Problem 14 - Applied Calculus

Question

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). $$ d(x)=120-0.16 x, \quad s(x)=0.08 x $$

Step-by-Step Solution

Answer

Equilibrium at \(x=500\); consumers' surplus: 20000; producers' surplus: 10000.

Step 1: Set Demand Equal to Supply

First, to find the market equilibrium, we need to set the demand function equal to the supply function. This involves solving the equation \(120 - 0.16x = 0.08x\).

Step 2: Solve for Equilibrium Quantity

Rearrange the equation from Step 1 to find \(x\) by adding \(0.16x\) to both sides: \[ 120 = 0.24x \]Now, solve for \(x\) by dividing both sides by 0.24: \[ x = \frac{120}{0.24} = 500 \] Thus, the market equilibrium quantity is \(x = 500\).

Step 3: Compute Consumers' Surplus

The consumers' surplus is calculated as the area between the demand curve and the market price, up to the equilibrium quantity. First, find the equilibrium price \(p\) by substituting \(x = 500\) into the supply function: \[ s(500) = 0.08 \times 500 = 40 \]Now, calculate the consumers' surplus: \[ \text{Consumers' Surplus} = \frac{1}{2} \times (120 - 40) \times 500 \]Where 120 is the price willing to pay at \(x = 0\) and 40 is the equilibrium price. \[ = \frac{1}{2} \times 80 \times 500 = 20000 \]

Step 4: Compute Producers' Surplus

The producers' surplus is the area between the supply curve and the market price, up to the equilibrium quantity. Here, it is calculated as: \[ \text{Producers' Surplus} = \frac{1}{2} \times 40 \times 500 \] Where 40 is the equilibrium price. \[ = \frac{1}{2} \times 40 \times 500 = 10000 \]

Key Concepts

Understanding the Demand Function

The demand function is crucial in market analysis as it represents the relationship between the price of a product and the quantity demanded by consumers. For the given equation, the demand function is represented by \( d(x) = 120 - 0.16x \). This can be interpreted as:

• Starting price consumers are willing to pay is 120 when the quantity is zero.
• As the quantity \( x \) increases, the price consumers are willing to pay decreases by 0.16 per additional unit.

This negative slope illustrates the law of demand: as price decreases, quantity demanded increases, and vice versa. Understanding the demand function helps determine the price consumers are willing to pay at various quantities, providing insight into consumer preferences and market behavior.

Exploring the Supply Function

The supply function shows how much of a product suppliers are willing to produce and sell at different price points. In this case, the supply function is \( s(x) = 0.08x \). Here's what this means:

• At zero production, the supply price is zero, suggesting no fixed cost to begin production.
• The positive coefficient of 0.08 implies that as more units are produced, the required price for each unit increases.

Supply functions often reflect the producers' costs and technological constraints. A positively sloped supply function like this signifies that higher prices are needed to incentivize higher production levels. It plays a key role in determining how suppliers respond to changes in market conditions.

Decoding Consumers' Surplus

Consumers' surplus represents the benefit consumers receive when they pay less for a product than they are willing to. At market equilibrium, it is the area above the equilibrium price and below the demand curve. Using the example given, the equilibrium quantity is 500, and the equilibrium price is 40. The consumers' surplus can be calculated using the formula:\[\text{Consumers' Surplus} = \frac{1}{2} \times (\text{Price at } x = 0 - \text{Equilibrium Price}) \times \text{Equilibrium Quantity}\]Plugging in the values, we get:\[\frac{1}{2} \times (120 - 40) \times 500 = 20,000\]Thus, the consumers' surplus of 20,000 represents the total added value or economic benefit consumers receive in this market scenario.

Unpacking Producers' Surplus

Producers' surplus is the extra benefit producers receive when they sell a product for more than their minimum acceptable price. It is the area below the equilibrium price and above the supply curve up to the equilibrium quantity.In the given example, the equilibrium price is 40, and the quantity is 500. The producers' surplus is calculated as:\[\text{Producers' Surplus} = \frac{1}{2} \times \text{Equilibrium Price} \times \text{Equilibrium Quantity}\]Inserting the values:\[\frac{1}{2} \times 40 \times 500 = 10,000\]The producers' surplus of 10,000 represents the gains producers make beyond the minimum they needed to bring the goods to market, reflecting the economic incentive for them in this market equilibrium.