Problem 16 - 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). $$ d(x)=360-0.03 x^{2}, \quad s(x)=0.006 x^{2} $$

Step-by-Step Solution

Answer

Market demand at \(x = 100\). Consumers' surplus is 29000; producers' surplus is 4000.

Step 1: Find the equilibrium point

To find the equilibrium point, we need to set the demand function equal to the supply function and solve for \(x\). This gives us:\[360 - 0.03x^2 = 0.006x^2\]Rearranging terms, we have:\[360 = 0.036x^2\]Solving for \(x\), we find:\[x^2 = \frac{360}{0.036}\]\[x^2 = 10000\]\[x = \sqrt{10000} = 100\]Therefore, the market demand is \(x = 100\).

Step 2: Calculate the consumers' surplus

The consumers' surplus is the area between the maximum price consumers are willing to pay (given by the demand curve) and the market price at the equilibrium point. The market price \(p\) at equilibrium is found by substituting \(x = 100\) into either function. Let's use the supply function:\[p = 0.006(100)^2 = 0.006 \times 10000 = 60\]The consumers' surplus is calculated as:\[\text{Consumers' Surplus} = \int_0^{100} (360 - 0.03x^2) \, dx - 100\times60\]First, compute the integral of the demand function from 0 to 100:\[\int_0^{100} (360 - 0.03x^2) \, dx = \left[360x - 0.01x^3\right]_0^{100} = 36000 - 1000 = 35000\]Thus, consumers' surplus is:\[35000 - 6000 = 29000\]

Step 3: Calculate the producers' surplus

Producers' surplus is the area between the market price at equilibrium and the supply curve. It is calculated as:\[\text{Producers' Surplus} = 100 \times 60 - \int_0^{100} (0.006x^2) \, dx\]To find this, compute the integral of the supply function from 0 to 100:\[\int_0^{100} (0.006x^2) \, dx = \left[0.002x^3\right]_0^{100} = 0.002 \times 1000000 = 2000\]Therefore, producers' surplus is:\[6000 - 2000 = 4000\]

Key Concepts

Demand Function

The demand function represents the quantity of a product that consumers are willing and able to purchase at various prices within a certain period. In the given exercise, the demand function is expressed as \( d(x) = 360 - 0.03x^2 \). This equation indicates that the quantity demanded decreases as the price increases, which is a common characteristic of demand functions. Understanding the demand function is crucial because it helps us gauge how consumers react to changes in prices. The constant (360 in this case) reflects the initial maximum quantity demanded when the price is zero, whereas the coefficient of the \(x^2\) term (\(0.03\)) shows how steeply demand drops as quantity increases. By setting the demand function equal to the supply function, we can determine the market demand or equilibrium point, which is where the amount consumers want to buy equals the amount producers want to sell.

Supply Function

The supply function indicates the quantity of a product that producers are willing to supply at different price levels. For the exercise, the supply function is given by \( s(x) = 0.006x^2 \). This function typically shows a direct relationship between price and quantity supplied, meaning that as prices rise, so does the quantity supplied. The function starts from zero quantity at zero cost, and the \(0.006\) coefficient shows the rate at which producers are able to increase supply with increasing prices. Unlike the demand function, the supply function is often concave upwards, capturing the increasing willingness of suppliers to provide more at higher prices. The intersection of the supply function with the demand function indicates the equilibrium price and quantity in the market, where suppliers are producing just as much as consumers are willing to buy at that price.

Consumers' Surplus

Consumers' surplus is a measure of the economic benefit consumers receive when they buy a product for less than the maximum amount they are willing to pay. In practical terms, it is the difference between what consumers are prepared to pay and what they actually pay for the total quantity purchased at the equilibrium price.For the demand function \(d(x) = 360 - 0.03x^2\) and the equilibrium point \(x = 100\), where the price \(p = 60\), the consumers' surplus is calculated by integrating the demand function from zero to the equilibrium quantity and then subtracting the rectangle area formed by the equilibrium price and quantity.- Consumers' surplus is given as: \[ \int_0^{100} (360 - 0.03x^2) \, dx - 100 \times 60 = 29000 \]This outcome signifies the total extra value that consumers gain from purchasing the product at market price rather than their maximum willingness.

Producers' Surplus

Producers' surplus represents the difference in revenue received by producers when they sell a product at a market price which is higher than the minimum price they are willing to accept to produce that quantity. It's a form of economic benefit to the producer.To figure out the producers' surplus for the supply function \(s(x) = 0.006x^2\) at the equilibrium quantity \(x = 100\) with a price \(p = 60\), we calculate the area above the supply curve and below the equilibrium price line.The producers' surplus is determined as follows:- \[ 100 \times 60 - \int_0^{100} (0.006x^2) \, dx = 4000 \]This value represents the total additional earnings producers receive by selling at the higher market price instead of the minimum they were willing to accept, reflecting their gains in participating in the market.