives:\[ 60 = 0.03x \]And by solving:\[ x = \frac{60}{0.03} = 2000 \]This means that 2000 units is the equilibrium quantity—at this level of production, the market is balanced.

Equilibrium Price

With the equilibrium quantity known, finding the equilibrium price becomes straightforward. The equilibrium price is the market price where the quantity supplied and quantity demanded are the same. To find this, we substitute our previously found equilibrium quantity (2000 units) back into either the demand or the supply equation.Using the demand equation:\[ P = 80 - 0.01 \times 2000 \]First, calculate the term involving quantity:\[ P = 80 - 20 \]Thus, the equilibrium price is $60. This price signifies the point at which market forces of supply and demand are balanced along the price axis, ensuring there is neither an excess supply nor a shortage.

Supply and Demand Equations

The foundation of any market analysis lies in understanding the supply and demand equations. These are mathematical expressions that represent how much of a good or service is available and how much of it people want at various price levels.

• The demand equation is given by \( P = 80 - 0.01x \). This formula shows that as the number of units (\(x\)) increases, the price (\(P\)) customers are willing to pay decreases.
• The supply equation is \( P=20+0.02x \). This raises with \(x\) because as more units are produced, suppliers require a higher price to cover costs and earn profit.

These equations together define a market's structure. By calculating where they intersect, economists determine pivotal market conditions such as equilibrium quantity and price, which balance production with consumer satisfaction.