Problem 62 - Precalculus with Limits

Question

Consumer Surplus and Producer Surplus In Exercises \(61-64,\) (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. $$\begin{array}{ll}{\text { Demand }} & {\text { Supply }} \\ {p=100-0.5 x} & {p=25+0.1 x}\end{array}$$

Step-by-Step Solution

Answer

The consumer surplus is 3906.25 and the producer surplus is 781.25.

Step 1: Graph the Demand and Supply Equations

Plot the given demand and supply equations on a graph with price \(p\) on the y-axis and quantity \(x\) on the x-axis. The demand curve will slope downwards and the supply curve will slope upwards. The intersection point of these curves represents the market equilibrium.

Step 2: Find the Market Equilibrium

Set the demand and supply equations equal to each other and solve for \(x\) to find the equilibrium quantity. We get \(100 - 0.5x = 25 + 0.1x\). Solving this equation gives \(x = 125\). Substitute \(x = 125\) into either the demand or supply equation to find the equilibrium price. Let's use the demand equation: \(p = 100 - 0.5*125 = 37.5\). Hence, the market equilibrium is at price \(p = 37.5\) and quantity \(x = 125\).

Step 3: Calculate the Consumer Surplus

The consumer surplus is the area of the triangle formed by the y-intercept of the demand curve (100), the x-axis, and the demand curve at the equilibrium quantity. Using the formula for the area of a triangle \(\frac{1}{2} * base * height\), the consumer surplus is \(\frac{1}{2} * (100 - 37.5) * 125 = 3906.25\).

Step 4: Calculate the Producer Surplus

The producer surplus is the area of the triangle formed by the y-intercept of the supply curve (25), the x-axis, and the supply curve at the equilibrium quantity. Using the formula for the area of a triangle \(\frac{1}{2} * base * height\), the producer surplus is \(\frac{1}{2} * (37.5 - 25) * 125 = 781.25\).

Key Concepts

Supply and Demand

Supply and demand are fundamental concepts in economics, essential to understanding how markets function. The demand curve represents the relationship between the price of a good and the quantity that consumers are willing to purchase. As the price decreases, consumers are usually willing to buy more, which is why the demand curve slopes downwards.

Conversely, the supply curve illustrates the relationship between the price of a good and the quantity that producers are willing to supply. As the price increases, producers are typically willing to supply more, resulting in an upward-sloping supply curve.

When depicting these on a graph, the intersection of the supply and demand curves indicates the market equilibrium, a critical point where the market clears and both consumers and producers are satisfied at a certain price and quantity.

Market Equilibrium

Market equilibrium occurs at the point where the quantity demanded by consumers is equal to the quantity supplied by producers. This balance is achieved without any surplus or shortage in the market.

To find the market equilibrium mathematically, you set the demand equation equal to the supply equation. In the problem provided, the demand equation is given by \[p = 100 - 0.5x\]and the supply equation by \[p = 25 + 0.1x\].Setting these equations equal leads to solving for the quantity (\[x\]) and price (\[p\]) at equilibrium.

For example, solving \[100 - 0.5x = 25 + 0.1x\]gives us \[x = 125\],which is the equilibrium quantity. Plugging this back into either equation provides the equilibrium price of \[p = 37.5\].At this point, the market is perfectly balanced without excess supply or demand.

Graphing Systems

Graphing systems of equations involves plotting both the demand and supply equations on a graph. This visual representation helps in understanding where the market equilibrium lies and the respective surpluses accrued.

When graphing, it's important to label your axes correctly, with price (\[p\]) on the vertical axis and quantity (\[x\]) on the horizontal. The downward-sloping curve corresponds to the demand equation (\[p = 100 - 0.5x\]) and the upward-sloping curve to the supply equation (\[p = 25 + 0.1x\]).

The point where these two curves meet represents the market equilibrium point. Above this point on the demand curve lies the consumer surplus, and below the equilibrium on the supply curve, we find the producer surplus. The areas of these triangles give a measure of the surpluses benefiting consumers and producers in the market.