Homework Assignment 3

Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC

Instructions:

You will scan your homework into a single PDF file to be upload.
Only legible assignments will be graded.
Late assignments will not be accepted.

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.

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 $\left(\left|\varepsilon_{Q^{d},P}\right|<1\right)$?

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.

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.

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 $p=40$.