Question

Consumer and Producer Surpluses Find the consumer surplus and the producer surplus for a product with the given demand and supply functions. Demand: \(p=\frac{60}{\sqrt{x^{2}+81}}\), Supply: \(p=\frac{x}{3}\)

Short answer

The exact values for the consumer and producer surpluses will depend on the numerical solution to the original equation, which requires some technical computations. However, the general approach is to find the equilibrium point, then integrate the functions as outlined in Steps 3 and 4.

Step-by-step solution

Determine the Equilibrium Price and Quantity

Find the quantity \(x\) where the demand equals the supply: \(\frac{60}{\sqrt{x^{2}+81}} = \frac{x}{3}\). This equation simplifies to \(180 = x\sqrt{x^{2}+81}\). Solve for \(x\) (hint: try squaring both sides and then isolating \(x\)).

Evaluate the Equilibrium Price

Once the equilibrium quantity is found, evaluate the matching equilibrium price by substituting the equilibrium quantity into either the demand or supply function.

Calculate Consumer Surplus

Consumer surplus is given by the integral from 0 to the equilibrium quantity of the demand function minus the equilibrium price: \(\int_0^{x} \left(\frac{60}{\sqrt{t^{2}+81}} - p\right)dt\).

Calculate Producer Surplus

Similarly, the producer surplus is the integral from 0 to the equilibrium quantity of the equilibrium price minus the supply function: \(\int_0^{x} \left( p - \frac{t}{3}\right)dt\).

Key concepts

Equilibrium Price

The equilibrium price is a foundational concept in economics, signifying the point where the quantity of goods producers are willing to supply equals the quantity consumers are willing to purchase. Understanding equilibrium allows us to analyze market conditions and anticipate how changes in supply and demand can affect prices.

When we look at the given demand function, which is \(p=\frac{60}{\sqrt{x^{2}+81}}\), and the supply function, \(p=\frac{x}{3}\), finding the equilibrium price involves setting these two equations equal to each other. This represents an intersection point on graphical representation – the supply and demand curves on a graph. Students often struggle with solving equations involving square roots and variables, as seen in this exercise's step 1.

To enhance understanding, it could be beneficial to create a visual representation of the process, using a graph to depict the supply and demand curves. This way, visually determining the point of intersection simplifies comprehension of the abstract concept of equilibrium. It's also crucial to mention that in practice, the equilibrium price is not always static; it can shift due to external factors affecting supply and demand, such as changes in technology or consumer preferences.

Demand and Supply Functions

Demand and supply functions are equations that economists use to represent the relationship between price and quantity for a product or service in the market. The demand function often shows how price decreases with increased quantity demanded, reflecting that consumers are generally willing to purchase more at lower prices. Conversely, the supply function typically shows an increase in price as quantity supplied increases, indicating producers' willingness to sell more at higher prices.

In the exercise, a specific form of the demand function is presented which can be intimidating due to the square root and the added constant within it. However, breaking it down helps clarify that it depicts how demand decreases as quantity (\(x\)) increases. The supply function is linear, suggesting a direct proportionality between price and quantity supplied.

For students to fully grasp these concepts, visual aids like graphs can be instrumental. Graphing both functions can reveal the nature of the supply and demand more intuitively. For example, a graph can show that, as the quantity increases, the demand line slopes downwards (less price, more demand), and the supply line slopes upwards (more price, more supply). These visualizations help to cement the understanding of how the functions interact within a market.

Economic Surplus Integration

Economic surplus integration is a critical concept in understanding the total benefit consumers and producers receive in a market. It comprises two parts: the consumer surplus and the producer surplus. The consumer surplus is the difference between the total amount consumers are willing to pay and the total amount they actually pay. In contrast, the producer surplus is the difference between what producers are paid for a good versus the minimum amount they would be willing to accept.

The calculation of these surpluses involves integrating the demand and supply functions, as seen in steps 3 and 4 of the provided exercise. While the idea of integrating to find areas under the curve can be difficult for some students, one improvement would be to use graphical methods or numerical integration when an analytical solution is not easily obtainable. It's helpful to present the surplus areas visually, typically shown as the area between the equilibrium price line and the demand or supply curve.

Students should be encouraged to use integration to find these areas precisely, but also to understand that these areas represent money left on the table – the consumer surplus is potential savings for consumers, while the producer surplus is potential extra revenue for producers. This tangible interpretation can make the abstract concept of surplus much more concrete and comprehensible.