Question

Imagine a market for \(X\) composed of four individuals: Mr. Pauper \((P),\) Ms. Broke \((B),\) Mr. Average \((A),\) and \(\mathrm{Ms}\). Rich \((R)\). All four have the same demand function for \(X\) : It is a function of income \((I), P_{x}\) and the price of an important subtitute \((Y),\) for \(X\) $$X=\frac{\sqrt{I P_{v}}}{2 P_{x}}$$ a. What is the market demand function for \(X\) ? If \(P_{x}=P_{y}=1, I_{F}=I_{h}=16, I_{A}=25,\) and \(I_{R}=100,\) what is the total market demand for \(X\) ? What is \(e_{x r_{x}}\) ? \(e_{x, p_{y}}\) ? \(e_{x, f} ?\) b. If \(P_{X}\) doubled, what would be the new level of \(X\) demanded? If Mr. Pauper lost his job and his income fell 50 percent, how would that affect the market demand for \(X ?\) What if Ms. Rich's income were to drop 50 percent? If the government imposed a 100 percent \(\operatorname{tax}\) on \(Y,\) how would the demand for \(X\) be affected? c. If \(I_{p}=I_{s}=I_{A}=I_{R}=25,\) what would be the total demand for \(X\) ? How does that figure compare with your answer to (a)? Answer (b) for these new income levels and \(P_{x}=P_{y}=1\) d. If Ms. Rich found \(Z\) a necessary complement to \(X\), her demand function for \(X\) might be described by the function $$X=\frac{I P_{Y}}{2 P_{X} P_{z}}$$ What is the new market demand function for \(X\) ? If \(P_{x}=P_{v}=P_{z}=1\) and income levels are those described by (a), what is the demand for \(X\) ? What is \(e_{x, P_{x}}\) ? \(e_{x, p_{y}}\) ? \(e_{x, 1} ? e_{x, p_{x}}\) ? What is the new level of demand for \(X\) if the price of \(Z\) rises to 2 ? Notice that Ms. Rich is the only one whose demand for \(X\) drops.

Short answer

Based on the given information, we found the market demand function, the total market demand, and calculated the elasticity coefficients for price, substitute price, and income. We also examined how changes in the prices and incomes affect the market demand for X, and incorporated a new demand function for Ms. Rich, finding the new market demand function and the effect on the demand due to changes in prices.

Step-by-step solution

Add the demand functions of the individuals

Since all individuals have the same demand function \(X= \frac{\sqrt{I P_{v}}}{2 P_{x}}\): Market Demand = \(X_P+X_B+X_A+X_R\) Where \(X_P, X_B, X_A, X_R\) are the demands of Mr. Pauper, Ms. Broke, Mr. Average, and Ms. Rich, respectively. ##Total Market Demand##

Plug in the values of prices and incomes

Given \(P_x = P_y = 1\), \(I_P = I_B = 16\), \(I_A = 25\), and \(I_R = 100\), we find the total market demand: \(X_P = \frac{\sqrt{16}}{2} = 2\) \(X_B= 2\) (same as Mr. Pauper) \(X_A = \frac{\sqrt{25}}{2} = \frac{5}{2}\) \(X_R = \frac{\sqrt{100}}{2} = 5\) Total Market Demand = \(X = 2+2+\frac{5}{2}+5=12\) ##Price Elasticity##

Calculate the \(e_{x, r_{x}}\)

Price elasticity \(e_{x, r_{x}} = (\frac{\partial X}{\partial P_x}) \cdot \frac{P_x}{X}\). Since \(X=\frac{\sqrt{I P_{v}}}{2 P_{x}}\), we can differentiate with respect to \(P_x\): \(\frac{\partial X}{\partial P_x} = - \frac{\sqrt{I P_{v}}}{2 P_{x}^2}\) \(e_{x, r_{x}}= - \frac{\sqrt{I P_{v}}}{2 P_{x}^2} \cdot \frac{P_x}{X} = -\frac{1}{2}\) ##Elasticity of the substitute price##

Calculate the \(e_{x, p_{y}}\)

Elasticity \(e_{x, p_{y}} = (\frac{\partial X}{\partial P_y}) \cdot \frac{P_y}{X}\). Since \(X=\frac{\sqrt{I P_{v}}}{2 P_{x}}\), we can differentiate with respect to \(P_y\): \(\frac{\partial X}{\partial P_y} = \frac{1}{2} \cdot \frac{I}{\sqrt{I P_{y} P_{x}^2}}\) \(e_{x, p_{y}} = \frac{1}{2} \cdot \frac{I}{\sqrt{I P_{y} P_{x}^2}} \cdot \frac{P_y}{X} = \frac{1}{2}\) ##Income Elasticity##

Calculate the \(e_{x, f}\)

Income elasticity \(e_{x, f} = (\frac{\partial X}{\partial I}) \cdot \frac{I}{X}\). Since \(X=\frac{\sqrt{I P_{v}}}{2 P_{x}}\), we can differentiate with respect to \(I\): \(\frac{\partial X}{\partial I} = \frac{1}{4} \cdot \frac{P_y}{\sqrt{I P_{y} P_{x}}}\) \(e_{x, f} = \frac{1}{4} \cdot \frac{P_y}{\sqrt{I P_{y} P_{x}}} \cdot \frac{I}{X} = 1\) ##b. Changes in Price##

Double the price of \(P_x\)

If \(P_x\) doubles to 2, we can plug this value back into the demand function: \(X=\frac{\sqrt{I P_{v}}}{4}\) From part a, we know the market demand for \(X\) when \(P_x = 1\). We can find the new market demand for \(X\) by substituting the original income values with the doubled price of \(P_x\). For Mr. Pauper and Ms. Broke: \(X = \frac{\sqrt{16}}{4} = 1\) For Mr. Average: \(X = \frac{5}{4}\) For Ms. Rich: \(X = \frac{5}{2}\) New total market demand for \(X\): \(X = 1+1+\frac{5}{4}+\frac{5}{2} = \frac{19}{4} = 4.75\) ##Changes in Incomes##

Decrease Mr. Pauper's income by 50%

If Mr. Pauper's income decreases by 50% to 8, his new demand for \(X\) would be: \(X_P = \frac{\sqrt{8}}{2} = \sqrt{2}\) New total market demand for \(X\): \(X = \sqrt{2}+2+\frac{5}{2}+5 = 11-\sqrt{2}\)

Decrease Ms. Rich's income by 50%

If Ms. Rich's income decreases by 50% to 50, her new demand for \(X\) would be: \(X_R = \frac{\sqrt{50}}{2} = \frac{5\sqrt{2}}{2}\) New total market demand for \(X\): \(X = 2+2+\frac{5}{2}+\frac{5\sqrt{2}}{2} = \frac{17}{2} + \frac{5\sqrt{2}}{2}\) ##Tax on Substitute##

Impose a 100% tax on \(Y\)

If the government imposed a 100% tax on \(Y\), the price of \(Y\) would become \(P_y = 2\). Substitute this value back into the demand function: \(X = \frac{\sqrt{2I}}{2P_x}\) Then, plug in the incomes and price of \(X\): Mr. Pauper and Ms. Broke: \(X = \frac{\sqrt{32}}{2} = 2\sqrt{2}\) Mr. Average: \(X = \frac{\sqrt{50}}{2} = \frac{5\sqrt{2}}{2}\) Ms. Rich: \(X = \frac{\sqrt{200}}{2} = 5\sqrt{2}\) New total market demand for \(X\): \(X = 2\sqrt{2}+2\sqrt{2}+\frac{5\sqrt{2}}{2}+5\sqrt{2} = 12\sqrt{2}\) ##c. Equal Incomes##

Calculate total demand for \(X\) with equal incomes

If \(I_P=I_B=I_A=I_R=25\), we can plug these values back into the demand function along with \(P_x = P_y = 1\): \(X_P = X_B = X_A = X_R = \frac{\sqrt{25}}{2} = \frac{5}{2}\) Total market demand for \(X\) = \(4 \times \frac{5}{2} = 10\) ##Changes with Equal Incomes##

Answer part b with new income levels

Since the total market demand for \(X\) with equal incomes is 10, we will analyze the impact of the price and income changes using this value. 1. If \(P_x\) doubled to 2: \(X = \frac{\sqrt{I P_v}}{4} = \frac{5}{4}\) per person Total market demand for \(X\) = \(4 \times \frac{5}{4} = 5\) 2. If Mr. Pauper's income falls 50% to 12.5, his new demand for \(X\) would be: \(X_P = \frac{\sqrt{12.5}}{2} = \frac{5\sqrt{1/2}}{2}\) New total market demand for \(X\): \(X = \frac{5\sqrt{1/2}}{2} + \frac{10}{2} - \frac{5}{2} = \frac{15}{2} + \frac{5\sqrt{1/2}}{2}\) 3. If Ms. Rich's income falls 50% to 12.5, then she is in the same situation as Mr. Pauper, and we can substitute their values like we did in part 2. 4. If the government imposed a 100% tax on \(Y\), and the price of \(Y\) becomes 2: \(X = \frac{\sqrt{2I}}{2P_x} = \frac{5\sqrt{2}}{2}\) per person Total market demand for \(X\) = \(4 \times \frac{5\sqrt{2}}{2} = 10\sqrt{2}\) ##d. New Demand Function for Ms. Rich##

Find the new market demand function

Ms. Rich's new demand function is \(X_R = \frac{I P_y}{2 P_x P_z}\), while for the three others, we keep the original demand function. New Market Demand Function: \(X = X_P + X_B + X_A + X_R\) ##New Values for the Market Demand##

Plug in the values

With \(P_x=P_y=P_z=1\), and the given incomes in part (a): Ms. Rich's new demand for \(X\): \(X_R = \frac{100}{2} = 50\) New total market demand for \(X\): \(X = 2+2+\frac{5}{2}+50 = 56+\frac{5}{2}\) ##New Price Elasticity##

Calculate the new \(e_{x, P_x}\) for Ms. Rich

Since \(X_R=\frac{I P_{y}}{2 P_{x} P_{z}}\), differentiate with respect to \(P_x\): \(\frac{\partial X_R}{\partial P_x} = - \frac{I P_{y}}{2 P_{x}^2 P_{z}}\) \(e_{x, P_x} = - \frac{I P_{y}}{2 P_{x}^2 P_{z}} \cdot \frac{P_x}{X_R} = - \frac{1}{2P_z} = -\frac{1}{2}\) ##New Elasticity of the Substitute Price##

Calculate the new \(e_{x, p_y}\) for Ms. Rich

Since \(X_R=\frac{I P_{y}}{2 P_{x} P_{z}}\), differentiate with respect to \(P_y\): \(\frac{\partial X_R}{\partial P_y} = \frac{I}{2 P_{x} P_{z}}\) \(e_{x, p_y} = \frac{I}{2 P_{x} P_{z}} \cdot \frac{P_y}{X_R} = \frac{1}{2P_z} = \frac{1}{2}\) ##New Elasticity of Complement##

Calculate the new \(e_{x, P_z}\) for Ms. Rich

Since \(X_R=\frac{I P_{y}}{2 P_{x} P_{z}}\), differentiate with respect to \(P_z\): \(\frac{\partial X_R}{\partial P_z} = - \frac{I P_{y}}{2 P_{x} P_{z}^2}\) \(e_{x, P_z} = - \frac{I P_{y}}{2 P_{x} P_{z}^2} \cdot \frac{P_z}{X_R} = -\frac{1}{2P_z} = -\frac{1}{2}\) ##New Level of Demand##

Calculate new level of demand for \(X\) if \(P_z\) rises to 2

If \(P_z\) rises to 2, Ms. Rich's new demand for \(X\) would be: \(X_R = \frac{100}{4} = 25\) The new total market demand for \(X\) would be: \(X = 2 + 2 + \frac{5}{2} + 25 = 29 + \frac{5}{2}\)

Key concepts

Individual Demand Function

In economics, an individual demand function helps us understand how a single consumer's quantity demanded for a good changes based on various factors such as income, price of the good, and the price of substitute goods. Each individual's demand can be expressed as a specific equation that reflects these influences.
For instance, if individuals have the same demand function, like the one given as \( X=\frac{\sqrt{IP_v}}{2P_x} \), it illustrates how demand for good \( X \) changes based on income \( I \), price of good \( X, \, P_x \), and price of a substitute \( P_v \).
This mathematical expression helps us aggregate individual demand into a market demand curve by summing up the demands of all consumers within the market. Understanding each function can predict how changes in economic conditions will affect overall demand.

Price Elasticity

Price elasticity measures how sensitive the quantity demanded of a good is to a change in its price. Specifically, it shows the percentage change in quantity demanded resulting from a 1% change in price. The formula to calculate price elasticity, \( e_{x, r_{x}} \), is given by the expression \((\frac{\partial X}{\partial P_x}) \cdot \frac{P_x}{X}\).
A key outcome of this calculation is determining whether a product is elastic, inelastic, or unitary. For example, if elasticity is greater than 1, the demand is elastic, meaning consumers are sensitive to price changes.
Conversely, if elasticity is less than 1, the demand is inelastic, indicating consumers are less affected by price changes. In our example, the price elasticity was \(-\frac{1}{2}\), which indicates an inelastic demand, leading us to infer that changes in price have a smaller impact on the quantity demanded of \( X \).

Income Elasticity

Income elasticity helps determine how sensitive demand for a good is to changes in consumer income. It is represented as \( e_{x, f} \) and is calculated through \((\frac{\partial X}{\partial I}) \cdot \frac{I}{X}\).
This measurement allows us to classify goods as normal or inferior based on how demand changes with income variations.
For instance, if the income elasticity of demand is greater than one, it is classified as a luxury, and demand increases more than proportionately to income.
If it's between zero and one, it's a normal good, meaning demand rises with income but less proportionately.
If it's negative, the good is considered inferior, with demand decreasing as income increases. In our example, the income elasticity was calculated as \( 1 \), indicating that good \( X \) is a normal good.

Substitute Goods

Substitute goods are products that can replace one another in consumption, meaning an increase in the price of one leads to an increase in demand for the other.
The concept of substitute goods is crucial in understanding cross elasticity of demand, denoted as \( e_{x, p_{y}} \), which measures how the quantity demanded for one good changes in response to price changes of a related good.
In the given scenario, the demand function \( X=\frac{\sqrt{IP_v}}{2P_x} \) reflects the role of \( P_v \) as the price of a substitute.
The cross elasticity \( e_{x, p_{y}} \) is typically positive; an increase in the price of a substitute, like any increase in \( P_y \), results in increased demand for good \( X \).
This relationship helps firms understand competitive dynamics and customer behavior in response to pricing strategies of substitute goods.