Problem Set 3: Constrained Optimization when Calculus Doesn't Work
Due Saturday, October 11 at 11pm on Gradescope
After Monday’s lecture, you should be able to do questions 3.1-3.3; after Wednesday’s class, you should be able to do the remaining three questions.
Sometimes all you want is chocolate
Suppose you have preferences over chocolate bars (good 1) and money spent on other things (good 2) which may be represented by the quasilinear utility function \(u(x_1,x_2) = 36 \ln x_1 + x_2\) Since “good 2” is money spent on other things, we’ll fix $p_2 = 1$ for this problem.
What bundle maximizes your utility if each chocolate bar costs $p_1 = 3$, and you have $m = 27$ to spend on chocolate bars and other things?
Suppose your income doubled. Would you buy twice as many chocolate bars? More than twice as many? Less? Explain.
Using graph paper, draw your budget line before and after the income increase; your optimal bundles from parts (a) and (b); and the indifference curves passing through those bundles. Be sure that all slopes in your graph are mathmatically accurate!
Three perfect substitutes
It’s pumpkin spice season again, and as we all know, Kai Ryssdal loves him some pumpkin spice. Kai’s utility function for pumpkin spice lattes (good 1), pumpkin spice cookies (good 2), and pumpkin spice cheesecake (good 3) may be represented by the utility function \(u(x_1,x_2,x_3) = 3x_1 + 2x_2 + 6x_3\)
Suppose $p_1 = 6$ and we observe Kai only buying pumpkin spice lattes. What must be true of $p_2$ and $p_3$?
Now suppose lattes double in price, to $p_1 = 12$, and we observe Kai buying some pumpkin spice cookies and some pumpkin spice cheesecake (and no lattes). Now what must be true of $p_2$ and $p_3$? (Hint: there are three things which must be true: one about $p_2$, one about $p_3$, and one about the relationship between them…)
Gross combinations
Jamal is throwing a party, and has $m = 600$ to spend on food. He can buy hot dogs (good 1) for $p_1 = 3$, or sushi (good 2) for $p_2 = 8$.
He imagines that people’s preferences over these two goods may be represented by the utility function \(u(x_1,x_2) = x_1^2 + 4x_2^2\)
What is the MRS of this utility function? Are Jamal’s preferences monotonic? Are they convex?
What combination of hot dogs and sushi would the Lagrange method find? (You don’t have to set up the Lagrangian; it’s sufficient to write down the tangency condition and the budget constraint.) Draw this bundle and the indifference curve passing through it in a diagram with Jamal’s budget line. Is this Jamal’s optimal choice? Why or why not?
The points (40,60) and (120,30) lie along Jamal’s budget line. Calculate his MRS at each of those points. In which direction is the “gravitational pull” operating at each of those points?
What is Jamal’s actual optimal choice? Illustrate it in a budget line/indifference curve diagram. (If you’re stuck: plug $x_2 = 75 - {3 \over 8}x_1$ into Jamal’s utility function to find his utility as a function of $x_1$, and plot that function in Desmos or some other program…what’s happening at the bundle you found in part (b)? At the two bundles from part (c)?)
Exercises 3.4-3.6 all use the budget constraints from Exercise 2.4 on the last problem set! Please refer to those questions for the setups, and feel free to refer to the solutions to that problem set as you answer those questions.
Optimization with a gift card
Note: this question was originally published with a different utility function, but the solutions were…inelegant, to say the least. Monstrous, to be more precise. I updated the problem on Saturday to the following utility function, which is much nicer and allows me to illustrate the four cases more clearly. Whichever function you used, you’ll get full marks just for following the correct procedure to solve the problem, even if you encountered algebraic horror along the way…
Consider Aliyah from Exercise 2.4(a). Suppose her preferences over Chuck E Cheese tokens (good 1) and money spent on other things (good 2) may be represented by the utility function \(u(x_1,x_2) = x_1 - {1 \over a}x_1^2 + 2x_2\) for some constant $a > 0$.
Characterize her optimal bundle as a function of $a$: that is, find $x_1^\star(a)$ and $x_2^\star(a)$.
Your answer to the previous question should have been a piecewise function with four cases. Pick four values of $a$ that illustrate those four cases, and draw her budget constraint/indifference curve diagram for each one.
Used stuff always goes for cheaper
Consider my problem from Exercise 2.4(b). Suppose now that my preferences over speakers and money may be represented by the Cobb-Douglas utility function \(u(x_1,x_2) = \alpha \ln x_1 + (1-\alpha) \ln x_2\) where speakers are good 1 and money for other Christmas presents is good 2.
For simplicity, assume speakers can be bought and sold as if they were a continuous good (i.e., not discrete).
What is my MRS at my endowment (i.e. the kink in the budget constraint at (10, 1000)? (The expression you derive will be a function of $\alpha$…)
For what range of $\alpha$ would I choose to neither sell any of my speakers for €100 nor buy additional speakers for €200?
Suppose I decide it’s optimal for me to sell exactly five of my speakers for €100 each to my friend Jeff. Draw a new diagram showing the indifference curves passing through my endowment point $(e_1,e_2)$ and my optimal choice $(x_1^\star,x_2^\star)$. What value of $\alpha$ would correspond to this being my optimal behavior?
The rest of this question is for students in 50Q only.
Write down the Lagrangian for this problem, treating the equations of the two segments of the budget lines as two nonnegativity constraints with Lagrange multipliers $\mu_1$ for the case where I sell speakers, and $\mu_2$ for the case where I buy additional speakers.
Suppose $\alpha = {1 \over 4}$. Explain how you would use the KKT method to solve for the optimal bundle. (You don’t actually have to go through the steps.) What are the values of $\mu_1$ and $\mu_2$ in this case?
Finally, let’s use KKT to replicate our findings from part b. One way to think about the optimal point being the kink (that is, the endowment), is that both budget constraints will bind at that point. Therefore, if it is optimal to be at the kink, we will have $\mu_1,\mu_2>0$. Write the first order conditions for $x_1,x_2$, and plug in the endowment point (10,1000) into these conditions. Solve for the lagrange multipliers $\mu_1,\mu_2>0$ in terms of $\alpha$. What does this say about the values of $\alpha$ where you would optimally consume your endowment bundle? How does this connect to your findings in part b?
The “Donut Hole”
Finally, consider Gunther from Exercise 2.4(c).
- For Econ 50: Suppose Gunther’s preferences over doctors’ visits and money spent on other goods could be represented by the perfect-substitutes utility function\(u(x_1,x_2) = ax_1 + bx_2\)
- For Econ 50Q: Suppose Gunther’s preferences over doctors’ visits and money spent on other goods could be represented by the perfect-complements utility function\(u(x_1,x_2) = \min \left\{ {x_1 \over a}, {x_2 \over b}\right\}\)
(You may assume $a > 0$ and $b > 0$.)