Question

Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function \(D(p)=500-10 p\) says that at a price of \(p=10,\) a quantity of \(D(10)=400\) units of the commodity can be sold. The elasticity \(E=\frac{d D}{d p} \frac{p}{D}\) of the demand gives the approximate percent change in the demand for every \(1 \%\) change in the price. (See Section 3.6 or the Guided Project Elasticity in Economics for more on demand functions and elasticity.) a. Compute the elasticity of the demand function \(D(p)=500-10 p\) b. If the price is $$ 12\( and increases by \)4.5 \%,\( what is the approximate percent change in the demand? c. Show that for the linear demand function \)D(p)=a-b p\( where \)a\( and \)b\( are positive real numbers, the elasticity is a decreasing function, for \)p \geq 0\( and \)p \neq a / b\( d. Show that the demand function \)D(p)=a / p^{b},\( where \)a\( and \)b$ are positive real numbers, has a constant elasticity for all positive prices.

Short answer

Question: Calculate the elasticity of the demand function \(D(p) = 500 - 10p\), the approximate percent change in demand when the price is \(12\) and increases by \(4.5\%\), and show whether the elasticity is a decreasing function for linear demand functions and if the demand function \(D(p) = a/p^b\) has a constant elasticity. Answer: 1. The elasticity of the demand function \(D(p) = 500 - 10p\) is given by \(E = (-10)\frac{p}{500-10p}.\) 2. The approximate percent change in demand when the price is \(12\) and increases by \(4.5\%\) is approximately -1.42%. 3. The elasticity is a decreasing function for linear demand functions. 4. The demand function \(D(p) = a/p^b\) has a constant elasticity.

Step-by-step solution

Part a: Compute the elasticity of the demand function \(D(p)=500-10 p\)

To begin with, we first recall that the elasticity \(E\) is defined as \(E=\frac{dD}{dp} \frac{p}{D}.\) Using the given formula \(D(p) = 500 - 10p\), we first find the derivative. To find \(\frac{dD}{dp}\), calculate the derivative with respect to \(p\): $$\frac{dD}{dp} = -10$$ Now, we can plug the derivative and the demand function into the formula for the elasticity: $$E = (-10)\frac{p}{500-10p}$$

Part b: Approximate percent change in demand

Given that the price \(p\) is \(12\) and increases by \(4.5 \%,\) we need to find the approximate percent change in demand. Start by calculating the original demand at the price of \(12\): $$D(12) = 500 - 10(12) = 380$$ Then, we adjust the price to \(p = 12 + 12 * 4.5\% = 12 + 0.54 = 12.54\). Now, we can find the new demand at the adjusted price: $$D(12.54) = 500 - 10(12.54) \approx 374.60$$ Finally, compute the approximate percent change in demand: $$\frac{D(12.54) - D(12)}{D(12)} \times 100\% \approx \frac{374.60 - 380}{380} \times 100\% \approx -1.42\%$$

Part c: Show that the elasticity is a decreasing function

We apply the same procedure as in part a to the linear demand function \(D(p) = a - bp\). First, find the derivative with respect to \(p\): $$\frac{dD}{dp} = -b$$ Now, use the derivative and the demand function in the formula for elasticity: $$E = (-b)\frac{p}{a-bp}$$ At this point, we need to show that the elasticity is a decreasing function. Compute the second derivative of the elasticity with respect to \(p\): $$\frac{d^2E}{dp^2} = \frac{2b^2}{(a-bp)^3}$$ Since \(a\) and \(b\) are positive real numbers, and \((a-bp)^3\) is always positive, we see that the second derivative is positive, making the elasticity a decreasing function for \(p \geq 0\) and \(p \neq a/b\).

Part d: Show that \(D(p) = a/p^b\) has a constant elasticity

Start by finding the derivative of the given demand function \(D(p) = a/p^b\) with respect to \(p\): $$\frac{dD}{dp} = -abp^{-b-1}$$ Next, insert the derivative and the demand function into the formula for the elasticity: $$E = (-abp^{-b-1}) \frac{p}{ap^{-b}}$$ Now, cancel out the "a" terms and simplify the expression: $$E = -abp^{-b-1}p^b = -bp^{-1}$$ This expression is a constant with respect to \(p\), indicating that the demand function \(D(p) = a/p^b\) has a constant elasticity for all positive prices.

Key concepts

Understanding the Demand Function

At its core, a demand function provides a mathematical model that describes the relationship between the price of a good or a service and the quantity demanded by consumers. In economic terms, it is written as a formula, such as \(D(p)=500-10p\), where \(D(p)\) represents the quantity demanded when the price is \(p\). This function allows us to predict how changes in price can affect demand. For instance, if the price increases, the model shows a corresponding decrease in demand, which reflects a basic principle in economics: as the price goes up, the quantity demanded generally goes down, all else being equal.

When using a demand function, it's important to keep in mind that it is a simplified model of reality. Actual demand can be influenced by many factors beyond price, such as consumer preferences, income levels, and the presence of substitute goods. However, demand functions provide a useful tool for examining the primary relationship between price and demand in theoretical scenarios and for making approximate predictions in real-world analysis.

Elasticity of Demand and Its Calculation

The concept of elasticity of demand is pivotal in understanding how sensitive the quantity demanded is to a change in price. Elasticity, denoted as \(E\), measures this responsiveness and is calculated using the formula \(E=\frac{dD}{dp} \frac{p}{D}\). A higher elasticity value indicates that demand is more sensitive to price changes, and vice versa.

In the exercise's step-by-step solution, we see that to find the elasticity of the demand function \(D(p)=500-10p\), we begin by deriving the first derivative \(\frac{dD}{dp} = -10\), then we apply the formula to calculate elasticity. The negative sign represents the inversely proportional relationship between price and quantity demanded – as one increases, the other decreases. Understanding this calculation is key for businesses and policymakers when considering price changes and their potential effects on sales volume and revenue.

Calculating Percent Change in Demand

Percent change in demand is a practical measure indicating how much the demand for a good or service has altered over a period or due to some change in conditions, like price. To calculate this, we take the difference in quantity demanded before and after the change, divide by the original demand, then multiply by 100 to get a percentage. Mathematically, it’s expressed as \(\frac{new demand - original demand}{original demand} \times 100\%\).

In our exercise example, when the price increases by 4.5%, we calculate the new demand at this price and then apply the formula to find that demand has decreased by approximately 1.42%. It’s important for students to grasp that percent change offers a way to quantify the impact of price fluctuations on demand in a relatable, understandable format, which can inform decision-making in business strategies and economic policies.