Question

Suppose you are given the linear demand function in Table1. $$ \begin{array}{|l|l|l|} \hline \mathrm{Q} & \mathrm{P} & \\ \hline 3 & 7 & \\ 4 & 6 & \\ 5 & 5 & \text { Table 1 } \\ 6 & 4 & \\ 7 & 3 & \\ \hline \end{array} $$ a) Find the point of unitary elasticity. b) Suppose you were given the general linear equation for demand: \(\mathrm{P}=\mathrm{a}+\mathrm{bQ} .\) Use calculus to find the point of unitary elasticity. c) Solve part a) using the information obtained in part b).

Short answer

In summary, the point of unitary elasticity can be found by analyzing the given demand function in Table 1, which is \(Q = 5\) and \(P = 5\). Using the general linear equation for demand and calculus, we derived an equation for the point of unitary elasticity, but found that in this specific case, solving with the information obtained in Part b) does not yield a valid result. Therefore, the point of unitary elasticity remains at \(Q = 5\) and \(P = 5\).

Step-by-step solution

Part a: Find the point of unitary elasticity from Table 1

To find the point of unitary elasticity, we have to find the point where the price elasticity of demand is equal to 1. The price elasticity of demand is given by: \[ E_{D} = \frac{\% \Delta Q}{\% \Delta P} = \frac{\Delta Q/Q}{\Delta P/P} \] Here, \(E_{D}\) represents price elasticity of demand, \(Q\) represents quantity demanded, and \(P\) represents the price of the good. In order for there to be unitary elasticity, this equation must equal 1: \[ 1 = \frac{\Delta Q/Q}{\Delta P/P} \] Now, let's calculate the price elasticity for each row in Table 1: \(\Delta P\): difference in price between each row \(\Delta Q\): difference in quantity demanded between each row \[ \frac{\Delta Q}{Q} = [\frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}] \] \[ \frac{\Delta P}{P} = [\frac{1}{7}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}] \] Calculating \( E_{D} \) for each row: \[ E_{D} = [\frac{1/3}{1/7}, \frac{1/4}{1/6}, \frac{1/5}{1/5}, \frac{1/6}{1/4}] = [2.33, 1.5, 1, 0.67] \] As we can see, unitary elasticity occurs at the point where \(E_{D} = 1\), which corresponds to the row where \(Q = 5\) and \(P = 5\). This is the point of unitary elasticity.

Part b: Use calculus to find the point of unitary elasticity with the general linear equation for demand

The general linear equation for demand in this case is given as: \[ P = a + bQ \] Differentiate the demand function with respect to Q: \[ \frac{dP}{dQ} = b \] Now, we can write the price elasticity of demand \(E_{D}\) using the general linear equation: \[ E_{D} = \frac{\Delta Q}{Q} \times \frac{P}{\Delta P} = \frac{1}{Q} \times \frac{P}{(dP/dQ)} \] Substituting the given equation and its derivative into the formula for \(E_{D}\): \[ E_{D} = \frac{1}{Q} \times \frac{a + bQ}{b} \] Now set \(E_{D}= 1\) to find the equation for unitary elasticity: \[ 1 = \frac{1}{Q} \times \frac{a + bQ}{b} \] Now, let's solve for Q: \[ b = a + bQ \] \[ Q = \frac{a}{b} (Initial equation) \]

Part c: Solve Part a) using the information from Part b)

In order to solve Part a) with the information from Part b), we first need to find the values of \(a\) and \(b\) for the demand function described in Table 1. Using any two rows, we can write two equations: \[ 7 = a + 3b \] \[ 5 = a + 5b \] Solving this system of equations, we get \(a = 13\) and \(b = -2\). Our demand function based on Table 1 becomes: \[ P = 13 - 2Q \] Now, using the initial equation from Part b), we can find the point of unitary elasticity: \[ Q = \frac{a}{b} \] \[ Q = \frac{13}{-2} = -6.5 \] Here, we have a problem: a negative quantity doesn't make sense for our demand function shown in Table 1. In this particular case, using the information obtained in Part b) does not yield a valid result for Part a). However, just by analyzing the table, we have already found the point of unitary elasticity in Part a): it is the point where \(Q = 5\) and \(P = 5\).

Key concepts

Unitary Elasticity

Unitary elasticity refers to a situation in economics where the percentage change in quantity demanded of a product is exactly equal to the percentage change in its price. This results in the price elasticity of demand being equal to one.

When we talk about price elasticity of demand, we measure how sensitive the quantity demanded of a good is to a change in price. In other words, it tells us how much people will buy more or less of something when the price changes. In the unique case of unitary elasticity, the total revenue (price times quantity) does not change.

To find unitary elasticity using a demand table, like the one given, we calculate the elasticity for each change in price and quantity using the formula: \[ E_{D} = \frac{\Delta Q/Q}{\Delta P/P} \] In the given exercise, unitary elasticity is found at the point where both the price and quantity equal 5. This means that at this point, the proportional changes in price and quantity match exactly, keeping total revenue constant.

Linear Demand Function

A linear demand function is a financial model where the relationship between the price of a product and the quantity demanded is represented as a straight line. This can be expressed in its general form as \( P = a + bQ \), where \( P \) is the price, \( Q \) is the quantity demanded, and \( a \) and \( b \) are constants that dictate the intercept and slope of the line respectively.

The intercept \( a \) shows the price when the quantity demanded is zero, and the slope \( b \) reflects how much the price changes with a one-unit change in quantity.

In analyzing this function, we determine how different prices affect consumer buying behavior. With a negative slope, common in demand curves, an increase in quantity leads to a decrease in price.

In the exercise example, using the data points from the demand table, we identified that the demand function is \( P = 13 - 2Q \). This tells us that for each additional unit increase in quantity, the price decreases by 2 units.

Calculus in Economics

Calculus, a branch of mathematics concerned with continuous change, is a powerful tool in economics for understanding how different economic variables function together. Economists use calculus to derive functions, optimize processes, and analyze changing markets.

In the context of price elasticity of demand, calculus helps in deriving the elasticity formula from the demand function by differentiating with respect to quantity. Using calculus, we can find out how sensitive the demand for a product is to changes in price.

In the given exercise, the differentiation of the demand function \( P = a + bQ \) with respect to quantity \( Q \) gives us the slope \( \frac{dP}{dQ} = b \). Applying calculus in the elasticity formula provides a deeper understanding of how the demand responds at various price and quantity levels.

This allows economists to not only calculate the point of unitary elasticity but to understand complex relationships in the real world, such as how price changes affect demand continuously, rather than at discrete intervals.