Problem 28 - 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) Demand: 70,000; Supply: 100,000; Price would decrease. (b) Equilibrium price is $80 and quantity is 80,000, confirming part (a).

Step 1: Calculate Quantity Demanded at $100

Using the demand curve equation \( q = 120,000 - 500p \), plug in \( p = 100 \) to find the quantity demanded:\[ q = 120,000 - 500 \times 100 = 120,000 - 50,000 = 70,000\]Thus, the quantity demanded is 70,000 units.

Step 2: Calculate Quantity Supplied at $100

Using the supply curve equation \( q = 1000p \), plug in \( p = 100 \) to find the quantity supplied:\[ q = 1000 \times 100 = 100,000\]Thus, the quantity supplied is 100,000 units.

Step 3: Determine Market Force Direction at $100

Compare the quantity demanded (70,000) and the quantity supplied (100,000). Since the quantity supplied is greater than the quantity demanded, there is a surplus in the market. This surplus will typically cause the market price to decrease.

Step 4: Find the Equilibrium Price

At equilibrium, the quantity demanded equals the quantity supplied. Set the demand and supply equations equal:\[120,000 - 500p = 1000p\]Solve for \( p \):\[ 120,000 = 1500p \ p = \frac{120,000}{1500} = 80\]The equilibrium price is \$80.

Step 5: Find the Equilibrium Quantity

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

Step 6: Verify Part (a) with Equilibrium Results

In part (a), there was a surplus at \\(100, causing the market price to fall. The equilibrium price is \\)80, where supply equals demand. Thus, our findings support that market forces drive the price down towards equilibrium, confirming that the surplus at \\(100 would indeed push prices closer to \\)80.

Key Concepts

Demand Curve

The demand curve is an essential concept in economics that illustrates the relationship between the price of a good and the quantity that consumers are willing to purchase. It typically slopes downwards from left to right. This indicates that as prices decrease, the quantity demanded increases and vice versa. This scenario is represented by the demand equation given in the exercise, where:

• The demand equation is represented as \( q = 120,000 - 500p \).
• It shows that at higher prices, like $100, the quantity demanded is lower as calculated to be 70,000 units.

In general, factors such as consumer preferences, income levels, and prices of related goods can shift the demand curve. However, for this exercise, we assume these factors remain constant. The curve's downward slope reflects the law of demand.

Supply Curve

Supply curves illustrate the relationship between a product's price and the quantity that producers are willing to supply. Unlike the demand curve, the supply curve usually slopes upwards. This implies suppliers are willing to produce and sell more as the price increases. In this exercise, the supply curve is given by:

• The equation for supply is \( q = 1000p \).
• At a price of $100, producers are willing to supply 100,000 units, which is higher than the quantity demanded at this price point.

Various factors can shift the supply curve, such as advancements in technology, changes in production costs, and the number of suppliers. However, this exercise assumes these factors are constant. The upward slope indicates that higher prices incentivize producers to supply additional quantities.

Market Surplus

Market surplus occurs when the quantity supplied exceeds the quantity demanded at a given price. It is a signal demonstrating an imbalance in the market. In the case example, at a price of $100:

• Demand is 70,000 units.
• Supply is 100,000 units, resulting in a 30,000 unit surplus.

Surplus pressures the price downward, as sellers might decrease prices to sell the excess inventory. As prices drop, the quantity demanded tends to rise, and a new balance might ultimately be struck at the equilibrium price, where there is no surplus or shortage. Understanding market surplus is key to predicting market adjustments.

Market Forces

Market forces refer to the economic factors affecting the availability and pricing of goods in a market. They are the invisible hand guiding the balance between supply and demand, pushing prices towards equilibrium. In this context:

• If the price is above equilibrium, like the $100 price point, a surplus occurs, driving the price down.
• This downward pressure continues until the market reaches equilibrium.

In this exercise, the equilibrium is achieved at a price of $80 with an equilibrium quantity of 80,000 units. Market forces naturally eliminate surplus or shortage, aligning supply with demand through voluntary adjustments in price. Understanding these forces is fundamental for predicting how changes in the economy might impact prices and quantities.