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“We talk directly about the slope of the production possibilities frontier. Let s think a a little bit about opportunity costs. And let s continue. The example that we ve working with here with our 100 people in our economy.
Each working 8 hours per day. And they have choices in how they can allocate their time for each hour that they work the person can either produce half of a gun or the person can produce 2 pounds of butter now economically speaking. We say that the cost of something is what one has to give up in order to get it and we introduced the notion of opportunity cost so i want to think about what the opportunity cost of butter is in this case. So we can do that but is thinking about this trade off.
And how the person is choosing to allocate their time here we notice that in order to get 2 pounds of butter. The person has to spend an hour making butter. But by definition. If the person is spending an hour making butter.
The person is not spending an hour making guns so in order to get 2 pounds of butter. We have to give up half of a gun because that s what the person would have been doing with that hour otherwise now we can just scale this up or down to think about what the opportunity cost of 1 pound of butter is and we notice that if the opportunity costs of 2 pounds of butter. It s half of them. Then the opportunity cost of 1 pound of butter since this is just 2.
Divided by 2 is just going to be 1 4. Of a gun because 1 4. Is just 1 2. So now we know that the opportunity cost of 1 pound of butter is 1 4.
Of a gun and we want to understand how that relates to the slope of the production possibilities frontier. The production possibilities frontier that corresponds to this setup is what we see here and we discussed before that it was in fact a straight line. So it should be reasonably easy to calculate the slope of this production possibilities frontier. Because we can say that the slope of a straight line.
It s just rise over run or alternatively change in y. Over change in x. But we can just then take two points on our production. Possibilities frontier and use them to calculate the slope.
So let s take these two points. Here and we ll say change in y and going from this point to this point. It s just negative 100 because we ve gone down by 100 alternatively. We can think about change in y is final minus initial in which case.
We can say it s just 200 300. Because we re counting this point. Here is the final and this is the initial and then we can think about our change in x. We want to make sure that we go in the same direction.
And we notice that our x went from 400 to 800. So again final. Minus initial is just going to give us 800 400. And this is just negative 100 over 400 or negative 1 4.
Now the first thing to notice is it shouldn t be surprising to you that we end up with a negative slope. Because we can see that the production possibilities frontier tilts to the left and lines that tilt to the left have negative slope. We can also notice here that this 1 4. The magnitude of the slope look strangely familiar because notice that we also saw a 1 4.
Over here when we were talking about opportunity house. So we can say it s the magnitude of the slope that this 1 4. Is in fact the opportunity cost of a pound of butter and in general. We can say that the absolute value of the slope is the opportunity cost of whatever is on the x axis here.
So in this case. The magnitude of the slope is the opportunity cost of a pond with butter. If you wanted to think about rather than thinking about the opportunity cost of a pound of butter you wanted to think about the opportunity cost of one gun one option would just be to redraw your production possibilities frontier with guns on the horizontal axis and butter on the vertical axis. Alternatively.
What what you could notice is that if the opportunity cost of a pound of butter is one fourth of a gun. Then the opportunity cost of one gun is four pounds of butter that they have a reciprocal relationship in that way and we ll see a little bit more about that later. But what you want to keep in mind. Here is you want to be very very clear then not only does the slope represent opportunity cost.
But the slope represents the opportunity costs specifically of this thing here like i mentioned before it s not always the case that production possibilities frontier czar straight lines and in fact a more reasonable assumption is that production possibilities frontier x. have this boat out shape. That we see here. The reason.
For this is because of the particular nature of the slope of this type of curve. So let s start by trying to understand what s going on with the slope here well notice when we re at the top of our curve our slope is pretty small and we can see this because i ve outlined two points here and we can look at the change in the variable on the y axis. Namely the change in guns that we need to see in order to see a particular change in pounds of butter and what we ll notice here is one more up here at the top of our production possibilities frontier. We only need a small reduction in the number of guns produced in order to get a pretty big increase in the amount of butter produced.
So we see when we re up at this part of the production. Possibilities frontier is we only need a small change in y to get a big change in x. Since our slope of a curve is change in y. Over change in x.
The magnitude of this change in y. Over change in x. Is going to be small on the other hand as we get down here. We notice that our curve gets a lot steeper as we get down here and we re getting towards a point.
Where we re producing a lot of butter and not a lot of guns. It actually takes a comparatively large decrease in the number of guns produced to get just a small increase in the amount of butter produced so here we can see that we need a large decrease in our y variable to get just a small increase in our x variable. And when we think about the slope. We end up with a big change in y divided by a small change in x.
Which means that at least in terms of magnitude. We re going to have a big absolute value of change in y over change in x. In general. When we see this boat out shape.
Here. We can think of increasing slope. As we go from the top of our curve to the bottom of our curve mathematically. The slope of a curve at a particular point is just the slope of a line tangent to the curve at that point.
So for example. We can see here that one more towards the top of our production possibilities frontier. Our tangent lines are pretty shallow meaning that at least in magnitude. They have a small slope conversely as we go down and to the right on our production.
Possibilities frontier. We see that our tangent lines are getting steeper in magnitude. So you can see is we re getting down here towards the bottom right we re seeing tangent line or slopes of the curve that are much larger in magnitude than we saw up here. Because the slope of the production possibilities frontier.
Represents. The opportunity cost of the thing on the x. Axis butter in this example. We can say that as we go down into the right on our production possibilities frontier.
This represents an increasing opportunity cost of butter as it turns out this sort of behavior actually makes intuitive sense. It s important to keep in mind that our production possibilities frontier represents. The set of all points. Where resources in an economy are being used efficiently.
So let s think about if we re at a point. Here. Where we re making all guns. And no butter in this particular example.
Let s think about what the most efficient way to switch over some resources to producing butter would be well. We d probably want to switch over those resources that are some combination of being the best at making butter and the worst at making guns right so here. If our society is making all guns and no butter. There are probably a lot of resources that are better at making butter than there are guns.
So we can just switch those over pretty easily and they can be pretty productive at making butter that represents why at this point. Here. We don t have to give up a whole lot of gun production in order to get a comparatively large amount of butter as we get down here more and more we re exhausting the resources that are comparatively good at making butter and at a certain point. We re forcing resources that are probably even better at making guns forcing them over it s making butter instead so down here in order to get from say this point here to a situation where we re making only butter.
We re taking resources that are better at making guns and forcing them to make butter instead. Which is why we end up giving up a lot of gun production in order to just get a little more butter. So this idea of efficiency and some resources being better at one thing than the other shows why it makes sense to consider this bowed out version of the production possibilities frontier. That said.
It s quite often the case that when you re given practice problems just for simplicity we deal with straight lines because it doesn t affect any of the ultimate conclusions that you re driving and it s just mathematically easier to work with. But it s important to remember this concept here because it is something that usually shows up in ” ..
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