Question

Demand functions and elasticity Economists use demand finctions to describe how much of a commodity can be sold at varying prices. For example, the demand function $D(p)=500-10p$ says that at a price of $p=10,$ a quantity of $D(10)=400$ units of the commodity can be sold. The elasticity $E=\frac{dD}{dp}\frac{p}{D}$ of the demand gives the approximate percent change in the demand for every $1\mathrm{\%}$ change in the price. (See Section 3.5 or the Guided Project Elasticity in Economics for more on demand functions and elasticity.) a. Compute the elasticity of the demand function $D(p)=500-10p$ b. If the price is $\mathrm{\$}12$ and increases by $4.5\mathrm{\%},$ what is the approximate percent change in the demand? c. Show that for the linear demand function $D(p)=a-bp$ where $a$ and $b$ are positive real numbers, the elasticity is a decreasing function, for $p\ge 0$ and $p\ne a/b$ 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: Compute the elasticity of the demand function, find the approximate percent change in demand, show that the elasticity is a decreasing function, and show that the demand function has constant elasticity. Answer: The elasticity of the demand function is $E=\frac{-10p}{500-10p}$. The approximate percent change in demand is $-1.42\mathrm{\%}$. The elasticity is a decreasing function for $p\ge 0$ and $p\ne a/b$. The demand function has constant elasticity $-b$ for all positive prices.

Step-by-step solution

(a. Compute the Elasticity of the Demand Function)

First, let's find the derivative of the demand function, $D(p)=500-10p$. We will use this to compute the elasticity. $dD/dp=-10$ Now, we can find the elasticity $E$ using the given formula: $E=\frac{dD/dp}{D/p}=\frac{-10}{(500-10p)/p}=\frac{-10p}{500-10p}$ So the elasticity of the demand function is $E=\frac{-10p}{500-10p}$.

(b. Approximate Percent Change in Demand)

Given that the price is $12andthepriceincreasesby4.5\mathrm{\%}$, we need to find the approximate percent change in the demand. First, let's find the new price and then find the new demand. Price $p_1=12$ New price $p_2=p_1\ast (1+0.045)=12\ast 1.045=12.54$ Now let's find the demand at both prices: Demand at $p_1$, $D(12)=500-10(12)=380$ Demand at $p_2$, $D(12.54)=500-10(12.54)\approx 374.6$ We can now find the percent change in demand: Percent Change in Demand $=\frac{D(p_2)-D(p_1)}{D(p_1)}\times 100=\frac{374.6-380}{380}\times 100\approx -1.42\mathrm{\%}$ So the approximate percent change in the demand is $-1.42\mathrm{\%}$.

(c. Show that Elasticity is a Decreasing Function)

For the linear demand function $D(p)=a-bp$, we first find the derivative: $dD/dp=-b$ Now, we find the elasticity, $E$: $E=\frac{-b}{(a-bp)/p}=\frac{-bp}{a-bp}$ To show that the elasticity function is a decreasing function, we need to find the derivative of $E$ with respect to $p$ and show that it is negative for $p\ge 0$ and $p\ne a/b$: $\frac{dE}{dp}=\frac{(-b)(a-bp)-(-bp)(-b)}{(a-bp{)}^2}=\frac{-ab+b^{2}p}{(a-bp{)}^2}$ Since $a$ and $b$ are positive real numbers, the numerator is negative for $p\ne a/b$. So, $E$ is a decreasing function for $p\ge 0$ and $p\ne a/b$.

(d. Show that the Demand Function has Constant Elasticity)

Given the demand function $D(p)=a/p^b$, we first find the derivative: $dD/dp=-ab{p}^{-(b+1)}$ Now, we find the elasticity $E$: $E=\frac{-ab{p}^{-(b+1)}}{(a/p^b)/p}$ $E=\frac{-ab{p}^{-(b+1)}}{a{p}^{-(b+1)}}$ $E=-b$ The elasticity is a constant, $-b$, for all positive prices, as it does not depend on the price $p$.

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is divided into two main parts: differential calculus, which studies the rate at which quantities change, and integral calculus, which looks at the accumulation of quantities. In the context of economics and the analysis of demand functions, calculus is used to determine the rate at which the quantity demanded of a good responds to changes in variables like its price.

In our example, calculus helped us find the derivative of the demand function, which provides the rate of change of demand with respect to price. Specifically, differentiation was used to find the slope of the demand curve at a given point. This slope, when utilized in the elasticity formula, tells us how responsive the quantity demanded is to price changes. Understanding this application of calculus is essential for anyone studying economics as it provides an analytical tool to measure and predict consumer behavior.

Price Elasticity of Demand

Price elasticity of demand is a measure used in economics to show how the quantity demanded of a good or service responds to a change in price. It is calculated by taking the percentage change in quantity demanded and dividing it by the percentage change in price. The elasticity of demand can be elastic, inelastic, or unitary. If the absolute value of elasticity is greater than 1, it is considered elastic, meaning consumers are quite responsive to price changes. Conversely, if it is less than 1, the demand is inelastic, implying consumers are not as responsive to price changes. At exactly 1, it is unitary elastic, indicating proportional responsiveness.

Understanding this concept is crucial for businesses as it affects pricing strategies and revenue. The example above illustrates how to calculate the elasticity of a linear demand function, providing insights into how sensitive the demand for a product is to changes in its price. This information is invaluable for making informed decisions about pricing strategies.

Percent Change in Demand

The percent change in demand is a way to express the change in the quantity demanded as a proportion of the initial amount demanded. It’s calculated by finding the difference between the new and the original quantity demanded, dividing by the original quantity demanded, and then multiplying by 100 to get a percentage. This metric is often used in tandem with price changes to analyze the effect of pricing on sales volume.

In the provided example, we calculated that a 4.5% price increase led to an approximate 1.42% decrease in demand, demonstrating a relatively inelastic response to the price change. This is a practical application of demand elasticity, helping to understand and forecast the impact of pricing decisions on sales volumes.

Elasticity of Linear Demand Functions

In the case of linear demand functions, which have the form $D(p)=a-bp$, elasticity changes as the price changes. However, it is important to note that for these functions, elasticity is always a decreasing function of price, meaning that as price increases, the demand becomes less responsive to further price changes. This characteristic stems from the constant slope of the linear demand curve. As price moves higher, the quantity demanded decreases, making the percentage change in quantity smaller relative to the percentage change in price.

By showing that the derivative of the elasticity with respect to the price is negative, we demonstrate that elasticity decreases as price increases. This is critical for pricing analyses: as prices go up, further price increases tend to have a smaller effect on reducing demand. For businesses, this implies that there is a diminishing return to raising prices, which can inform pricing tactics and revenue optimization strategies.