Let's go on now and talk about growth. So just in general, from a business point of view, growth is a pretty essential idea, you hope your business grows over time in some fashion. Growing customers, growing the amount of product that you produce, etc. And so we'd like to think about some models that are purposefully constructed to help us understand and capture growth processes. And so examples that I say could be the number of customers that you have at time t, t is going to change as we''ll go into the future, hopefully the number of customers changes. It might be the revenue of your company in a particular quarter, called a quarter q, and as we go forward, as time rolls forward, your subsequent quarters, you'll be interested in your revenue growth. And [COUGH] it might be how an investment is growing over time, you know you have money, you choose to invest it, how is that investment growing? You're planning for retirement for example, so hopefully your retirement savings is growing. We want to have models for growth, now it's possible that a linear model could be helpful for growth. There's no reason why something couldn't grow in a linear fashion, but there are some alternatives to linear models for growth processes. Now I would call a linear model an additive one because we're adding on the same amount of the output for each one unit increment in the input or the x value, so they're additive. Now an alternative for things growing in an additive fashion, when we add on an absolute amount in each time period, is a proportionate one. And a proportionate increase, I really mean a percent increase, so from each period we don't go up by an absolute amount; we go up by a proportionate amount. So I might say every year our savings grows by 5%, not by $5 thousand or 5 thousand rupees, but rather by 5%. In terms of a salary, you might want to describe your salary changes not in an absolute sense, you know my salary went up by a thousand dollars, but in a percent way, for example, my salary went up by 3% this year. And so we often think of changes in terms of percentages, and if we change by a constant percent increase from one time period to another then for now I'll call that proportionate growth, and that's to be contrasted to additive growth. So to compare the two sorts of growth processes, a linear versus a proportional one, I'm going to talk about interest for a little while. Now interest is ideally what happens when you invest some money, but interest comes in two flavors. It is sometimes called simple and other times compound interest. And so let's see what happens to money when it grows according to a simple interest model. So let's say you start off with $100, that's often termed the principal, principal investment, and at the end of every single year you're going to earn 10% simple interest on the initial $100. Now 10% of 100 is 10 so that means after year one you've got an extra $10, and so if you have a look in the table, you start off year zero is now the principal of 100, after the first year you got $110. Now because it's simple interest, in the second year you're only going to earn interest on the principal, which 10% of 100 is still 10, so you go up by another $10 in year two, and likewise in year three, another $10, so 130, 140. And I went out to 10 years, by which time your 100 will have grown to 200. So every year the investment grows by exactly the same amount, that is $10, so that's additive, it's the same amount in each period. I'm going to contrast that now to what we term compound interest. So with compound interest, we're going to start off with the same amount of money, in this example $100, we'll still call it the principal, but we're going to now earn compound interest. And what compound interest means is that the interest itself earns interest in subsequent years. Simple interest was when the interest was only on the principal, compound interest is when you're earning interest on the interest. Now this gives rise to a very different growth process because now every year we're going to go up by 10%, not of just the initial amount, but of the current amount. So in year one were in the same place, we go up 10% of 100, and that's $10, so we've got $110 in year one. Here's the key idea though, that in year two we're going to earn 10% on the $110, not on the 100, so 10% of 110 is 11, and if we add 11 to 110, we get 121. So notice now in year two, we got $121 rather than $120. In year three, because we're earning 10% interest on $121 it turns out that we have got an investment valued $133.10, which is more than $130. And if we go all the way out to year ten we end up with $259.37, as compared to the $200, had we been earning only simple interest. So you can see that when the interest itself is earning interest, we're looking at a different sort of process. Now as we go from one period to the next, we're not going up by the same absolute amount, $10, rather we're going up by the same relative or proportionate amount, we're going up by 10% each time. So you're getting one cell in the table by multiplying the previous cell just by 1.1, so you could implement such growth process very easily within a spreadsheet, the sort of thing that spreadsheets are ideal for. But that's compound interest versus simple interest. Now I've drawn the growth of these two investments on a graph so you can contrast them. The brown step function at the bottom is what happens to your $100 when it's earning simple interest, and so we really thought of these as steps, as in physical steps. What's going on is that each step is of the same width and exactly the same height, that's sort of how we make stairs typically. When we compound the interest, you can see how the growth function is, we're growing in a different fashion with the compound interest, the blue step function. If you were to try to walk up those steps you would find that the width of each step is the same, but the height is increasing because it's going up by 10% from the previous height every time. So that would be a pretty tricky sequence or set of steps to walk up eventually, because the steps are going to get higher and higher and higher each time. Because we are going proportionality rather than in an additive or linear fashion, so you can see the differences, and in fact, the blue curve is an exponential function, and the brown is linear. All right, so that illustrates the difference between a linear growth and a compound or proportional growth.
