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Economics: Deriving Algebraic Equations for PPF

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About this Lesson

  • Type: Video Tutorial
  • Length: 22:00
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 235 MB
  • Posted: 03/29/2010

This lesson is part of the following series:

Economics: Full Course (269 lessons, $198.00)
Economics: Introduction to Economic Thinking (18 lessons, $33.66)
Economics: Production Possibilities (3 lessons, $8.91)

This video lesson will teach you how to model out the PPF (production possibilities frontier) using algebraic expressions and equations. Taught by Professor Tomlinson, this video lesson was selected from a broader, comprehensive course, Economics. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/economics. The full course covers economic thinking, markets, consumer choice, household behavior, production, costs, perfect competition, market models, resource markets, market failures, market outcomes, macroeconomics, macroeconomic measurements, economic fluctuations, unemployment, inflation, the aggregate expenditures model, banking, spending, saving, investing, aggregate demand and aggregate supply model, monetary policy, fiscal policy, productivity and growth, and international examples.

Steven Tomlinson teaches economics at the Acton School of Business in Austin, Texas. He graduated with highest honors from the University of Oklahoma and earned a Ph.D. in economics at Stanford University. Prof. Tomlinson's academic awards include the prestigious Texas Excellence Teaching Award given by the University of Texas Alumni Association and being named "Outstanding Core Faculty in the MBA Program" several times. He has developed several instructional guides and computerized educational programs for economics.

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In these next two lessons, we're gong to introduce and explore one of the economists' favorite ideas, comparative advantage. And we're going to show how two people with differing abilities can cooperate to increase their wealth. We're going to be using the tool that we developed in the last lecture, the production possibilities frontier. And we're going to first derive the production possibilities frontier for a cleaning service run by Bernie. In the next lesson, we'll be introducing a potential cleaning partner for Bernie and show how cooperation can increase the wealth of both cleaners. But first, let's look at Bernie.
Bernie has a service that provides two cleaning chores: Bernie scrubs rooms and Bernie sweeps rooms. The first thing that we want to do is explore Bernie's technology; that is, what is Bernie able to do? Well, let's look. Bernie takes 20 minutes to scrub one room, Bernie takes 10 minutes to sweep one room. In economics, we call these numbers the unit labor requirements, how much time it takes for Bernie to perform each of these tasks. And the unit labor requirements describe Bernie's productivity. Notice that Bernie can sweep faster than he can scrub. So 20 minutes to scrub or 10 minutes to sweep.
Now let's suppose that Bernie is going to work for 1 hour. How much can Bernie accomplish? Well, if he devotes an hour to scrubbing, then we can find out how many rooms he scrubs by dividing 1 hour, or 60 minutes, by the time that it takes Bernie to scrub one room. So 60 minutes divided by 20 minutes is equal to three rooms scrubbed in 1 hour. If Bernie devotes 1 hour to sweeping, on the other hand, divide 60 minutes by 10 minutes per room swept and you find that Bernie can sweep six rooms in one hour. So we'll use these numbers then to represent Bernie's productivity; that is, if Bernie spends all of his time scrubbing, in an hour he can scrub three rooms. If Bernie spends an hour sweeping, he can sweep six rooms.
Now, of course, if Bernie has an hour to work, he needn't do all sweeping or all scrubbing. In fact, he can do any combination of the two. So what we'll do now is derive a production possibilities schedule for Bernie by looking at different combinations of sweeping and scrubbing that are possible for Bernie in 1 hour. Remember Bernie can spend 10 minutes sweeping or 20 minutes scrubbing and divide up his hour any way he likes.
Well, let's look at some numbers that work for Bernie. And here we have a production possibility schedule, much like the one that we looked at last time for rice and wheat. Bernie can spend an hour scrubbing and scrub three rooms. That leaves no time for sweeping, so he sweeps zero rooms. If Bernie decides that he wants to sweep one room, that's going to take 10 minutes. Subtract 10 minutes from 60 minutes to leave 50 minutes. Bernie has 50 minutes now to spend scrubbing. And if you divide 50 minutes by 20 minutes per room scrubbed, you find that Bernie has time to scrub two and a half rooms. If Bernie wants to sweep two rooms, however, that's going to take 2 10 or 20, leaving only 40 minutes for scrubbing, so that he can scrub a total of two rooms, and so forth. If he wants to sweep three rooms, that leaves him time to scrub one and a half, if he wants to sweep five rooms, that leaves him time to scrub only half of a room, and if e spends all of his time sweeping, so that he spends no time scrubbing, he can sweep a total of six rooms. This is the production possibilities schedule for Bernie. Notice, as we look at these number, we are already getting a taste of Bernie's opportunity cost. Every room that Bernie sweeps causes him to give up one-half of a room scrubbed. The opportunity cost for Bernie of sweeping one room is one-half of a room that he doesn't scrub.
Well, the next logical step is to take the information from the production possibilities schedule and represent it in a graph. Here's how that would work. I'll move these numbers over to the board, so that you have them for reference, and I'll put up some axes that we can use to draw a production possibilities curve for Bernie's business. Let's put sweeping up here on the vertical axis and let's put scrubbing on the horizontal axis. So here's sweeping and here's scrubbing. Now, let's plot numbers from Bernie's production possibilities schedule into this diagram. And we see that, let's see here, 1, 2, 3, 4, 5, 6 rooms is the maximum that he can sweep and 1, 2, 3 is the maximum that he can scrub, so it looks like my chart here is going to work. I'm going to calibrate my axes a little bit differently, just for the sake of making things nice and neat. I'm going to let each of the hash marks represent two rooms swept, so here's one room, 2, 3, 4, 5, 6. And I'm going to let each of the hash marks down here on the bottom represent one room scrubbed, so here's 1, 2, 3, 4 and so forth. I could go ahead and put the numbers in to calibrate the axis, so let me do that, just to make things nice and neat. Here's 2, 4, 6, 8 and so forth, and down here we have 1, 2, 4, 5, 6, 8 and so on.
Now, let's put in the points from the schedule into the diagram so that we can plot the production possibilities curve. Let's start here with if we scrub three rooms, then we can sweep zero, so go to three rooms scrubbed and zero room swept and we get this dot right here on the production possibility frontier, three scrubbed, zero swept. If we sweep one room, then we are going to be only able to scrub two and a half rooms, so we're going to get a point like this one right here. If we decide to sweep two rooms, then we're going to be able to scrub two rooms, so that gives us a point like this one and so forth. Just keep plotting the points. If we are going to scrub one and a half rooms, that leaves us time to sweep three and if we scrub one, that leaves us time to sweep four. One-half leaves us time to sweep five. And if we don't do any scrubbing at all, we can sweep six rooms.
Well, there you have it. There's the production possibilities frontier for Bernie. Bernie can sweep anywhere from zero to six rooms. He can scrub anywhere from zero to three rooms and if he divides his time, he can achieve any combination that's on this line. Let's now imagine that Bernie can divide his time even more finely between sweeping and scrubbing and we'll be able to connect those dots and get a straight line. This is what Bernie's production possibilities frontier looks like.
Now let's notice a few things about Bernie's production possibility frontier. First, the vertical intercept represents the maximum number of rooms that Bernie can sweep if he spends a whole hour sweeping. The horizontal intercept represents the maximum number of rooms that Bernie can scrub if he spends all of his time scrubbing. The downward slope reminds us of scarcity. Bernie's time is scarce. If he wants to scrub more rooms, he has to give up rooms swept. There are points outside the production possibilities frontier, representing Bernie's limitations. He only has an hour, so he couldn't, say, sweep six rooms and scrub six rooms. That would take too much time. There are also points inside the frontier if Bernie were using his time inefficiently.
So the intercepts tell us about the maximum amount of sweeping or scrubbing that Bernie could possibly get from an hour's worth of time. What about the slope of the line? What does that tell us? The slope, as usual, tells us about the opportunity cost of scrubbing, measured in terms of sweeping; that is, if Bernie wants to scrub one additional room, if want to increase scrubbing from, say, two rooms to three rooms, that is a horizontal movement in this space. That means we would have to reduce the amount of sweeping that Bernie does. That would be a vertical movement in the space. And if we take the amount of sweeping that Bernie gives up to do an extra room's worth of scrubbing, that gives us the opportunity cost of scrubbing the room. So the rise, which, in this case, would be a change in Bernie's sweeping, a reduction in the amount of sweeping that Bernie is able to do, over the run, which would be the increase in scrubbing, is the slope of the production possibilities frontier. In this case, in order to have a change in scrubbing equal to one room, we get a change in sweeping equal to -2 rooms. For every room that Bernie scrubs, he gives up two rooms that he could have swept. The slope of this line is -2 and it's the opportunity cost of scrubbing for Bernie, measured in terms of the sweeping that he can't do. Of course, this is logical. It takes him 20 minutes to scrub a room, divide that by 10 minutes that it takes to sweep a room, and you see that he could be sweeping two rooms in the time that it takes him to scrub one. And that's the slope of this line. The opportunity cost is the slope of the production possibilities frontier.
Notice that we have a line here, so we can describe it with an algebraic equation. And, in this case, the equation for this line is going to be, first of all, we have the variable on the vertical axis, so I can write this sweeping - that's the thing we're going to be describing with our line - sweeping is going to be equal to - the next thing you do is write down the vertical intercept. Six rooms swept, this is the maximum number of rooms that Bernie can sweep if he spends a full hour sweeping, minus the slope of the line, which, in this case, is 2, multiplied by the horizontal axis variable. The horizontal axis variable is scrubbing. So here's the formula for the line: intercept - slope x-axis variable. The amount of sweeping that Bernie does is equal to six rooms, the maximum that he can sweep, minus 2 times the number of rooms that he scrubs, because any time he scrubs a room, that takes 20 minutes, and that's time enough to sweep two rooms. So the vertical intercept is the constant in this equation and the slope, -2, is the opportunity cost of scrubbing, measured in terms of sweeping that Bernie isn't able to do. When he scrubs a room, he gives up two rooms' worth of sweeping. There's the equation for Bernie's production possibilities frontier.
Well, it seems kind of arbitrary that we've put sweeping on the vertical axis and scrubbing on the horizontal axis. And it is, in fact, quite arbitrary. We could have just as easily put scrubbing on the vertical axis and sweeping on the horizontal. Well, I'd like to try to make that point clear now by taking the production possibilities frontier and showing you how to convert it into the other way of expressing things, with scrubbing up top and sweeping on the horizontal.
Here's what I'm going to do, I'm going to put this line in for my production possibilities frontier and I'm going to secure it with tape. And what I'm about to do is simply transpose the axes. I'm going to just convert this diagram from one with scrubbing on the horizontal to one with scrubbing on the vertical. Bernie still has the same productivity, no matter how we express it. Bernie can still do the same combination of sweeping and scrubbing, but I'm just going to change the way I write it. And it's going to be as easy as picking this graph up and flipping it over, so that the scrubbing axis is vertical and the sweeping axis is horizontal. I'll have exactly the same expressions. So, let me do that then. I'll pick scrubbing up and put it on the vertical. I'll pick sweeping up and put in on the horizontal, and then I'll flip my diagram over, so that the scrubbing information is on the vertical and the sweeping information is on the horizontal. Now notice I've got a bit of a mess here, because I still have all the markings that I did for my previous graph. But if you'll bear with me and let me tape my production possibilities curve down to the axis firmly, and if you'll let me clear my pad here, so that I get rid of the old writing, voile! We've got the new production possibilities frontier with sweeping on the horizontal axis and scrubbing on the vertical axis. The vertical intercept is now the maximum amount of scrubbing that Bernie can do if he scrubs for an hour. And, as we've seen, that's equal to three rooms. The horizontal intercept is the maximum amount of sweeping that Bernie can do if he sweeps for an hour and, as we've seen, that's six rooms. The formula for this new equation is going to be given by this expression. Bernie's production possibilities frontier, expressed this way with scrubbing on the vertical and sweeping on the horizontal, has exactly this mathematical expression. 3 is the vertical intercept or the maximum number of rooms he can scrub in an hour, and - is the opportunity cost. Anytime Bernie sweeps an extra room that takes 10 minutes. 10 minutes divided by 20 minutes, the amount of time that it takes to scrub a room, gives us . Anytime Bernie sweeps a room, he's giving up one-half of a scrubbed room, because it takes half as long to sweep a room as it does to scrub the room.
So there you have it, Bernie's production possibilities frontier. Scrubbing is equal to three rooms max minus one-half the opportunity cost times each room that he sweeps. Now, let me show you something about the relationship between those two production possibilities frontier equations, and this is what's interesting.
Notice that the original equation was sweeping is equal to 6 - 2 times scrubbing. Well, let's rewrite this equation and move scrubbing over to the left-hand side, so that we have an equation that has scrubbing measured in terms of sweeping. Well, here's what I'll do, I'll move this -2 scrubbing to the left-hand side, and that's equal to 6 minus sweeping. So all I did was move this expression to the right-hand side and this expression to the left-hand side. It looks like I need to make sure that I change the sign when I move it over. Then let's divide both sides by 2, and I get scrubbing = 3 - sweeping.
There you have it. The production possibilities frontier can be written either way, in terms of sweeping or in terms of scrubbing. This is the vertical intercept, the maximum amount of scrubbing and this, -, is the opportunity cost.
Here's one more interesting thing to note. Up here, this -2 was the opportunity cost of scrubbing, measured in terms of sweeping. If Bernie wants to scrub a room, he gives up two rooms that he could have swept. Down here, this - is the opportunity cost of sweeping, measured in terms of scrubbing. If Bernie wants to sweep a room, he gives up one-half of a room that he could have scrubbed. Now, this is the thing that's important to notice: the opportunity cost of scrubbing, measured in terms of sweeping, is the reciprocal of the opportunity cost of sweeping, measured in terms of scrubbing. If Bernie wants to sweep a room, he has to give up half of a room that he could scrub. If Bernie wants to scrub a room, he has to give up two rooms that he might otherwise sweep. See, the opportunity costs are reciprocals of each other, and that makes mathematical sense, because whenever you're moving and dividing, you're going to wind up with the reciprocals. But it even makes good intuitive sense. It takes 10 minutes to sweep, it takes 20 minutes to scrub. So if you scrub, you're giving up 20 minutes divided by 10 minutes, two rooms you could have swept. If you want to sweep, that takes 10 minutes, so that means you're giving up 10 minutes divided by 20 minutes, one-half of a room that you could have scrubbed. The opportunity cost of sweeping, measured in terms of scrubbing, is the reciprocal of the opportunity cost of scrubbing, measured in terms of sweeping. And that's really what it means to pick this diagram up and flip it around. You wind up with exactly the same line, only now you're measuring the axes differently. You're measuring the different variables on the different axis, so that the intercepts are switched around, and the slope of the new line is the reciprocal of the slope of the original line. The original line, with sweeping on the vertical, had a slope of -2. The new line, with scrubbing on the vertical, has a slope of -.
Now, there's one more thing that we ought to notice about this picture before we begin to apply it to the bigger question of comparative advantage and gains from trade. That's this: Bernie's production possibilities curve is a straight line. What does that tell us about Bernie's technology, his abilities? Remember the production possibilities frontier from our previous example with rice and wheat was bowed outwards. The outward-bowed production possibilities frontier reminded us that not all resources are equally well suited to all uses. Some land is wet, it's good for rice, some land is dry, and it's good for wheat. Bernie, on the other hand, apparently has time and talent that's equally well suited to sweeping or scrubbing. There is no outward bow to Bernie's production possibilities frontier, because there is increasing opportunity cost. Anytime Bernie wants to sweep an extra room, it's going to take him 10 minutes. Anytime Bernie wants to scrub an extra room, it's going to take him 20 minutes. These are the assumptions that we've used to describe Bernie's technology. Well, because Bernie's costs are constant in this way, because all of his time and talent is equally well suited for sweeping and scrubbing, Bernie's opportunity cost is constant. The constant opportunity cost of one-half of a scrubbed room for every room that you want to sweep, or two swept rooms for every room that you want to scrub, gives us a straight line production possibilities frontier.
So, a quick summary: a bowed-outward production possibilities frontier, a concave production possibilities frontier is for a case where the technology has increasing opportunity cost, and that's because not all resources are equally well suited to all uses. But a straight line production possibilities frontier is for a case where the opportunity cost is constant. And that will be the case when all resources are equally well suited to the production of either of the goods or services.
So now we've given a thorough description of Bernie's technology. We started with his unit labor requirements, then we looked at what he was able to do with 1 hour of his time. Then we looked at a schedule of all the possibilities of sweeping and scrubbing that Bernie could turn out. Then we drew a picture to represent his production possibilities, and we saw that it was a straight line, representing constant opportunity costs. Then we showed that we could represent the production possibilities with either good on either axis, and what we found was that the opportunity cost of one, in terms of the other, was the reciprocal of the opportunity cost of the other, in terms of the original good, -2 versus -.
Now, these are pretty technical points. In the next lesson, we're going to apply these technical points as we look at potential cooperation between Bernie and a business partner, Anne, as they seek to increase the wealth of their cleaning businesses.
Introduction to Economic Thinking
Production Possibilities
Deriving an Algebraic Equation for the Production Possibilities Frontier Page [4 of 4]

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