[SOUND] One way out of this gridlock is to restrict preferences that we're trying to aggregate by democratic means. And I'm going to show you now that in certain cases, and very likely, and very fortunately, these are the cases which are commonly observed in public economic decision making, in decision making involving taxes, and public expenditures, democratic aggregation of even vastly polarized preferences is possible. And these are the cases when preferences are so-called single-peaked. Before I go into details let me explain what being single-peaked means. Suppose that the choice variable is measured along a certain axis, and there we have higher and lower values of this choice variable, you will see the picture in a second, but let me just show it to you on this slide. What I mean is suppose that preferences of an agent over this choice variable are as follows. First, they increase in this variable then they reach a peak. And then they decrease over the very same variable. Such preferences are called single-peaked. It is possible, for example, that preferences are increasing all along or that preferences are decreasing all along. This will also be single peak preferences. In this case, peaks will be either at this or this range of the policy interval. What rule out though would be preferences which first go up, and then go down. There is a drop and then there is an increase, and then there is another peak and so forth. This will not be single peak preferences. So, let's assume that what we're trying to aggregate are single-peaked preferences. And let me show you now how the single-peaked preferences naturally occur in choices of taxes and public expenditures. Suppose that there are several agents, and suppose that these agents are different from each other, so we want try to aggregate diverse preferences. Lets start from a situation where agents are different from each other, in terms of their appreciation of the public good. The utility function is Ui(x,G), and X is private consumption, and G is the public good, and this utility function is of the falling form. So, agents differ from each other in this coefficient alpha_i. High alpha means that this public good is highly valuable for the agent. Lower alpha means that the very same public good is of a low value for the agent, and there might be people with intermediate levels of alpha as well. Suppose that we have n agents, and just for, to be specific, let's assume that the first agent has the lowest valuation of the public good, in other words alpha1 is the smallest, and the last agent n has the highest valuation of the public good, and alpha_n is the highest. This utility function involves a neoclassical function alpha phi of G that we saw quite a number of times before. And that means that every agent, irrespective of her alpha, appreciates this public good. And if the provision of this public good increases, all else equal, this agent is better off. and, however, the marginal utility of this public good declines as the provision of public good rises over here in this kind of function here. Now, let's suppose that agents have to agree about the level of public good and to include just one single policy variable here because normally in situations such as this one we talk about taxes and expenditures, in our case about taxes and public good provision. But suppose that there are n agents, and we want to share it evenly the burden of funding this public good among the agents. And that means that every agent has to pay a lump sum tax. And the tax rate here is tau=G/n. Tax rules are the same for every agent. Everyone has to pay the same amount, and lets see what agents have to say about that. If we agree about this funding rule, then the only choice variable now is G. And every agent has the following utility function as a function of G alone. This function involves this agent's initial wealth, which is w_i minus the tax that this agent has to pay, the flat tax, the lump sum tax, G over n, and plus the gains from this public good. So, lets have a look as to how this single variable reduced for utility function depends on the provision of public good. So, what we have to do is to deduct from this curve this curve G over n, and for simplicity sake I ignored w_i for the time being, and this difference is as follows. First, as you see, it's positive. And then it raises its maximal level here, and then beyond this point it declines, and at this point where these two curves intersect each other the difference is zero it becomes negative for the down. And this is the graph of the function w_i of G. And as we see, this function is indeed single-peaked, which is a great news. And it is single-peaked because it's the difference between single-peaked, to be more technically precise, concave function, and a linear function. So, we have a single-peaked function over which describes preferences of a given agent. And qualitatively, this function will be the same for every agent. Although the peak of this function G_i star will be different for different agents depending on how much they value the public production input. Now, what would be the optimal level of public good provision from agent i's perspective, from this agent's point of view? This optimal level is denoted by G_i star. Let's make sure that we understand each other at this point. No agent decides unilaterally the level of public good provision. G is not a choice variable of this agent. G is public choice variable, but that certainly does not preclude this agent from contemplating what level of G he would prefer over any other alternative. And this is G_i star. So, if you write the first order condition of this maximization problem, V_I(G) maximized over G, this first ordered condition will lead to the following equation from which we can calculate the level of public good provision, optimal from the perspective of agent i. And, as you see, the marginal return, the marginal utility I should say, to this public good will be in inverse relationship to the appreciation of this agent of this public good. So, higher alpha leads to lower phi'(G) and because phi'(G) is a decreasing function, higher alpha predictably leads to higher optimal levels of public good provision. We agreed that agent n has the highest valuation of this public good. And as a result he would prefer a very high level of public good provision. Some agent I between one and N has an intermediate valuation of this public good. And his preferred provision would be here. And, finally, the first agent has the lowest appreciation of the public good, and therefore she would opt for fairly low optimal provision of the public good. So, we see here a political conflict, and the question is if democracy can reconcile this conflict. And as I will be showing to you very shortly, the answer is positive. Simply because preferences are single-peaked. But before I do that, let me discuss with you very briefly another possible cause, another possible reason of preference polarization. Here, in this first model, we assume that preferred choices of agents are different because they differ in their evaluation of the public good. But there might be other factors of polarization of preferences. One is that was just discussed, and this is diversity of preferences, of a utility function, I should say. Another one would be the economic inequality, would be the differentiation in wealth. And let's discuss that very briefly. To make this discussion more informative let's assume that this time around the public good is not funded by a lump sum tax. In fact, it's funded by a proportional wealth tax. In other words, every agent has to pay a certain portion of her wealth to fund the provision of the public good. And in that case, the contribution of agent i towards the provision of public good will be as follows, where w_i is the wealth of that particular agent, and w bar is the average wealth. G is the amount of public good to be funded, and n is the number of agents. And that clearly shows us that in that case more wealthy agents would be required to make a greater contribution to the provision of the public good. Now, how that is going to affect agent's preferred choices of the public good. Let's have a look at the problem that every agent has to solve, to define what would be the optimal public good provision. In her case, in this case, I assume that the utility functions are the same. So, I assume away the diversity of preferences which was a source of political conflict in the previous model. I assume, however, that levels of wealth are different, and this is another source of preferred choices polarization. Please notice that we have two effects here which are at work. The first one is that wealthy agents are, well, indeed they are wealthier. And, therefore, all else equal, they would prefer more, they would prefer more of the public good. They are "buying" this public good. And they buy that public good at a certain price. We ignore that the wealth also affects the price. The wealth is affecting the income. And the income effect, which is a very well known effect in the consumer choice theory, tells us that, all else equal, wealthier agents would opt for more of the public good because they can afford to fund more of the public good, simply because they are wealthier. But, on the other hand, as we just observed, wealthier agents pay higher share of this public good. And from the consumers choices perspective they simply buy, and again "buy" is a metaphore. They "buy" this public good G at a higher price (relatively higher price). So, we have a price effect, and if something is more expansive, if price of something goes up then the demand for this something goes down. And the price effect works in the opposite direction to the income effect. The overall impact of these two effects - the income effect and the price effect - depends on the form of the utility function. For example, just to be a little bit more technical, if there is no income effect, if the utility function is linear in x, then of course the income effect will disappear. And the price effect will dominate, and, therefore, wealthier agents would prefer lesser of the public good. This combination could be different in the case of other utility functions, but the bottom line here is that economic inequality, diversity of wealth is another possible cause of polarization of agents' preferences over the public goods provision. Now, I would like now to come back. Just for simplicity sake to the first model where preferences of agents are different. And suppose that the public good is, as we assumed before, funded by a lump sum tax. And we just agreed that every agent has, that all of these agents have single-peaked preferences, and that preferred levels of public good provision increase in the appreciation of agent of this public good. In other words, there is a monotonic association with an alpha_i and G_i. And, therefore, if we put together profiles of all agents on a single chart, we have this picture. This is the profile of the, of course it's an approximate picture, but it captures the important features of this situation, and this is all that I need. This is the preference profile of the first agent, with the lowest appreciation of the public good. And her preferred level of funding of this public good will be G1 star. Second, third, five of them altogether for simplicity's sake. And this is the final one. And the preferred level of funding of public good for this agent is G_n star. So, lets see what would be an outcome of democratic aggregation of these single-peaked preferences. Such outcome should survive every attempt to be defeated by democratic means. By defeating it by democratic means I mean, suggesting an alternative, such that there would be a majority of agents that would prefer this alternative to the proposed option. And let's say, for example, if this option G_n star, which is the optimal choice of the agent with the highest valuation of this public good, would be such a democratic winner. Well, it won't be because I would suggest, for example, this alternative a little bit to the left of G_n star. And, please, notice that every agent but the the last one would prefer this alternative to this one. So, all agents but the last one form a majority that would defeat this alternative and prefer this one. Why? Because all of their preferences are descending over G in this range, and this is because this preferences are single-peaked. So now you see why it's so important to have preferences single-peaked. So, this alternative is not natural outcome of democratic aggregation because it can be defeated by democratic means. So what is an outcome of such democratic aggregation? Quite obviously, it's this one. Let's select the so-called medium agent. Now, the agent is medium if the number of agents or, to be more precise, of preferred choices of agents to the left of this agent and to the right of this agent is the same. This is called the median agent. And in our case there are five agents. So, the third agent is the median one because the number of peaks of other agents preferences to the left, this and this, and the right, are equal to each other. They're equal to two. And quite naturally would say, okay, and what would happen if the number of agents is not odd, as it is here, if the number is even. Well, in that case, as you can easily figure out, that every level of public good provision which will be between the two peaks in the middle, would be medians. But to keep things simple, let me assume that the number of agents is even. So, this is the median agent, and this is the optimal choice of the median agent. And notice that this choice cannot be defeated by any democratic majority. Why? Well, very simply because if I would attempt to deviate from this option, say moving here, then I would say that the agents to the right of the median, these two agent would disagree because they would be worse off. In the case of moving from here to here, simply because their preference is single-peaked. So, every attempt to deviate from this median choice to the left will be defeated, or blocked, or prevented by majority comprising the median agents, and all the agents to the right of the median agent. And vice versa, if I attempt to move to this direction, to increase the provision of the public good, then the median agent would be opposed to that because I'm moving away from her optimum. But also, all the agents to the left of median will also be against this move. And in this case, this alternative will be blocked by a majority which is comprised of the median agent and the agents to the left. So, this optimal choice of the median agent is indeed a natural outcome of democratic aggregation of individual preferences. And this choice is called the Condorcet Winner, oftentimes Condorcet Champion because it wins democratically over any other alternative. So, the good news so far is that democracy is a workable mechanism of decision making, of public decision making over public economics policy variables such as taxes and public goods. So, we were able, at least in some cases when preferences are single-peaked, to overcome one of the two problems of democracy as a political regime when it comes to fiscal policy making. That is that democracy leads to gridlocks. In this case, there wouldn't be any gridlock. This choice will be the one which is supported by democratic procedure of individual preference aggregation. But how about economic efficiency of this choice? Well, unfortunately it's not guaranteed and, in fact, it's quite likely it won't be economically efficient. Let's try to understand why. What is economical efficient in this case is what is Pareto optimal. And it's easy to check that in this situation Pareto optimality is equivalent to something that maximizes the aggregated wealth of the society. And if I aggregate these individual utility functions, and, again, if I ignore w_i because these are constants, the outcome of this aggregation is as follows. And, therefore, the maximization of this expression leads to the following social optimal level of public good provision, where alpha bar is the average value of alpha. In fact, the democratic choice would entail a different level, in general, of public good provision which would be found from this equation. And because alpha-bar is very possibly different from alpha-median these will be two different variables. And, as a result, democratic choice will not be optimal. Here are illustrations. Suppose that you have several agents, five: alpha one, alpha two, alpha three, alpha four and alpha five. Now, the average level of alpha, which should be our guidance if we want to achieve social optimum, will be somewhere right here, and the median one is here. So, the median is less than average, and, as a result, there will be two little public goods, and the tax rate will be too low in comparison with what is the social optimum. And here is another situation where the average level of health would be somewhere here, but the median is here, and, as a result, in that case we'll have too much public goods and too high a tax rate. Neither will be socially optimal. Now, another thing that might be problematic when we talk about democratic decision making in relation to public economic policies, taxes, and public expenditures is that even if the democratic choice is socially optimal, we can not rule out, for example, that alpha-median would be equal to alpha-bar, alpha average. Suppose that alpha is somewhere over here. The very same picture tells us there will be quite a few people unhappy about this democratic choice because for all agents, say the median one, the democratic will be far away from what they would find optimal. These two agents will be unhappy because from their perspective there will be too high tax rate and there will be a lot of public good that they really don't need that much. And for these agents quite the opposite, they would want to have more of this public good because they valued it much higher than what the median agent does. So, even we have reached or could reach overall economic efficiency there will be a whole lot of disagreement, and displeasure, and disappointment in the society because interests of agents are different from each other. And what democracy produces, despite of its possible economic efficiency, will not be acceptable and satisfactory from the point of view of every given agent. And, of course, we can come up with similar illustrations when the source of diversity of preferences is not the difference in utility functions but the difference in income and, perhaps, some other variables. We can conclude by stating that when preferences are homogeneous (in other words, when, for example, in the case of utility functions, alpha_i close to each other or income levels are close to each other), in such cases democracy delivers public policies which are a) very likely socially efficient and b) they are close to what is preferred by every individual voter. Such policies would approximate social optimum, and they would not cause political conflict. On the other hand, preference disparity, differentiation, diversity could sway public policies away from social optimum. And even if they don't do that, they would still leave many voters disappointed over the democratic outcome. And, therefore, it should come as no surprise that in the modern world democracies are shown to be much more proper, prosperous, politically stable in more homogenous nations and societies, and vice versa. Democracies in nations and societies which are polarized culturally, socially, economically, religiously, and what not oftentimes are prone to be unstable, and they might not deliver efficient policy outcomes.