### Discussion Paper

## Abstract

Economists traditionally tackle normative problems by computing optimal policy, i.e., the one that maximizes a social welfare function. In practice, however, a succession of marginal changes to a limited number of policy instruments are implemented, until no further improvement is feasible. I call such an outcome a “restricted local optimum”. I consider the outcome of such a tatonment process for a government which wants to optimally set taxes given a tax code with a fixed number of brackets. I show that there is history dependence, in that several local optima may be reached, and which one is reached depends on initial conditions. History dependence is stronger (i.e. there are more local optima), the more complex the design of economic policy, i.e. the greater the number of tax brackets. It is also typically stronger, the greater the interaction of policy instruments with one another — which in my model is equivalent to agents having a more elastic labor supply behavior. Finally, for a given economy and a given tax code, I define the latter’s average performance as the average value of the social welfare function across all the local optima. One finds that it eventually sharply falls with the number of brackets, so that the best performing tax code typically involves no more than three brackets.

## Comments and Questions

see attached file

1. I should indeed refer to the papers by Guesnerie.

2. I want to point out that the main point of the paper is to illustrate how the kauffman idea that greater complexity and greater interaction between units leads to "rugged landscapes", i.e. a large number of local optima, can ...[more]

... be applied to economics in a context where the local interactions between units come from rational behavior by optimizing agents rather than being exogenous. I use the tax brackets example because there is a natural counter part to the interaction between adjacent units, which is the elasticity of supply by agents. So the tax example is just one example, I actually believe the argument is more relevant for other kinds of reforms, but it is easier (at least for me) to understand and spell these effects out in this context.

3. Moving the kink points allows us to change the tax schedule locally; on the other hand, moving the marginal tax rate for one bracket would in fact imply changing the whole tax schedule for the upper brackets. To avoid that, one could change the marginal tax rate for the next bracket and let the tax schedule unchanged for subsequent brackets, but that is exactly what changing disposable income at the kink while leaving other kinks unchanged achieves. So the government does focus on marginal tax rates, but it can only change the tax schedule for two consecutive brackets at the same time.

4. As the paper's focus is on a positive theory of taxation, I was not careful in relating the SW function to individual utilities. And one may argue that the disutility of labor is not oberved to the social planner. But I can try to run simulations with a utilitarian SW function.

see attached file

I can try to run simulations with the steepest ascent algorithm. In principle, this should not affect the number of local optima, but in practice the number of local optima that may be found with that algorithm can be different from what is done in the paper.