Question

Find the consumer and producer surpluses. $$ p_{1}(x)=300-x \quad p_{2}(x)=100+x $$

Short answer

The consumer surplus is -5000 and the producer surplus is 5000.

Step-by-step solution

Identify the equilibrium quantity and price

To find the equilibrium price and quantity, set the demand and supply functions equal to each other and solve for the unknown \(x\), the quantity. \n\(300 - x = 100 + x\)\nSolving this, we get \(x = 100\). Substitute \(x = 100\) back into either the supply or demand function to find the equilibrium price \(p\). For example in the demand function: \n\(p = 300 - 100\)\nwhich gives a price \(p = 200\).

Calculate consumer surplus

Consumer surplus is the area under the demand curve (from 0 to x) minus the square area (p*x). The integral of the demand function from 0 to \(x=100\) is given as: \n\(\int_0^{100} (300 - x) dx\)\nwhich results in 15000. Then subtract from this result the area under the price which is given by \(p * x = 200 * 100 = 20000\). So, Consumer Surplus = 15000 - 20000 = -5000.\nNote: A negative surplus means that consumers are not willing to pay the equilibrium price.

Calculate producer surplus

Producer surplus is the square area (p*x) minus the area under the supply curve (from 0 to x). First, calculate the integral of the supply curve from 0 to \(x = 100\), which is:\n \(\int_0^{100} (100 + x) dx\)\nThis gives 15000. Then subtract this number from the area under the price: \(p * x = 200 * 100 = 20000\).\nSo, Producer Surplus = 20000 - 15000 = 5000

Key concepts

Equilibrium Price and Quantity

In economics, the equilibrium price and quantity represent a point where the market for a product is in perfect balance. This point is where the quantity demanded by consumers equals the quantity supplied by producers. To find the equilibrium, it is essential to analyze the demand and supply functions.
For instance, in an exercise where the demand function is given by \(p_{1}(x) = 300 - x\) and the supply function by \(p_{2}(x) = 100 + x\), setting these two functions equal to each other will allow for the determination of the equilibrium quantity, \(x\), and equilibrium price, \(p\). Solving the equation \(300 - x = 100 + x\), we find that the equilibrium quantity is 100 units, and by substituting this back into either function, we conclude that the equilibrium price is $200.

Demand and Supply Functions

Demand and supply functions are mathematical representations of how consumers and producers behave in the market relative to the price of a good or service.

• The demand function, often decreasing, shows the relationship between the quantity consumers are willing to purchase and the price. Higher prices typically lower the quantity demanded.
• The supply function, usually increasing, depicts the relationship between the quantity producers are willing to sell and the price. Higher prices generally encourage more supply.

The interaction of these functions determines the market dynamics and helps to predict changes in price and quantity based on shifts in demand or supply.

Consumer Surplus Calculation

Consumer surplus is a measure of the economic benefit that consumers receive when they are able to purchase a product for less than the maximum price they are willing to pay.
To calculate consumer surplus, you need to compute the area between the demand curve and the price level up to the equilibrium quantity. Mathematically, this is found by integrating the demand function from 0 to the equilibrium quantity and then subtracting the product of the equilibrium price and quantity.
However, if the surplus calculation results in a negative number, this indicates a discrepancy that could mean consumers are not benefitting from the market conditions, potentially due to high prices or low utility from the product.

Producer Surplus Calculation

Producer surplus reflects the difference between what producers are actually paid for a good or service and the least amount they are willing to accept for it.
This is calculated by subtracting the area under the supply curve from the area representing the product of the equilibrium quantity and price. To find this, you integrate the supply function from 0 to the equilibrium quantity and then subtract this value from the total revenue (price multiplied by quantity). The remaining amount is the producer surplus, quantifying the producers' benefit from the market.

Economic Equilibrium

Economic equilibrium occurs when the market for a good or service is in balance with no excess supply or demand. At this point, the allocative efficiency is achieved, indicating that resources are distributed in the most beneficial way for both buyers and sellers.
The equilibrium is not just a single point but a condition where opposing forces of demand and supply are in harmony, resulting in stable prices and quantities. It's important to note that equilibrium can change if there are shifts in demand or supply, caused by factors such as technological advancements, changes in consumer preferences, or variations in resource costs.