The Envelope Behavior

There's a neat little theorem in economic theory, called the envelope theorem. It is a mathematical result that describes the behavior of solutions to optimization problems. I will give an intuitive description of what it's about.

Optimization basically means finding a solution that best satisfies some set of conditions. Some conditions are inviolable, meaning that a solution must satisfy these conditions. These are called constraints. Other conditions characterize how good a particular solution is, these are called objectives. The classical example in economics is the problem of allocating income to purchase consumption goods. In this case, the optimization problem concerns finding the set of goods to purchase that makes a consumer the happiest. Whatever measurement for happiness is the objective that we are trying to maximize, and the income of the consumer gives a constraint in the sense that we cannot buy more goods than we can afford.

The envelope theorem characterizes how the optimal solution changes if we change the conditions imposed on the optimization problem. That is to say, the theorem describes how a consumer's optimal consumption choices may change if we altered how much he prefers different goods or if we change his income.

Take for example a price change in a good x. We hope to understand how a consumer's best choice of consumption changes given this change in price. There are two effects at work: the direct effect of the change in price on happiness. (Perhaps after a price reduction, a consumer feels the happiness he gains from an extra unit of x now outweighs the price of x, causing more consumption of good x). There is also an indirect effect, which affects other parameters of the optimization problem, which themselves affect consumer choice. For example, when the price of x decreases, one can afford more of x, so the consumer's income has effectively increased in that he is now less constrained in what sets of goods he can purchase. Income itself has an effect on consumption choice, so here, the changing price of x acts indirectly on happiness through income as a mediator.[2]

The envelope theorem tells us that as long as we are looking at the optimal solution, the changes of the solution based on changes in any parameter of the problem depends only on the direct effect of the parameter in question. The indirect effect is accounted for by the optimization process. This is to say, if we look at a consumer's optimal choice after the price change in x, the consumer accounts for how indirect effects of say price on income affect his choices, so that his observed optimal choice after the price change is attributable to only the direct effect that prices have on his happiness.

This theorem seems to hint at a certain dynamism in the process of optimization. The optimized behavior is able to in a sense "smooth out" the indirect effects from changing parameters. This smoothing effect that eliminates indirect effects is, to be sure, not present when we are not talking about optimal choices. Indeed, when we are not talking about optimal choice, the indirect effects from changing a parameter is usually nonzero, so it is curious that the procedure of optimization is somehow synonymous with acting in a way that counteracts the indirect effects from changing parameters, something that we did not really set out to control.

There is a practical side to this story. If we adopt the economics point of view of human behavior, then any equilibrium in society (which is code for everyday behavior on "good days") can be interpreted as being at a local optimal point on the scale of social welfare. We can view public policy as changing the parameters of the societal optimization problem, to which people must respond optimally to arrive at a new equilibrium, which, when public policy goes right, is at a point of greater social welfare than before.

There is always a concern of perverse incentives in policymaking, which basically means when a policy encourages people to do exactly opposite of what was intended. For example, in 1990, Mexico City passed a law that attempted to reduce air pollution and congestion in the city by limiting when cars can be used. They limited travel based on the last digit of cars' license plates, so during half of the week, the odd numbered cars can go on the road, and during the other half, the even numbered cars are allowed. This policy was so inconvenient that it caused some people to buy two cheaper, more polluting cars so that they could travel every day, which ironically exacerbated the congestion and pollution problems.  (Eskeland & Feyzioglu, 1995)

The problem of perverse incentives reminds me of what the envelope theorem tells us. While as policymakers, we would like to just implement a law that we hope people will obey in a straightforward way, the problem is that new laws in effect only change the constraints as to what people can do, and after people have optimized their choices given these new constraints, their optimal solution may be very far from the direction that the policymakers had intended the people to go. This is like how through optimization, the process itself is dynamic enough to adjust to indirect effects and produce a surprising result that we have not done anything to achieve.

The general human ability to optimize in unexpected ways, is the bane of well-intentioned policymakers everywhere. Much like how the envelope theorem predicts there are side-effects to the mathematical optimization problem, we should always keep in mind the potential for human behavior to produce side-effects that policymakers simply cannot foresee.


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1. H should be a professional motivator. He has mastered a good combination of peer pressure and physical intimidation to force people to write blog posts. So here I am, writing again after almost a year's hiatus. It's not that I have forgotten about the blog. More that I haven't figured how to put a lot of my ideas into words. But this exercise basically forces me to write, so we will see how this goes! Anyway, H, I know you're reading this. Do consider a career in professional motivation.

2. Mathematically, the effects from changing parameters are represented by partial derivatives. Say an objective L depends on parameters p (price) and M (income). Here, the income also depends on price, so we have M(p). Then The total effect of price on the objective ∂L/∂p has two components: the direct component dL/dp and the indirect component from chain rule: (∂L/∂M)*(dM/dp). The envelope theorem states that when we evaluate these partials at the optimal choice, the indirect effect always equals zero.

Updating late at night

Means I am unmotivated to write good updates. Therefore I will try for teh next three days to update before 6 PM.

G's Drafts

I've been updating this or the Other Blog once every day in a personal groove thing. Tonight I couldn't think of anything to write because instead of thinking of things to write I bummed around with my friends. While bashing my head and trying to decide whether or not I should phone it in or not, I had a realization. G's last update over 15 (!) posts ago. You know how this is supposed to be a collaborative blog? I do now.

I know G hasn't forgotten about this since I see his drafts embedded between my posts. To shame him into actually finishing them, I will list them all right here.

The Envelope Behavior

Is Dishwashing an Example of Coasian Bargaining?

Momenergy: A Homeopathic Interpretation

Scav, an introduction

Does Our Name Define Us?

 We are Our Names

Monopole Song

Personal and Professional Life

Collective Psyche

Debunking the 1=2 Proof

Smartphones

 

If G does not write at least one of these by Sunday I will do them all for him.
And I will ensure they are all terrible.

Contacts

In about a month many of my friends will be graduating. It's a bittersweet thing. On one hand, they now get to enter real life. On the other hand, they now have to face real life. Of course, I don't give a damn about their trials and tribulations. Doing that would make me a Real Person, and as a UChicagoan I traded all of my empathy for the ability to rant about comedy theory for 24 hours straight. Their graduation affects me in the regard that I'll never talk to them again.

It's a simple problem. People only talk if they're in contact. Tautological. But contact is hard to maintain. People use facebook, but it's hard to make that meaningful. Calling and email is better, but nobody uses that. I should, because it forges powerful connections, but GUESS WHAT: unless I have a responsibility to email, I won't email!

Oh wait.

What if I made it a responsibility to email? Not that emailing becomes a chore, but that I have to do something I want to. Not tricking myself, but building additional structure into my life to make something I want to be important more important. So I'm going to slowly over time build an emailing list, where I must email certain people on certain days. I can email other days, but I must email those days too. Going to start tomorrow with a couple of names, then add from there.

Backwards Ideas

Most people get ideas first as ideas, and then they refine them, and then they title them. Last week I noticed something odd. I first come up with titles, and then try to fit ideas to them. I'm wondering if this leads to different types of ideas, or at least different modes of realizing them.

Giving Stuff

Normally I try to keep the snarking out of this blog because it's tricky to make it sound funny, and because it's too easy to mistake amused hatred for poisonous hatred. But I want to talk about something similar, so I guess I'll push the rant through, too.

We've got this thing called Dream, Believe, Smile, a club that tries to make campus a happier place. They do this by putting inspirational posters up everywhere.

I hate them so much.

Slacktivists, the lot of them. Seeing a poster doesn't make me think "I'm happy!" It makes me think "They're too lazy to actually bother." Making a person feel better involves making a small connection, like a conversation, a compliment, even a smile. The important thing is that you have to show you care. A DBS poster does not do that. It's too sterile.

Now I'd think that with this logic I'd be okay with Free Hugs Day. But for some reason I hate that too. I Think it's because offering free hugs to passerbies looks less like "I want to brighten your day by hugging you" and more like "I want to validate myself by hugging you." Hugging literally takes like two seconds of your time and doesn't show a connection if there's nothing else setting it up. It's just a cheap way of getting other people to hug you.

But I still like the idea behind DBS: we have the ability to make other people happier and we should put that power to good use. I just think that DBS and FHD are terrible ways to do that. Instead, I like to give out free food.

This has a couple of advantages over the hug method. First of all, I make my own candy. There's the whole "Have some of my time and energy" deal. And it's pretty novel to have homemade candy. Second, it's a lot more unusual to get free candy than to get a free hug, which makes it stick more. Finally, food is delicious.

Of course these can't be the main reasons, because I've had equal success giving away storebought candy and fruit and stresses balls. It's something intrinsic to giving, and even more specifically to giving stuff. Unlike hugging, giving stuff is a zero sum game. Not only do you gain, I lose. I think this is why giving is so powerful. I want to make your day better so much that I'm willing to sacrifice a tiny part of myself in the process. Sure, I can make like 400 candies in less than an hour, but the sentiment is still there. By consciously losing out in the exchange, I've imbued the giving with a lot more meaning.

Anyway, that's how I resolve the cognitive dissonance between my all-consuming hatred for DBS and FHD and my fondness for giving stuff away. We have roughly the same idea, just different implementations. And I think my implementation goes way further.

Word Puzzles

I love word games. Not all of them, though; things like scrabble and bananagrams never held much appeal to me. Mostly I like the ones that focus on properties of words. Word puzzles are even better. I find it incredibly fun to search for words with a given property, whether etymological, definitional, or even just structurally. Plus you can play them in your head anytime, anywhere. Here are some of my favorites. My best answers are linked.

Shortest word with all five vowels: http://tinyurl.com/7s5cpnu

Longest word without any vowels: http://tinyurl.com/88vadcg

Longest word with only one vowel: http://tinyurl.com/6upsokt

Longest word with only one type of vowel: http://tinyurl.com/6wfbmb (yeah, I'm kinda fixated on vowels)
 
Longest word with alternating vowels and consonants: http://tinyurl.com/894pped (if we can count y as a vowel)

Longest word with only one consonant: http://tinyurl.com/6pafg2o

I need to find some new ones. Any ideas?