There are situations where marginal tax rates of 100% or nearly 100% may be justified. Three models will be sketched, using indifference curves. One, which makes unusual assumptions about preferences for labour, can justify income subsidies of low incomes with implicit marginal tax rate of 100%. The second, assuming high substitutability between consumption and work, might justify marginal tax rates approaching 100% on the highest incomes. The last, with competition between skilled workers (such as sportsmen or inventors) for market share, might justify marginal rates of 100% on high incomes of a particular type. The assumptions under which these conclusions follow may not hold in actual economies, but they might sometimes. In any case, extreme results, and the reasons for them, can help us understand how incentives work and their implications for taxation.

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Going to do some welfare economics. You’ve been lined up for it now. But this is old fashioned welfare economics. It’s public finance, of course. What am I going to do? I’m going to ask you to think about some very simple models, highly simplified. So since I’m going to be coming out with some pretty strong conclusions, I think I want to share with you my sense that these are not exactly realistic models. They’re models that are intended to let us all get a real feel of what kinds of conclusions might come out if we were to be more careful about the specification of the real world. But the only way you can get any insight into what kinds of arguments might work or not work is by doing it in simple models that you can really understand first. That’s what this is about. So it’s a simple model, pretty standard. It’s a population of people and they have utility functions. Well, you’ve just been learning about utility functions. Just take it for what it’s supposed to mean. And there are 3 variables that go into this utility function: consumption, work done, which if you’re going to apply this at all, W is going to be wage income. So it’s the reward of the work or labour that people do. And the type of the individual. This is the unobservable element. Some people are very clever, able to do really difficult jobs without a lot of trouble. And other people find that really quite difficult. But you can’t write the tax system in a way that assumes you know that kind of information. Now this afternoon I’ll be talking a bit about why you can’t do this. But I mean of course you realise this is true that the way the tax system works is it takes information that readily observable on the basis of the trades that people do, their interactions in markets and so on, specifically their before tax income measured by their total wage income in this case. And taxes are applied to that so that you can spend some part of W or maybe more than W if you get a subsidy. I promised subsidies as well as taxes in the title. So we’re certainly going to have that. And that’s what gets spent on consumption. T is type, it’s not time. This is not a model where any time dimension comes in. So we’re not into dynamic public finance here. We’re just trying to get at one particular feature that is going to be central in any kind of discussion of what the tax system should look like. Which is one in which there is both before tax income and after tax income and a variety of people, heterogeneous population of people, absolutely essential. But otherwise we’re going to be as simple as possible. Then we’re going to assume a number of very special features. I will keep referring to this functional relationship between before tax income W and after tax income C, work and consumption, as being the budget constraint. Everybody faces the same budget constraint. That’s the effect of type being unobservable but we’re not bringing it in. Oh, of course things are not quite that simple. And I will later point out there are a number of ways in which we really would want to use type ideally and could. Our question is: How can we maximise total utility, sum of total utility? Well, of course we haven’t fully specified the model yet because we haven’t said what people can do. But first I want you to get a sense of how this looks if we put it in a picture. I suspect this is familiar to many of you but these are 3 indifference curves, obviously not indifference curves of the same person. We’ve got a typical indifference curve of a low skilled person, a medium skilled person and a high skilled person. I don’t mean anything about the intrinsic worth of these people. It’s just measured by the nature of their preferences. This is the background picture that I’m going to be exploiting. I can’t use this picture to prove any results but I think you’ll see that seen in context it can be very suggestive for dealing with various interesting cases. That’s what I want to do. Now what I haven’t been clear about or I didn’t say explicitly, you may have read it on the screen, is that the total amount of consumption in this economy has to be equal to the total amount of wages. And I’m actually going to be drawing diagrams where some part of total wage income of total work is used by the government for fixed purposes. Because if you’re trying to fit this sort of model to actual national income data, of course you find that that’s generally quite a large item. So I think it kind of helps our intuition to be a little realistic on this. Now, the first thing we want to appreciate, terribly simple point really, is that since people are going to be choosing all with the same budget constraint, each person will, provided of course they are standard economic persons who choose rationally, they will be at a point on some particular indifference curve which has that indifference curve tangential to the budget constraint. So that’s the picture we’re going to see. I’m going to draw things now with a particular assumption which I found was enormously helpful for getting reasonable regularity into behaviour. And an assumption I think is perfectly reasonable in most contexts. And that is that if you look at a couple of arbitrarily chosen indifference curves of 2 different people, they will cross at most once. That’s a rather geometric statement. But it says that it’s meaningful to compare people by whether they find work harder or easier. So this parameter T is really describing something that of course I said was ability. And that's exactly what we had in the picture that I did there. By suggesting indifference curves would typically look like that, I was immediately trying to get you to take it for granted that there would be single crossing. And now I’m saying the budget constraint, that C function is going to be a curve which will be tangential to the indifference curves that correspond to peoples actual choices. That’s what we’ll see in the next picture. Oh, wherever it is. There it is. We’ve got a consumption function which is... It is red and I don’t know whether you can possibly see that it’s red. But I’m going to call it red. Nice easy picture to play with. The surprising thing is that although it could be almost anything as soon as we start making interesting assumptions it will suggest interesting results. Well, I’m going to look at 3 special cases. Or I have looked at 3 special cases in preparing this talk. Whether I will get to case 3 depends very much on a clock that I have facing me here. And I’m not at all sure that I’ll get to case 3. And really it would make me quite happy to think you were all terrible eager to know what do I have up my sleeve for the final little bit. Yet another bit of stuff in inventing which seems have been rather a theme at this particular meeting. But back to the first 2 cases. The first case is going to change the look of the indifference curves by allowing that actually at least up to a point people like working, people want jobs. Well you do, I do. So that’s some of us at least. And the second example I’m going to talk about is one very extreme case where consumption and work are perfect substitutes one for the other. You may think that the answer there is going to be dead easy. Not a bit of it. Then the third case, you see, is how you would like to tax inventors. I think these are all for various reasons, perhaps it would be time to indicate quite interesting cases. Well this is the general picture that we’ve got. By the way that 45 degree line going up there, at least I claim its 45 degrees, is the production function that the average of C and W, let’s say the average of the points that are chosen by people, has to lie on that line. But don’t struggle to imagine it. That is the sort of thing you do on your own late at night at the desk to try to get a sense of what’s going on. Alright what about enjoying work? Well, it’s not just a matter of theoretical interests because there’s quite a lot of evidence from these pleasure and happiness studies that not having a job is really very bad for most peoples’ happiness. That ought to be reflected in the indifference curves that we use. Here of course I’m extending it beyond the question of having a job and not having a job to how much work you actually do. And it seems to me at least plausible and this is the assumption I’m going to follow out that we should take an interest in indifference curves of this kind of shape. In other words working really hard, really long hours is a bit more than we’d like to do unless there was some financial reward to doing it. But that we might well be prepared to do a certain amount of work even without financial reward. Now you can see what question I’m going to ask. What should the shape of the optimal consumption function, budget constraint be if people typically have indifference curves like that? We constrain the budget constraint to be non-decreasing. I’ll comment on why. I first did the problem without making that constraint and I was getting absurd answers. And I’m going to have to assume that at the lower end of the distribution things kind of tail off. Well, actually you may know that all data kind of suggests that, that really there are relatively few people with tiny productivity. Actually there are probably some people with negative productivity but I won’t pursue that particular example today. What will we find? That the optimal shape is like that. There’s a flat bit for the lowest levels of before tax income, in other words marginal tax rate of 100%. The presence of total marginal tax rates of 100% has kind of shocked people for as long as I’ve been an economist. If you work these things out for the British economy at least, I think it’s always been true throughout that period that at the lowest incomes the marginal tax rate for people with low incomes is 100%. It turns out that that may not be unjustified provided of course that you think this assumption that up to a point people like working is correct. It doesn’t follow simply from the picture. You do have to do some work to check as you perhaps realised when I said I had to assume that the distribution of types was thin at the bottom. But not all that thin, just realistically thin. It has to be at least negative exponential. The implementation of this you would think would be quite difficult. It’s actually quite surprising that in fact it’s possible for things to be set up with 100% marginal tax rate because you’d think that this would stop the labour market working properly at the lowest levels. You would think that you have to be able to attract workers by offering a higher wage. Otherwise there’s no way in which the market could work to clear properly. And 100% marginal tax rate means that doesn’t work until you move beyond that range. But at least it suggests that in the model a high marginal tax rate would be right. Now the reason I put in a constraint but said that the tax rate couldn’t be more than 100% was that of course because the model would have said it should be higher than 100%. And it curls round the indifference curve of the lowest ability people. Well there you are. Let’s carry on. Tell you about the next example, one with extreme substitutability which I looked into partly because I think extreme cases are quite illuminating even if they are unrealistic. I think it’s always right in discussing optimal income taxation. That you start with the case where people have totally inelastic substitution of consumption for labour. In that case of course it’s true that the optimum has a nearly 100% marginal tax rate for everybody. You can get the first best if people don’t wish to substitute one for the other. That means of course that they had inverted L shaped indifference curves. It’s a nice easy case and it’s as well to understand that it’s true. And starting from that case intuition may well make you think that the greater substitutability between consumption and labour, the less would be the marginal tax rates. So if you wanted to mount an argument for having low marginal tax rates, it would seem that you should look to see whether there might be evidence that people are quite easy about substituting consumption for labour. Let’s say have indifference curves that are pretty much straight lines. Perhaps not that extreme because you’d think well the extreme case must be equally easier. The answer must be zero marginal tax rate, so something like that. Well, now I’ve written here an example of what a utility function might look like to have perfect substitutability. I mean it’s not intrinsically unbelievable and you have to look into your own psyches to decide whether there might be some truth in that. I really wasn’t prepared to pursue this on the assumption that there was no upper bound to how much people could work. So that in fact I’ve used a picture like this. Typical indifference curves for the 3 types that I spoke of although I have in mind a whole continuum of types of people. I’m talking about here. And we’ve got consumption and work as before. So obviously at that upper bound suddenly people’s elasticity of substitution changes from infinity to zero at that corner on the indifference curves. But I think of the corner on the indifference curve as coming at a very high number of hours per year, per lifetime. And it’s then a question. Are you going to find that everybody goes there? Now I want you first to concentrate on the straight line part of these indifference curves. Let’s imagine that we’re getting pretty close to saying people are completely straight line indifference curves. Well in that case remember that the budget constraint is going to be a lower envelope to the indifference curves that people choose to be on. A lower envelope to straight lines is necessarily concave. That’s to say it is necessarily the case that the slope of the budget constraint keeps decreasing or at least not increasing as you increase the quantity of wages. Actually this is what things should look like. I don’t just mean at the optimum. I mean that you’re always going to be able to represent things this way. Now of course it’s possible that everybody should choose to work at the maximum number of hours possible. And this may make you begin to really worry about the nature of the assumption. It’s very hard to push through the idea of high substitutability between consumption and labour for the whole range as you can see. So for a big range here you have the marginal tax rate increasing. The marginal tax rate is 1 minus the derivative of the budget constraint here. But then you have the possibility that things may turn around at the top. Now at that point of course I should enter into some algebra but I don’t have the time left to do that. And you don’t want me to do it. But it turns out that certainly with central kinds of cases, utility function for example, that the right answer is that the marginal tax rate keeps increasing as wages go up. So for all types working less than the maximum budget constraint is going to be concave. So we’re going to have this increase all the time. And certainly that happens at the optimum. And in the optimum you even have an increasing marginal tax rate. Right up to the very top. On the whole I think one should use models where there is no upper bound to the type, the ability of people, because clearly there’s a great deal of uncertainty about where the top might be. And that’s best captured by having a probability distribution for the types of people which has no upper bound. So that would then lead to the conclusion that asymptotically the marginal tax rate would be 100% for very high incomes if you have this high substitutability. But it’s true that at the very top of the ability scale people are indeed working as much as they can. That’s what the mathematical analysis shows. That was my second example. And again you see what I’m after here is to show how good is the intuition you can get from simply looking at a rather simple graph although you do have to get into the business of a bit of calculation to do it. But now that makes one wonder. How do things vary through the whole range? It shouldn’t be too surprising to conclude that there aren’t going to be really nice results about, say, monotonicity of the optimal marginal tax rate in the elasticity of substitution between consumption and labour. There should be a warning certainly not to rely too much on very simple intuition when you’re thinking about these problems. But let me use the time I have left to talk about my third example which is one that in the end I’ve decided it would be harder to... I would need longer than this to show graphically quite what is happening. But I have a graph that may help you to understand what’s happening. Here I describe a rather different model although it’s within the same general framework as we’ve had already. The easiest, although a rather strange model is to suppose there’s a population of only 2 people but the government in setting the tax system doesn’t know the types of the 2 people. What the 2 people are, are inventors? This is an economy where the only thing that happens is that people try to create very productive technology. And one of them may succeed better than the other. I ignore the case where they both happen to come out with exactly the same technology described by productivity. Of course, so there’s only one winner. And if there’s only one winner what happens to the other one? Well the other one is going to have to rely on a government hand out of some kind to live. So what is the government got to choose? It’s got to create a budget constraint which says if you get nothing, if you were the loser in this inventing competition, you will get C of zero. There is a minimum level on the vertical access. We’ve got that picture here as C nought. So here I’m calling the 2 people, perfectly reasonable in this context, Alpha and Beta. Alpha of course is the somewhat more skilled inventor. Beta is the less skilled inventor. But remember we don’t know in advance what Alpha and Beta are. It’s just the distribution of the possibilities of Alpha and Beta. What could anyone do? Concentrate on the Beta type. There’s going to be an indifference curve that shows for type Beta the combinations of C and W. Remember now this time that C is going to be what that person gets if anything. And W is going to be some measure of the quality of the invention which you could think of as being total earnings from selling the invention after you get the patent. That will vary of course. There’s going to be a right answer. That broken line by the way as you will have guessed is supposed to indicate what the optimal constraint might be. And where it intersects zero utility curve, zero is the least utility that anyone is prepared to accept. So the most that the Beta type would be prepared to do given the broken line consumption constraint is given by the intersection of the zero utility curve, the zero indifference curve with the constraint that we impose. Well then given that what is the Alpha type going to do? Well the Alpha type... There’s no point in doing W less than the Beta type would have done. So instead in this particular model the Alpha type may do exactly what the Beta type would have done. Just keep him out. Or he may do more. He will look at the consumption function and the budget constraint and decide whether to do more. My claim is, and that’s what I don’t have time to show you the reasons for, is that, provided in the distribution, the joint distribution of Alpha and Beta, Alpha and Beta tend to be fairly close together. They don’t have to be all that close. And sometimes they could be quite far apart. It is in fact optimal to have budget constraint that is flat like this broken line in its upper range. In other words marginal tax rates should be zero. But I think although that’s a very peculiar sounding model... Of course it’s actually my attempt to give you a simpler version of a model in which there are many types and where they can move into a different kind of activity than inventing. So something that will sound a bit more realistic. But I thought the spirit of the lecture I’m giving now was to be that we can use very simple and unrealistic models to show possibilities that you might not otherwise have thought. Could this possibly apply then? Well this isn’t going to be an argument for doing it for the whole economy because not everybody is an inventor. But at the same time I would say that amongst high income people a lot of them are making their income as a result of something like direct competition with others. The most extreme cases would be people like tennis players who play in tournaments and if you had a prize system that the winner takes all. That would be exactly what I’ve described and called inventing. So there may be categories. Now the interesting thing is that these may be categories that you could define. You could think in terms of having special tax rates for people in certain fields or professions. Now you may say that’s politically impossible. That’s exactly the kind of thing people have been discussing about limiting bankers salaries, particularly in Europe. So maybe it’s a possibility. Thank you very much. Applause.

# Sir James A. Mirrlees (2014)

## Some Interesting Taxes and Subsidies

# Sir James A. Mirrlees (2014)

## Some Interesting Taxes and Subsidies

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