Problem 15
Question
Beth enjoys skydiving and is willing to pay \(p\) dollars per jump for \(x\) jumps, where \(p=D(x)=7.5 x^{2}-60.5 x+254\) a) Find Beth's consumer surplus if she makes 2 jumps. b) Suppose the supply function for Aero Skydiving Center is given by \(p=S(x)=15 x+95 .\) Find the producer surplus if the center sells Beth 2 jumps. c) Find the equilibrium point and the consumer and producer surpluses at this point. Assume that Beth makes no more than 5 jumps. d) Explain what the equilibrium point represents to both Beth and Aero Skydiving Center.
Step-by-Step Solution
Verified Answer
Beth's consumer surplus for 2 jumps: $8\; dollars; Producer surplus: $5\; dollars. Equilibrium point: $\approx2.93\, jumps, \$138.95$. At equilibrium, consumer surplus: $\approx21.44$, producer surplus: $\approx40.44\; dollars$. It balances Beth's desires with Aero's supply.
1Step 1: Finding Consumer Surplus for 2 Jumps
The consumer surplus is the area between the demand curve and the actual price paid for 2 jumps. The price Beth is willing to pay for 2 jumps is \(D(2) = 7.5 \times 2^2 - 60.5 \times 2 + 254\). Calculate this value to find the maximum price willing to pay. Then, calculate the actual price using \(S(2) = 15 \times 2 + 95\). Consumer surplus is given by the formula: \(CS = \frac{1}{2} \times (D(2) - S(2)) \times x\) where \(x = 2\).
2Step 2: Calculating Producer Surplus for 2 Jumps
Find the price received by the Aero Skydiving Center using the supply function \(S(x)\). At 2 jumps, calculate \(S(2) = 15 \times 2 + 95\) to find the minimum selling price. The producer surplus is the area between the actual price received and the supply curve. Use \(PS = \frac{1}{2} \times (S(2) - C(x)) \times x\), where \(C(x)\) is usually the cost or zero if not specified.
3Step 3: Finding the Equilibrium Point
To find the equilibrium point, solve for \(x\) where the demand equals the supply, \(D(x) = S(x)\). Set \(7.5x^2 - 60.5x + 254 = 15x + 95\), solve the quadratic equation to find the equilibrium \(x\). Then substitute \(x\) back into either \(D(x)\) or \(S(x)\) to find the equilibrium price.
4Step 4: Calculating Consumer and Producer Surplus at Equilibrium
Substitute the equilibrium quantity back into the demand and supply functions to find \(D(x)\) and \(S(x)\) at equilibrium. Use the consumer surplus formula: \(CS = \frac{1}{2} \times (D(e) - P_{eq}) \times e\), and the producer surplus formula: \(PS = \frac{1}{2} \times (P_{eq} - S(e)) \times e\) where \(e\) is the equilibrium quantity.
5Step 5: Understanding the Equilibrium Point
The equilibrium point represents the quantity and price at which the quantity of jumps Beth wants to buy equals the amount Aero Skydiving Center is willing to sell. At this point, there is no surplus or shortage in the market, maximizing satisfaction for both Beth and the Aero Skydiving Center.
Key Concepts
Equilibrium PointDemand and Supply FunctionsQuadratic EquationsEconomic Surplus Analysis
Equilibrium Point
In economics, the equilibrium point is crucial as it represents a state where the demand for a product equals the supply. This means that the quantity consumers are willing and able to purchase perfectly matches the quantity that producers are willing and able to sell.
Finding the equilibrium requires solving a system of equations where the demand and supply functions are equal. In the case of Beth's skydiving adventure, we reach equilibrium by setting the demand function equal to the supply function: \( 7.5x^2 - 60.5x + 254 = 15x + 95 \). By solving this quadratic equation, we determine the equilibrium quantity of jumps, which represents the point of maximum efficiency in the market.
At equilibrium, the market "clears," meaning there are no excess supply or demand. This balance ensures optimal resource allocation, benefiting both consumers and producers. For Beth and the Aero Skydiving Center, reaching equilibrium means Beth enjoys her jumps at a reasonable price, while the center maximizes its profits.
Finding the equilibrium requires solving a system of equations where the demand and supply functions are equal. In the case of Beth's skydiving adventure, we reach equilibrium by setting the demand function equal to the supply function: \( 7.5x^2 - 60.5x + 254 = 15x + 95 \). By solving this quadratic equation, we determine the equilibrium quantity of jumps, which represents the point of maximum efficiency in the market.
At equilibrium, the market "clears," meaning there are no excess supply or demand. This balance ensures optimal resource allocation, benefiting both consumers and producers. For Beth and the Aero Skydiving Center, reaching equilibrium means Beth enjoys her jumps at a reasonable price, while the center maximizes its profits.
Demand and Supply Functions
Demand and supply functions are fundamental tools in analyzing market behavior. They represent the relationship between price and the quantity demanders are willing to purchase and suppliers are willing to provide.
The demand function for Beth is given by a quadratic equation: \( D(x) = 7.5x^2 - 60.5x + 254 \), which shows how much she is willing to pay for each subsequent jump. It's important because it helps predict how her willingness to pay changes as the number of jumps increases.
The supply function for the Aero Skydiving Center is linear: \( S(x) = 15x + 95 \), indicating the minimum price the center is willing to accept per jump based on the number sold. By analyzing these functions, both Beth and the center can forecast and strategize their decisions in the marketplace.
The demand function for Beth is given by a quadratic equation: \( D(x) = 7.5x^2 - 60.5x + 254 \), which shows how much she is willing to pay for each subsequent jump. It's important because it helps predict how her willingness to pay changes as the number of jumps increases.
The supply function for the Aero Skydiving Center is linear: \( S(x) = 15x + 95 \), indicating the minimum price the center is willing to accept per jump based on the number sold. By analyzing these functions, both Beth and the center can forecast and strategize their decisions in the marketplace.
Quadratic Equations
Quadratic equations form an essential part of economic analysis, especially in this scenario involving the demand function. A quadratic equation is typically in the form: \( ax^2 + bx + c = 0 \). These equations have specific characteristics, such as parabolic curves, which can open upwards or downwards, affecting the nature of the solution.
In Beth's case, solving the quadratic equation \( 7.5x^2 - 60.5x + 254 = 15x + 95 \) involves finding the value of \( x \) (the number of jumps) at which the demand equals the supply. This step often requires rearranging terms to achieve the standard quadratic form for solving purposes. Understanding quadratic equations is essential in predicting how changing parameters (like price and quantity) affect the equilibrium and market dynamics.
In Beth's case, solving the quadratic equation \( 7.5x^2 - 60.5x + 254 = 15x + 95 \) involves finding the value of \( x \) (the number of jumps) at which the demand equals the supply. This step often requires rearranging terms to achieve the standard quadratic form for solving purposes. Understanding quadratic equations is essential in predicting how changing parameters (like price and quantity) affect the equilibrium and market dynamics.
Economic Surplus Analysis
Economic surplus is a measure of the extra benefits obtained by participants in a market. It's divided into two main types: consumer surplus and producer surplus.
Consumer surplus arises when consumers like Beth pay less for a good than what they are willing to pay. Graphically, it's the area between the demand curve and the market price up to a certain quantity. For Beth, this means she gains extra satisfaction from paying less than her maximum willingness for her skydiving jumps.
Producer surplus refers to the additional earnings the producer receives by selling at a price higher than the minimum they are willing to accept. For the Aero Skydiving Center, this surplus is the area between the supply curve and the market price.
Analyzing economic surplus helps visualize the overall gains from trade in the market, reflecting economic efficiency. Surplus calculations are crucial as they demonstrate how resources are being allocated and the resultant market welfare.
Consumer surplus arises when consumers like Beth pay less for a good than what they are willing to pay. Graphically, it's the area between the demand curve and the market price up to a certain quantity. For Beth, this means she gains extra satisfaction from paying less than her maximum willingness for her skydiving jumps.
Producer surplus refers to the additional earnings the producer receives by selling at a price higher than the minimum they are willing to accept. For the Aero Skydiving Center, this surplus is the area between the supply curve and the market price.
Analyzing economic surplus helps visualize the overall gains from trade in the market, reflecting economic efficiency. Surplus calculations are crucial as they demonstrate how resources are being allocated and the resultant market welfare.
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