Problem 8 - 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. \(D(x)=1000-10 x, \quad S(x)=250+5 x\)

Step-by-Step Solution

Answer

(a) Equilibrium point: (50, 500). (b) Consumer surplus: $12,500. (c) Producer surplus: $6,250.

Step 1: Find the Equilibrium Point

The equilibrium point occurs where the demand price equals the supply price, which means solving for when \( D(x) = S(x) \). Plug in the expressions: \[ 1000 - 10x = 250 + 5x \] Solve this equation for \( x \). First, add \( 10x \) to both sides:\[ 1000 = 250 + 15x \] Subtract 250 from both sides:\[ 750 = 15x \] Divide by 15:\[ x = 50 \] Now substitute \( x = 50 \) back into either \( D(x) \) or \( S(x) \) to find the price at equilibrium:\[ D(50) = 1000 - 10(50) = 500 \] Therefore, the equilibrium point is \( (50, 500) \).

Step 2: Calculate Consumer Surplus

Consumer surplus is the area between the demand curve and the equilibrium price level, from the y-axis to the equilibrium quantity. It is given by:\[ ext{Consumer Surplus} = \int_0^{50} (D(x) - 500) \, dx \]This becomes:\[ \int_0^{50} (1000 - 10x - 500) \, dx = \int_0^{50} (500 - 10x) \, dx \]Evaluate this integral:\[ \begin{align*} &\left[ 500x - 5x^2 \right]_0^{50} \ &= (500(50) - 5(50)^2) - (500(0) - 5(0)^2) \ &= (25000 - 12500) - 0 = 12500 \end{align*} \]Thus, the consumer surplus is \( 12500 \) dollars.

Step 3: Calculate Producer Surplus

Producer surplus is the area between the supply curve and the equilibrium price level, from the y-axis to the equilibrium quantity. It is calculated by:\[ ext{Producer Surplus} = \int_0^{50} (500 - S(x)) \, dx \]Substitute \( S(x) \):\[ \int_0^{50} (500 - (250 + 5x)) \, dx = \int_0^{50} (250 - 5x) \, dx \]Evaluate this integral:\[ \begin{align*} &\left[ 250x - \frac{5}{2}x^2 \right]_0^{50} \ &= (250(50) - \frac{5}{2}(50)^2) - (250(0) - \frac{5}{2}(0)^2) \ &= (12500 - 6250) - 0 = 6250 \end{align*} \]Thus, the producer surplus is \( 6250 \) dollars.

Key Concepts

Consumer Surplus

Consumer surplus represents the benefit or gain that consumers receive when they are able to purchase a product for a price that is less than the highest price they are willing to pay. It reflects the extra value consumers get from paying less than what they might have expected. In economic terms, it's the area between the demand curve and the actual market price up to the quantity sold.

In our exercise, the consumer surplus can be visualized as the area under the demand curve, above the price level, up to the equilibrium quantity. This can be calculated through integration. The formula for consumer surplus when using calculus is:

• \[ \int_{0}^{Q_e} \left(D(x) - P_e\right) \, dx \]

Here, \(D(x)\) is the demand function, and \(P_e\) is the price at equilibrium. For our specific problem, the equilibrium quantity \(Q_e\) is 50 units, and the equilibrium price \(P_e\) is \(500. By performing the integral, we can find that the consumer surplus is \)12,500. This means consumers gain an economic benefit worth $12,500 in total by buying the product at the equilibrium price.

Producer Surplus

Producer surplus is the counterpart to consumer surplus from the seller’s perspective. It measures the difference between what producers are willing to accept for a good or service versus what they actually receive—the market price.

• Producer surplus is the area above the supply curve and below the market price.
• It indicates the producers’ gain from selling goods at a price higher than their minimum acceptable price—cost of production.

In the given exercise, the producer surplus calculation involves finding the area under the market price line and above the supply curve up to the equilibrium quantity. Calculating this involves integration, using:

• \[ \int_{0}^{Q_e} \left(P_e - S(x)\right) \, dx \]

Where \(S(x)\) represents the supply function, and \(P_e\) is the equilibrium price, \(500 in this case. Using the equilibrium quantity of 50, we find the producer surplus to be \)6,250. This reflects the total extra earnings producers achieve over the specified supply cost.

Demand and Supply Functions

Understanding demand and supply functions is crucial in finding the equilibrium point in any market. Let's break down what these functions represent:

• **Demand Function \(D(x)\):** This function depicts the relationship between the quantity of a product demanded by consumers and the price of the product. In the given exercise, the demand function is \(D(x) = 1000 - 10x\). This means for every unit increase in quantity, the price consumers are willing to pay decreases by \(10.
• **Supply Function \(S(x)\):** This equation illustrates the relationship between the quantity of a product supplied by producers and its price. Here, the supply function is \(S(x) = 250 + 5x\), suggesting that as more units are supplied, the price the producer is willing to accept increases by \)5 per unit.

Finding the equilibrium point involves setting these two functions equal to each other because this is the point where the quantity demanded by consumers equals the quantity supplied by producers. Solving \(1000 - 10x = 250 + 5x\) gives us the equilibrium quantity \(x = 50\), and substituting back gives the equilibrium price \(P_e = 500\). This is the point where the market clears, meaning supply equals demand.