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The demand function is a fundamental concept in economics that shows the relationship between the price of a good and the quantity demanded. In our example, the demand function is given by \[ q = 1000 - 20p \]This equation tells us how many Honolulu Red Oranges are sold each week depending on the price \( p \). Here, \( q \) is the quantity demanded, and \( p \) is the price per orange.

• The negative sign before \( 20p \) indicates an inverse relationship between the price and quantity demanded. Simply put, as the price increases, the quantity demanded decreases.
• The constant \( 1000 \) represents the maximum quantity demanded when the price is zero, although this is merely a theoretical maximum as prices can't be negative.

By understanding this demand function, we can analyze how changes in price will affect sales, which is crucial for businesses to set the right prices and achieve desired sales levels. The demand function is often the first step in more complex economic calculations, such as determining price elasticity of demand.

Maximizing revenue is a key goal for businesses, and it requires finding the price point where revenue is highest. Revenue is calculated by multiplying the price (\( p \)) by the quantity demanded (\( q \)). In this instance, the revenue function \( R(p) \) is derived from the demand function. That is,
\[ R(p) = p \times q = p \times (1000 - 20p) = 1000p - 20p^2 \]To find the price that maximizes revenue:

• We compute the derivative of the revenue function \( \frac{dR}{dp} \) and set it to zero. This gives us the price at which revenue is either maximized or minimized.
• Solving \( 1000 - 40p = 0 \) yields \( p = 25 \), indicating that at \(25 per orange, revenue hits its peak.

Plugging \( p = 25 \) back into the revenue function, we find the maximum revenue, which is \)12,500. This critical analysis helps businesses in setting optimal prices that enhance profitability over time. Ensuring prices align with revenue goals requires understanding both consumer behavior (as captured by the demand function) and how price adjustments influence revenue.

Elastic demand refers to a situation where the quantity demanded of a good changes significantly when there is a change in price. This concept is quantified using the price elasticity of demand (PED), calculated as:\[ PED = \frac{dq}{dp} \times \frac{p}{q} \]Using our demand function, \( q = 1000 - 20p \), and analyzing at \( p = 30 \), we found \( PED = -1.5 \).

• A PED of -1.5 indicates elastic demand since its absolute value is greater than 1. This means that the percentage change in quantity demanded is greater than the percentage change in price.
• Elastic demand suggests that consumers are sensitive to price changes. For example, if the price rises, the drop in quantity demanded is large relative to the price increase.

Understanding whether demand is elastic or inelastic (less responsive to price changes) aids businesses in pricing strategies. For products with elastic demand, like the oranges in this example, even small price changes can lead to significant changes in sales volume. Thus, businesses must carefully consider pricing decisions to not adversely affect their revenue.