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Chapter 3: Problem 25

Find the equilibrium point for the following pairs of demand and supply functions. \(D(p)=1600-53 p\) \(S(p)=320+75 p\)

Short Answer

Expert verified
The equilibrium price is \( p = 10 \) and the equilibrium quantity is \( q = 1070 \).

Step by step solution

01

Set Demand Equal to Supply

To find the equilibrium point, set the demand function equal to the supply function: Now, we can solve for the price.
02

Solve for the Price

Set the given functions equal to each other and solve for \( p \): This equation simplifies to: Combine like terms: Solving for \( p \): Thus, \( p = 10 \).
03

Find the Equilibrium Quantity

Substitute the equilibrium price \( p = 10 \) back into either the demand or supply function. We'll use the demand function: So, the equilibrium quantity is \( q = 1070 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

demand function
A demand function represents how the quantity of a product consumers are willing to buy varies with its price. The formula typically appears as \(D(p) = a - bp\), where:
  • \(D(p)\) is the demand at price \(p\),
  • \(a\) is the intercept indicating the maximum quantity demanded when the price is zero,
  • \(b\) is the slope indicating the rate of change in demand with respect to price.
In our example, the demand function is \(D(p)=1600-53p\). Here, 1600 is the quantity demanded when the price is zero, and 53 is the decrement in demand for every unit increase in price. Notice the negative sign suggests that demand decreases as price increases, which is known as the law of demand. Understanding the demand function is crucial for interpreting how different prices affect the market demand for a product.
supply function
A supply function shows how the quantity of a product that producers are willing to sell varies with its price. It is usually formulated as \(S(p) = c + dp\), where:
  • \(S(p)\) is the supply at price \(p\),
  • \(c\) is the intercept indicating the minimum quantity supplied when the price is zero,
  • \(d\) represents the rate of supply change with respect to price.
In the given problem, the supply function is \(S(p)=320+75p\). Here, 320 is the quantity supplied when the price is zero, and 75 is the increase in supply for every unit increase in price. The positive relationship between price and quantity supplied reflects the law of supply, which indicates that higher prices motivate producers to supply more. Grasping the supply function helps you understand how various prices influence the amount of product available in the market.
solving equations
To find the equilibrium point where demand equals supply, we need to set the demand function equal to the supply function and solve for the price. This involves solving a linear equation. Here's how it works:
  • First, set the given demand \(D(p)\) equal to supply \(S(p)\): \(1600 - 53p = 320 + 75p\).
  • Combine like terms by bringing all terms involving \(p\) to one side and constants to the other side: \(1600 - 320 = 75p + 53p\).
  • Simplify this equation to get: \(1280 = 128p\).
  • Finally, solve for \(p\) by dividing both sides by 128: \(p = 10\).
Now that we've found the equilibrium price \(p = 10\), substitute this back into either the demand or supply function to find the equilibrium quantity, which gives \(q = 1070\). Thus, the market is in equilibrium at a price of 10 with a quantity of 1070.

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