For non-linear demand curves, calculus is used to find the price elasticity at a specific point. This is done by taking the limit of the elasticity formula as the change in price (ΔP) approaches zero: $ε = \lim_{\Delta P \to 0} -\frac{P}{Q} \frac{\Delta Q}{\Delta P}$. This limit yields a definition of elasticity based on the derivative of the direct demand function, $Q = g(P)$, which is expressed as $dQ/dP$ or $g'(P)$. The resulting formula is $ε = -\frac{P}{Q} \frac{dQ}{dP}$. Because quantity (Q) is a function of price ($Q=g(P)$), the elasticity formula can be expressed entirely as a function of price by substituting the demand function for Q. Due to the Law of Demand, the derivative of the demand function, $dQ/dP$, is negative, meaning the price elasticity of demand (ε) is typically a positive value.

Defining Point Price Elasticity Using the Derivative of the Demand Function

The Law of Demand states that there is an inverse relationship between the price of a product and the quantity that consumers demand. When the price is lower, more consumers are willing to purchase the product, resulting in higher demand. Conversely, as the price rises, the quantity demanded falls. This principle is the reason demand curves are expected to slope downward.

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A product's demand curve, also known as a demand function, illustrates the relationship between the quantity of items consumers will purchase and every possible price point. [4, 7] A key feature of demand curves is their downward slope, which signifies that as a product's price increases, the quantity consumers demand for it falls. This inverse relationship also means that when a product's available quantity is low, it can be sold at a high price. The curve itself is derived from the willingness to pay (WTP) of each consumer, showing the number of consumers whose WTP is at or above any given price.

Demand Curve

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

CORE Econ - The Economy 2.0: Microeconomics

Willingness to Pay (WTP) represents the maximum price an individual is prepared to pay for a good, reflecting their personal valuation of it. This valuation is subjective and can differ significantly among consumers based on factors such as how important they consider the good, their work ethic, and their available financial resources. A consumer will only purchase a product if its price is less than or equal to their WTP. [1, 2, 7]

Willingness to Pay (WTP)

Gregory King was a 17th-century English official who compiled extensive statistical data on the population and wealth of England. This data became the empirical basis for the Davenant-King Law of Demand.

Gregory King (1648–1712)

Charles Davenant, an English politician from the 17th century, is credited with publishing the Davenant-King Law of Demand in 1699. He utilized the data compiled by Gregory King to articulate the relationship between the supply of a commodity and its price.

Charles Davenant (1656–1714)

Published in 1699 by Charles Davenant based on data from Gregory King, this law articulated the relationship between a crop's harvest size and its market price. It specifically analyzed the price of corn, calculating, for example, that a 10% shortfall in the harvest would result in a 30% price increase.

The Davenant–King Law of Demand

Law of Demand

This diagram illustrates two distinct, parallel demand curves, labeled D and D'. The vertical axis represents the price in pounds, from 0 to 12,000, while the horizontal axis indicates the quantity of consumers, from 0 to 100. The lower demand curve, D, is a downward-sloping line that connects the points (0, 6000) and (60, 0). The upper demand curve, D', is also a downward-sloping line, linking the points (0, 10000) and (100, 0).

A Graph Showing Two Alternative Demand Curves (D and D')

A firm's pricing decision is significantly influenced by the slope of the demand curve, which indicates how sensitive consumer demand is to price changes. If the demand curve is steep, it signifies that a price increase will not cause a substantial decrease in sales. This allows the firm greater power to raise its price without losing many customers.

Impact of Demand Curve Steepness on Pricing Power

The price elasticity of demand is a measure of the responsiveness of consumer demand to a change in price. It is formally defined as the percentage change in quantity demanded that occurs in response to a 1% increase in the product's price. For convenience, this value is typically expressed as a positive number. Demand is considered 'elastic' if the elasticity value is greater than 1, and 'inelastic' if it is less than 1.

Price Elasticity of Demand

A consumer survey is a method for estimating product demand, which is especially valuable for new products like cereals or space flights where historical sales data is unavailable. The technique involves asking a large sample of consumers about their willingness to pay at different price points. By analyzing the proportion of respondents willing to purchase at each price, it's possible to construct the entire demand curve for the product.

Estimating a Demand Curve via Consumer Surveys

The data for the Apple Cinnamon Cheerios demand curve originates from a 1996 study by economist Jerry A. Hausman. [7] The study, titled 'Valuation of New Goods Under Perfect and Imperfect Competition,' was included in the book 'The Economics of New Goods,' published by the University of Chicago Press on pages 207-248. [7]

Source of Demand Curve Data for Apple Cinnamon Cheerios (Hausman, 1996)

For simplicity in economic models, both supply and demand curves are frequently depicted as straight lines. In reality, they are more likely to be non-linear. The precise shape of these curves depends on the distribution of valuations among market participants—that is, how willingness to pay (for demand) or willingness to accept (for supply) varies from person to person.

Linearity of Supply and Demand Curves as a Simplification

Figure 8.7 shows the market demand curve for bread, representing the total quantity demanded by all consumers at various prices. Fundamentally, this curve also illustrates the willingness to pay (WTP) of the consumers for each unit of the good. The curve has a downward slope, indicating that as the price falls, more consumers are willing to purchase the product.

The Market Demand Curve for Bread (Figure 8.7) and Consumer Willingness to Pay

This activity presents a scenario involving a hat shop to analyze the consequences of a price adjustment. Initially, the shop sells 20 hats weekly for \$10 per hat. After raising the price to \$12, weekly sales decrease to 15 hats. The task is to use this data to evaluate a series of statements and identify the correct ones.

Activity: Analyzing a Hat Shop's Price Change

The inverse demand function, written as $P = f(Q)$, defines price as a function of quantity. [3] This representation aligns with the standard graphical display of a demand curve, with price on the vertical axis and quantity on the horizontal axis. [1] It specifies the maximum price a firm can set to sell a given quantity of its product. While the direct demand function is used to express price elasticity in terms of price, the inverse demand function is useful for expressing elasticity entirely in terms of quantity.

Inverse Demand Function: Price as a Function of Quantity

The direct demand function, denoted as $Q = D(P)$, expresses the quantity demanded (Q) as a function of price (P). Consistent with the downward-sloping nature of the demand curve, this is a strictly decreasing function. It is derived by rearranging the inverse demand function, $P = f(Q)$, to solve for Q. Thus, the function $D(P)$ is the mathematical inverse of the inverse demand function $f(Q)$. This direct functional form is necessary for the mathematical determination of the market equilibrium price by equating it with the direct supply function.

Direct Demand Function: Quantity as a Function of Price (Q = D(P))

Aggregate demand represents the total expenditure on all goods and services produced within an economy at a given time. [1] It is a macroeconomic concept that sums up the demand from all sectors of the economy. [2, 5]

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