You see how we brought -- last time, we talked about further out indifference curves make you happier.
Today, we talked about the fact that you're limited by your budget. So we have the furthest indifference curve you can get to is gonna be, definitionally, at the tangent of the indifference curve and the budget constraint.
And, once again, that gives you -- we realize we don't want to measure utils, but, just for mathematical, for mathematical purpose, that gives utility at the tangency of square root of 18, OK? At that point, you are choosing six cookies and three pizzas. That is the choice you are making. That is the best off you can get given your budget. And, to see this, let's talk about some other points and why they're not better, OK? Let's talk about point A. Why isn't point A better? Why isn't it better to have two-- maybe you just-- maybe you like cookies a lot and don't like-- or like pizza a lot and don't like cookies that much. How can we say that point D is better than point A? Yeah?
AUDIENCE: Because point D is on a higher indifference curve.
JONATHAN GRUBER: It's on a higher indifference curve. So point D dominates point A because it's a higher indifference curve. Well, fine. Same person, by that logic, why not choose point E?
AUDIENCE: It's above the budget. JONATHAN GRUBER: Yeah, you can't afford it. So the bottom line is you can see graphically why the tangency is the best you're going-- is the best you're going to do. OK, likewise, point C you wouldn't choose. Point C has the same slope.
It has the same slope as point D. In other words, the slope is minus 1/2 at point C. You've drew a line tangent to point C. The slope will
be minus 1/2, just like it is at point D, but you wouldn't be spending all your money. So you wouldn't choose that point either. Yeah?
AUDIENCE: What if you have just three indifference curves so there is none that hit the tangent? Do you just go for one that's like the most tangent I guess?
JONATHAN GRUBER: We're going to come to-- we're going to-- well, first of all, we're not going to have discrete indifference. We could have lines, and the lines could end up-- you could end up lying along. You could end up lying along a budget constraint for example. Or you could have-- you could even have utility functions, which just touch a budget constraint at one extreme or another. And we'll talk about those cases.
Yeah? AUDIENCE: So utility function go through lines and the budget constraint, right?
JONATHAN GRUBER: Yeah.
AUDIENCE: Isn't this just Lagrange [INAUDIBLE]??
JONATHAN GRUBER: Well, let's come to the math then.
OK, let's come to the mathematical derivation. So that's the graphic. So let's come to the math, OK? Now, always a bit of a tightrope act when I'm doing math up here on the board, so bear with me, OK? But the key thing is the math of constraint optimization is all about the marginal decision. Remember, it's hard to say how many cookies you want. It's easier to say should I have the next cookie, OK? It's about constraint optimization. And what we want to ask is we essentially want to compare how do you feel about trading off pizzas versus cookies versus what will the market let you do in sort of trading off pizzas versus cookies. That is the optimum is going to occur when we set your marginal rate of substitution, which, remember, we defined as minus MUc over MUp, equal-- I'm going to get rid of this-- equal to your marginal rate of transformation, which we defined as minus pc over pp. And this is the fundamental equation of consumer choice. If you understand this equation, you can solve virtually every consumer choice problem I'll give you, OK? That basically, at the optimum, the ratio of marginal utilities equals the ratio prices. That is the rate at which you want to trade off pizza for cookies is the rate at which the market will allow you to trade off pizza for cookies, OK? Basically, it's saying the ratio of the benefits. Think of this as the benefits and this as the costs. Think of the MRS as the benefits. It's what you want. MRT is the costs. It's where you're constrained. You want to set the ratio of the benefits equal to the ratio of the costs, OK? Now I find it actually easier to think of it this way. If you just rearrange terms, you can write it as MUc over pc equals MUp over p sub p. I like this way of writing it because I call this the bang for the buck equation. What this is saying, your marginal happiness per dollar should be equal. This is sort of the happiness per dollar spent on cookies. This is the happiness per dollar spent on pizza. And you want those to be equal. You want the bang for the-- you want to put your next dollar where it's going to make you happiest, OK? And so, basically, think of that as your bang for your buck. So, for example, suppose you were in a position where the MRS was greater than the MRT. You're in a position where the marginal utility of cookies-- and I'm getting rid the negatives. There's negative on both sides. So I'm just going to get rid of the negatives, OK? The marginal utility of cookies over the marginal utility of pizza was greater than the price of cookies over the price of pizza, OK? That is the slope of the indifference curve was greater than the slope of the budget constraint. This is the slope of the indifference curve. OK, this is slope of the indifference curve. This is the slope of the budget constraint. In absolute value, the slope of the indifference curve is greater in absolute value than the slope of the budget constraint, OK? That would be true at points like point A, point A where you intersect-- where you basically intersect from above the budget constraint by the indifference curve. So a point like point A has a steeper slope of the indifference curve than does the budget constraint. What that says is intuitively-- and, once again, I want you to understand the intuition-- the rate at which you are willing to give up, the rate at which you are willing to give up cookies for pizzas-- I'm sorry. Let me say it-- let me say it a better way. The marginal benefit to you of another cookie relative to another pizza is higher than what the market will charge you to turn pizza into cookies. Let me say it again. The marginal benefit to you of another cookie, which is this-- this is how much more you want the next cookie relative to how much more you want the next pizza-- is greater than what the market is going to charge you to trade in your pizza for cookies. Therefore, you should trade in your pizza for cookies, OK? So let's say this mathematically. At a point like A, point A, OK, you have your marginal utility for pizza is the derivative of the utility function with respect to the number of slices of pizza. It's the marginal utility. It's derivative of the utility function. So it's dU dp, which is equal to 0.5 times C over square root of P times C, OK? And, at point A, at point A, we had two cookies and five pizzas. At point A, P was five. C was two. OK, that's true of point A. So we can evaluate the marginal utility dU dp, which equals 0.5 times C over square root of P times C. So that's 1 over the square root of 10. That's the marginal utility of the next slice of pizza. The next slice of pizza makes you 1 over square root of 10 happy. Once again, that number is meaningless. So we only care about it in ratios. So we need the ratio. So let's do the marginal utility of cookies. That's dU dC, which is 0.5 times P over square root of P times C, which is 2.5 over the square root of 10, OK? So the marginal utility of pizza is 1 over square root of 10. Marginal utility of cookies is 2.5 over the square root of 10. Therefore, your marginal rate of substitution is minus 2.5. Remember, marginal rate of substitution is MUc over MUp. So your marginal rate of substitution is minus 2.5. What does that mean? Can anyone tell me what that means? Your marginal rate of substitution is 2.5. What does that mean? That is a meaningful concept. Utils are not, but that is. Yeah, say it loudly so we can hear.
AUDIENCE: You're willing to trade-- you're willing to trade two pizzas for one cookie.
JONATHAN GRUBER: You're willing to trade. Exactly, you're willing to give up 2.5 slices of pizza for one cookie. That's what that number means. And that is a meaningful number. That's not an ordinal. That's cardinal. We can use that. You are willing to give up 2.5 slices of pizza to get one cookie. What is the market asking you to give up? How much pizza do you have to give up to get one cookie? Half a slice. You are happy to give up 2 and 1/2 slices of pizza to get a cookie, but the market is saying we'll let you have a cookie for half a slice of pizza. So what should you do?
AUDIENCE: Trade.
JONATHAN GRUBER: Eat less pizza. Eat more cookies. That will unambiguously make you happier. And that's why you should move from point A towards point D. OK, that's the intuition, OK? You basically want to trade pizza for cookies until these things are equal.
Indeed, I'd like you to go home and do the same math starting at point B. If you do the same math starting at point B, you'll find the MRS is much below 1/2. That is, at that point, you are happy to give up tons of cookies to get pizza because, jeez, you've got 10 cookies and one slice of pizza. You'd give up tons of cookies to get pizza. But the market says you only have to give up two cookies to get pizza. So you'll happily do it, and you move back towards point D. And that's sort of in a bundle sort of the intuition and math and graphics of how we do constrained optimization. OK, that is hard and very important. Questions about that? Don't hesitate to ask. OK, that is hard and very important. If you understand this, you're sort of done with consumer theory, OK?
This is sort of the core of what consumer theory is all about. It's all about this balancing act. The whole course is fundamentally all about one equation, which is marginal benefits equals marginal costs, OK? Everything we do is going to be about weighing
the marginal benefit of an activity against its marginal costs. If we take the next step, what's the benefit?
And what's the cost? Well, here the marginal benefit is the MRS. The marginal cost is the MRT. We want to set them equal.
And this sort of example I hope explained why, OK? So that is how we think about constrained choice.
Now I want apply it. I want to apply it by looking at the example of food stamps, OK?
Now food stamps are not actually called food stamps anymore. When I was a kid, they were called food stamps. It's basically a program the government has
that provides money for individuals to buy food if they're low income. Essentially, we have in the US what's called the poverty line.
And I'll talk a lot more about this at the end of the class,
but the poverty line is essentially a measure of what's a minimum level of resources you need to live in America.
The poverty line for an individual is about $14,000. OK, for a family of four, it's about $28,000.
How you feel about that number obviously is going depend on where you're from. If you're from Boston, you'll say that's insane. If you're from some rural part of the country,
you think, yeah, that's poor, but manageable. OK, we'll talk later about the poverty line, what's good and bad about it.
But, in any case, if you're below the poverty line in America, roughly speaking, you get help with buying food. And that comes through a program we now call SNAP.
It used to be called food stamps. I've got to update my notes. Supplemental Nutrition-- I don't know.
I know the N is for nutrition. OK, so, basically, what the SNAP program does
is it gives you a debit card. If you qualify on income grounds, you get a debit card, and that debit card can be used to buy food and food only, OK?
So you essentially get a debit card from the government that you can use to buy food if you're poor enough.
And they give you sort of a fixed amount every month, and that amount can be used to purchase food.
So here's the question. Why go through this rigmarole? Why not just give people cash?
This fancy thing, if we want to give poor people money, why don't you just give them money? And we're going to-- I don't want the answer yet, OK?
What I want to do is show you graphically how we think about the trade-off, and then we'll come to the answer. So hold your thoughts.
So let's actually graph how we think about food stamps. Let's go to figure 3-5A.
And let's start with a cash transfer. So here's the setup. Imagine people start with an income of $5,000.
That's super poor, OK? $5,000 is their whole family income for the year, OK?
And let's say all they can spend it on is food or shelter. Remember, as this gentleman pointed out, in life, there's more than two goods, but it makes it a lot easier to have two goods. So imagine this case. Your two goods are food and shelter. And, actually, quite frankly, if you're that poor, that probably is the only two goods you have to-- you can worry about at that level of income. OK, it's food and shelter. So you $5,000 to devote to food and shelter. So you have some original budget line, which is labeled there original budget line, that runs from 5,000 in food to 5,000 in shelter. And then you can have some of in between, some along the way, OK? Now let's say we give someone $500 in cash.
Obviously, this graph is not to scale, OK? It looks like you're doubling his income, but it's only $500.
This just sort of makes it easier, a not to scale graph. Let's say we give someone-- we say to them, look, you're poor.
We're going to give you $500 in cash. Well, now all we've done is shift out your budget constraint from 5,000 to 5,500.
OK, we've shifted out your budget constraint from 5,000 to 5,500. What does that do to your choices?
Well, consider two different types of people. Person y, OK, they used to be on indifference curve I0.
They used to spend almost all their income on food and not a lot on shelter. They were probably homeless, OK? So they spent all their money on food and were basically homeless.
Now what do they do? Well, they spend a little more on food and a lot more on shelter. Maybe now they get-- you know, $400 still doesn't buy you much shelter. They spend a little more, OK? Maybe, a night a week, they can get shelter, OK? So, basically, that's what they do. That's their constrained optimization. We're not saying it's right or wrong. This is not normative economics. It's positive. The positive thing is, given their utility function, they move from point y1 to y2. Now imagine someone like individual x. They're different. Their tastes are such that they don't need to eat. They just want to have shelter.
So they're up at point x1 initially. And you give them that $500, and they spend just a little bit more of it on food and even more of it on shelter. They just love their shelter, OK? And they're just super-- they're super Weight Watchers. They don't eat, OK? So, basically, they move from x1 to x2. Once again, not normative right or wrong, it's just these are feasible choices people could make given the opportunity set with which they're faced. And that's what happens when you give them the $500 in cash. Questions about what I did here on this graph alone?
AUDIENCE: Like, even if like you gave them money specifically for food, couldn't they then just reallocate their other money?
JONATHAN GRUBER: OK, that's a good point. We'll come back to that. That's time out if you're not a sports fan. OK, so we will come back to that. And, in fact-- OK, but do people understand what the cash transfer is, how it works?
OK, now let's go to SNAP. And let's say, with SNAP, instead of giving them $500, we'll give them the debit card. Instead of handing them a $500 check, we give them a debit card with $500 on it that can only be used on food.
How does this affect their budget constraint?
Now we see where budget constraints start to get interesting and fun and the kind of challenges you're going to face in this course in drawing budget constraints. The original budget constraint continues to be the original budget line running from 5,000 to 5,000. The new budget constraint is this kinked line that runs from 5,000 on the y-axis to the point x2 at 5,000 on the y-axis. So it starts at 5,000 on the y-axis, 0 on the x-axis. There's a flat line that goes to 5,000 on the y-axis, 500 on the x-axis. And then it slopes down parallel to the original budget constraint to 5,500. Can someone explain to me why that's the new budget constraint?
AUDIENCE: You can't spend a negative amount. So you can't spend like negative amounts of your non-food-stamp money on food.
JONATHAN GRUBER: Exactly, you have-- we are forcing you to spend at least $500. Compared to cash, where you can do whatever the hell you want, we are forcing you to spend $500 of your money on food. Coming to the question back there, it doesn't have to be a specifically labeled 500. It can be any 500. But we're forcing you to spend at least $500 on food. Well, what does that do to your choices? Well, for person y, it makes no difference whether they get cash or whether they get food stamps. Now the person, light blue shirt, turquoise shirt, asked that question. Why does it make no difference? Yeah? Why does it-- whatever, greenish, I don't know, yeah, you. Why does it make no difference for person y if I give him food stamps or cash?
AUDIENCE: He's already spending a lot of his money on food. So any money he gets he can just reallocate differently so he can spend some of the money he would have used on food on shelter.
JONATHAN GRUBER: Exactly, he can just reallocate his money, OK? That's exactly right. So, for person y, there's no difference. Look, they're already spending, what, $4,900 on food. You give him a thing labeled $500 for food. It's not going to affect their life. They'll just take 500. They'll just spend-- they'll just treat it as $500 more in cash. They're indifferent. So nothing affects them. But what about person x?
Well, person x, remember, the dashed portion of this budget constraint is from the old cash example. And the dotted indifference curve is what they would have chosen with cash. Remember, person x with cash would have chosen to still spend less than $500 on food. Even when you gave them $500, they still only spent $300 on food. So we are forcing them to not be on their preferred budget constraint. Rather, we're forcing them down to point x2, which is they'll spend the minimum they can on food, but the minimum is $500, OK? We are forcing them down to point x2. Now why do I say forcing them? Why do I know for sure they are being forced, that they're less happy at x2 than they would have been when they gave them the cash? How do I know that for sure?
AUDIENCE: They're at a lower indifference curve.
JONATHAN GRUBER: Exactly. Think of it this way. The fundamental-- one of the important things is people always get to the point that makes them happiest, OK? We call it the robustness of economic equilibria. People get to the point that makes them happiest. They want-- they always before had the choice of spending $500 on food, and they chose not to. Therefore, if you force them to spend $500 on food, they must be less happy, OK? Think of it that way. They always could have spent $500 on food. They didn't. Therefore, in forcing them, you're making them less happy, OK? So they are worse off, OK? They are forced to spend. They'd rather spend some of that money and find a nicer place to live, but we're not letting them. We're making them buy food, OK? Do people-- I don't want-- I just want to know if people understand the graphics here and the conclusions I drew. OK, now why? Why are we doing this? Why would you-- they're better off with cash. Why would we force them to have food? Yeah?
AUDIENCE: Say because what makes-- what puts people on the highest indifference is just what makes them happiest, but not necessarily what makes them like live the longest or like have the best health So, perhaps, like if you never spend money on food, and then you die, that would be really bad.
JONATHAN GRUBER: OK, but, basically, what you're saying is you know better than the guy. Let me-- I'm not accusing you. I'm just saying, look, if people knew best, maybe they'd like to just like have a nice house and die, OK? If people knew best, then there'd be no reason to do this. The reason to do this is because we think they don't know best. So, for example, let's change the label on the y-axis, just a small change. Let's cross out shelter and write cocaine. [LAUGHTER] OK? Well, in that case, maybe we don't feel so bad about forcing the guy to buy food instead of cocaine, OK? In other words, this a program which might make sense if we are paternalistic. Now we're getting into normative economics, paternalistic.
If we think that people won't necessarily make the right decisions for themselves, then it may be worth actually making them worse off because they're not worse off. Their perceived benefits are worse, but they don't know what they're doing, OK? Now you can see why-- I hope you can sort of immediately see why this concept makes economists a little nervous because why do we know what they want better than they do, OK? So it makes people a little bit nervous, economists a little bit nervous, and a lot of people a little bit nervous to say, gee, maybe they're just happier doing cocaine. And how do we know that that's the wrong way for them to spend their resources? Yeah?
AUDIENCE: Well, like can't you look at it from the perspective of like this is taxpayer money, right? So then aren't you also just factoring in how the taxpayer wants to spend their money and then their indifference curve and all their information?
JONATHAN GRUBER: That's a very good point. Now but there's sort of two points there. First of all, if the taxpayers' goal is to help poor people, then why shouldn't you make them as happy as possible, right? If tax-- why am I giving money to this poor guy? Because I'm sad his poor. But, what you're saying, I'm not actually that sad he's poor. I'm sad he's not eating. If you're really just sad he's poor, then you should give him money. If what you're sad about is, gee, I don't like how he's living-- I don't like his-- I'm sad he can't have better food to eat, sad at the place he lives. Then you're starting to impose your preferences, but let's be important. That's imposing your preferences. Yeah?
AUDIENCE: I feel like the indifference curve only goes for happiness or like contentedness, but, really, the point of SNAP isn't really with contentedness or happiness, but rather like what would be to a more sustainable life.
JONATHAN GRUBER: Well, that's a related point of the taxpayer. If the taxpayer cares about, look, we want a healthy populace that's going to live a long time and be productive and pay taxes, then that would be a reason to do this. But, once again, I want to emphasize, OK, this is paternalism. If you really just care what makes people happiest, you should give them cash, OK?
So that raises two questions, OK? First of all, first question-- yeah?
AUDIENCE: So how about like negative [INAUDIBLE].. Because, for example, if we pump a lot of money-- if we allow people to spend a lot on shelter,
that's not really going to help people. It would just make the real estate developers rich. And say the amount of shelter is kind of
fixed, but like the amount of food that eaten [INAUDIBLE].. So, if we let people spend more money on food-- JONATHAN GRUBER: Yeah, yeah, so, basically, that's
a great question. And, in general, we're going to-- I'm going to answer a lot of those questions with the same cheat this semester, which is we're going to assume the markets are perfectly
functioning. So there's no-- you're imposing sort of a market failure. If there's no market-- once there's market failures,
all bets are off. But, with no market failure and no paternalism, you'd want to give them cash.
So this raises an important question. Do food stamps actually increase food purchases?
First of all, there's two reasons why they might not. Reason one is everybody could be like y.
x is sort of a silly case, right? You're going to die if you eat that little. And food stamps aren't that much.
They're maybe like $3,000 a year. Everybody is going spend $3,000 on food. So the first issue is the first reason why food stamps may not matter is that, in fact, everybody is spending at least that amount. Everybody is like y, and nobody is like x. What's another reason why it might not matter? What's a way people could get around food stamps? Yeah? AUDIENCE: Buy food with food stamps and sell it. JONATHAN GRUBER: Yeah, they could set up a black market where they, essentially, say, look, I only want $2,000 of food. The government is making it worth $3,000. I'll buy my extra $1,000 of food, and I'll sell it to people who do want it. And I'll end up still eating $2,000 worth of food. So we actually want to know do food stamps actually increase food consumption in practice. Are they making a difference? Well, actually, we've run an experiment on this, OK? We're going to talk in this class a lot about empirical results in economics. This class is mostly going to be a theoretical class. That is we'll talk about models and ideas. But we're also-- since, basically, I'm an empirical economist, we're going to talk about empirical economics, which is results and testing the theories we develop. Empirical economics, here's a great example of empirical economics is we set up a theoretical model.
You always want to start with the theory, but the theory sometimes has predictions, which are uncertain. Here we have an uncertain prediction from theory about whether food stamps will affect food purchases or not. So let's test it. And the way we test it is we actually have run food stamps cash out experiments where we literally take guys on food stamps and give them cash instead and watch what happens to their consumption before and after. It's a real randomized trial. We literally flip a coin. Heads, you keep your food stamps. Tails, we replace those food stamps with an equal amount of cash. Then we watch what happens. What happens is that people spend about 15% less on food when you give them cash instead of food stamps. That is food stamps is forcing people to spend about 15% more on food than they would like to unconstrained by the cash. Yeah? AUDIENCE: Yeah, this gets you into the behavior of [INAUDIBLE]. I remember reading an experiment like, if you have the price of gas go down, the actual like amount of money spent on gas is constant. And this might translate to food stamps because like food stamps are like explicitly on food.
JONATHAN GRUBER: Yeah, you know, that's a great question. And that's you're asking about richer theory, richer theory. And I'm telling you that I'm going to give you the empirical evidence. So, whatever the theory is, the empirical evidence tells you what happens. And there's different explanations for why.
So the empirical evidence is that, basically, the price of our paternalism is 15%, OK? We are making people, effectively, 15% worse off. We're making them spend 15% more food than they want to. So is it worth it? Well, actually, the evidence is starting to pour in that it might not be worth it because there's starting to be a lot of experiments where we're giving people just cash, especially in developing countries. In developing countries, the answer seems to be just giving people cash makes them better off, that actually, especially in developing countries, people use the cash in productive ways. So, for example, they have a series of evaluation programs where they've given people cash, mostly in developing countries, in Africa in particular, some in the US.
And they find that people spend relatively little of that on drugs and alcohol, but they actually tend to spend it productively. And, in fact, they found, in developing countries, this often provides valuable resources for individuals to start businesses. So they ran experiment Uganda where a nonprofit company randomly offered a group of women $150, which is huge relative to their income. That's 50% to 100% of annual income in Uganda, $150. And what they found was, after two months-- after 18 months, these women had used that money to start businesses. And that actually raised their earnings. That actually effectively doubled their earnings. From that one injection of cash, it led them to actually double their annual earnings, OK? So that leads one to think that maybe we should stop being paternalistic and just give cash. Unfortunately, if you're a reader of policy websites like
I am, the best one of which is vox.com-- it's a great website-- they had an article just the other day pointing out how they actually followed these women up nine years later. And, nine years later, the effect had totally gone away. So the story isn't quite necessarily positive, but it's not negative. They're not worse off, but it looks like, at least what in the short run made them better off, well, that effect fades over time. But the bottom line is, at this point, I think the evidence is sort of probably in favor of being less paternalistic and just giving people cash, but that runs into a lot of difficulties in terms of our concerns about how people will spend it. So let me stop there. We will come back on Monday, and we'll talk about how we actually go from this stuff to the demand curves we started the class with.
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