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.
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