I have chosen to focus my mathematical exploration on applications of Calculus in Business situations. To begin with, I was looking for an interesting real life situation I could base my investigation on. Having lived in an economically well developed country like Germany for almost my whole life, the accessibility to a wide range of products and their varying appeal to the consumer are subconsciously part of my daily life.
The fact that some products are enormously demanded by society whereas others aren‘t that successful on the market gave me the idea to investigate how demand is influenced by outside factors. Thereby I discovered the concept of Price Elasticity of Demand which is useful in indicating the responsiveness of the demand of a certain good to a change in its price. I thus decided to explore the different levels of Price Elasticity of demand, namely elastic, inelastic and unit elasticity, and their effect on revenue by means of both an exponential and a quadratic demand function.
Finally I applied the acquired knowledge to a highly demanded and very popular product in Germany, which is coffee, and modeled its change in demand dependent on varying prices as well as outside factors such as brand loyalty and income.
Table of Contents
- Rationale
- Introduction
- Price Elasticity of Demand
- Connecting E(p) and R(p)
- Exponential Demand Function
Objectives and Key Themes
The objective of this mathematical exploration is to investigate the concept of Price Elasticity of Demand and its effect on revenue. The exploration aims to determine whether a product's price is elastic, inelastic, or of unit elasticity by exploring different demand functions and modeling their optimum price and maximum revenue. This concept is then applied to a real-life situation.
- Price Elasticity of Demand
- Relationship between Price Elasticity and Revenue
- Different types of demand functions (exponential and quadratic)
- Optimum pricing and maximum revenue
- Real-world application (e.g., coffee)
Chapter Summaries
Rationale: This section explains the author's motivation for choosing this topic, focusing on the application of calculus in business situations. The author highlights the observation of varying product demand in Germany, leading to an investigation into Price Elasticity of Demand and its impact on revenue using exponential and quadratic demand functions. The study ultimately applies these findings to a popular German product—coffee—modeling its demand fluctuations based on price changes and external factors like brand loyalty and income.
Introduction: The introduction establishes the significance of price fluctuation in daily life and introduces Price Elasticity of Demand as a key indicator of the relationship between price changes and consumer purchasing behavior. It emphasizes that while price increases generally lead to decreased demand, various factors such as substitute product availability, product indispensability, and brand loyalty influence this relationship. The introduction also highlights the connection between Price Elasticity of Demand and revenue, emphasizing the importance of modeling demand functions to predict the impact of price changes on both demand and revenue for businesses. The main aim of the exploration—to analyze Price Elasticity of Demand and its effect on revenue to determine a product's elasticity type (elastic, inelastic, or unit elasticity)—is clearly stated.
Price Elasticity of Demand: This section defines Price Elasticity of Demand (E(p)) as the ratio of the percentage change in quantity demanded to the percentage change in price. It explains the formula for calculating E(p) and discusses the three levels of demand: elastic (|E(p)| > 1), inelastic (|E(p)| < 1), and unit elasticity (|E(p)| = 1). Each elasticity level is defined and explained, highlighting the sensitivity of demand to price changes at each level. The section clarifies that because an increase in price generally leads to a decrease in demand (and vice-versa), E(p) will usually have a negative value. The absolute value of E(p) is used to categorize elasticity.
Connecting E(p) and R(p): This section focuses on the relationship between Price Elasticity of Demand (E(p)) and revenue (R(p)). It derives an equation connecting the two by implicitly differentiating the revenue function R(p) = pq with respect to price (p). The resulting equation demonstrates how the rate of change in revenue (dR/dp) depends on the quantity demanded (q) and the Price Elasticity of Demand (E(p)). The section then analyzes the implications of different elasticity levels on the change in revenue (dR/dp) with respect to changes in price, concluding that elastic demand leads to a decrease in revenue when the price increases, inelastic demand leads to an increase in revenue when the price increases, and unit elasticity leads to a relatively unchanged revenue with price increases.
Exponential Demand Function: This chapter delves into the application of an exponential demand function, D(p) = Ae^(-kp), to model the relationship between price and demand. The author demonstrates how to calculate the Price Elasticity of Demand (E(p)) using this function and how to determine the optimum price and maximum revenue based on the elasticity. The conditions for elastic, inelastic, and unit elasticity are explored within the context of this specific demand function.
Keywords
Price Elasticity of Demand, Revenue, Demand Function, Elastic Demand, Inelastic Demand, Unit Elasticity, Optimum Price, Maximum Revenue, Exponential Function, Quadratic Function, Calculus, Business Applications.
Frequently Asked Questions: A Mathematical Exploration of Price Elasticity of Demand
What is the main topic of this mathematical exploration?
This exploration investigates the concept of Price Elasticity of Demand and its effect on revenue. It aims to determine whether a product's price is elastic, inelastic, or of unit elasticity by exploring different demand functions and modeling their optimum price and maximum revenue. A real-life application is also included.
What are the key themes explored in this document?
The key themes include Price Elasticity of Demand, the relationship between Price Elasticity and Revenue, different types of demand functions (exponential and quadratic), optimum pricing and maximum revenue, and a real-world application (using coffee as an example).
What are the objectives of this mathematical exploration?
The objective is to understand Price Elasticity of Demand and its impact on revenue. This involves analyzing different demand functions to determine the optimum price and maximum revenue, and categorizing the elasticity of a product as elastic, inelastic, or unit elastic.
What is Price Elasticity of Demand (E(p))?
Price Elasticity of Demand is defined as the ratio of the percentage change in quantity demanded to the percentage change in price. It helps determine the sensitivity of demand to price changes. |E(p)| > 1 indicates elastic demand, |E(p)| < 1 indicates inelastic demand, and |E(p)| = 1 indicates unit elasticity.
How is Price Elasticity of Demand related to Revenue (R(p))?
The exploration derives an equation connecting E(p) and R(p) through implicit differentiation of the revenue function R(p) = pq. This shows how the rate of change in revenue depends on quantity demanded and Price Elasticity of Demand. Elastic demand leads to decreased revenue with price increases, inelastic demand leads to increased revenue with price increases, and unit elasticity leads to relatively unchanged revenue with price increases.
What types of demand functions are used in this exploration?
The exploration uses both exponential (D(p) = Ae^(-kp)) and quadratic demand functions to model the relationship between price and demand. The analysis demonstrates how to calculate E(p), optimum price, and maximum revenue using these functions.
What is the real-world application discussed in this exploration?
The exploration applies the concepts of Price Elasticity of Demand and revenue optimization to the coffee market in Germany, modeling demand fluctuations based on price changes and other factors like brand loyalty and income.
What are the chapter summaries provided in this document?
The document provides chapter summaries for the Rationale (author's motivation and study design), Introduction (significance of price fluctuation and Price Elasticity of Demand), Price Elasticity of Demand (definition and types of elasticity), Connecting E(p) and R(p) (relationship between Price Elasticity and Revenue), and Exponential Demand Function (application of exponential demand function to model price-demand relationship).
What are the keywords associated with this exploration?
Keywords include Price Elasticity of Demand, Revenue, Demand Function, Elastic Demand, Inelastic Demand, Unit Elasticity, Optimum Price, Maximum Revenue, Exponential Function, Quadratic Function, Calculus, and Business Applications.
- Quote paper
- Stefanie Mücka (Author), 2014, Price Elasticity of Demand and its effect on Revenue, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/276575
