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Chapter 1: Problem 6

The demand curve for a quantity \(q\) of a product is \(q=\) \(5500-100 p\) where \(p\) is price in dollars. Interpret the 5500 and the 100 in terms of demand. Give units.

Short Answer

Expert verified
5500 is the maximum demand at $0 price (units), and 100 represents demand decrease per $1 price increase (units per dollar).

Step by step solution

01

Understand the Demand Function

The demand function given is \( q = 5500 - 100p \). This equation expresses the quantity demanded \( q \) as a function of the price \( p \). The goal is to interpret the coefficients 5500 and 100 in this context.
02

Interpret the Constant Term 5500

The constant term 5500 in the demand function \( q = 5500 - 100p \) represents the maximum quantity demanded when the price \( p \) is zero. In other words, if the price is zero, the quantity demanded would be 5500 units. The units for this number are units of the product.
03

Interpret the Coefficient -100

The coefficient -100 in front of the \( p \) term represents the rate of decrease in the quantity demanded for each one dollar increase in price. Thus, for every dollar increase in the price, the quantity demanded decreases by 100 units. This is often referred to as the price sensitivity or the change in demand with a change in price.

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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 is a mathematical representation that shows the relationship between the price of a product and the quantity demanded by consumers. The given demand function is \( q = 5500 - 100p \). This tells us how the quantity demanded \( q \) changes with respect to the price \( p \).
  • The first part of the function, "\( 5500 \)", is the intercept, indicating the initial demand level with zero pricing.
  • The second part, "-100p", shows how much demand drops if the price goes up.
Demand functions help in understanding consumer behavior. They're crucial for businesses to set prices strategically. In this case, the demand decreases linearly with every increment in price.
Quantity Demanded
Quantity demanded refers to the total number of units that consumers will buy at a specific price. In the function \( q = 5500 - 100p \), the quantity demanded depends directly on the price \( p \). If you imagine this as a graph, the x-axis represents the price, and the y-axis represents the quantity demanded. The equation shows a downward slope, meaning demand decreases as price increases.
  • At a price of $0, the maximum potential demand is 5500 units.
  • For each dollar increase in price, the demand drops by 100 units.
This reflects the typical behavior where consumers tend to buy less of a product as it becomes more expensive.
Price Sensitivity
Price sensitivity, or elasticity, measures how responsive the quantity demanded is to a change in price. In the demand function \( q = 5500 - 100p \), the "-100" coefficient represents this sensitivity.
  • Each dollar increase in price leads to a 100-unit decrease in quantity demanded.
  • It shows strong price sensitivity, indicating consumers react significantly to price changes.
Understanding price sensitivity helps businesses and economists predict how changes in pricing will affect demand. A high sensitivity suggests that small changes in price can lead to large changes in demand, impacting revenue and market strategy.

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Most popular questions from this chapter

Interest is compounded annually. Consider the following choices of payments to you: Choice 1: $$\$ 1500$$ now and $$\$ 3000$$ one year from now Choice 2: $$\$ 1900\( now and $$\$ 2500\) one year from now (a) If the interest rate on savings were \(5 \%\) per year, which would you prefer? (b) Is there an interest rate that would lead you to make a different choice? Explain.

A sporting goods wholesaler finds that when the price of a product is \(\$ 25\), the company sells 500 units per week. When the price is \(\$ 30\), the number sold per week decreases to 460 units. (a) Find the demand, \(q\), as a function of price, \(p\), assuming that the demand curve is linear. (b) Use your answer to part (a) to write revenue as a function of price. (c) Graph the revenue function in part (b). Find the price that maximizes revenue. What is the revenue at this price?

If you deposit $$\$ 10,000$$ in an account earning interest at an \(8 \%\) annual rate compounded continuously, how much money is in the account after five years?

A company is considering whether to buy a new machine, which costs $$\$ 97,000$$. The cash flows (adjusted for taxes and depreciation) that would be generated by the new machine are given in the following table: $$ \begin{array}{c|c|c|c|c} \hline \text { Year } & 1 & 2 & 3 & 4 \\ \hline \text { Cash flow } & \$ 50,000 & \$ 40,000 & \$ 25,000 & \$ 20,000 \\ \hline \end{array} $$ (a) Find the total present value of the cash flows. Treat each year's cash flow as a lump sum at the end of the year and use an interest rate of \(7.5 \%\) per year, compounded annually. (b) Based on a comparison of the cost of the machine and the present value of the cash flows, would you recommend purchasing the machine?

Use the variable \(u\) for the inside function to express each of the following as a composite function: (a) \(y=\left(5 t^{2}-2\right)^{6}\) (b) \(P=12 e^{-0.6 t}\) (c) \(C=12 \ln \left(q^{3}+1\right)\)
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