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.

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.

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?

Project: Wordblasting

Two days ago I complained about how it's really hard to reach out to people in Facebook. Yesterday I realized I had a bunch of small writing ideas that couldn't carry their own project, but might be fun to share. Today I realized that G was never going to update Momenergy ever again. This gave me an idea.

I'm going to start updating Momenergy again with these bits and scraps of rants. Each one I will also send to three random people on Facebook and see how they react. The rants probably won't be available on general Facebook, but they'll always be accessible here.

Have two completely written out, one on MC Escher and one on metahumor. Going to upload them both over the next two days.

DEATH Redacted

It has come to my attention that under the challenge rules G had ten days from the start of reading the post to solve the problem, and that given G's lack of internet communication he probably has yet to read the post.

G has not won DEATH.

DEATH

G has won DEATH.

A Challenge to G

I come back from my job in Chile and find out he made only two posts since I left. I'm sure G has an excuse like "but I had so much wooooooork", but I had a similar workload as him and spent all my free time hiking around and shit and still managed to throw up 30+ posts. Obviously G needs a little motivation. Therefore I send a CHALLENGE!

Often when I was flying around I'd be stuck in a seat I didn't like, a middle seat or an aisle seat. Now I know that there are a lot of people who'd prefer aisle seats just as I prefer window ones. The way the system is set up you can't switch with them. You can pick a different available seat if the plane isn't completely full, but there's no way for two people to use the system to trade seats. This isn't a problem if the two people are friends, or can talk to each other. But how can we arrange trades between strangers? How do I know that the person two rows behind me would gladly switch his window seat for my aisle? As the systems currently work, you can't.

So the CHALLENGE: G, you must work out an adequate system that would ensure that, after all seats are assigned, people can swap seats with strangers. If you succeed, I will owe you a pie of my choosing. If you fail, DEATH.

By reading this, G, you implicitly agree that you will accept this CHALLENGE. You have 10 days.