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Mathematics in Business and Economics

Limits and Calculus

The next stage in our mini-tour is to look at the role of calculus in economics. For many students, this is where things move from one conceptual level to another and things get really tricky. Again, our advice is to do lots of practice questions to see the different approaches and to ensure that you build the confidence that is bred from success.

To Infinity and Beyond:

Limits and infinity

One of the core characteristics of calculus is the idea of the 'limit' and 'infinity'. This often proves a difficult concept for students who find maths hard because it is so abstract. In the same way that it is very difficult to conceptualise what a million pounds looks like or to get one's head around the notion that the government spend £500 billion a year. The limit and infinity are difficult to come to terms with. What is needed therefore is some way of putting these concepts into context. The following explanation is an attempt to do that before we get into the numbers side of things.

What is infinity?

It is very difficult to conceptualise what we mean by infinity. For example, if we look at the fraction 1⁄3 it is reasonable to conceptualise what we mean by it. Get a cake, cut it into three equal parts - not much problem there. But if we express 1⁄3 as a decimal we get a slightly different problem. The answer is 0.3 recurring - if we continued the calculation, i.e. by dividing 3 into 1 we would continue for evermore. The same problem arises with other common numbers - try pi for example (22⁄7) or the square root of 2.

Let us take pi. What is the true value of it? We use the number 3.14 to express pi but that is not 100% accurate - far from it. 3.142 is more accurate - we are closer to the real value of pi but still a long way away from it. How about 3.14285714 - that is much closer but still an infinite number of figures away from its true value. Same thing with the square root of 2. The square root of 2 is a number somewhere between 1 and 2. How do we know this? Well, the square root of 1 is 1 and the square root of 4 is 2. The square root of 2, therefore, must lie somewhere between these two extremes.

We can narrow things down a bit further, 1.5²= 2.25 so we now know the number we are looking for is between 1 and 1.5. If we continue on this trial and error process we can narrow down the number further and further. We can find that the square root of 2 must be less than 1.45 (2.1025), less than 1.44 (2.0736) and so on until we get to 1.41 (1.9881). So is the square root of 2 1.41? Obviously not. So lets go a bit further.

Is the number we are looking for 1.419? No, that gives us 2.013561. We could continue in this vein until we get to the figure 1.414 but again that would not be the square root of 2 because 1.414² is 1.999396. But we are close - aren't we? Well let's take the next digit. Is 1.4149 the square root of 2? No because that equals 2.00194201. If we move through the digits from 1.4149 down to 1.4141 we could see that the next digit in the sequence is 1.4142, which when squared is 1.99996164 - so getting closer.

We could carry on this process for evermore and get ever closer to the square root of two but never quite getting any definitive number - there would just be an ever increasing sequence of 9s after the 1. with some other number after the last 9! What we can say is that we are approaching the limit which represents the square root of 2.

The same principle can be applied to gradients on curves. The basic formula for finding a gradient on a curve is the change in y (Δ y) divided by the change in X (Δ x). But this is only an approximation - the gradient of a curve changes at every point. What you now have to imagine is an infinite number of points on the gradient - a bit like the infinite number of figures in the calculation for the square root of 2.

Look at the diagram below.

The point on the curve we are interested in finding out is C. Like the case with the square root of 2 above, we know that the gradient will lie between the value of the gradient at A and the gradient at point B. As an estimate, the gradients can be found by the formula given above and shown on the graph (ΔY/ΔX) If we took the gradient further down from A (say at point A1) we would get closer to the answer and if we took the gradient at point B1 we would have a further narrowing down of the range - just like we did when we calculated the square root of 2. We could continue the process getting ever closer to point C and continue to narrow down the likely answer. This is what we mean by the 'limit'. We can say that the gradient of C is only one true finite number. But it may be impossible to find that number because we could get infinitely close to it but never quite get there. It is written thus:

lim f(x) = Q
where Q is the limit of f(x) as X approaches a

Another way that is often used to help describe the principle of a limit is to think of velocity. Imagine a bullet being fired from a gun. We could measure the distance the bullet travelled and the time it took and calculate the average speed of the bullet over that time period. The graph for this relationship is shown below.

What if we wanted to find out the speed the bullet was traveling at a precise point in time (point P). The speed of the bullet would be the gardient of the curve at point P precisely. To do this we could take the average gradient between two points - P and Q in this example - and this would be an approximation similar to the early approximation we made of the square root of 2 being 1.41. However, if we moved point Q progressively closer to P, the gradient would get nearer and nearer to the gradient at point P. In this example we are dealing with speed. Speed presents a bit of a problem because to represent speed at a point in time the bullet must be traveling some distance since it makes no sense to say that the bullet travels no distance in no time at all! So what we are looking for is a close approximation to its speed at a specific point in time and to do that we use this idea of limits - how fast is the bullet travelling at a point in time which is infinitely small?

The graph represents this relationship; the gradient of the line at points P, Q1 is closer to that of point P than Q; the gradient of the line at P, Q2 is closer still and so on in the same way that 1.412 was closer to the square root of 2 than 1.41 and 1.414 was closer than 1.41 and 1.4142, etc. closer still.

What we have just been calculating is the derived function - the gradient of the tangent to a curve at the point where P and Q coincide. From this a number of rules can be found which will be used in our discussions on differentiation. The derivative is an important concept in economics as it is used to measure the rate of change of a function at a particular point. It might seem a little absurd to think about anything in human behaviour happening at a single point in time but if we remember that in economics we are concened with decision making the concept becomes a little easier to grasp.

Limits and margins are closely related. Think of income taxes, for example. The government might be interested to know what might happen to the amount of hours worked if they decided to raise the basic rate of income tax from 22% and 40% to 23% and 41%. At the time of writing the point at which someone moves from paying 22% to paying the current 40% higher rate is £31,400. That means that every £1 earned over £31,400 would now be taxed at 40p in the pound rather than 22p. What effect does this have on incentives especially if the rate were to increase? Would people work the same number of hours, work less because they do not see it being worth their while putting in the extra hours if the government are going to take more of their income or will they work more hours to make up for the fact that they are being taxed more heavily?

The number of hours worked therefore could be seen as a function of the tax rate, i.e. is dependent upon the tax rate. Using a model that may have been built around previous observations on human behaviour with regard to tax and hours worked, the Treasury could make a reasonable estimate of what might happen to hours worked and thus output, as a result of increasing tax rates. Again, nobody is suggesting that these models or the equations that are used within them are 100% accurate but they do provide a basis for informing decision making.

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