The Guild Wars 2 economy as a whole has been fabricated by individuals using gold sinks and gold taps. The main things I am thinking about are what would cause the rate of growth of the economy as a whole to have the brakes applied as time goes on? Now that I have described what exponential growth and logarithmic growth are and what the differences between them are we can start to talk about why the Guild Wars 2 economy grows logarithmically versus the US economy which grows exponentially. Why does the Guild Wars 2 Economy grow Logarithmically? This is also why it is interesting! So, why would the two be different? That is the main difference between the two and a difference between the GW2 economy and the US economy. So, exponential functions grow at an ever increasing rate, and logarithmic functions grow at an ever slowing rate. As t gets bigger this clearly gets smaller as any fraction would when you increase the bottom term. You can see that by looking in the equation for the term that depends on t. This means that as time goes on our logarithmic function slows down in growth. On the other hand if you look at the derivative of the logarithmic formula you will see that as t gets larger the value of the derivative actually gets smaller. You can see that this term will clearly get larger as t gets larger. Look at the function and see that the only term that depends on t, is b t. The rate of change itself is actually increasing exponentially. This means that as time goes on our exponential formula increases the rate at which it grows. Quickly examining the simplified derivative of the exponential formula and you can see that as the value of t gets bigger so does the value of the derivative. Since this logarithm is being taken on the number b, which is also constant, the logarithm of this number will always be the same as well. For more details see the natural logarithm. For the specifics, ln is a special type of logarithm where the base is equal to the number, e. This substitution just makes the formula easier to read because ln(b) is a constant term. That is why I made the substitution C = ln(b). Really the only thing you need to know is that ln(b) will always be the same for any value of the functions main variable time, t. Firstly, let me explain the ln(b) term that appears in both expressions. You can see the two formula are drastically different. The derivative of the exponential formula is Īnd the derivative of the logarithmic formula is, Without getting side tracked into calculus, all you need to know is that the derivative of a formula merely describes that formula's rate of change from one time to the next. The primary difference between the two can been seen by looking at their respective derivatives. Where x(t) is the value of the formula as some time, t, the initial value is a, and b is a base of the logarithm and governs the rate of growth much like it does in the exponential function. If something is growing logarithmically than its growth is described by the formula, On the other hand we discussed that logarithmic growth is the exact opposite of exponential growth in the way that division or subtraction is the opposite of multiplication and addition. Where, a is the initial value, and b is the rate of growth. To quickly recap, in the last article we found that if something is growing exponentially, than the value of that "something", x(t), at some time in future, t, is given by the formula The blue line is exponential growth and the purple logarithmic This graph shows the opposite nature of these two functions. In part one of the series I mainly answered the questions, what does it mean for something to grow exponentially? and what does it mean for something to grow logarithmically? Today I would like to tackle the final question I presented in the first part, why might wealth grow logarithmically in GW2?
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