Problem 32 - Applied Calculus

Question

The demand curve for a product is given by \(q=\) \(120,000-500 p\) and the supply curve is given by \(q=\) \(1000 p\) for \(0 \leq q \leq 120,000,\) where price is in dollars. (a) At a price of \(\$ 100,\) what quantity are consumers willing to buy and what quantity are producers willing to supply? Will the market push prices up or down? (b) Find the equilibrium price and quantity. Does your answer to part (a) support the observation that market forces tend to push prices closer to the equilibrium price?

Step-by-Step Solution

Answer

(a) Consumers want 70,000, producers supply 100,000, surplus pushes price down. (b) Equilibrium is at $80 and 80,000 units; prediction in (a) confirmed.

Step 1: Compute Quantity Demanded

Use the demand equation to find quantity demanded when the price is \(100. Substituting \)100 for \( p \) in the demand equation \( q = 120,000 - 500p \), the calculation becomes: \[ q = 120,000 - 500 \times 100 \] which simplifies to \( q = 120,000 - 50,000 = 70,000 \). So, the quantity demanded is 70,000 units.

Step 2: Compute Quantity Supplied

Use the supply equation to find the quantity supplied when the price is \(100. Substituting \)100 for \( p \) in the supply equation \( q = 1000p \), the calculation becomes: \[ q = 1000 \times 100 \] which simplifies to \( q = 100,000 \). So, the quantity supplied is 100,000 units.

Step 3: Determine Market Price Movement

Compare the quantity demanded and supplied at $100. Since the quantity demanded (70,000) is less than the quantity supplied (100,000), there is a surplus in the market. According to market forces, prices tend to decrease when there is a surplus.

Step 4: Find Equilibrium Price

Set the demand equation equal to the supply equation to find the equilibrium price. Therefore: \[ 120,000 - 500p = 1000p \] Solving for \( p \), we have: \[ 120,000 = 1500p \] \[ p = \frac{120,000}{1500} = 80 \] The equilibrium price is $80.

Step 5: Find Equilibrium Quantity

Substitute the equilibrium price back into either the demand or supply equation to find the equilibrium quantity. Using the supply equation: \[ q = 1000 \times 80 = 80,000 \] Therefore, the equilibrium quantity is 80,000 units.

Step 6: Verify Market Price Prediction

Check if the prediction in part (a) aligns with the equilibrium found. In part (a), the surplus at $100 suggests the price should decrease. This aligns with the equilibrium price found in part (b) which is $80, supporting the observation that market forces push prices towards equilibrium.

Key Concepts

Demand Curve

The demand curve represents the relationship between price and quantity demanded by consumers. It typically slopes downwards to the right, illustrating the law of demand: as prices decrease, consumers tend to buy more. In this case, the demand curve is represented by the equation \( q = 120,000 - 500p \). Here, \( q \) is the quantity demanded, and \( p \) is the price.

This linear equation shows that as the price increases, the quantity demanded decreases. For example, when the price is $100, substituting into the equation gives \( q = 70,000 \), meaning consumers are willing to buy 70,000 units at this price.

Supply Curve

The supply curve showcases the relationship between price and quantity supplied by producers. It generally slopes upwards to the right, indicating the law of supply: as prices increase, producers are willing to supply more of the product. For this problem, the supply curve is \( q = 1000p \).

Here, as the price goes up, the quantity that producers are willing to supply also increases. At a price of \(100, the supply equation becomes \( q = 100,000 \). This means that producers are ready to supply 100,000 units when the price is \)100.

The direction of the supply curve emphasizes producers' sensitivity to price changes, with higher prices incentivizing more production.

Market Surplus

When there is a difference between quantity supplied and quantity demanded at a certain price level, this results in a market surplus or deficit. For example, at a price of $100, consumers demand 70,000 units, but producers supply 100,000 units. This creates a surplus of 30,000 units.

A market surplus occurs when the quantity supplied exceeds the quantity demanded. Surpluses typically lead to downward pressure on prices, as producers attempt to sell off excess inventory.

In this scenario, the surplus indicates that the market forces will likely push the price down to eliminate the excess supply, which should guide prices towards equilibrium.

Equilibrium Price

The equilibrium price is the price at which the quantity supplied equals the quantity demanded. It is a crucial point where the supply and demand curves intersect. In the given problem, setting the demand equation equal to the supply equation \( (120,000 - 500p = 1000p) \) allows us to solve for the equilibrium price.

Solving this yields an equilibrium price of \(80. This is the price at which the market is perfectly balanced, with no surplus or shortage.

Understanding equilibrium helps us predict how market forces will act to stabilize prices when they deviate from this point, as seen with the adjustment from \)100 to $80.

Equilibrium Quantity

Equilibrium quantity is the number of goods or services that are supplied and demanded at the equilibrium price. Once we have the equilibrium price, substituting it back into either the supply or demand equation will give the equilibrium quantity.

Here, using the supply equation \( q = 1000 imes 80 \), we find that the equilibrium quantity is 80,000 units. At this point, there is no incentive for price movement as the market is in balance.

Finding the equilibrium quantity confirms how a market reaches a stable state, where the interests of suppliers and consumers align, creating a harmonious economic environment.