Homework Assignment 3

Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC

Only two randomly-chosen questions will be graded.

Question 1:

Suppose there are 90 consumers in the market with two goods $x$ and $y$. There are two types of individuals A and B. 60 of the consumers are type A and the 30 of them are type B. Type A consumers have a utility function of $u(x,y) = x^{1/3}y^{1/3}$ and each of them has an income of $m_A$. Type B consumers have a utility function of $u(x,y) = x^{2/3}y^{1/3}$ and each of them has an income of $m_B$.

  1. Find the individual demand for type A for good $x$ and $y$ as a function of $m_A, p_x$, and $p_y$.
  2. Find the individual demand for type B for good $x$ and $y$ as a function of $m_B, p_x$, and $p_y$.
  3. Find the market demand for good $x$ as a function of incomes and prices.
  4. Find the market demand for good $y$ as a function of incomes and prices.
  5. Suppose price of good $x$ is 2 and the price good $y$ is 2. Further suppose $m_A = 60$ and $m_B = 120$. Find the market demand for good $x$ and good $y$ at the given prices and incomes.

Solutions 1:

  1. From the general formula of Cobb-Douglas demand function: $$ \begin{align} x_A(p_x, p_y, m_A) = \frac{a}{a+b} \frac{m_A}{p_x} = \frac{1}{2} \frac{m_A}{p_x} \\ y_A(p_x, p_y, m_A) = \frac{b}{a+b} \frac{m_A}{p_y} = \frac{1}{2} \frac{m_A}{p_y} \\ \end{align} $$ We can also use tangency condition to solve for the demand for the goods x and y.

  2. $$ \begin{align} x_B(p_x, p_y, m_B) = \frac{a}{a+b} \frac{m_B}{p_x} = \frac{2}{3}\frac{m_B}{p_x} \\ y_B(p_x, p_y, m_B) = \frac{b}{a+b} \frac{m_B}{p_y} = \frac{1}{3}\frac{m_B}{p_y} \\ \end{align} $$

  3. $$ \begin{align} X(p_x, p_y, m_A, m_B) = 60 \times x_A(p_x, p_y, m_A) + 30 \times x_B(p_x, p_y, m_B) = 30 \frac{m_A}{p_x} +20 \frac{m_B}{p_x} \end{align} $$

  4. $$ \begin{align} Y(p_x, p_y, m_A, m_B) = 60 \times y_A(p_x, p_y, m_A) + 30 \times y_B(p_x, p_y, m_B) = 30 \frac{m_A}{p_y} +10 \frac{m_B}{p_y} \end{align} $$

  5. $$ \begin{align} X(p_x, p_y, m) = 30 \frac{60}{2} + 20 \frac{120}{2} = 900 + 1200 = 2100 \\ Y(p_x, p_y, m) = 30 \frac{60}{2} + 10 \frac{120}{2} = 900 + 600 = 1500 \end{align} $$

Question 2:

Suppose that daily demand for breakfast sandwiches at a local store is given by the following:$Q^{d}=16-4P$

  1. What is the formula for the own price elasticity of demand as a function of price? In other words, please provide a formula for the price elasticity of demand where the only variable on the right-hand-side is price.

  2. What is the price elasticity of demand for breakfast sandwiches at the price of $1, 2, 3.5,$ and $3.75$, respectively:

  3. As you can see, the price elasticity is different depending on the values of prices at which it is evaluated. For what price is the price elasticity of demand equal to one? In other words, for what price is demand unit elastic?

  4. For what range of prices is demand elastic $\left(\left|\varepsilon_{Q^{d},P}\right|>1\right)$? For what range of prices is demand inelastic $\left(\left|\varepsilon_{Q^{d},P}\right|<1\right)$?

Solutions 2:

  1. $ \varepsilon_{Q^{d},P} = \frac{\partial Q^{d}}{\partial P}\cdot\frac{P}{Q^{d}}= -4\cdot\frac{P}{16-4P}=\frac{-P}{4-P} $
  2. Plug different levels of $P$'s into the formula in Q2.1. The answers are, in absolute value, $1/3, 1, 7,$ and $15$, respectively.

  3. From part 1 we have, $ \varepsilon_{Q^{d},P} = \frac{-P}{4-P} $ $$\begin{align} \implies -1&=\frac{-P}{4-P} \\ \implies 4-P&=P \\ \implies 2P&=4 \\ \implies P&=2 \end{align}$$

  4. For the given demand function, demand becomes more elastic as the price rises. Therefore, demand is elastic for $P \in (2,4]$. Conversely, for $P \in [0,2)$, it is inelastic. Note that elasticity is zero at $P=0$.

Question 3:

Suppose we have a market with 200 individuals with preferences over two goods, $x$ and $y$.

  1. Consider that all 200 individuals have the utility function $U=x^{1/3}y^{2/3}$ and that each individual has the same income and is subject to the same prices. Calculate the market demand for $x$ as a function of income and prices.

  2. Find the own price elasticity of demand for good $x$. Is the elasticity constant or is it changing at different values of $P_x$?

  3. Suppose now that we know $m=150, P_{x}=4,$ and $P_{y}=3$. What is the market demand for $x$? This should be a number.

  4. Consider now that 100 individuals $($type A$)$ have the utility function $U=x^{1/3}y^{2/3}$ and the other 100 $($type B$)$have the utility function $U=x^{2/3}y^{1/3}$. Each individual still has the same income and is subject to the same prices. Calculate the market demand for $x$ as a function of income and prices.

  5. Suppose now that we know $m=210, P_{x}=6$, and $P_{y}=4$. What is the market demand for $x$? This should be a number.

Solutions 3:

  1. $x = \frac{1}{3}\frac{m}{P_{x}}$ using the Cobb-Douglas demand formula also written in the Q1 above. This provides us with the individual demand. Since everyone is the same, market demand is just: $Q_{x}^{d}=\sum_{i=1}^{200}x_{i}=200 \times x=\frac{200}{3}\frac{m}{P_{x}}$

  2. $\frac{\partial Q^d_{x}}{\partial P_x} \frac{P_x}{Q^d_x} = -\frac{200}{3} m (\frac{1}{P^2_x}) \Big(\frac{P_x}{\frac{200}{3} \frac{m}{P_x}} \Big) = -1$. Cobb-Douglas preferences features a constant elasticity of demand. More generally, any demand function that has a form of $Q = Ap^{\varepsilon}$ has a constant elasticity of demand.

  3. We use the formula from above: $Q_{x}^{d}=200 \times \frac{1}{3}\times\frac{150}{4}=2,500$

  4. We already know type A individuals will have the individual demand: $x_{A}=\frac{1}{3}\frac{m}{P_{x}}$. Type B individuals will have the individual demand: $x_B = \frac{2}{3}\frac{m}{P_{x}}$. Market demand, therefore, is: $Q_{x}^{d}=\sum_{i=1}^{200}x_{i}=100\left(\frac{1}{3}\frac{m}{P_{x}}\right)+100\left(\frac{2}{3}\frac{m}{P_{x}}\right)=\frac{300 m}{3 P_{x}}=\frac{100 m}{ P_{x}}$

  5. Using the formula from above, $Q_{x}^{d}=100\times\frac{210}{6}=3,500$

Question 4:

Suppose we have two people in the market: A, B. Their utility functions are $U_A=min ${$ x_1,x_2 $}$ $, $U_B=min ${$ x_1,\frac{x_2}{2} $}$ $. Solve their optimal choices and then find market demand.

Solutions 4:

For consumer A: The optimal choice is given by $x_1^* = x_2^*$ and then, in the budget constraint we get $$\begin{align} p_{1}x_{1}^{*} + p_{2}x_{1}^{*} &= m_{A} \\ \implies x_{1}^{*} &= \frac{m_{A}}{p_{1} + p_{2}} \\ \implies x_{2}^{*} &= \frac{m_{A}}{p_{1} + p_{2}} \end{align}$$ And for consumer B: The optimal choice is given by $x_1^* = \frac{x_2^*}{2} \implies 2x_1^* = x_2^* $ and, then in the budget constraint we get $$\begin{align} p_{1}x_{1}^{*} + 2p_{2}x_{1}^{*} &= m_{B} \\ \implies x_{1}^{*} &= \frac{m_{B}}{p_{1} + 2p_{2}} \\ \implies x_{2}^{*} &= \frac{2m_{B}}{p_{1} + 2p_{2}} \end{align}$$ Market Demand for good 1 $ = \frac{m_{A}}{p_{1} + p_{2}} + \frac{m_{B}}{p_{1} + 2p_{2}} $ And, market Demand for good 2 $ = \frac{m_{A}}{p_{1} + p_{2}} + \frac{2m_{B}}{p_{1} + 2p_{2}} $

Question 5:

Suppose we have two people in the market for good x: Adam, Barbara. Their demand functions are as follows: $x_A = max ${$ 0, 20 - 2p $}$ $, $x_B = max ${$ 0 , 30 - 2p $}$ $. Find market demand and the price elasticity of demand at $p=40$.

Solutions 5:

For Adam: $$\begin{align} x_{A} \ \text{at P = 40} = max(0, 20-80) = max(0, -60) = 0 \end{align}$$ For Barbara: $$\begin{align} x_{B} \ \text{at P = 40} = max(0, 30-80) = max(0, -50) = 0 \end{align}$$ So, the market demand = 0. Elasticity of demand $ = 0$ because the demand curve is vertical when $p>15$, which is perfect inelastic.