Saturday, November 8, 2008
Friday, November 7, 2008
Wednesday, November 5, 2008
Monday, November 3, 2008
The big question facing today's politicians is, how was Joe able to provide a surplus with limited resources?
What we found out was that through a change in technology, Joe was able to increase production.
Looking at the graph, we can see that Joe's garden under normal conditions would produce an amount of garden produce indicated at S1 (supply state one). You know, without this reminder, that this is a static analysis, under what we've previously described as the condition of ceterus paribus. So we need to understand why the curves (straight lines are curves too!) show different levels of output.
Under S1, Joe planted his rows at 48 inch intervals. Keeping row interval constant, what are the possible changes in supply described by these slanted curves? Rainfall could be an important determinant of growth. The amount of time spent keeping weeds down matters. The number of sunny days. Extreme weather, like hail, for instance. Insect infestations. Disease.
That's why I've labeled the y-axis "normal". Some growth conditions can be better, some worse, but for the purpose of our analysis, we'll simply look at normal conditions. Some of these conditions are outside of Joe's control. Although he can do things to mitigate them. Some of these conditions are dependent upon Joe. Does Joe spray for disease and pest control? Does Joe take the time to hoe his rows?
The level to which Joe's efforts rise to meet the changing demands within his garden place will have impact on the productivity of his garden. So let's change the conditions of ceterus paribus so that all these variables are set to "normal" and institute our technology change. Row width.
When we look at the chart below, we see that when we hold all other variables constant, as we change row width we see an increase in production. From 48 inch rows, to 46 inch, 40 inch, all the way down to 24 inch rows. And as we reduce the row width, what we see is an increase in the amount of garden vegetables produced by Joe's garden. Until we get to 22 inch row width. Under our conditions of "all other things being equal" (ceterus paribus) we find that Joe is no longer able to increase his rate of production simply by decreasing his row width. That is, the rate of growth in production is not the same as a rate of change as it was when we were changing row widths up to the point where we went from 24 inch rows to 22 inch rows.
Let's get mathy.
The product transformation curve shows you a line--or curve--that is basically straight for most of its existence. If you're comfortable with geometry, you can describe this line as being essentially a 45 degree line starting at the origin. (The origin is where x and y are equal to zero.)
As the curve moves from left to right, the line describes changes in output based upon changes in technology inputs. When you have a 45 degree line, it's pretty simple to describe what is happening, and what the relationship is between the change in technology and the change in output (garden production). In our case, technology is an input, carrots are our output. And as we have increased the use of row-density to create greater outputs, we have found that the change in output is directly caused by this new technology.
Or, the rise in production is equal to the run of technology. With a 45 degree line, we can say that the curve is described by the statement that productivity is determined by the change in the rise of output as it relates to the run of technology. Or, the slope of the curve is determined by the ratio of increase of rise to increase of run.
If it has been awhile since you've heard the expression "rise over run", perhaps now you can intuit what your math teacher was trying to explain. The "slope" of a curve, whether the curve is straight up-and-down, or--as in our case--is 45 degrees to the origin, or flat; all these different curves are described as have a particular slope. In our case, the change in rise over run is constant up to the 22 inch row level. That means our slope has a ratio of 1:1. For every increase in technology (decreasing row width...sorry, but I just figured you'd go "huh"? "Increasing decreasing?" Yeah. Our technology was changing row width. As we increase the amount of row narrowing, we increase our use of technology, while decreasing row-width. Our discovery of row-width technology leaves us with this kind of verbal confusion. Learn to live with it. Following difficult to understand arguments will make you a better person. Trust me.) we have a concurrent and equal increase in output.
The Change in Tech (x) is equal to the Change in Output (y). Our ratio is 1:1. This is the "product" of our slope calculation. Just as Joe's garden has a product--veggies!--mathematical equations have products, too. I'm going to try to avoid confusing you when I refer to mathematical products by referring to them as "mp" parenthetically, in a hope that we can steer clear of confusion until you get to the point where you're comfortable with me referring to products as the results of mathematical statements. There's a pretty good convention out there that allows you to infer a lot about the math that's going on in a discussion of math. When you hear the word "sum", you're talking about adding and subtracting. When you hear the word "product", you're talking about multiplication and division. When you work with variables, like x and y, your always talking about products since you end up multiplying the variable with its associated constant, even when you don't see the constant. You don't show "1" as a constant all the time. It's just additional work, and it's there even if you don't see it. If there is a variable, like x or y, the statement--or equation--results in a product.
So how would a mathematician show you the equation where the result is a 45 degree line?
If a change in 2 technology units results in the change in 2 output units, he would show his work like this:
In ratio form, it would appear as 2:2.
In equation form, we would replace the colon with a dividy thingy. "/" Or, with "÷". So, it would appear as
2 ÷ 2 =
And, the answer for all these equations is 1. When the slope of a line runs at a 45 degree angle, the change in the rise is equal to the change is the run. And at any point along a 45 degree angle, you're going to find out that the slope of the angle--at each and every point along the line--is equal to 1.
Add 3.5 units of tech? You get 3.5 units of increased output. 3.5:3.5 = 1.
3.5 ÷ 3.5 = 1.
3.5 = 1.
Not all slopes need to equal 1, though. What would happen to the shape of the curve if we got 2 units of output with every 1 unit of technology?
Well, something like this happens on our product transformation curve. Do you see that the curve no longer continues at a 45 degree angle at 24 inch row width? What has happened?
It's a phenomenom described by economists as "diminishing marginal returns". Before this point, marginal returns were constant. (In our case, at a one to one ratio. Constant marginal returns are also possible at other descriptions of slope. When the ratio is one to one, the rate of increase is simply described as 1. When the ratio is two to one (2:1), the rate of increase is simply described as 2. "Rates" refer to ratios. You can apply this understanding of rates and ratios when you think about mortgage "rates" or passbook savings account interest "rates".
When you look at our transformation curve, you see that output still increases when you reduce row width below 24 inches, but the rate of that increase has slowed. The slope of the curve is flatter. Or, it is increasing, only at a decreasing rate. We are experiencing diminishing marginal returns.
When you listen to the guys who work on Wall Street, or to dull economists in the classroom, you keep hearing them refer to something as the "margin". You make money on the margin. You lose money on the margin.
This can be confusing to newcomers to economic thought and discussion, because you've also heard of "buying and selling on the margin". Margin is a word used by finance guys, but they are pretty clear--through practise--what margin they're talking about. They use words like naked puts, long puts, short calls to describe their intent. They trade stocks that they own, even though they've only made a deposit on their ownership transaction. If you're an investor who uses these investment practises, you probably dread hearing about a margin call. For finance guys, margin calls represent the Day of Reckoning.
In our purveyance, margins simply represent the changes in output (or price) when other things change. Just like Joe adopted a new technology--row width--we saw an associated change in marginal output. To a point. And that point is referred to as a turning point.
Can you see how many turning points there are in the graph above? (Let me put down here.)
Turning points occur when the rate of change changes. As long as the slope (rise over run) increases, the increase in the output curve increases. We refer to this as having a "positive" slope. A flat, horizontal line has neither positive nor negative slope. Can you think of a number that is neither positive nor negative? If you can, you can with confidence state the slope of that curve.
It is this one, simple point that drives competent financial news analysts to drink.
If you were a producer of goods, where would you find your optimal production output on this curve?
And before we leave for the day, let's take a look at our first graph.
Joe started out supplying a certain amount of garden output, S1. What happens if Joe is lazy? Or Joe simply decides to work less and invest less? Wouldn't these conditions be enough to reduce Joe's garden output to S2? But, if we start at S1 under conditions of ceterus paribus and introduce new, row-width technology, we see a shift in output to S3.
Now, compare the Supply Curve with the Product Transformation Curve. The shape of this curve is really a better representation of what's happening with Joe and his garden. While we are pursuing a static, microeconomic analysis of Joe's Garden, the transformation curve is actually a better guide for us in understanding the real world.
If we were in Joe's Garden State S1, what would be the things Joe could do to increase his production?
He could spend more time taking care of pests. He could spend more on irrigation. If sunlight was below normal, he could wrap some of his crops to help them retain heat. He could add another Scarecrow. All things he would normally do, but in this case, wanting to boost production, he does a lot more of the same old thing. What the shape of the transformation curve shows us is, whether he does a lot more in these terms to increase his production, he is probably going to hit a point where a little more hoeing, or a little more watering, will result in smaller and smaller marginal returns. Smaller increases in marginal output.
If Joe was an experienced, intelligent farmer, can you see a point at which Joe would refrain from "doing anything more"? Of course you can. It will be at that point where the slope of the curve becomes flat--even if for just a moment. This point is referred to as the point of maximum point of return. You increase the returns on your investment, whether it's through hoeing, watering or pest abating until you get no further marginal increase in product output as possible.
How much wood would a woodchuck chuck, if a woodchuck could chuck wood? He would chuck as much wood as is possible, without decreasing marginal returns.