Transcription
How's the class? So today we're going to be talking about utility theory in chapter 20.
The concept of utility in economics, it's essentially the want-satisfying power of a good or service. Or another way you can look at it is your level of happiness. Um, now, this word "utility," it's not necessarily implying usefulness or practicality as we think of the word utility. It's got a lot of usefulness, we can, we can use it for a lot of things, that's what we think of utility. Actually, it's strongly dependent on individuals' tastes and preferences. And essentially, when it comes to utility, consumers will seek the highest possible satisfaction. So essentially, we're talking about utility maximization or utility optimization, which is essentially a lot of economics, right? Economics is all about making decisions, and how do we maximize or optimize our outcome given our relatively scarce resources?
Okay, so when we talk about happiness or utility, we're going to be talking about it in terms of a concept of a util. A util is a representative unit by which utility is measured. So we're essentially going to be converting happiness into a point scale. And this concept of a util is a point, a point on our happiness scale. So as we add numbers, remember economics, we talked about the gold star ideas in economics, we talked about opportunity costs, we talked about supply and demand, and I also talked about marginal decision-making. So marginal utility.
Economics is a marginal decision-making process, meaning we make decisions on the margin. We decide what we're going to do next. Remember we talked about the word marginal, you can substitute words like additional, next, or change. So marginal utility, it's the change in total utility due to a one-unit change in the quantity of a good or service consumed. So marginal utility equals change in total utility with respect to our divided by a change in quantity consumed. Now, if I were to write that in a little more efficiently, generally I write it like: MU is a change in total utility with respect to the change in quantity. Now, this change in quantity, because we're talking about incremental steps in economic decision-making, usually this denominator of a marginal when it comes to a change in quantity, it's going to be a change in one unit. But this is essentially how I like to kind of write it out on notes if I tend to be writing it out for a class.
So here are some observations regarding utility theory. Marginal utility, or your change in utility, your change in happiness, or your additional happiness, falls as more is consumed. And then marginal utility, your change in happiness, equals zero when total utility is at its maximum. So this marginal utility falls as more is consumed, it's essentially kind of saying that as you repeat an action over and over again, the benefits of it start to diminish. So it takes us to this concept of diminishing marginal utility, or the law of diminishing marginal utility, as you'll also see it stated as. As more of any good or service is consumed, its additional benefit declines, or sometimes they say its additional benefit eventually declines. So increases in total utility become smaller and smaller as more is consumed during a given time period.
So one thing I want you to note is that we're going to be looking at two concepts: marginal utility and total utility. They're going to be tied into each other, and it's going to be important to see our total utility, but we're going to be focusing on marginal utility for our decision-making. In other words, how much of a good or service do we consume? And we're going to be adding things like prices as we go to help make our decision a little more concrete in terms of maximizing our happiness.
So, so here's a look at, uh, pizza and marginal utility. So like we mentioned, as you consume a good more and more, you're going to have reducing amounts of happiness. What we don't see here is we don't see that marginal, we're just looking at total utility. So as you notice in our slices of pizza, as we go down or increase our consumption of pizza, we're going to see our total happiness is going to be increasing, which is good, right? We're going to eat, be eating one pizza piece of pizza, two, three, four, and we're going to get happier and happier and happier as we go. But what sometimes people overlook is the very important marginal changes, the changes as we consume one more. So we're going to add that concept in of marginal utility, and I'm going to add a column here between these two. And as we can see, going from zero to one, we're going to get an additional happiness, going from zero happiness prior to consuming one piece of pizza. Then we have one piece of pizza, and then we're going to have marginal happiness, a change in our marginal happiness, going from zero to 50. So the total utility at that point, because we've only had one piece of pizza, is going to be 50. And then we're going to have a second piece of pizza. This is our marginal piece at this moment. We're adding, we're changing the total amount, we're adding it, and so we saw a jump in our total utility from 50 to 90. So that's great, our total happiness is going up. But one thing to note is how much did it go up by? It went from 50 to 90. So our marginal utility of this second piece of pizza is equal to 40. So our marginal utility is going down despite our total utility going up. Then we're going to go to our third piece of pizza, and now we're going to jump up in our total happiness from 90 to 120. So we can see what that third piece of pizza contributed. It contributed 30 units. And then as we go out to another piece of pizza, to our fourth piece of pizza, then we can see going up another 10 units, and then seven units. And then actually, that sixth pizza pizza looks like we hit our maximum. Now we're stuffed, we're full, we almost feel uncomfortable, we're loosening our belts or whatever at our other meal. So we're going backwards in total happiness now. So instead of going up in our total utility from 137, now we went down. So now we had a minus 17 marginal utility. So our marginal can actually get negative, and that's when the point when you know you went too far, you went too far once you went past five because you took yourself backwards in happiness.
Another look. Now we can look at our total utility getting it off of the marginal. So before we had total and we got our marginals. Now we have our marginals and now we can get our total utility. Sorry, let me. So now I'm going to add a column for total utility. And so going from zero to one, we got 50 units of happiness. So our total utility going from zero, now we have 50. Then our second piece, we added 25. So now we have a total utility of 75. Our third piece of pizza, now, well, it didn't add anything. We didn't go backwards, but we're no happier. But once we eat that fourth piece of pizza, now it sent our total happiness backwards, and we should have stopped earlier.
So all we did so far is we laid out a point scale and how we're going to deal with this concept of happiness, how we're going to get some numbers to it, and how we're going to analyze the changes in happiness. Now we're going to be looking at optimizing our consumption choices because this is really about the consumer's perspective. After this, we're going to switch, the chapters are going to switch to looking at things from the firm's perspective. But right now, we're looking at the consumer's perspective.
So the consumer optimum is going to be based on a choice of a set of goods and services that maximizes the level of satisfaction for each consumer, subject to a limited income. This is the very important condition. Remember, economics is all about maximizing our happiness in terms of our optimizing our outcome based on scarce or limited resources. So our limited resource as an individual will be our limited income.
So optimizing consumption choices. A consumer's income should be allocated so that the last dollar spent, the marginal dollar on each good purchased, yields the same amount of marginal utility. So if we're talking about a marginal decision-making concept, a marginal decision-making concept, we're really looking at the next action that we take. So it's all about what do we do next? When you talk about marginal decision-making, it's what do we do next? How do we spend our last dollar? How do we spend our next dollar? That's essentially what we're going to be looking at from here.
So looking at the end result, what you're looking, what you're seeking to do is achieve this optimum equality. The parity between these ratios is what you're trying to achieve as you go through your rounds of purchases, trying to determine how you spend your scarce resources. So for me, again, I simply like to write this out as: Marginal utility of good A, the additional happiness of good, good A, with respect to that price of good A. We're seeking to achieve a bundle of consumption goods, goods A, B, and C, such that: Marginal utility of A with respect to the price of A is equal to the marginal utility of B with respect to the price of B, which is equal to the marginal utility, marginal utility of C with respect to the price of C. So we're looking, remember the change, we're looking to our next purchase to achieve this parity between these ratios, and this will help us optimize our consumption happiness bundle.
So let's take a look at a situation where we are dealing with a budget of $13 a week that we have to spend, and we're only looking to spend it on two goods in this consumption bundle. And we're looking at, uh, your phones, and maybe you get, uh, happiness out of sports applications as well as game applications. Both of these are goods that will give you satisfaction. Now, we're trying to figure out what level we should consume, given the price of the sports app being $1 and the price of the game app being $2. If we didn't have prices associated with this, then all we'd be looking at is which marginal utility gives us the greatest happiness. But now we're trying to determine it based on its relative price so we can maximize our happiness given our scarce resources.
So if you recall, we're trying to maximize our consumption bundle given that solution where we have the marginal utility of A with respect to the price of A is equal to the marginal value of B with respect to the price of B. So we want to see things in these ratios over here. And so why don't we go ahead and add those columns, and then that will help us determine how we want to allocate our resources to optimize our happiness?
Okay, so there we have added the marginal utility for the sports apps with respect to the price and the marginal utility for the games app with respect to the price. So now we have, uh, an idea where we get the most bang for the buck, essentially. So for starting out with $13, I have $13 to spend. Where do I spend my money on the first app? I'm going to look at the ratio. So this is the highest ratio, 1200. I'm going to, I'm going to get the most bang for my buck out of spending it on a, uh, the sports app. And so my first dollar will go there. So up from 13, then it'll go down to 12 because I spent $1. And then after that, I got to figure out where I'm going to spend my money next. And now the highest ratio possible again will be leading to this 1000. So that's another dollar spent. So now I'm going down from 12 in my budget to 11. And then a thousand. The next step I could take is I could take a step of a third sports app, which would get me 800 in that ratio, or I could go to the game app, and that 850 is higher than the 800. So I will spend my money there. I get more bang for the buck there. That was a $2 app. So no, my budget, uh, allows, I only go to have $9 left. And from 850, then I'll go back to the sports app because 800 is greater than the 700 that I could get from the game app. So that's another dollar. So now I have $8 left in my budget. And then from there, I go so forth and so on. That gives me six, five, and so forth and so on. So I would figure this out. And then once I'm done, I would tally up how many sports apps I have and how many games apps I have, and I would figure out, and that would give me the most efficient use of my limited income to maximize my happiness. So if we continue this through, it looks like my optimum bundle of happiness based on my $13 limited income would yield five sports apps and four game apps to make me happy. And essentially looking at the price, if it's a dollar each, that's $5. And then $2 on the other side, so $8. So that's $13. So yes, the total equals to $13. I exhausted my income, my budget, and that would this mix gives me the total happiness possible.
Okay, so I hope that made sense. Now going to, uh, something called the substitution as well as the real income effect. The substitution effect is a behavior thing. So it's the tendency to substitute cheaper commodities for more expensive commodities. And so when a relative price increase occurs, we're going to change your consumption bundle. We're going to substitute away from higher relatively higher priced goods and substitute our consumption towards relatively lower priced goods. And then there's the real income effect. And it's essentially the change in consumers' purchasing power when price changes. When a price change occurs, essentially the real income effect says that because, for example, if price goes down for a good, you have relatively more income in a sense, you can purchase more goods and services. Or what it is here, as price increases, you can purchase less. So your real income decreases.
Now we're going to bring in something called indifference curves. Indifference curves, it's a curve composed of a set of consumption alternatives where each point on the curve yields the same amount of satisfaction to a given consumer. A consumer does not prefer one point over another on the curve. In other words, they are indifferent to each and every consumption bundle on the curve. They'll be equally happy on a given indifference curve. So essentially, they are indifferent.
So an indifference curve will be set up looking like this, where we have the two goods that we could consume, A and B, maybe apples and basketballs. And this is the general shape of an indifference curve. It is convex to the origin. It has a negative slope, but one thing you'll notice, it's generally a curvilinear slope where we have at different points on it, we're going to have different slopes. And as you go move away to from the origin and consumption here, you're going to see that the slope, the rise over the run, flattens.
Okay, now the indifference curve also, the nature of these is that there's a whole bunch of indifference curves within our diagram, and they're all uniform. There's going to be no overlap. So all I'm doing is really taking a really, a few of the potential indifference curves on here. It really would just be a black mess of indifference curves, making a totally black opaque. All I did was pick out a bunch. And our object is to find the indifference, the highest is to achieve the highest indifference curve farthest from the origin because that indicates the highest level of happiness. So essentially, these are ordinal levels of measurement. We're talking about order. So it's really not, it's, it's not cardinal in a sense where it has actual meaning. All we're trying to do is get it this concept of, of well, where we want to be. We want to be at the highest possible indifference curve. So that's, uh, what we're trying to achieve in this exercise, or in this understanding.
So now, some additional concepts regarding indifference curves. The shape is always curved due to this concept of diminishing marginal utility. So this diminishing marginal utility means as you consume more and more of a good, then its additional happiness decreases. So when you're looking at it relative to other goods, as we consume more and more of B, then our happiness goes down. So look what, in a sense, look at this slope here, this higher rise in terms of good B versus good A, the run. You notice that compared to over here, good B relative to good A, the slope, the rise over the run, is different. Now, remember, they're equally happy on this indifference curve. That of these two points that I picked out on, look, supposed to be on the same indifference curve. So good B, now in terms of to be equally happy, there's a lot of B that could be traded for just a little A. Whereas over here, very little B for a lot of A. Because over here, we're consuming a lot of B, so it becomes less pleasure-inducing for us. And A, because of its scarcity, it becomes more so. As we go down on this side, on the horizontal axis, we're consuming a lot at this point. We're consuming a lot of good A right here at this point, relative to very little good B over here at this point on the vertical axis. So that's saying there that the level of happiness, the, the bundle, consumption bundle to keep us equally happy, we need a lot of good A relative to the now scarce good B. So because B is scarce, it's going to give us a lot more additional happiness. So that's how this marginal rate of substitution and so forth, uh, can be considered in this. Sorry, I, I brought in a concept we didn't mention yet. Shape is always curved due to diminishing marginal utility. It always has a negative slope. So that negative ratio, that negative B to A is the trade-off concept that, that hopefully you're from, you're thinking of as we look at these ratios. And the shape is convex to the origin.
Now, this is the concept that I was trying to bring in: the marginal rate of substitution. The change in the quantity of one good that just offsets a one-unit change in the consumption of another good. So total satisfaction or total utility remains constant. So those, those ratios on the different points on that curve, those represent that marginal rate of substitution, the, the trade-off that has where they offset one good offsets the other to maintain a constant total utility.
So using indifference curves and something called the budget constraint, you maximize utility given a budget constraint when the indifference curve's tangency point is equal to the slope of the budget constraint. Actually, we'll get back to that slide. I just jumped to.
So considering a budget constraint, suppose our budget is $100 and we have two goods, uh, good A and good B, where good A's $10 each and $20 each. So if we had $100 and we spent all of our money on good A, then let's say two, four, six, eight, ten, then we spend it all on 10 units of good A. $100 times 10, I mean, 10 times 10 units is $100. That's our entire budget. But if we spend all of our budget on good B at $20 each, then we would have only, uh, we would have five units of B at $20 each times five, that's our entire budget. And our, like, just like our, just like our production possibilities frontier with a straight line, that becomes our budget constraint. This is what we're limited to.
Now, when we add this concept of indifference curves, remember we want to achieve the highest indifference curves. And in terms of the indifference curves, there's going to be a bunch in this diagram, right? So what we want to do is achieve the highest one. And the highest one is the one where the indifference curve just touches the budget constraint at one point, and that would be I star right there. So with this, we just combined the two ideas of a budget constraint and indifference curves. And, uh, essentially, what we're looking to do is reach the highest indifference curve. Sure, we could be higher, we could be at, I, I'll call it I2, or we could be at I3, and, but it's unobtainable based on our budget constraint. For us to reach I2 or I3, we would need an increase in our budget such that the budget constraint shifts to the right, outwards, away from the origin. And if it does, then we can start achieving higher levels of happiness, higher indifference curves.
Finally, what we're going to be doing is a process where we're going to be deriving the demand curve. And so what we're going to be doing is using the budget constraint, indifference curves, and price changes to derive that demand curve. So the process that we're going to be doing is: first, identify the quantity where utility is maximized. So like we just did, we're going to have a budget constraint and indifference curves. Next thing we're going to do is we're going to go from our starting point, and then we're going to change the price of the good in question. Uh, we can increase it, we could decrease it. We'll see, we'll see what direction we go. Actually, we're going to redraw the budget constraint with the new price, and then we're going to identify the new quantity with the new indifference curve, and then we're going to relate it to the graph of the demand curve. And so once we do that, we should have completed this exercise of deriving the demand curve. Now, there's going to be some imprecision in this. We're going to overlook certain things, like whether or not we're needing to determine an inferior or normal good. We're going to assume normal goods, uh, for for the situation, and we'll go from there.
Okay, so to start the process, let's look at two goods where we have sandwiches and gasoline in our consumption bundle mix, and we have a budget of $300. So the first thing we're going to do is we're going to set up our budget constraint. So if we have over here on the horizontal axis, sandwiches, and on the vertical, gasoline, with our $300 budget, and gasoline at $2 a gallon, and sandwiches at $3 each. If we spent our entire budget of $300 on sandwiches at $3 each, we'd be able to buy 100 sandwiches. On the other hand, at $2 a gallon for gasoline, it's definitely not Hawaii, at $2 a gallon for gasoline. If we spent our entire budget of $300, we'd be able to purchase 150. And there, now we have our budget. Oops, there, now we have our budget constraint.
So now let's just say our exercise here, we want to derive the demand curve for sandwiches if the price of sandwiches goes down from $3 each down to $2 each. And so if the price goes down to $2 each, then if we spend all of our money on sandwiches, I'm just going to pick one point out here, and I'm going to call that 150 sandwiches. Then we have a new budget constraint. Now, in terms of the budget constraint, uh, it only going to pivot out to the right with a situation like this. Sorry, the lines aren't very straight. I apologize for that. That's the best I can do under these conditions. Anyway, so now what we did was we changed the price for sandwiches, and now we see the budget constraint changes, and it pivoted out to the right like that. Now, what we didn't do was to add the indifference curves to show what point of consumption we're actually going to be at in terms of gasoline and sandwiches. Sandwiches is what we're really concerned with because we're looking for the demand curve for sandwiches. But now we're going to add in the indifference curves to show each one where we're at our optimum level of happiness given the situation of prices at each step. So again, I apologize for the quality of the hand-drawn stuff here, but anyway, now I added the optimum solution at our starting point with the original prices, and that gives us I1. So that tangency point is right about there. I'm going to eyeball it and say that's about 75 units of gasoline and about 40 sandwiches. But then we had a price change. The price for sandwiches went down, and now because of the pivoting of the indifference curve, now we can achieve a higher tangency point at a higher level indifference curve, which indicates a higher level of happiness. So I'm just going to again sort of eyeball it, and I'm going to choose this point here, call this I2. And so now we have a new level of consumption. So maybe it looks like we have this many units, we'll say 80 units of gasoline, and I'm going to call it 60 sandwiches. Now, with this change, so we saw the effects of a price change leading us to a higher level of happiness. Well, how is it that we derive the demand curve based off of this, based off of this information? Well, essentially, we have all that we need. We have two levels of prices and two levels of quantities. So what I'm going to do now, then, is I'm going to add the demand curve underneath this one. So here I have on the horizontal axis, the quantity of sandwiches, which is essentially what we have here. We have the quantity of sandwiches being produced or consumed given the prices. So we have 40 and 60. I'm going to just match it up through this. Okay. And even though it's not on the upper diagram, we do have the price change. We have prices that went from $3 down to $2. And so at $3, we were consuming 40 sandwiches. At $2, we were consuming 60 sandwiches. So with that price-quantity combination, now we have our demand curve. Now, if I wanted to change the price of gasoline and to derive the demand curve for gasoline, ideally, if I like to have it stacked like this and have the quantities of 40 here and 60 match up to 40 and 60 here, and then just have these changed and then add in, then ideally, if gas was in question, I'd want to put gas on the horizontal axis for both diagrams. So it'd be the quantity of gas on the lower diagram, and then gas on the horizontal axis of the upper diagram. But anyway, that's the process that we just discussed in the PowerPoint slide. This kind of wraps up. This had a lot of different ideas in this one example. I hope this made sense. Please feel free to contact us if you have any questions. Stay healthy, and I look forward to talking to you all soon. Take care. Aloha.