Problem 12 - Calculus and its applications

Question

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=\frac{100}{\sqrt{x}}, \quad S(x)=\sqrt{x}\)

Step-by-Step Solution

Answer

(a) Equilibrium point: (100, 10) (b) Consumer surplus: 1000 (c) Producer surplus: 333.33

Step 1: Equilibrium Point

The equilibrium point occurs where the demand price equals the supply price. This means solving the equation \(D(x) = S(x)\). Given \( D(x) = \frac{100}{\sqrt{x}} \) and \( S(x) = \sqrt{x} \), we equate them:\[ \frac{100}{\sqrt{x}} = \sqrt{x} \]By multiplying both sides by \(\sqrt{x}\) to eliminate the fraction, we get:\[ 100 = x \]The equilibrium quantity is \(x = 100\). Substituting \(x = 100\) back into either \(D(x)\) or \(S(x)\) will yield the equilibrium price, \(p_e\). Choosing \(S(x)\), we have:\[ p_e = \sqrt{100} = 10 \]Thus, the equilibrium point is \((100, 10)\).

Step 2: Consumer Surplus Calculation

Consumer surplus is the area between the demand curve and the price level, up to the equilibrium quantity.The formula for consumer surplus is:\[ CS = \int_0^{Q_e} D(x) \, dx - (p_e \times Q_e) \]Substitute the known values, where \( Q_e = 100 \) and \( p_e = 10 \):\[ CS = \int_0^{100} \frac{100}{\sqrt{x}} \, dx - (10 \times 100) \]The integral \( \int \frac{100}{\sqrt{x}} \, dx \) is solved as follows:\[ \int \frac{100}{\sqrt{x}} \, dx = 200\sqrt{x} + C \]Evaluate from 0 to 100:\[ \left[ 200\sqrt{x} \right]_0^{100} = 200\sqrt{100} - 200\sqrt{0} = 200 \times 10 = 2000 \]Hence, consumer surplus is:\[ CS = 2000 - 1000 = 1000 \]

Step 3: Producer Surplus Calculation

Producer surplus is the area above the supply curve and below the price level, up to the equilibrium quantity.The formula for producer surplus is:\[ PS = (p_e \times Q_e) - \int_0^{Q_e} S(x) \, dx \]Substitute the known values, where \( Q_e = 100 \) and \( p_e = 10 \):\[ PS = (10 \times 100) - \int_0^{100} \sqrt{x} \, dx \]The integral \( \int \sqrt{x} \, dx \) is solved as follows:\[ \int \sqrt{x} \, dx = \frac{2}{3}x^{3/2} + C \]Evaluate from 0 to 100:\[ \left[ \frac{2}{3}x^{3/2} \right]_0^{100} = \frac{2}{3} \times 100^{3/2} = \frac{2}{3} \times 1000 = \frac{2000}{3} \approx 666.67 \]Thus, producer surplus is:\[ PS = 1000 - \frac{2000}{3} = \frac{1000}{3} \approx 333.33 \]

Key Concepts

Consumer Surplus

Consumer surplus is a measure of the benefit consumers receive when they purchase a product at a market price lower than the maximum price they are willing to pay. Imagine you're ready to pay $15 for a gadget but find it for $10. The $5 difference is your gain or surplus.

In mathematical terms, consumer surplus is represented as the area under the demand curve above the equilibrium price and up to the equilibrium quantity. This requires calculating the integral of the demand function from zero to the equilibrium quantity and then subtracting the total revenue paid by consumers, which is the equilibrium price times the equilibrium quantity.

• Demand function: Reflects consumers' willingness to pay
• Equilibrium price: The price at which the market clears
• Consumer surplus: Area between demand curve and price level

In our exercise, we determined that the equilibrium point is at quantity 100 units with a price of $10. By solving the integral of the demand function, we calculated the total area under the curve up to 100 units and found the consumer surplus to be $1000.

Understanding consumer surplus helps businesses and policymakers ensure that markets function efficiently and consumers retain purchasing power.

Producer Surplus

Producer surplus is the flipside of the economic benefits, focusing on the producers or sellers in the market. It reflects the difference between what producers are willing to accept for a product versus what they actually receive. For example, if a producer is willing to sell a shirt for $8 but sells it for $10, the $2 surplus is extra gain.

This surplus is visually represented as the region above the supply curve and below the equilibrium price, up to the equilibrium quantity. The larger the producer surplus, the more satisfied producers are with market prices.

• Supply function: Indicates the price sellers are willing to accept
• Producer surplus: Area above the supply curve and below price level
• Revenue difference: Equilibrium price minus minimum accepted price

In the exercise, with an equilibrium price of $10 and quantity of 100, the producer surplus is calculated by taking the difference between total area from producers' income and the integral of supply function. Here, we found the producer surplus to be approximately $333.33.

Comprehending producer surplus is critical for producers to strategize pricing and for assessing overall market efficiency.

Demand and Supply Function

The demand and supply functions are foundational to understanding market dynamics. Each function serves to determine how prices and quantities of goods shift based on consumers' and producers' behaviors.

**The Demand Function:**The demand function, usually denoted as \(D(x)\), represents consumers' readiness to purchase varying quantities of a good at different prices. It shows an inverse relationship: as price lowers, demand generally increases.

In the exercise, we used \(D(x) = \frac{100}{\sqrt{x}}\), which exemplifies this principle since as \(x\) increases, \(D(x)\) falls.

**The Supply Function:**Conversely, the supply function \(S(x)\) expresses how much producers are willing to sell at different prices, depicting a direct relationship: higher prices can lead to more goods being supplied.

Our exercise defined \(S(x) = \sqrt{x}\), highlighting that as quantity increases, acceptable selling prices also rise.

• Equilibrium occurs where demand equals supply.
• Understanding functions can aid in predicting market outcomes.
• Shocks in demand or supply can disrupt equilibrium, affecting prices and quantities.

Knowing how demand and supply functions interact provides insights into price changes, market health, and helps make informed business decisions.