Question

A company determines that the price \(p\) of a product can be modeled by \(p=70-\sqrt{0.02 x+1}\), where \(x\) is the number of units of the product demanded per day. Describe the effect that raising the price has on the number of units demanded.

Short answer

Increasing the price of the product will lead to a decrease in the number of units demanded per day.

Step-by-step solution

Understanding Given Function

The given function \(p=70-\sqrt{0.02 x+1}\) is a price-demand model, it describes the relationship between the price and the demand. It is a decreasing function which means as the value of x (demanded units) increases, the price p decreases.

Interpreting the effect of raising the price

If the price p is increased, i.e., if we are moving upwards on the y-axis, it would mean we have to move to the left on the x-axis since it's a decreasing function. This means the demand (x) will decrease if price p is increased, thus there are less units demanded per day.

Key concepts

Decreasing Function

A decreasing function is a mathematical representation where an increase in one variable leads to a decrease in another. In the context of the price-demand model, as you move along the graph of the function, if the product's price decreases, the demand for units typically increases.
In simpler terms, a decreasing function in economics, like our model, means when the price of a product goes down, consumers tend to buy more of it. Conversely, as the price rises, the number of units demanded falls. This is the core idea behind why this function is described as decreasing.

• When the graph slopes downward as it moves from left to right, it indicates a decreasing function.
• The mathematical expression for this model, specifically, is given by the function: \( p=70-\sqrt{0.02x+1} \).
• Understanding how the graph changes helps predict consumer behavior when prices fluctuate.

Price Elasticity of Demand

Price elasticity of demand is an important concept in economics that measures how the quantity demanded of a product changes in response to a change in price. If demand is elastic, a small price change leads to a significant change in the quantity demanded.
In our exercise, the function tells us about the sensitivity of demand to changes in price. Since the function \( p=70-\sqrt{0.02x+1} \) is decreasing, it suggests an inverse and elastic reaction to price changes.

• Understanding elasticity helps businesses decide how to price their products.
• A highly elastic demand means consumers will react significantly to price changes.
• Inelastic demand would mean price changes have little effect on demand.

The elasticity can determine how pricing strategies are approached to maximize revenue without vastly lowering demand.

Inverse Relationship in Economics

An inverse relationship in economics is a situation where two variables move in opposite directions. In our price-demand model, as the price increases, the demand decreases, which is a classic example of an inverse relationship.
This concept is fundamental in understanding consumer behavior. Knowing that an increase in price leads to decreased demand can help businesses in setting prices strategically.

• It illustrates how consumer preferences change as economic conditions like price fluctuate.
• This relationship helps in predicting trends in customer purchasing patterns.
• By quantifying this relationship, firms can optimize profits and market reach.

Grasping the inverse relationship is crucial for developing effective business strategies that align with economic principles.