Problem Set 2: Constrained Optimization when Calculus Works
Due Saturday, October 4 at 11pm on Gradescope
Exercises 2.1 and 2.2 may be done after Monday’s lecture; 2.3 and 2.4 correspond to Wednesday; and the rest are based on Friday’s lecture.
At a minimum, you should do 2.1, 2.3, 2.5, and 2.6. Students in 50Q should do 2.2 and 2.8. The most important exercises are the last ones!
Monotonicity and Convexity
Sarah enjoys eating both apples (good 1) and oranges (good 2). Her preferences over these two goods are strictly monotonic and strictly convex. Remember the formal definition of convexity: if preferences are strictly convex, then if a consumer is indifferent between two bundles, she prefers a convex combination of those bundles to either bundle.
Sarah is choosing between the following bundles:
| Bundle | Apples | Oranges |
|---|---|---|
| $A$ | 2 | 6 |
| $B$ | 6 | 6 |
| $C$ | 4 | 4 |
| $D$ | 2 | 2 |
| $E$ | 6 | 2 |
Is each of the following statements true or false? Briefly explain.
Because $B$ has more of both goods than $D$, it must be that $B \succ D$.
Because $C$ is a convex combination of $B$ and $D$, it must be that $C \succ D$ and $C \succ B$.
Because $B$ has the same number of apples as $E$, it is possible that $B \sim E$.
Because $C$ has more oranges than $E$, but fewer apples, it is possible that $C \succ E$ or $E \succ C$.
Because $C$ is a convex combination of $A$ and $E$, it must be that $C \succ E$ and $C \succ A$.
Suppose Sarah tells you she’s indifferent between bundles $A$ and $E$. Using that information, sketch a possible indifference map showing the indifference curves passing through each of the five bundles.
Sarah’s friend Katrina also likes apples and oranges, but her preferences aren’t quite the same: she prefers more fruits than less, but only cares about the total number of fruits she eats, not which kind they are. (Therefore, for example, she would be indifferent between eating 4 apples, 4 oranges, or 2 of each.) Repeat part (b) for Katrina and draw her indifference curves passing through each of the five bundles. Are her preferences strictly monotonic? Are they strictly convex?
One Function to Rule Them All
The Constant Elasticity of Substitution (CES) utility function takes the form \(u(x_1,x_2) = \left(x_1^r + x_2^r\right)^\frac{1}{r}\) where the parameter $r$ measures the “substitutability” of goods 1 and 2.
Solve for the marginal rate of substitution (MRS) of this utility function.
What is the MRS when $r = 1$? What kinds of preferences does this represent in that case?
What is the MRS when $r = 0$? What kinds of preferences does this represent in that case? Can you write an equivalent utility function (i.e., one that represents the same preferences) using a different functional form? Note that if you actually try to find $MU_1$ and $MU_2$ when $r = 0$, you need to use L’Hôpital’s rule…but that’s beyond the scope of this assignment. It’s fine for this case to just plug $r=0$ into the expression for the MRS you found in part (a).
(For 50Q only.) What is the MRS, in the limit, as $r \rightarrow -\infty$? What kinds of preferences does this represent in that case? Can you write an equivalent utility function (i.e., one that represents the same preferences) using a different functional form?
Changing Budget Sets
Each of the following scenarios describes a “before” and “after” situation. In each case, using a single diagram clearly indicate:
- the consumer’s budget line before the change $(BL_1)$ and after the change $(BL_2)$
- the set of bundles, if any, that were initially available to the consumer but are no longer available after the change
- the set of bundles, if any, that were initially unavailable to the consumer but are now available after the change
How far would you go for bananas?
- Before: Jasmine has $m = 16$ dollars to spend on apples (good 1) and bananas (good 2). At her local store, apples cost $p_1 = 2$ dollar per pound, and bananas cost $p_2 = 4$ dollars per pound.
- After: Jasmine’s local store closes! She finds a different store that sells bananas for $p_2^\prime = 2$ dollars per pound. (It also sells apples for $p_1 = 2$.) However, she has to take the bus there, which costs €4, reducing her fruit budget to $m^\prime = 12$ dollars.
Scarce resources, unlimited desires
- Before: Chris has $m = 60$ dollars per month to spend on cell data (good 1) and used DVDs of 1980’s movies (good 2). Cell data costs $p_1 = 2$ dollars per GB and used DVDs cost $p_2 = 4$ dollars per DVD.
- After: Chris buys an unlimited data plan for €20 per month; this allows him to consume as much cell data as he wants.
Effect of a gas tax
- Before: Kakraba is a cab driver who has $m = 70$ per day to spend on gas (good 1) and other things (good 2). Gas costs $p_1 = 2$ per gallon. Note: in this problem “good 2” is “dollars spent on other things” (i.e. things other than gas). Sometimes this kind of good 2 is referred to as a “composite good.”
- After: The government passes a gas tax, raising the price of gas to $p_1^\prime = 5$ dollars per gallon. However, to offset the pain this causes cab drivers, they also give cab drivers a cash subsidy of €30 per day, therefore raising Kakraba’s income to $m^\prime = 100$ dollars per day.
Kinked budget sets
Sometimes a budget set isn’t a simple straight line. Here are some examples of budget sets with kinks that you’ll encounter during your studies of economics. In each, clearly draw the budget constraint. Hint: when applicable, think of the extreme cases in which someone consumes as much good 1 as they can, or as much good 2 as they can, and what’s happening at any “kink” in the constraint.
A token gift
- Good 1: Chuck E Cheese tokens
- Good 2: Money spent on other things
- Scenario: Aliyah got birthday presents from two aunts. One gave her €25 in cash, and the other gave her a €30 gift certificate to Chuck E Cheese for her to buy tokens. Each Chuck E Cheese token costs €0.25.
Used stuff always goes for cheaper
- Good 1: Sonos speakers
- Good 2: Dollars to spend on other goods
- Scenario: Sonos speakers are really awesome; I recently bought 10 of them. After I plugged them in, I realized I might not need all 10. However, if I want to sell them, since I took them out of the box, I could only get €100 for each one. If I wanted to buy more, additional speakers would cost €200 apiece. I currently have €1000 of disposable income; so you can think of me currently having an “endowment” of (10 Sonos speakers, €1000 in cash).
The “Donut Hole”
- Good 1: Doctor’s visits per year
- Good 2: Money spent on other things per year
- Scenario: Gunther has an annual disposable income of €3200. His health insurance plan has the following structure: he can go to the doctor 6 times a year with no copay. If he goes to the doctor more than 6 times, he has a copay of €200 per visit for the next 8 visits. After that, his copay drops to €100 per visit for any additional visits.
Cobb-Douglas Optimization with the Lagrange Method
Consider the Cobb-Douglas utility function \(u(x_1,x_2) = x_1^{\alpha}x_2^{1-\alpha}\) where $0 < \alpha < 1$.
Use the Lagrange method to find the utility-maximizing bundle, as a function of $\alpha$, if $p_1 = 6$, $p_2 = 2$, and $m = 48$.
Interpret your results. How do the optimal choices of $x_1$ and $x_2$ change as $\alpha$ changes?
Plot the budget line, indifference curve, and optimal bundle if $\alpha = \frac{ 1 }{ 4 }$.
Quasilinear Optimization with the Lagrange Method
Consider the function \(u(x_1,x_2) = 20 \ln x_1 + x_2\)
Use the Lagrange method to find the maximum of this function, subject to the constraint \(x_1 + x_2 = m\) as a function of the constant $m$: that is, find $x_1^\star(m)$ and $x_2^\star(m)$. You may assume that $m > 0$.
Now suppose we were to add another constraint that $x_1 \ge 0$ and $x_2 \ge 0$. In other words, the goal now is to find the highest value of $u(x_1,x_2)$ along the segment connecting the points $(0,m)$ and $(m,0)$. What are the solution functions $x_1^\star(m)$ and $x_2^\star(m)$ now?
Lagrange with Linear Functions
Suppose you were trying to maximize the perfect-substitutes utility function \(u(x_1,x_2) = ax_1 + bx_2\) subject to the constraint \(x_1 + 2x_2 = 12\) where $a$ and $b$ are strictly positive constants.
Set up the Lagrangian of this problem, and find its first-order conditions.
In your own words, explain why the Lagrange method will never work to find a unique solution to this problem.
Suppose, as in the last question, that we further constrain the solution to be in the first quadrant (i.e., $x_1 \ge 0$ and $x_2 \ge 0$). What would be the solution to the maximization problem, as a function of $a$ and $b$?
Can’t Buy Me One (Harmonica)
This question is only required for 50Q students.
Jake and Elwood are looking to buy some instruments as they launch their new band. They need to buy some number of harmonicas ($x_1$) and saxophones ($x_2$) for their performances.
While Ray’s Music Exchange is perfectly willing to sell these goods in continuous quantities ($x_1=3.14$ harmonicas is perfectly acceptable), they have a somewhat annoying store policy: every customer must buy at least 2 harmonicas in order to make purchases at the store. That is, Jake and Elwood can only walk out the door with a bundle where $x_1\ge 2$. Ray has set a price of $p_1$ for harmonicas and $p_2$ for saxophones. You can assume that Jake and Elwood will choose to purchase positive quantities of both goods.
Jake and Elwood have $m$ dollars to spend, and gain utility \(u(x_1, x_2) = x_1^\alpha x_2^{1-\alpha}\) from purchasing the two goods.
What is the objective function in this setting? What are the constraints?
(Hint: for the sake of later subproblems, you might want to take a certain monotonic transformation of the utility like we discussed in section to make the algebra easier.)