Homework Assignment 2

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:

Luke's choice behavior can be represented by the utility function $u(x_1,x_2) = x_1 + x_2$. The prices of $x_1$ and $x_2$ are denoted as $p_1$ and $p_2$, and his income is $m$.

  1. Draw at least three indifference curves and find its slope $($i.e. MRS$)$. Is the MRS changing depending on the points of $(x_1,x_2)$ at which it is evaluated, or constant?

  2. Draw a budget constraint assuming that $p_1 < p_2$. Find the optimal bundle $ (x_1^{*},x_2^{*}) $ as a function of income and prices.

  3. Draw a budget constraint assuming that $p_1 > p_2$. Find the optimal bundle $ (x_1^{*},x_2^{*}) $ as a function of income and prices.

  4. Draw a budget constraint assuming that $p_1 = p_2$. Find the optimal bundle $ (x_1^{*},x_2^{*}) $ as a function of income and prices.

Question 2:

Consider a utility function $u(x_1, x_2) = 4x_1 + 3x_2$.

  1. What is the optimal bundle with $p_{x_1}$, $p_{x_2}$, and income $m$?
  2. What is the optimal bundle with $p_{x_1} = 2$, $p_{x_2} = 3$, and income $60$?

Question 3:

Paris has a utility function over berries $($denoted by $B$ $)$ and chocolate $($denoted by $C$ $)$ as follows: $$ U(B,C) = 6 ln(B) + 3 ln(C) $$ The price of berries and chocolate is $p_B$ and $p_C$, respectively. Paris's income is $m$.

  1. What preferences does this utility function represent?
  2. Find the $MRS_{BC}$ as a function of $B$ and $C$ assuming $B$ is on the x-axis.
  3. Find the optimal bundle B and C as a function of income and prices using the tangency condition.
  4. What is the fraction of total expenditure spent on berries and chocolate out of total income, respectively?
  5. Now suppose Paris has an income of ${$}$300. The price of a container of berries is ${$}$20 and the price of a chocolate bar is ${$}$10. Find the numerical answers for the optimal bundle, by plugging the numbers into the solution you found in Q3.3.

Question 4:

Consider a general utility function $U(x_1, x_2)$. Let's now solve for the optimal bundle generally using the Lagrangian Method.
1. Write down the objective function and constraint in math.
2. Set up the Lagrangian Equation.
3. Fnd the first derivatives.
4. Find the first order conditions. What's the interpretation for $\lambda$?
5. Rearrange them to get the tangency condition.

Question 5:

Lorelai's choice behavior can be represented by the utility function $$ u(x_1,x_2) = 0.9ln(x_1)+ 0.1x_2.$$ The prices of both $x_1$ and $x_2$ are ${$}$10 and she has an income of ${$}$80.

  1. What preference does this utility function represent?
  2. Drawing four indifference curves using www.desmos.com, www.econgraphs.org or another computer tool of your choice.
  3. Find the marginal utility of $x_1$ and $x_2$. What is the maximum number of $x_1$ so that $MU_{x_1}$ is bigger than or equal to $MU_{x_2}$?
  4. Given her income of ${$}$80, how many units of $x_1$ can she buy? Would she buy any positive number of $x_2$ in light of the answer from Q5.3? Find the optimal bundle.
  5. Suppose instead her income is ${$}$100. Would she buy any positive number of $x_2$? Find the optimal bundle using the tangency condition.
  6. Find the optimal bundle with an income of ${$}$110 using the tangency condition. What happens to the current consumption amount of $x_1$ compared to the consumption of $x_1$ when income was ${$}$100?
  7. Will the optimal bundle be the same or different if the utility function were $u(x_1,x_2) = 9 ln(x_1) + x_2$? What if the utility function were $u(x_1,x_2) = x_1^9 \times exp(x_2)$? Explain.

Question 6:

Jess has the utility function $U(x_1,x_2) = min ${$ 2 x_1, 2 x_2 $}$ $. The price of $x_1$ is $p_{x_1}$, the price of $x_2$ is $p_{x_2}$, and his income is $m$.
1. Find Jess’s optimal bundle $x_1^{*}$ and $x_2^{*}$ as a function of $p_{x_1}$, $p_{x_2}$, and $m$.
2. What's the proportion of consumption amounts between $x_1^{*}$ and $x_2^{*}$? In other words, find $\frac{x_1^{*}}{x_2^{*}}$.
3. Suppose instead the utility function is $U(x_1,x_2) = min ${$ 0.5 x_1, x_2 $}$ $. Without solving for the optimal bundles, what's the proportion of consumption amounts between $x_1^{*}$ and $x_2^{*}$, i.e. $\frac{x_1^{*}}{x_2^{*}}$?

Question 7:

Logan has preferences over olives $(x_1)$ and ice creams $(x_2)$. He prefers to eat them separately but not together, which is represented by: $$ U(x_1, x_2) = x_1^2 + x_2^2.$$ Also suppose his income is ${$}$200 and the prices of olives and ice creames are $p_{x_1} = {$}1$ and $p_{x_2} = {$}2$, respectively.

  1. Is this convex preferences or concave preferences?
  2. Solve for the bundle that satisfis the tangency condition.
  3. What's the level of utility using the bundle you just solved in Q7.2?
  4. Consider spending all of your income on olives $(x_1)$. What's the utility from this bundle?
  5. Which bundle does Logan prefer between the bundle in Q7.2 and Q7.4? Does the tangency condition lead to the optimal bundle?
  6. What's the optimal bundle when price of olives is ${$}$3.

Question 8:

Consider a utility function $u(x_1, x_2) = \sqrt{x_1} + \sqrt{x_2}$.
1. What is the optimal bundle with $p_{x_1}$, $p_{x_2}$, and income $m$?
2. $($optional$)$ Would the corner solution where all income is spent on single good be possible?