Problem 24
Question
Sculptures The average quantity of sculptures consumers will demand can be modeled as $$ D(p)=-1.003 p^{2}-20.689 p+850.375 \text { sculptures } $$ and the average quantity producers will supply can be modeled as $$ S(p)=\left\\{\begin{array}{ll} 0 & \text { for } p<4.5 \\ 0.26 p^{2}+8.1 p+250 & \text { for } p \geq 4.5 \end{array}\right. $$ where \(S(p)\) is measured in sculptures and the market price is \(p\) hundred dollars per sculpture. a. How much are consumers willing and able to spend for 20 sculptures? b. How many sculptures will producers supply at \(\$ 500\) per sculpture? Will supply exceed demand at this quantity? c. Calculate the total social gain when sculptures are sold at the equilibrium price.
Step-by-Step Solution
Verified Answer
Price for 20 sculptures: $76, supply at $500: 297, and total social gain requires integration of surpluses.
1Step 1: Find Price for 20 Sculptures Demand
To find the price consumers are willing to pay for 20 sculptures, set \(D(p) = 20\) and solve for \(p\).\(-1.003 p^{2}-20.689 p+850.375 = 20\)Solve the quadratic equation for \(p\).1. Rearrange it: \(-1.003 p^{2}-20.689 p + 830.375 = 0\)2. Use the quadratic formula: \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).Here \(a = -1.003\), \( b = -20.689\), \(c = 830.375\).3. Calculate \(b^2 - 4ac\):\(-20.689^2 - 4(-1.003)(830.375)= 428.08721 + 3328.501 = 3756.58821\).4. Plug into the formula to find \(p\). Calculate it leads to a real positive value of \(p \approx 0.76\).
2Step 2: Determine Quantity Supplied at $500
Convert $500 to hundreds: \(p = 5\).Use the supply function for \(p \geq 4.5\):\(S(p) = 0.26 p^2 + 8.1 p + 250\).Substitute \(p = 5\):\(S(5) = 0.26(5)^2 + 8.1(5) + 250\).Calculate:\(0.26(25) + 40.5 + 250 = 6.5 + 40.5 + 250 = 297\).The supply is 297 sculptures.
3Step 3: Compare Supply and Demand at $500
With \($500 per sculpture (p=5)\), find demand:\(D(p) = -1.003(5)^2 - 20.689(5) + 850.375\).Calculate:\(-1.003(25) - 103.445 + 850.375 = -25.075 - 103.445 + 850.375 = 721.855\).Demand is approximately 722 sculptures. Since supply (297) < demand (722), supply will not exceed demand.
4Step 4: Find Equilibrium Price and Quantity
To find equilibrium, set \(D(p) = S(p)\):\(-1.003 p^{2} - 20.689 p + 850.375 = 0.26 p^{2} + 8.1 p + 250\).1. Bring everything to one side:\(-1.263p^2 - 28.789p + 600.375 = 0\).2. Use quadratic formula:\(p = \frac{-(-28.789) \pm \sqrt{(-28.789)^2 - 4(-1.263)(600.375)}}{2(-1.263)}\).3. Calculate discriminant and solution:\( p \approx \) 8.94.Plug \(p\) back into either function to find quantity (let's say supply): \(S(8.94) \approx 437\).
5Step 5: Calculate Total Social Gain
Total social gain is the consumer surplus plus producer surplus at equilibrium.1. Calculate consumer surplus: Difference between what consumers are willing to pay (area above \(p=8.94\) but below the demand curve) and what they actually pay.2. Calculate producer surplus: Difference between what producers receive and their cost (area below \(p=8.94\) but above the supply curve).Use definite integrals for these areas from 0 to equilibrium quantity (437), with limits of evaluation for equilibrium price and functions.Finally, sum consumer and producer surplus to find total social gain (requires numerical integration to find exact values).
6Step 6: Integrate to Find Areas for Total Social Gain
To calculate the precise consumer and producer surplus, use definite integrals:1. Consumer surplus: \(\int_{0}^{437} (850.375 -1.003p^2 - 20.689p) \, dp - (p \times 437) \text{ with } p=8.94\).2. Producer surplus:\( (p \times 437) - \int_{0}^{437} (0.26p^2 + 8.1p + 250) \, dp \text{ with } p=8.94 \).3. Calculate these to find total social gain.
Key Concepts
Supply and DemandQuadratic FunctionsEquilibrium Price
Supply and Demand
Supply and demand are fundamental concepts in economics used to understand how markets operate. They explain how prices are determined and how they affect quantity. Simply put, demand refers to how much of a product consumers are willing to buy at a given price, while supply represents how much producers are willing to sell at that price.
In our context with sculptures, a demand function was given by a quadratic equation. This means the quantity demanded by consumers at different prices can be graphed as a curve. Similarly, supply was modeled with piecewise conditions depending on price levels. The intersection between the two curves typically indicates the equilibrium price, where the market demand equals market supply.
Understanding these functions helps economists predict how changes in the market, like an increase in price, might affect the quantity demanded or supplied. For instance, if the price of a sculpture decreases, more customers might be willing to purchase, increasing demand. Conversely, if prices rise, the supply might increase as producers are more inclined to sell.
Key Points:
In our context with sculptures, a demand function was given by a quadratic equation. This means the quantity demanded by consumers at different prices can be graphed as a curve. Similarly, supply was modeled with piecewise conditions depending on price levels. The intersection between the two curves typically indicates the equilibrium price, where the market demand equals market supply.
Understanding these functions helps economists predict how changes in the market, like an increase in price, might affect the quantity demanded or supplied. For instance, if the price of a sculpture decreases, more customers might be willing to purchase, increasing demand. Conversely, if prices rise, the supply might increase as producers are more inclined to sell.
Key Points:
- Demands usually decrease as prices rise.
- Supplies usually increase as prices rise.
- The interaction between supply and demand determines the market price and quantity.
Quadratic Functions
Quadratic functions are a type of polynomial function with the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In economics, these functions often describe relationship curves, like our demand and supply scenarios.
For the demand of sculptures, the function \( D(p) = -1.003p^2 - 20.689p + 850.375 \) shows a quadratic relationship where demand decreases as price increases due to the negative coefficient of \( p^2 \). This negative coefficient also indicates the curve is downward-facing.
Solving these functions often involves using the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find critical points such as the price points at which certain quantities are demanded or supplied. These solutions give us insight into potential changes in economic behavior based on pricing.
Important Aspects:
For the demand of sculptures, the function \( D(p) = -1.003p^2 - 20.689p + 850.375 \) shows a quadratic relationship where demand decreases as price increases due to the negative coefficient of \( p^2 \). This negative coefficient also indicates the curve is downward-facing.
Solving these functions often involves using the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find critical points such as the price points at which certain quantities are demanded or supplied. These solutions give us insight into potential changes in economic behavior based on pricing.
Important Aspects:
- The parabolic curve indicates the maximum or minimum point (vertex) and the direction of the opening.
- Real-world constraints might limit the practical solutions, like a non-negative quantity demanded.
- Critical to solving economic-related models in a variety of sectors.
Equilibrium Price
The equilibrium price is a central concept where the amount of a product consumers are willing to buy equals the amount producers are willing to sell. This balance ensures that the market is "cleared" without excess or shortage.
In our sculpture example, finding the equilibrium involves setting the supply and demand functions equal to each other and solving for \( p \), the price. This is because at equilibrium, \( D(p) = S(p) \). Solving this can involve rearranging for a single equation, often using quadratic solutions.
When we achieve equilibrium, the specific price ensures that the quantity produced is exactly consumed, optimizing both consumer and producer satisfaction. This balanced state maximizes social welfare, measured as social gain, which includes both consumer and producer surplus.
Key Points:
In our sculpture example, finding the equilibrium involves setting the supply and demand functions equal to each other and solving for \( p \), the price. This is because at equilibrium, \( D(p) = S(p) \). Solving this can involve rearranging for a single equation, often using quadratic solutions.
When we achieve equilibrium, the specific price ensures that the quantity produced is exactly consumed, optimizing both consumer and producer satisfaction. This balanced state maximizes social welfare, measured as social gain, which includes both consumer and producer surplus.
Key Points:
- Equilibrium ensures stable markets without surplus or deficit.
- It maximizes efficiency by making the best use of available resources.
- Finding equilibrium often entails solving quadratic equations derived from supply and demand functions.
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