[BLANK_AUDIO] Okay. So, we saw that there are some very severe limitations on practical use of lump sum taxes, and these limitations are due to asymmetric information. And we concluded that the way out of this restriction would be to link taxes to something which is, at least in principle, in theory, observable and verifiable. Come income taxes. Why income taxes are realistic, why they are practical? Because individual's income in modern societies is oftentimes, in principle, observable to tax authorities. If it's your wages, then if you are employed by a company which is officially registered, this company has to report the tax authorities your employment income. If you receive some other income, for example, if you receive interest from a bank or dividends from investments, this is also reportable to the government, and, as a result, again in theory, governments have the ability to learn a whole lot about individual's incomes. Unless individuals engage in tax evasion, which is a criminal activity, and is punishable by law or unless there is a large informal sector in economy where people's incomes are unreported and non-taxable, and this is one reason of why economists with the big informal sector are relatively poor economies because governments in such economies cannot collect revenues to make this economies more prosperous. So, let's suppose that an agents' income is publicly observable, and this income is z. This is pre-tax income. A portion of this income has to be paid to the government, and it has to be paid to the government according to a certain function, T of z. And this function is oftentimes called tax schedule. Your consumption, or post-tax income, x is the difference between z and T of z. And, of course, the big question is how you calculate T of z, how you calculate tax schedule. Tax base is clear, it's individual's income. What is tax schedule? There is a variety of tax schedules in a modern world. And the simplest one, perhaps, is what is known as linear or flat income tax, whereby your tax is just a certain percentage of your pre-tax income. Russia is an example of countries which have flat income tax. And the personal income tax rate in most cases, there are some exceptions, is 13%. It's a very easy, simple tax schedule to administer. It has certain limitations though, but it might still be an optimal choice, and most likely it was the case in Russia. So, there were reasons to introduce this particular type of tax system. But let's talk a little bit more about that. More generally, the tax schedule is a certain function which is non-linear. And it's important to keep in mind the distinction between different types of taxes. Taxes could be progressive or regressive. Here is a definition of what is a progressive tax. Tax is progressive if tax liabilities per income which is the difference T of z over z is monotonically increasing. It means that the more wealthy an agent becomes, the larger portion of this agent's income will be paid as taxes. Notice that for linear, flat income tax T of z over z is constant. There is another way to, to describe progressive taxes which is slightly different from the definition I gave you before. So, these two definitions are not equivalent but they convey the same idea. And the second definition is that what is known as the marginal tax rate T prime of z which is the derivative of T of z is also monotonically increasing. What is the marginal tax rate? Suppose you make an extra $100, marginal tax rate shows how much of that amount will be paid as taxes. And if an income tax is progressive, then this marginal tax rate would be increasing as well. If you have look at actual tax schedules in modern world, in developed nations, then real life tax schedules work approximately as follows. There are certain thresholds of individuals income, most of the time we're talking about annual income. If your income is below the lowest of this thresholds, usually you pay no taxes. This is what is known as tax credit. It's a certain minimal amount of income where no tax liability ensue. If your income is above this threshold, if you are in this interval, then you pay taxes according to this rate which is fairly low. Then if income exceeds this threshold, your tax rate goes up, and then, there are two or three what is known as tax brackets. These intervals are known as tax brackets. There are several tax brackets and in the end you hit the highest tax rate, highest tax bracket and if your income is above this highest threshold and beyond, then you pay taxes at the fairly high marginal rate. And again in developed nations, this top marginal tax rate could be fairly high. Russians enjoy a remarkably low personal income tax rate of 13%, and elsewhere in the world you can easily find marginal tax rates which might be close to 50%, sometimes they're above 50%. But anyway, this is how progressive taxes work in modern countries. Let's talk a little bit how to choose a particular income tax schedule because obviously there is a variety of ways to calculate tax liabilities, and even if you confine yourself to this simple schedule with several brackets, the question is how to choose these brackets, and what would be a marginal tax rates in those certain brackets. This is an outcome of fairly complex political choice. And this political choice is usually something that reflects redistributional considerations. Societies want to reduce inequality, and to reduce inequality you have to reallocate tax liability so that wealthy agents pay higher share of their income than agents which are poor. And as you saw, agents with really low income pay no taxes. In fact, sometime they can get even subsidies. So, in choosing a tax schedule, you have to first be aware of the political pressure towards redistribution which explains progressive taxation. But for an economist, there is another important rationale of choosing tax schedules. And that again, has to do with informational asymmetry and tax payers incentives. You can manipulate with tax schedules, so that it enables you to address the asymmetric information problem, and I can show you very briefly how and why. [SOUND]