Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Find the consumer surplus and producer surplus for the pair of demand and supply equations. Supply \(p=125+0.0006 x\) Demand $$p=600-0.0002 x$$

Step-by-Step Solution

Answer

The short answer will depend on the numerical values obtained for the equilibrium price and quantity, consumer surplus, and producer surplus. As such, an exact short answer cannot be provided without solving the equations.

Step 1: Find the equilibrium

To find the equilibrium price and quantity, set the supply equation equal to the demand equation: \(125 + 0.0006x = 600 - 0.0002x\). Solve this equation to find the equilibrium quantity, \(x\). Then, substitute \(x\) back into either the supply or demand equation to find the equilibrium price, \(p\).

Step 2: Calculate the consumer surplus

The consumer surplus is the area between the demand curve and the price level up to the equilibrium quantity. It can be found by calculating the area of the triangle formed by the price axis, the quantity axis, and the demand curve. The formula for this is \(0.5 * (p_{max} - p_{equilibrium}) * x_{equilibrium}\), where \(p_{max}\) is the maximum price that consumers are willing to pay (the y-intercept of the demand curve), \(p_{equilibrium}\) is the equilibrium price, and \(x_{equilibrium}\) is the equilibrium quantity.

Step 3: Calculate the producer surplus

The producer surplus is the area between the supply curve and the price level up to the equilibrium quantity. It can be found by calculating the area of the triangle formed by the price axis, the quantity axis, and the supply curve. The formula for this is \(0.5 * (p_{equilibrium} - p_{min}) * x_{equilibrium}\), where \(p_{min}\) is the minimum price that producers are willing to accept (the y-intercept of the supply curve), \(p_{equilibrium}\) is the equilibrium price, and \(x_{equilibrium}\) is the equilibrium quantity.

Key Concepts

Equilibrium Price and Quantity

When we talk about markets, the equilibrium price is where the amount of goods consumers are willing to buy equals the amount that producers are willing to sell. This point is crucial because it determines the price and quantity of goods in a market without any external interference. Finding the equilibrium involves setting two important functions against each other: the demand equation and the supply equation.

The exercise given illustrates such a scenario, where the intersection of these equations reveals the balance point of the market. To find this critical point where supply equals demand, you'd solve an equation set like the one provided in the solution steps. The equilibrium quantity (\(x_{equilibrium}\) and price (\(p_{equilibrium}\) are determined, and this is where no consumer or producer has an incentive to change their behavior - buyers will buy exactly as much as sellers are willing to sell, at a price agreeable to both.

Demand and Supply Equations

Demand and supply equations are mathematical models that describe consumers' buying behavior and producers' selling behavior, respectively. In the provided exercise, the demand equation is \(p = 600 - 0.0002x\), which shows the price \(p\) that consumers are willing to pay for different quantities \(x\) of a product. As the quantity increases, the price consumers are willing to pay decreases, which illustrates the law of demand: an increase in price leads to a decrease in quantity demanded.

On the flip side, the supply equation \(p = 125 + 0.0006x\) describes the price producers will set for their product based on the quantity. As the quantity increases, so does the price. This reflects the law of supply, which states that an increase in price results in an increase in the quantity supplied. By balancing these two equations, markets strive to reach an equilibrium.

Surplus Calculation

Now, onto a favorite topic among economists, the 'surplus' calculation. This aspect involves understanding consumer surplus and producer surplus. Consumer surplus is the difference between what consumers are willing to pay for a good or service and what they actually pay at the market price. It represents the benefit consumers receive when they purchase goods at a price lower than their maximum willingness to pay. Using the formula mentioned in the step-by-step solution, \(0.5 * (p_{max} - p_{equilibrium}) * x_{equilibrium}\), we can actually quantify this benefit.

Producer surplus, conversely, is the difference between the market price and the lowest price at which producers would still be willing to sell their goods. It shows the benefit to producers, as they are able to sell their goods at a price higher than the minimum they'd accept. The calculation, \(0.5 * (p_{equilibrium} - p_{min}) * x_{equilibrium}\), similarly gives us the means to measure this advantage. In the context of the exercise, both consumer and producer surplus calculations capture the essence of the benefits that trade provides to both sides of the market.