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Chapter 3: Problem 28

Show that the demand function \(D(p)=a / p^{b},\) where \(a\) and \(b\) are positive real numbers, has a constant elasticity for all positive prices.

Short Answer

Expert verified
Show that the demand function \(D(p)=a / p^{b}\) has a constant elasticity for all positive prices. Solution: 1. Differentiate the demand function with respect to price: \(\frac{dD(p)}{dp}=-\frac{ab}{p^{b+1}}\) 2. Calculate the price elasticity of demand: \(E=-\frac{ab}{p^{b+1}} \times \frac{p^{b+1}}{a}\) Simplify the equation to get: \(E=-bp\) 3. The price elasticity of demand is \(E=-bp\), which depends only on the constant \(b\) and is constant for all positive prices. Thus, the demand function \(D(p)=a / p^{b}\) has a constant elasticity for all positive prices.

Step by step solution

01

Differentiation of the demand function

To find the price elasticity of demand, we first need to differentiate the demand function with respect to price. The demand function is given by: $$D(p)=\frac{a}{p^b}$$ Applying power rule of differentiation, we get: $$\frac{dD(p)}{dp}=-\frac{ab}{p^{b+1}}$$
02

Price elasticity of demand

Next, we need to find the price elasticity of demand, which is defined as: $$E=\frac{\frac{dD(p)}{dp}}{\frac{D(p)}{p}}$$ We have already found \(\frac{dD(p)}{dp}\), and the demand function \(D(p)\) is given. So we can substitute these values into the formula: $$E=\frac{-\frac{ab}{p^{b+1}}}{\frac{\frac{a}{p^b}}{p}}$$ Simplifying the above equation, $$E=-\frac{ab}{p^{b+1}} \times \frac{p^{b+1}}{a}$$ $$E=-bp$$
03

Conclusion

We have now found the price elasticity of demand for the function to be \(E=-bp\). Since \(b\) is a constant (a positive real number), the price elasticity of demand is constant for all positive prices, as it depends only on the value of \(b\). Therefore, the demand function \(D(p)=a / p^{b}\) has a constant elasticity for all positive prices.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Demand Function Differentiation
The differentiation of a demand function is a fundamental step in determining how sensitive the quantity demanded is to changes in price. In the context of the exercise, the demand function is given as
\[ D(p) = \frac{a}{p^b} \]
where \( a \) and \( b \) are positive real numbers and \( p \) represents the price. The differentiation of this function with respect to price, using calculus, is crucial for analyzing the price elasticity of demand. By applying calculus principles, specifically the power rule of differentiation, we can calculate the derivative of the demand function. Calculus thus proves to be a valuable tool in economics for understanding market dynamics and consumer behavior.
Constant Elasticity
Elasticity in economics refers to the responsiveness of one variable to changes in another variable. The concept of constant elasticity refers to a situation where the elasticity is the same, regardless of the price level. In the given exercise, after differentiating the demand function, we arrived at an expression for elasticity that is a multiple of the price \( p \), specifically
\[ E = -bp \]
Since \( b \) is a constant, the rate of change of quantity demanded in response to a change in price does not depend on the price itself but only on this fixed exponent. This characteristic is known as constant elasticity because the responsiveness, or elasticity, does not vary with the price鈥攊t remains constant over different price levels.
Power Rule of Differentiation
The power rule is a basic rule in calculus that is used to differentiate functions of the form \( x^n \), where \( n \) is any real number. According to the power rule, the derivative of such a function is \( nx^{n-1} \). This rule streamlines the differentiation process, especially for polynomial functions. In the exercise solution, the power rule is applied to the demand function equation
\[ D(p) = \frac{a}{p^b} \]
after rewriting the fraction as \( ap^{-b} \). The derivative with respect to price is then found as \[ -\frac{ab}{p^{b+1}} \]
This essential calculus technique enables economists and students alike to analyze various economic functions swiftly and accurately.
Calculus in Economics
Calculus is an indispensable mathematical tool in the field of economics. It allows economists to model and analyze the behavior of economic agents under various conditions. For instance, it facilitates the measurement of how changes in price affect supply and demand, which can then be used to make informed decisions about pricing, production, and policy-making. In the exercise, calculus helps us determine the elasticity of demand鈥攁 key concept that measures how the quantity demanded reacts to a change in price. Using calculus, we not only differentiate the demand function to find the rate of change but also apply those derivatives to calculate elasticity, optimal pricing, and other crucial economic variables.

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