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Chapter 5: Problem 14

For each demand function \(d(x)\) and supply function \(s(x)\) : a. Find the market demand (the positive value of \(x\) at which the demand function intersects the supply function). b. Find the consumers' surplus at the market demand found in part (a). c. Find the producers' surplus at the market demand found in part (a). $$ d(x)=120-0.16 x, \quad s(x)=0.08 x $$

Short Answer

Expert verified
Market demand is at 500 units, with consumers' surplus of 20000 and producers' surplus of 10000.

Step by step solution

01

Set up and solve the equilibrium

To find the market demand where the demand and supply intersect, set the demand function \( d(x) \) equal to the supply function \( s(x) \): \( 120 - 0.16x = 0.08x \). Solve for \( x \) by first combining like terms: \( 120 = 0.24x \). Divide both sides by 0.24 to isolate \( x \): \( x = \frac{120}{0.24} \). Calculate \( x \) to find the equilibrium quantity.
02

Calculate the equilibrium quantity

Divide 120 by 0.24 to find \( x \): \( x = 500 \). So, the market equilibrium, or the market demand, occurs at \( x = 500 \).
03

Calculate the equilibrium price

Substitute the equilibrium quantity \( x = 500 \) back into either the demand function or the supply function to find the equilibrium price. Using the supply function: \( s(500) = 0.08 \times 500 = 40 \). The equilibrium price is \( P = 40 \).
04

Calculate Consumers' Surplus

Consumers' surplus is the area above the price line and below the demand curve at the equilibrium quantity. The formula for consumers' surplus is \( \text{CS} = \frac{1}{2} \times ( ext{Base} \times ext{Height} ) \). The base is \( 500 \) and the height is the difference between the intercept of the demand curve (120) and the price (40): \( \text{CS} = \frac{1}{2} \times (500 \times (120-40)) = \frac{1}{2} \times 500 \times 80 = 20000 \).
05

Calculate Producers' Surplus

Producers' surplus is the area below the price line and above the supply curve at the equilibrium quantity. The formula for producers' surplus is \( \text{PS} = \frac{1}{2} \times ( ext{Base} \times ext{Height} ) \). The base is \( 500 \) and the height is the equilibrium price (40) because the supply curve starts from zero: \( \text{PS} = \frac{1}{2} \times (500 \times 40) = \frac{1}{2} \times 500 \times 40 = 10000 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Demand Function
The demand function is essential in understanding consumer behavior in a market. It's a mathematical relationship between the quantity of a product demanded and its price. For our exercise, the demand function is represented as \( d(x) = 120 - 0.16x \). Here, \( x \) is the quantity demanded, and the function shows that as the quantity \( x \) increases, the price \( d(x) \) decreases.

Key points to remember about the demand function:
  • The intercept (120 in this case) is the maximum price consumers are willing to pay.
  • The negative slope indicates the inverse relationship between quantity and price.
In practice, when you set demand function equal to another function, like supply, you're solving for the market equilibrium, which is crucial for determining consumers' surplus and producers' surplus.
Supply Function
The supply function represents the relationship between the quantity of a product that producers are willing to sell and the price. In our example, the supply function is given by \( s(x) = 0.08x \). Unlike the demand function, the supply function typically has a positive slope, indicating that as the price increases, producers are willing to supply more of the product.

Characteristics of the supply function include:
  • A direct relationship between quantity and price, reflected by a positive slope.
  • The origin of the supply curve, in this case from the point \( 0 \).
Finding where the supply and demand functions intersect lets us determine the market equilibrium - the point where the quantity supplied meets the quantity demanded at a certain price.
Consumers' Surplus
Consumers' surplus is a crucial concept in economics that measures the benefit consumers receive from purchasing goods at a price lower than what they are willing to pay. In our example, consumers' surplus can be visualized as the area between the demand curve and the price level up to the market equilibrium quantity.

To calculate the consumers' surplus:
  • Identify the intercept of the demand curve (120) and the equilibrium price (40).
  • Calculate the area of the triangle formed above the price level and below the demand curve using the formula: \( \frac{1}{2} \times \text{Base} \times \text{Height} \).
  • Substitute the equilibrium quantity (base = 500) and the difference between demand curve intercept and equilibrium price (height = 80) into the formula.
  • Consumers' Surplus = \( \frac{1}{2} \times 500 \times 80 = 20000 \).
This measure demonstrates the additional satisfaction consumers gain from buying products at a market price lower than their maximum willingness to pay.
Producers' Surplus
Producers' surplus represents the extra revenue that producers earn from selling their goods at a market price higher than the minimum they would be willing to accept. In the exercise, this is shown as the area below the equilibrium price level and above the supply curve up to the equilibrium quantity.

Steps to calculate producers' surplus include:
  • Recognize that the supply curve starts from zero, and the equilibrium price is 40.
  • Use the same triangle area formula: \( \frac{1}{2} \times \text{Base} \times \text{Height} \).
  • Plug in the equilibrium quantity as the base (500) and the equilibrium price as the height (40).
  • Producers' Surplus = \( \frac{1}{2} \times 500 \times 40 = 10000 \).
This calculation signifies the economic benefit producers receive when the market price exceeds their minimum sell price, making production more profitable.

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